Emiliano Lorini

Cristiano Castelfranchi

The cognitive structure of surprise: looking for basic principles

We develop a conceptual and formal clarification of the notion of surprise as a belief-based phenomenon by exploring a rich typology. Each kind of surprise is associated with a particular phase of the cognitive processing and involves particular kinds of epistemic representations (representations and expectations under scrutiny, implicit beliefs, presuppositions). We define two main kinds of surprise: mismatch-based surprise and astonishment. In the central part of the paper we suggest how a formal model of surprise can be integrated with a formal model of belief change. We investigate the role of surprise in triggering the process of belief reconsideration. There are a number of models of surprise developed in psychology of emotion. We provide several comparisons of our approach with those models.

Introduction

Surprise is the automatic reaction to a mismatch. It is a (felt) reaction/response of alert and arousal due to an inconsistency (discrepancy, mismatch, non-assimilation, lack of integration) between an incoming input and our previous knowledge, in particular an actual prediction or a potential prediction. It invokes and mobilizes resources at disposal of an activity for a better epistemic processing of this strange information (attention, search, belief revision, etc.), but also for coping with the potential threat. Surprise is aimed at solving the inconsistency and at preventing possible dangers (the reason for the alarm) due to a lack of predictability and to a wrong anticipation. Moreover, there are different kinds and levels of Surprise. There is a first-hand surprise -the most peripheral one, just due to the perceptual mismatch between what the agent sees and its sensory-motor expectations; while the deeper and slower forms of surprise are due to symbolic representations of expected events, and to the process of information integration with previous long-term knowledge and explanation of the perceived data [START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF]. This is surprise due to the implausibility of the new information. Low level predictions are based on some form of statistical learning, on frequency 1 and regular sequences, on judgment of normality in direct perceptual experience, on the strength of associative links and on the probability of activation [START_REF] Kahneman | Norm theory: comparing reality to its alternatives[END_REF] or on mental simulation. On the other hand high level predictions have many different sources: from analogy ("The first time he was very elegant, I think that he will be well dressed") and, in general, inferences and reasoning ("He is Italian thus he will love pasta"), to natural laws, and -in social domain -to norms, roles, conventions, habits, scripts ("He will not do so; here it is prohibited"), or to Theory of Mind ("He likes Mary, so he will invite her for a dinner; He decided to go in vacation, so he will not be here on Monday"). In this work we are mainly interested in the analysis of those forms of Surprise which involve symbolic high-level representations of expected events or objects and a recognized event or object in the external world. We are not going to analyze those forms of surprise due to the mismatch between low-level sensory expectations and a raw perceptual input (raw sensor datum). We restrict our analysis to those forms of cognitive surprise involving an already perceived and recognized object or event. 1 In order to account for the process of cognitive recognition we have developed in our complementary work [START_REF] Lorini | The unexpected aspects of surprise[END_REF] an abduction-based procedure of explanation assessment and selection. This procedure has the function of returning the best explanation of the data obtained by the sensors. We have shown in [START_REF] Lorini | The unexpected aspects of surprise[END_REF] that this selected explanation can mismatch with some pre-existent cognitive representation and therefore be responsible for the generation of surprise. In this work we do not introduce any abduction-based procedure of explanation selection and we simply assume that an agent can directly perceive and recognize an object or event without interpreting the perceptual raw sensor data by means of some abductive procedure.

In section 2 a formal logic of beliefs and probabilities is introduced. This simple logic is developed in order to provide formal representations of several kinds of mental attitudes. We provide definitions for beliefs and expectations of an agent. Moreover we characterize the notion of scrutinized expectation, i.e. the expectation on which the agent is focusing its attention and that the agent tries to match with the perceptual data. We introduce: 1) the special kind of mental operation retrieve with the function of introducing a new expectation into the mental Test (scrutiny) space of the agent; 2) the special action of perceiving some fact with the function of modifying the agent's perceptual data. In section 3 we analyze two different kinds of surprise which involve all informational mental states characterized in section 2. We argue that these forms of surprise are the basic forms of surprise in cognitive systems involving symbolic high-level representations of expected events. 2 1. Mismatch-based surprise (given the conflict between a perceived fact and a scrutinized representation). I'm actively checking whether a certain event is happening, that is I have an endogenous anticipatory explicit representation of the next input and I attempt to match the incoming data against it. If there is a mismatch (conflict) between the two representations there is surprise. The intensity of this form of surprise is a function of the probability assigned to the expectation conflicting with the perceived fact.

2. Astonishment or surprise in recognition. I perceive a certain fact and recognize the implausibility of this. The recognition of implausibility of the perceived fact can based on two different kinds of mental processes.

(a) On one side, after perceiving a certain fact ϕ that I was not actively expecting, I can retrieve from my background knowledge the probability of the event and conclude that "I would not have expected that event". The intensity of astonishment is a function of the probability assigned to the negation of the perceived fact (¬ϕ).

(b) On the other side after perceiving a certain fact ϕ I infer from my explicit beliefs the negation of the perceived fact.

We will argue that the previous typology of Surprise is based on the characterization of different kinds of informational mental states. 3 The following figure 1 summarizes them.

According to our view an agent has a representation under scrutiny (a focused expectation) and this must be distinguished from all those accessible representations and expectations in background (at an unconscious and automatic level). This distinction between expectations and beliefs under scrutiny and background expectations and beliefs looks similar to Kahneman & Tversky's distinction [START_REF] Kahneman | Variants of uncertainty[END_REF] between active expectations and passive expectations. According to Kahneman & Tversky the former occupy consciousness and draws on the limited capacity of attention; the latter kind are available at a mere automatic and effortless level. Passive expectations could be the product of priming. Moreover, an agent looks at the world and acts in the world within the assumption of a presupposed complex mental framework, of a given presupposed frame (or script) which represents its unproblematic interpretation of the context of the situation where its action and perception are situated and which supports all (focused and background) expectations. Thus when presupposing to be entered in a restaurant the agent can reasonably expect to perceive a waiter, some tables and so on... Expectations and beliefs under scrutiny, background expectations and beliefs and presupposed frame are members of the general set of explicit informational mental states.

The last category of informational mental states is the category of implicit expectations and beliefs, that is all those (potential) beliefs and expectations that can be inferred from explicit beliefs and expectations [START_REF] Ortony | Surprisingness and expectation failure: Whats the difference[END_REF][START_REF] Levesque | A logic of implicit and explicit belief[END_REF].

Figure 1: Ontology of informational mental states

In section 4 we try to build a bridge between our model of surprise and the theory of belief change. We start by extending the formal language and introducing update processes whose function is the modification of beliefs and expectations of the agent after the perception of a new fact. We extensively investigate the role of surprise in triggering belief change and we provide cognitive principles governing this kind of cognitive phenomenon.

Let us sum up here the major claims of our work. Surprise is a very relevant belief-based phenomenon, with mental and behavioral aspects. In order to understand it, it is necessary to model the relationships between basic properties of beliefs (like their strength, their being explicit or implicit, their being passively assumed or actively tested) and surprise kinds and dimensions. It is a beliefbased phenomenon because it is based on an actual or potential prediction formulated on the basis of the other beliefs, and because one of its main effect is the revision of our assumptions: the new data must be assimilated in our knowledge base, and our beliefs (base of our wrong prediction) must be revised. We claim that in the literature there are very good studies and claims on surprise: like the idea that it depends on expectations, or the claim that its intensity depends on the "unexpectedness" of the stimulus [START_REF] Ortony | Surprisingness and expectation failure: Whats the difference[END_REF][START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF], the claim that one can deal with this in terms of information or probability, or the claim that there are different kinds of expectations -active versus passive, explicit versus implicit -which produce different kinds of surprise. However, we claim that more careful distinctions and clear characterizations of surprise are needed. We present here a 'theory driven' account of surprise, an analytical cognitive model which allows us to predict and distinguish different levels and kinds of surprise, not necessarily already discriminated in the empirical researches. Sometimes even common sense concepts look much richer: for example, the concept of "astonished" is not identical to the concept of "surprised", or to the concept of "being disoriented". We try to provide some principled and precise distinctions of these different levels and kinds of surprise and to formalize some relevant properties of them. It is not for example the same kind of surprise when we immediately recover from it, while saying "What a stupid I am! It is obvious! I should have expected this", and when we remain in a strong and long suspended situation, unable to realize/accept and understand what has happened. Of course, we take into account some important psychological models [START_REF] Ortony | Surprisingness and expectation failure: Whats the difference[END_REF][START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF][START_REF] Reisenzein | Reactions to surprising events: A paradigm for emotion research[END_REF], which are very relevant and interesting, and also currently implemented models in AI [START_REF] Macedo | Modelling forms of surprise in an artificial agent[END_REF]. But they are still quite poor and simplified. They for example do not clearly distinguish between the surprise relative to the invalidation of a strong anticipated expectation, and the surprise relative to the degree of "unexpectedness" of the new incoming data. The two processes are -in our model -related and partially complementary, but not identical.

A complex view of surprise and of its nature and functions is necessary to understand the phenomenon. We do not model all its aspects. We do not investigate the experiential, phenomenal character of surprise [START_REF] Reisenzein | The subjective experience of surprise[END_REF]: surprise as a felt signal. 4Moreover we do not model functional aspects of surprise (alert, learning, etc. ) except those related to belief reconsideration (Section 4). As some psychologists have stressed [START_REF] Meyer | An experimental analysis of surprise[END_REF][START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF] surprise can culminate in a process of belief change.

In this work we want to try to suggest some interesting ways a formal and computational model of surprise viewed as a belief-based phenomenon can be integrated with a formal model of belief change. Indeed belief revision theory has been mostly focused on the problem of finding general principles characterizing the process of belief change, but has completely neglected to account for the causal precursors of this kind of process.

Formal bases

We define in this section the formal language with the related syntax and semantic. We use a logic of probabilistic quantified beliefs with a semantics similar to the semantics given in [START_REF] Fagin | Reasoning about knowledge and probability[END_REF]; [START_REF] Halpern | Reasoning about uncertainty[END_REF]. We add to the basic formal language the standard dynamic operator for talking about actions. Moreover, we use special formal constructs to denote the representation under scrutiny (or under test) of a given agent and the agent's perceptual data collected by its sensors. We characterize two special kinds of actions: the mental operation retrieve which has the function of moving new information into the scrutiny (test) space; -the action perceive which has the function of modifying the agent's perceptual data. The main function of the formalism is to disambiguate the relevant concepts and notions of our model of surprise.

Syntax

The primitives of the formal language are the following:

• A set of atomic actions AT = {a, b, ...}.

• A set of propositional variables Π = {p, q, ...}.

The set P ROP = {ϕ, ψ, ...} is the set of propositional formulas defined by the closure of Π under the Boolean operations ∧ and ¬. On one hand OBS is the set of perceptual actions defined as the smallest set such that:

• if ϕ ∈ P ROP then observe(ϕ) ∈ OBS.
On the other hand RET R the is the set of retrieve mental operations defined as the smallest set such that:

• if ϕ ∈ P ROP then retrieve(ϕ) ∈ RET R.
ACT = {α, β, ...} is the set of actions which is defined as the smallest set such that:

• AT ⊆ ACT ; • OBS ⊆ ACT ; • RET R ⊆ ACT ;
• if α and β ∈ ACT then α; β ∈ ACT (sequential composition).

Our language L SURP is given by the following rule in extended Backus-Naur Form: reads "ϕ is the representation that the agent is scrutinizing"; Datum(ϕ) reads "ϕ is a datum perceived by the agent".

Φ ::= p|¬Φ|Φ 1 ∧ Φ 2 |BelΦ| [α]
Propositional formula ϕ such that T est(ϕ) should be considered the content of the expectation that the agent is actually scrutinizing and comparing and matching with the incoming input data. On the other hand propositional formula ϕ such that Datum(ϕ) should be considered a datum obtained by the agent's sensors. With perceptual datum we mean here something similar to the notion of datum given in [START_REF] Rescher | Plausible reasoning[END_REF]. A perceptual datum is in our vocabulary some piece information gathered by the agent's sensors which is a candidate for becoming a belief of the agent. 5 It will be shown below that both perceptual data and scrutinized representations play a crucial role within the surprise processing. Moreover we use the following abbreviations:

α Φ = def ¬ [α] ¬Φ; n i=1 d i P (Φ i ) ≥ c = def d 1 P (Φ 1 ) + ... + d n P (Φ n ) ≥ c d 1 P (Φ 1 ) ≥ d 2 P (Φ 2 ) = def d 1 P (Φ 1 ) -d 2 P (Φ 2 ) ≥ 0 n i=1 d i P (Φ i ) ≤ c = def n i=1 -d i P (Φ i ) ≥ -c n i=1 d i P (Φ i ) < c = def ¬( n i=1 d i P (Φ i ) ≥ c) n i=1 d i P (Φ i ) > c = def ¬( n i=1 d i P (Φ i ) ≤ c) n i=1 d i P (Φ i ) = c = def n i=1 d i P (Φ i ) ≤ c ∧ n i=1 d i P (Φ i ) ≥ c

Semantics

We define with M the class of models of the form

M =< W, B, R 0 , R 1 , R 2 , P, T EST, DAT A, π >
where each element of the tuple is defined as follows.

• W = {w, w , w , ...} is a non-empty set of possible worlds.

• B is a mapping B : W -→ 2 W associating a set of possible world B(w) to each possible world w. The elements in B(w) are the alternative (worlds) that the agent considers possible at world w.

• R 0 , R 1 , R 2 are mapping 1. R 0 : AT -→ (W -→ 2 W ) 2. R 1 : OBS -→ (W -→ 2 W ) 3. R 2 : RET R -→ (W -→ 2 W )
associating sets of possible worlds R a 0 (w), R observe(ϕ) 1

(w) and R retrieve(ϕ) 2 (w) to each possible world w. Those worlds w' such that w ∈ R a 0 (w), w ∈ R observe(ϕ) 1 (w) and w ∈ R retrieve(ϕ) 2 (w) are respectively those worlds which are achievable from w by doing the atomic action a, achievable by doing the action of perceiving ϕ and achievable by doing the operation of retrieving the expectation that ϕ from the background level. 5 The need for the distinction between data and beliefs has been addressed by several other authors (see [START_REF] Paglieri | Data-oriented belief revision: Towards a unified theory of epistemic processing[END_REF]. For instance [START_REF] Tamminga | Expansion and contraction of finite states[END_REF] advocated the need for two levels of explanation in dealing with belief revision, namely information (data) and beliefs. This leads him to describe belief revision as a two steps process: first, information revision, managed by applying a paraconsistent monotonic logic of first-degree entailment; second, belief extraction, that takes care of assuring nonmonotonicity, consistency, and closure under logical consequence. In Tamminga's work, the main focus is placed on inconsistency at the level of information (data) vs. consistency at the level of beliefs.

• P is a function which associates with each world w in W a probability space P (w) = (W w , X w ) where:

-W w ⊆ W is called sample space; -X w is a probability function defined on W w such that X w : W w -→ [0, 1] and ∀w ∈ W w ∈Ww X w (w ) = 1
• TEST is a (test) function T EST : W -→ P ROP6 which assigns a propositional formula to each possible world. This function returns the representation that the agent is scrutinizing at a certain world, i.e. the representation on which the agent is focusing its attention and that the agent matches with the perceptual data.

• DATA is a (perception) function DAT A : W -→ P ROP which assigns a propositional formula to each possible world. The function returns the datum obtained by the agent 's sensors at world w.

• π : Π -→ 2 W assigns a set of worlds to each propositional variable.

Here we suppose that B, TEST, DATA, every R a 0 , every R observe(ϕ) 1

and every R retrieve(ϕ) 2 are partial functions.

We use the following two notational abbreviations:

• (Domain): ||Φ|| Ww = {w ∈ W w |M, w ϕ}; • (Probability of a Domain): X w (||Φ|| Ww ) = w ∈||Φ|| Ww X w (w ).
Truth conditions

• M, w p ⇐⇒ w ∈ π(p) • M, w ¬Φ ⇐⇒ not M, w Φ • M, w Φ 1 ∧ Φ 2 ⇐⇒ M, w Φ 1 and M, w Φ 2 • M, w BelΦ ⇐⇒ ∀w if w ∈ B(w) then M, w Φ • M, w d 1 P (Φ 1 )+...+d n P (Φ n ) ≥ c ⇐⇒ d 1 X w (||Φ 1 || Ww )+...+d n X w (||Φ n || Ww ) ≥ c • M, w T est(ϕ) ⇐⇒ ϕ = T EST (w) • M, w Datum(ϕ) ⇐⇒ ϕ = DAT A(w) • M, w [α] Φ ⇐⇒ ∀w if w ∈ R α (w) then M, w Φ
where R α (w) is defined according to the following (1), ( 2), ( 3) and (4).

1. R a (w) = R a 0 (w); 2. R observe(ϕ) (w) = R observe(ϕ) 1 (w); 3. R retrieve(ϕ) (w) = R retrieve(ϕ) 2 (w); 4. R α;β (w) = (R β • R α )(w).

Basic properties and definitions

For what concerns the probabilistic fragment of our logic we inherit all axioms and inference rules given in [START_REF] Halpern | Reasoning about uncertainty[END_REF]; [START_REF] Fagin | Reasoning about knowledge and probability[END_REF]. Soundness and completeness of this deductive system for a logic of belief and probability has been proved. The axiom system is made of three different kinds of axioms and inference rules. Axioms and inference rules are given for: 1) propositional reasoning and Bel modal operator 7 ; 2) for reasoning about probability 8 ; 3) for reasoning about linear inequalities 9 . Moreover we suppose here that believing that Φ holds implies that the maximum value of probability is assigned to Φ. Formally:

(Incl Bel/P rob ) BelΦ → (P rob(Φ) = 1).
This axiom requires that the set of worlds which are considered possible by the agent in an arbitrary world w is a superset of the sample space with respect to the arbitrary world w:

• for every w ∈ W W w ⊆ B(w).
With respect to the dynamic component we use standard axioms from dynamic logic. We take the axioms and inference rules of the basic normal modal logic for the dynamic operator and the standard axiom for sequential composition:

(K) [α] (Φ → Ψ) ∧ [α] Φ → [α] Ψ ([α] -N ecessitation) From Φ infer [α] Φ (Composition) [α] [β] Φ ←→ [α; β] Φ.
Moreover we suppose the following axioms for atomic actions and perceptual actions:

(Det At ) a Φ → [b] Φ (P erc 1 ) ϕ ←→ observe(ϕ) (P erc 2 ) [observe(ϕ)] Datum(ϕ) 7 a) All instances of propositional tautologies; b) Modus ponens: from Φ and Φ → Ψ infer Ψ; c) K-axiom for Bel: Bel(Φ → Ψ) ∧ BelΦ → BelΨ; d) Bel-Necessitation: From Φ infer BelΦ. 8 a) Nonnegativity: P (ϕ) ≥ 0; b) Probability of Truth: P ( ) = 1; c) Additivity: P (Φ 1 ∧ Φ 2 ) + P (Φ 1 ∧ ¬Φ 2 ) = P (Φ 1 ); d) Equivalence: From Φ 1 ←→ Φ 2 infer P (Φ 1 ) = P (Φ 2 ).
9 See [START_REF] Fagin | Reasoning about knowledge and probability[END_REF].

(P erc 3 ) observe(ϕ) Φ → [observe(ϕ)] Φ.

(P erc 1 ) says that: 1) it is always possible for the agent to perceive ϕ if ϕ is true in the external world and 2) if it is possible for the agent to perceive ϕ then ϕ is true in the external world. (P erc 2 ) says that after ϕ is perceived, ϕ becomes a perceptual datum that is the action of perceiving ϕ moves a new datum ϕ into the data space of the agent.

(P erc 3 ) guarantees that perceptual actions are deterministic. (Det At ) guarantees that atomic actions follow the same path.

We note that the previous axioms correspond to the following semantic constraints:

• for every w ∈ W if w ∈ R a 0 (w) and w ∈ R b 0 (w) then w = w ; • for every w ∈ W R observe(ϕ) 1 (w) = ∅ if and only if M, w ϕ; • for every w ∈ W if w ∈ R observe(ϕ) 1 (w) then ϕ = DAT A(w ); • for every w ∈ W if w ∈ R observe(ϕ) 1 (w) and w ∈ R observe(ϕ) 1 (w) then w = w .
Finally we suppose that the following are valid properties of retrieve mental operations:

(Retr 1 ) retrieve(ϕ) → ¬T est(ϕ) (Retr 2 ) [retrieve(ϕ)] T est(ϕ) (Retr 3 ) retrieve(ϕ) Φ → [retrieve(ϕ)] Φ.
(Retr 1 ) says that if it is possible for the agent to retrieve ϕ then ϕ is a a representation which is not actually scrutinized. (Retr 2 ) says that after ϕ gets retrieved, ϕ is scrutinized by the agent. Thus the mental operation of retrieving ϕ has the function of modifying the mental setting of the agent, by moving a new representation ϕ into the test space of the agent. Finally (Retr 3 ) guarantees determinism for retrieve mental operations. We note that the previous axioms correspond to the following semantic constraints:

• for every w ∈ W if ϕ = T EST (w) then R retrieve(ϕ) 2 (w) = ∅; • for every w ∈ W w ∈ R observe(ϕ) 2 (w) then ϕ = T EST (w ); • for every w ∈ W if w ∈ R retrieve(ϕ) 2 (w) and w ∈ R retrieve(ϕ) 2 (w) then w = w .
We call SURPRISE the logic axiomatized by the axioms and inference rules for probabilities and beliefs given in [START_REF] Halpern | Reasoning about uncertainty[END_REF]; [START_REF] Fagin | Reasoning about knowledge and probability[END_REF] and discussed above, the axiom Incl Bel/P rob , the previous axioms and inference rules for actions in general and the special axioms for atomic actions, perceptual actions and retrieve mental operations. We call SURPRISE models the set of models M Surp ⊆ M satisfying all the semantic constraints imposed in this section and write |= Surp ϕ if ϕ is valid in all SURPRISE models. Moreover we write Surp ϕ if ϕ is a theorem of SURPRISE.

Having defined retrieve mental operation and formulated their properties we can characterize the notion of background expectation (or background belief). A background (or passive) expectation is in our vocabulary an expectation whose content is available and accessible by means of a retrieve mental operation, that is a background expectation is an expectation whose content can be mentally retrieved. Formally:

Background(ϕ) = def retrieve(ϕ) .
The present distinction between expectations and beliefs under scrutiny of the form T est(ϕ) and background expectations and beliefs of the form Background(ϕ) looks similar to the distinction given in psychology between active expectations and passive expectations [START_REF] Tversky | Support theory: a nonextensional representation of subjective probability[END_REF][START_REF] Kahneman | Variants of uncertainty[END_REF]. According to Kahneman & Tversky the former "occupy consciousness and draws on the limited capacity of attention"; the latter kind are "automatic and effortless". Passive expectations can be either permanent, such as categories and assumptions about the external world, or temporary, such as the priming effects in psychological experiments.10 3 Kinds of Surprise

Mismatch-based surprise and astonishment

It is the objective of having an operational and cognitively plausible model of surprise which gives rise to the need to introduce and exploit the notion of representation under scrutiny (representation to be tested). Indeed we want to model realistic cognitive systems which process input data and which are focused on a small portion of their internal information state. The purpose of this section is to clarify the distinction between mismatch-based surprise (there is a recognized conflict between the agent's input data and the agent's representation under scrutiny) and astonishment. While the notion of mismatch-based surprise is an operational notion and is associated with a recognized logical conflict between the incoming information and a representation under scrutiny, astonishment is in our view the response to the recognized implausibility of the input data. When I am astonished about something, I cannot believe what I see and this presupposes that I'm trying to believe, I'm trying to find an explanation for what I see, but I'm suspended. Astonishment seems to be due to a difficulty, to a delay due to this process of integration, of accounting for, which in this case is not automatic and fast, not immediately successful. We cannot in fact believe something just putting it in our belief base; we must check about consistency (especially if there are reasons for suspecting some inconsistency). If the actual input generates an intense astonishment then it means that the input is unexpected and rather unpredictable from my actual beliefs. If I have to accept it, I have to adjust my beliefs in such a way that they can account for this unexpected event. Generally in order to cope with an intense astonishment, I need a deep and large revision of my well consolidated beliefs.

Example 1.

Consider a person being in a terrible delay. He needs to take a train from Florence to Rome at 8:00 a.m. It is 7:56 a.m. and the guy is still running to reach the Florence station. Finally he arrives at the station at exactly 8:00 a.m. and checks whether the train for Rome is standing in the station. At the moment of the perceptual test the agent has the representation of the train for Rome standing in the station explicit in his mind and attributes a high probability to this fact. When the agent perceives the train for Rome in not standing the agent gets very surprised since the incoming representation (logically) conflicts with the explicit representation of the train for Rome is standing in the station and the probability assigned to the fact the train for Rome is standing in the station is very high. This kind of Surprise is what we call mismatch-based surprise.

Example 2. It is 5:50 p.m. and Bill is working in his office when Mary phones Bill and tells him: "I will come to your office at 6 p.m.! Wait for me there!". After Mary's call Bill decides to stop working and to rest until Mary will arrive. Bill expects with high probability that Mary will knock on the door of the office at 6 p.m. and focuses his attention on this. It is 5:53 p.m. and suddenly someone knocks on the door. Bill opens the door and sees that a policeman is standing in front of the door. There is not logical conflict between the scrutinized representation Mary will knock on the door of the office at 6 p.m. and the perceived fact a policeman knocks on the door of the office at 5:53 p.m (indeed Mary knocks on the door at 6 p.m. is not inconsistent with a policeman knocks on the door at 5:53 p.m). Thus there is not mismatch-based surprise. But Bill gets very astonished by perceiving the fact a policeman knocks on the door of the office at 5:53 p.m.. Indeed Bill retrieves the information concerning a policeman knocking on the door of the office at 5:53 p.m. from his background knowledge and recognizes the implausibility of the perceived fact given what he knows ("I wouldn't have expected to perceive a policeman knocking on the door of my office!").

Let us consider more carefully the two notions of mismatch-based surprise and astonishment from a qualitative and quantitative point of view. We want to specify the mental configurations associated with these two emotional responses and to provide the criteria to quantify them (to measure their intensity).

Definition 1: Mismatch-based Surprise (given the conflict between a perceived fact and a scrutinized representation). The cognitive configuration of mismatchbased surprise relative to the mismatch between a perceptual datum ψ and a scrutinized representation ϕ is defined by the following facts:

1. ψ is the agent's perceptual datum; 2. ϕ is the representation scrutinized by the agent and 3. the agent believes that ϕ and ψ are incompatible facts.

Formally: M ismatchSurprise(ψ, ϕ) = def Datum(ψ) ∧ T est(ϕ) ∧ Bel(ψ → ¬ϕ).
Definition 2: (Retrieval-based) Astonishment. The cognitive configuration of (retrievalbased) Astonishment relative to a perceptual datum ψ is defined by the following facts:

1. ψ is the agent's perceptual datum; 2. the agent can retrieve from its background knowledge either the expectation ¬ψ or the expectation that ψ that is, either the expectation that ¬ψ or the expectation that ψ is "mentally" available at a background level.

Formally:

Astonishment(ψ) = def Datum(ψ) ∧ (Background(¬ψ) ∨ Background(ψ)) 11
Definition 3: Intensity of Mismatch-based Surprise (given the conflict between a perceived fact and a scrutinized representation). The mismatch-based surprise relative to the mismatch between a perceptual datum ψ and a scrutinized representation ϕ has intensity equal to (or higher than) c if and only if the probability assigned to the scrutinized expectation that ϕ (invalidated by the perceived fact ψ) is equal to (or higher than) c. Formally:

IntensityM ismatchSurprise(ψ, ϕ) ≥ c = def M ismatchSurprise(ψ, ϕ)∧P (ϕ) ≥ c IntensityM ismatchSurprise(ψ, ϕ) > c = def M ismatchSurprise(ψ, ϕ)∧P (ϕ) > c IntensityM ismatchSurprise(ψ, ϕ) = c = def M ismatchSurprise(ψ, ϕ)∧P (ϕ) = c.
Definition 4: Intensity of (Retrieval-based) Astonishment. The (retrieval-based) astonishment relative to a perceptual datum ψ has intensity equal to (or higher than) c if and only if the probability assigned to ¬ψ (the negation of the perceived fact) is equal to (or higher than) c. Formally:

IntensityAstonishment(ψ) ≥ c = def Astonishment(ψ) ∧ P (¬ψ) ≥ c IntensityAstonishment(ψ) > c = def Astonishment(ψ) ∧ P (¬ψ) > c IntensityAstonishment(ψ) = c = def Astonishment(ψ) ∧ P (¬ψ) = c.
According to definitions 3 and 4 the intensity of (retrieval-based) astonishment is equal to the probability assigned to the opposite of the perceived fact (we can call it degree of unexpectedness of the perceived fact as in [START_REF] Ortony | Surprisingness and expectation failure: Whats the difference[END_REF] whereas the intensity of mismatch-based surprise is equal to the probability assigned to the formula invalidated by the perceived fact.

Let us discuss some formal properties of (retrieval-based) astonishment and mismatchbased surprise.

Proposition 1.12

|= Surp IntensityM ismatchSurprise(ψ, ϕ) = c∧(Background(ψ)∨Background(¬ψ)) → IntensityAstonishment(ψ) ≥ c
The previous proposition says that if the agent is surprised by the mismatch between the perceptual datum ψ and a scrutinized expectation that ϕ and this surprise has intensity c then if the agent has either a background available expectation that ψ or a background available expectation that ¬ψ then the intensity of astonishment is higher than c. Therefore (retrieval-based) astonishments are by nature more intense than mismatch-based surprises. The reader should also note that the two dimensions of surprise are not necessarily complementary (the sum of the two is not necessarily equal to 1). Indeed I could be surprised with intensity 0.5 by the mismatch between the perceptual datum ψ and the scrutinized expectation that ϕ and be astonished with intensity 0.7 by the recognized implausibility of ψ. Thus the two kinds of surprise are both qualitatively and quantitatively different.

Often mismatch-based surprise and (retrieval-based) astonishment occur together after having perceived a certain fact ψ. According to proposition 1 the intensity of (retrievalbased) astonishment is higher than the intensity of mismatch-based surprise. Consider next scenario.

Example 3. Imagine a person walking along the Thames. The person is scrutinizing whether there is the tower of London (ϕ) and is attributing a high probability to this fact. Suddenly the person turns the eyes toward the river and perceives there is a whale (ψ) (see the recent facts in London). The person gets surprised because of the recognition of the incompatibility between ψ and ϕ. Indeed the person believes that ψ → ¬ϕ.

But he also gets highly astonished. Indeed the person recognizes (after having retrieved from his background knowledge the information about ψ) the implausibility of the fact there is a whale (or even stronger the impossibility of the fact there is a whale). The intensity of the astonishment is equal to the probability assigned to ¬ψ.

Inference-based astonishment

In the previous paragraph we have defined astonishment the kind of surprise which involves a recognized implausibility of a perceived fact ϕ. We have assumed that the recognition of implausibility of the perceived fact ϕ is based on the mental availability of either the expectation that ϕ or the expectation that ¬ϕ. Indeed according to definition 2 retrieval-based astonishment concerns those background passive expectations that the agent can retrieve from the background level. As noticed by Ortony & Partridge [START_REF] Ortony | Surprisingness and expectation failure: Whats the difference[END_REF] surprise can also arise from an inconsistency between an implicit passive expectation and the input proposition. With implicit expectations they mean all those facts that can be inferred from the explicit beliefs by few and simple deductions (see Figure 1).

We think that Ortony & Partridge's distinction is relevant for a model of surprise and that in order to implement it formally we should relax the assumption of logical omniscience of the agent. In order to do it formally we should identify in the complete set of beliefs a subset of this and call it the set of explicit beliefs (or belief base as in the tradition of Belief Revision13 ). This is the set of beliefs that the agent can use to make inferences and which is not closed under classical inference14 . Given the set of explicit beliefs we could define implicit (passive) beliefs as all those beliefs which can be inferred from the elements of the belief base (and which are not members of the belief base).

Having defined a set of Explicit Beliefs and a set of Implicit Beliefs, we can make more precise our definition of Astonishment. Indeed we can account for the astonishment due to a recognized conflict between a post-hoc belief or expectation (a belief which is inferred from the explicit beliefs and which was implicit before the perception) and the incoming input data: we call it Inference-based Astonishment.

Since the distinction between explicit and implicit belief is not formally specified under the present analysis we only give here a verbal characterization of Inference-based Astonishment.

Definition 5: (Inference-based) Astonishment. The cognitive configuration of (Inferencebased) Astonishment relative to a perceptual datum ψ is defined by the following facts:

1. ψ is the agent's perceptual datum (something perceived by the agent);

2. the agent can infer and effectively infer ¬ψ from its explicit beliefs (when ¬ψ was the content of an implicit belief before the perception).

We should also consider as a matter of completeness all those cases of post-hoc reconstruction of the probability of the perceived event. This would allow us to generalize definition 5. In those cases while attempting to assimilate/integrate the perceived datum the agent "derives" that the event is not so probable (this is different from inferring some fact which is incompatible with the perceived fact). While asking to himself: was this unpredicted event/datum predictable? it reconstructs the probability of the event and conclude that "I would never had expected that". Therefore the intensity of inference-based astonishment relative to the perceived fact ψ must depend on the probability assigned to ¬ψ (the higher the probability assigned to ¬ψ, the more intense the astonishment).

We have provided two different notions of astonishment. On one side (definition 2) after perceiving ψ there is a simple retrieval of either the expectation that ψ or the expectation that ¬ψ when either the expectation that ψ or the expectation that ¬ψ is mentally available at the background (passive) level. On the other side (definition 5) either the negation of the perceived fact is inferred from the explicit beliefs or there is a post-hoc reconstruction of the probability of the perceived fact (a probabilistic inference). In both cases some mental operation must be done in order to make the agent aware of the implausibility of the perceived fact. We conclude this section by summarizing our basic ontology of on-line surprise (whose cognitive configuration is obtained during the perceptual phase and before an eventual belief reconsideration). In our view at least three species must be considered: surprise based on the mismatch between a representation under scrutiny and an incoming input (definition 1), retrieval-based astonishment (definition 2) and inference-based astonishment (definition 5).

Some comments

Let us stress more in this section the main differences between our approach and Ortony & Partridge's approach by making explicit the main important issues that are neglected in their model and that our model tries to clarify.

Ortony & Partridge's model does not capture in our view the important distinction between the previous two kinds of astonishments (retrieval-based astonishment and inference-based astonishment). Ortony & Partridge's model is only focused on inferencebased astonishment and completely neglects to account for the other important kind.

In inference-based astonishment, the subject did not in fact derive the prediction/expectation that ¬ϕ before perceiving ϕ (the prediction is just potential and implicit in its mind). While attempting to assimilate/integrate the new data he infers from his explicit beliefs the opposite. Therefore the mental operation involved in this kind of astonishment is an inferential action15 : it transforms some potential and implicit expectation (or belief) into an explicit and scrutinized one. This is exactly the content of the informal definition 5 given in section 3.2.

In retrieval-based astonishment on the contrary, when perceiving ϕ a pre-existent expectation that ϕ (or a pre-existent expectation that ¬ϕ) is available (it can be retrieved from the background level even without a constructive inferential process). Indeed in our view an agent has always a certain number of accessible beliefs and expectations in background (at an unconscious and automatic level) and these expectations and beliefs in background must be distinguished from the representation under scrutiny formally identified as a test formula (see Figure 1 and section 2.3). When perceiving ϕ, retrieval-based astonishment may simply arise from the automatic retrieval of either the background probabilistic expectation that ϕ or the background probabilistic expectation that ¬ϕ. Therefore the mental operation involved in retrieval-based astonishment is a retrieve mental operation which transforms some background expectation (or belief) into a scrutinized expectation. This is exactly the content of the formal definition 2 given in section 3.1.

In our view Ortony & Partridge's model does not capture this distinction between 1) surprise arising from the recognition of implausibility of the perceived fact due to an inferential process from my explicit beliefs and 2) surprise arising from the recognition of implausibility of the perceived fact due to a retrieval of a background expectation. The incompleteness of Ortony & Partridge's model is due to the lack of distinction between background expectations and representations on one side and implicit expectations and beliefs on the other side (indeed they only account for the second kind). This distinction is relevant in our approach and it gives us the possibility to articulate a richer typology of surprise.

Moreover, in our model there are two parallel components and paths for surprise, and there are two parameters that we should take into account in order to quantify surprise (see figure 2 below).

(i) I can have an expectation under scrutiny whose content is ϕ (the expected event or entity): when this prediction is invalidated, happens to be wrong, this means that I perceive something different. In other word there is an input datum ψ mismatching with ϕ. Even nothing is something: also the absence of any object when I was expecting and scrutinizing ϕ, that is the fact that ϕ does not happens (¬ϕ) is in any case an unpredicted/unexpected input datum which invalidates the representation under scrutiny ϕ.

(ii) Having perceived ψ, the expectation that ψ (or the expectation that ¬ψ) is available at the background and unconscious level (or the expectation that ¬ψ is inferred from explicit beliefs and expectations).

We claim that, on the one side, surprise is a function of the probability of the invalidated fact under scrutiny (ϕ); while on the other side it is a function of the probability of the perceived fact ψ . On the one side the more certain was my scrutinized expectation, the more probable is ϕ, the more surprise I am 16 (see definition 3 in section 3.1).

On the other side the more unpredictable, the more unexpected ψ (the more expected ¬ψ), the more astonished I am (see definition 4 in section 3.1 as well as the generalization of definition 5 which deals with probabilistic inference). To distinguish these two facets, components, and processes we have proposed to use for the former case the term mismatch-based surprise (the signal of the invalidation of the expectation under scrutiny), and the term astonishment (either retrieval-based astonishment or inferencebased astonishment) for the latter case. Ortony & Partridge seem to consider surprise only the second phenomenon and path. Indeed according to their model intensity of surprise only depends on the degree of "unexpectedness" of the perceived fact 17 .

But not always the surprise processing involves the two paths. Indeed one can be surprised by some perceived fact ψ one did not expect without having to expect and test something else which is evaluated to be incompatible with ψ: not necessarily the astonishment presupposes the mismatch-based surprise. Moreover one can be surprised by some perceived fact ψ which is evaluated to be incompatible with some scrutinized fact ϕ without having to be astonished by the recognized implausibility of ψ: not necessarily a mismatch-based surprise entails an astonishment as a felt reaction. 16 At a meta-level too we might say that the mismatch was unexpected. 17 The same criticism can be addressed toward all those computational models which claim that surprise is simply a function of unexpectedness of the incoming input and which neglect the dimension strength of the invalidated expectation [START_REF] Macedo | Modelling forms of surprise in an artificial agent[END_REF][START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF][START_REF] Ortony | Surprisingness and expectation failure: Whats the difference[END_REF]. Other models based on Information theory claim that Surprise is a function of the distance between prior probabilities and posterior probabilities after the conditioning on the set of perceived data (see [START_REF] Baldi | Surprise: A shortcut for attention[END_REF] for instance). For the same reasons we believe that this last approach is incomplete since it is unable to provide qualitative distinctions inside the surprise phenomenon.

Surprise and belief change

As some psychologists have stressed [START_REF] Meyer | An experimental analysis of surprise[END_REF][START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF] surprise can culminate in a process of belief change. The aim of the following analysis is to suggest some interesting ways a formal model of cognitive surprise can be integrated with a formal model of cognitive belief change. Formal approaches to belief revision are mainly interested in finding rationality principles and postulates driving belief change (this is for instance the main purpose of the classical AGM theory [START_REF] Alchourron | On the logic of theory change: Partial meet contraction and revision functions[END_REF]. All those models implicitly assume that when the agent perceives some fact ϕ the perception is always a precursor of a belief change with ϕ. Thus the main problem with AGM theory is a missed identification of the precursors of belief change. Our attempt here is to clarify under what conditions belief revision should be triggered after having perceived a certain fact. We claim that surprise plays a crucial role in triggering this kind of process and that it is implausible to assume that realistic cognitive agents revise their beliefs with ϕ every time they perceive a new fact ϕ. Realistic and non-omniscient cognitive agents are situated in complex environments where many tasks must be solved. Since accurate belief revision and update require time and considerable computational costs, realistic cognitive agents need some mechanism which is responsible: 1) for signaling the global inconsistency of the knowledge base with respect to the incoming input and 2) for the revision of beliefs and expectations of the agent. One of the adaptive functions of surprise is exactly this. Belief change in cognitive agents is triggered by very surprising incoming input. The intensity of surprise relative to the incoming input "signals" to the agent that things are not going as expected and that the knowledge of the environment must be reconsidered. Indeed wrong beliefs generally lead to bad performances and to failure in the intention and goal fulfillment. On the other hand resource bounded cognitive agents do not generally reconsider their beliefs and expectations when the input data are not recognized to be incompatible or implausible with respect to their pre-existent knowledge. Indeed it is not convenient for the survival of the agent to update or reconsider beliefs every time a new fact is perceived. When the world flows as expected and we are not aware of the inadequacy of our knowledge of the world, we do not need to criticize and reconsider this knowledge. Indeed reconsidering beliefs after every perception would strongly interfere with the agent's ongoing performance and would continuously divert its attention away from its intentionally driven activity. A model of cognitive belief change should be able to account for this trade-off between extensive belief change triggered by surprise and belief change avoidance when perception does not generate surprise.

Dealing with unexecutable updates

In [START_REF] Kooi | Probabilistic dynamic epistemic logic[END_REF] a combination of the dynamic epistemic logic of [START_REF] Gerbrandy | Bisimulations on Planet Kripke[END_REF][START_REF] Gerbrandy | Reasoning about information change[END_REF] with the probabilistic logic of Fagin & Halpern is given. This combination results in a probabilistic dynamic epistemic logic where it is possible to talk about beliefs and probabilities as well as information change for beliefs and probabilities. In this probabilistic extension of Gerbrandy's logic of information update the symbol ϕ! is introduced. ϕ! is the process of updating beliefs with an arbitrary sentence ϕ. 18 The aim of this section is to suggest a way to modify the framework given in [START_REF] Kooi | Probabilistic dynamic epistemic logic[END_REF] and [START_REF] Gerbrandy | Bisimulations on Planet Kripke[END_REF]; [START_REF] Gerbrandy | Reasoning about information change[END_REF] in order to investigate the role of surprise in information change.

In order to do this we must import update processes into our formal language L SURP . UP D is the set of update processes defined as the smallest set such that:

• if ϕ ∈ P ROP then ϕ! ∈ UP D.
We call L SURP+ the new extended language with update processes. The new language L SURP+ is given by the following rule in extended Backus-Naur Form:

Φ ::= p|¬Φ|Φ 1 ∧ Φ 2 |BelΦ| [α] Φ|d 1 P (Φ 1 ) + ... + d n P (Φ n ) ≥ c|T est(ϕ)|Datum(ϕ)| [ϕ!] Φ where p ∈ Π, ϕ ∈ P ROP , α ∈ ACT , d 1 , ..., d n , c are real numbers and ϕ! ∈ UP D.
Semantics of formulas in L SURP+ is the same semantics given for formulas in L SURP (see section 2.2). We only need to provide semantics for formulas of the form

[ϕ!] Φ. This is given next. 19 • M, w [ϕ!] Φ ⇐⇒ ∀(M , w ) if (M , w ) ∈ R ϕ! (M, w) then M , w Φ
We suppose that R ϕ! (M, w) is defined according to the following definition 6.

Definition 6. Given a model M =< W, B, R 0 , R 1 , R 2 , P, T EST, DAT A, π >, a world w ∈ W and a propositional formula ϕ ∈ P ROP we suppose that

EITHER R ϕ! (M, w) = ∅ OR R ϕ! (M, w) = ( M ϕ , w ϕ ).
Moreover we suppose that M ϕ and w ϕ are defined as follows.

1. w ϕ = w.

2. M ϕ =< W, B ϕ , R 0 , R 1 , R 2 , P ϕ , T EST, DAT A, π > where P ϕ (w) = ( W ϕ w , X ϕ w ) and W ϕ
w , X ϕ w , B ϕ are defined according to the following (a), (b) and (c).

18 Richer logics of information update have been proposed. In Baltag et al.'s logic of information update [START_REF] Baltag | The logic of public announcements, common knowledge and private suspicions[END_REF] for instance complex communicative actions are described in terms of action models, which stand for complex events that carry information for agents. Different kinds of informational scenarios in a multi-agent setting can be described in this logic. For instance we can describe scenarios where not all agents have the same observational access to what is happening in reality. In van Benthem et al. (2006) probabilities are added to Baltag et al.'s framework in order to reason about probabilistic information in a multi-agent setting and to describe how belief and probability update is affected by the reliability of the source of information. 19 We generalise here the standard approach of dynamic logic where actions are interpreted as transitions between worlds in a Model (see also [START_REF] Van Linder | Seeing is believing (and so are hearing and jumping)[END_REF][START_REF] Meyer | A logical approach to the dynamics of commitments[END_REF]. Indeed we interpret update processes as transitions between pairs (Model, world). (c) for all w ∈ W and w ∈ W ϕ w : X ϕ w (w ) = Xw(w ) Xw(||ϕ|| Ww ) .

According to definition 6 updating with ϕ either cannot be performed or yields an updated model M ϕ which differs from the original model only with respect to the accessibility relations for Bel modal operator and the probability functions. When an update with ϕ is successfully performed the original model M is transformed into the updated model M ϕ in such a way that for all worlds w all alternatives that an agent considers possible where ϕ does not hold are removed and worlds where ϕ does not hold are removed from the sample space W w . Moreover for all worlds w the probability function is redefined according to Condition 2(c).

In [START_REF] Kooi | Probabilistic dynamic epistemic logic[END_REF]; [START_REF] Gerbrandy | Reasoning about information change[END_REF]; [START_REF] Gerbrandy | Bisimulations on Planet Kripke[END_REF] it is assumed that an update with ϕ is always executable i.e. the authors suppose that R ϕ! (M, w) is never empty and always yields the updated model M ϕ . Thus these theories of information update assume that the formula ϕ! is valid. Here we suppose that belief update is triggered only under certain specific preconditions. This implies that in our view R ϕ! (M, w) may be empty (see definition 6). This is the most striking difference between our version of belief update and standard versions of it. We will try to characterize the necessary preconditions for belief update in the next section and to investigate the role of surprise in the process.

The first relevant aspect to verify is whether the previous model transformation guarantees that the semantic constraints given in section 2.3 are preserved. This is indeed the case.

Lemma 1. If M is a SURPRISE model then M ϕ is a SURPRISE model too.
Let us suppose in a way similar to [START_REF] Kooi | Probabilistic dynamic epistemic logic[END_REF] that an update with ϕ can be performed only if the agent does not assign zero probability to ϕ. This assumption is made explicit in our framework by the next postulate:

(NotZero Upd ) ϕ! → P (ϕ) > 0.
This property corresponds to the following semantic constraint:

• for every w ∈ W if R ϕ! (M, w) = 0 then X w (||ϕ|| Ww ) > 0. 20
You should notice that under this requirement formula Belϕ → [¬ϕ!] ⊥ becomes valid, that is if an agent believes that ϕ holds then an update with ϕ cannot be executed. This implies that under the present framework belief revision with inconsistent information is left unspecified. 21We also postulate that an agent has always epistemic access to all executable updates that is, if an update with sentence ϕ is executable then the agent believes that the update with ϕ is executable. Formally:

(Access Upd ) ϕ! → Bel ϕ! .
This property corresponds to the following semantic constraint:

• for every w ∈ W if there is a w such that w ∈ B(w) and R ϕ! (M, w ) = 0 then R ϕ! (M, w) = 0.

Before starting to investigate some formal consequences of our definition of belief update we provide the following definition of objective formula.

Definition 7. We define the set of objective formulas OBJ = {o 1 , o 2 , ...} as the smallest set such that:

• if ϕ ∈ P ROP then ϕ ∈ OBJ (propositional formulas are objective formulas);

• if ϕ ∈ P ROP then T est(ϕ) and ¬T est(ϕ) ∈ OBJ (test formulas and negation of test formulas are objective formulas);

• if ϕ ∈ P ROP then Datum(ϕ) and ¬Datum(ϕ) ∈ OBJ (perception formulas and negations of perception formulas are objective formulas);

• if o 1 ∈ OBJ and α ∈ ACT then [α] o 1 and α o 1 ∈ OBJ.
We can now prove that the principles summarized in the following Lemma 2 are sound given the semantics of update processes (definition 6).

Lemma 2.

(Upd 1 ) ϕ! Φ → [ϕ!] Φ (Upd 2 ) o 1 → [ϕ!] o 1 where o 1 is an objective formula (Upd 3 ) Bel(ϕ → ϕ! Φ) → [ϕ!] BelΦ (Upd 4 ) ϕ! BelΦ → Bel(ϕ → [ϕ!] Φ) (Upd 5 ) ( n i=1 d i P (ϕ ∧ ϕ! Φ i ) ≥ cP (ϕ)) → [ϕ!] ( n i=1 d i P (Φ i ) ≥ c) (Upd 6 ) ( ϕ! ( n i=1 d i P (Φ i ) ≥ c) → ( n i=1 d i P (ϕ ∧ [ϕ!] Φ i ) ≥ cP (ϕ)) (Upd 7 ) [α] ϕ! Φ → [ϕ!] [α] Φ
(Upd 1 ) establishes that belief updates are deterministic. According to (Upd 2 ) the truth value of an objective formula does not change after a belief update. (Upd 3 ), (Upd 4 ), (Upd 5 ) and (Upd 6 ) describe how beliefs and probabilities change after an update. According to (Upd 7 ) the effects of an update process on a model are independent from the fact that the update process may be executed after or before a sequence of actions (a sequence where each element is either an atomic action or a perceptual action or a retrieve mental operation).

Finally we can precisely define our extended logic of surprise with update processes. We call SURPRISE+ the logic axiomatized by the axioms and inference rules of the logic SURPRISE (see section 2.3) plus the previous nine principles for update processes (Upd 1 )-(Upd 7 ), (Access Upd ) and (NotZero Upd ). Moreover we write Surp+ ϕ if ϕ is a theorem of SURPRISE+.

We are able to prove by the seven principles summarized in Lemma 2 and the previous postulates Access Upd and Incl Bel/P rob (section 2.3) that two compact reduction principles for beliefs and updates on side and probabilities and updates on the other side follow from the axiomatic system of our logic. These two principles are similar to Gerbrandy's reduction principle for beliefs and updates [START_REF] Gerbrandy | Reasoning about information change[END_REF][START_REF] Gerbrandy | Bisimulations on Planet Kripke[END_REF]) and Kooi's reduction principle for probabilities and updates [START_REF] Kooi | Probabilistic dynamic epistemic logic[END_REF]. These results are summarized in the following theorem.

Theorem 1.

(Upd 8 ) Surp+ Bel(ϕ → [ϕ!] Φ) ∨ [ϕ!] ⊥ ←→ [ϕ!] BelΦ (Upd 9 ) Surp+ ( n i=1 d i P (ϕ∧[ϕ!] Φ i ) ≥ cP (ϕ))∨[ϕ!] ⊥ ←→ [ϕ!] ( n i=1 d i P (Φ i ) ≥ c)
Several interesting properties of update processes follow from theorem 1 and the principles given in Lemma 2. Let us consider only some of them. According to (Upd 10 ) after an update with ϕ the agent believes that ϕ holds. According to (Upd 11 ) for each m-level nested belief that o 1 holds (where o 1 is an objective formula), the m-level nested belief is preserved after a belief update. (Upd 12 ) shows the strong similarity between updating with propositional formulas in our framework and classical bayesian updating.22

Surprise-based belief update

We have noticed in the previous section that a relevant difference exists between the present approach to belief update and some standard approaches [START_REF] Kooi | Probabilistic dynamic epistemic logic[END_REF][START_REF] Gerbrandy | Reasoning about information change[END_REF][START_REF] Gerbrandy | Bisimulations on Planet Kripke[END_REF]. Differently from standard approaches we have supposed that belief update is triggered only under certain specific preconditions and is not always executable. The aim of this section is characterize some of those necessary preconditions for belief update and to show that surprise plays a crucial role in triggering this mental process. We begin with the formalization of our general intuition by supposing that two necessary preconditions for belief update are expressed by the following two additional principles (NecT rig1) and (NecT rig2).

(NecT rig1) ϕ! → Datum(ϕ) 23 (NecT rig2) ϕ! ∧T est(ψ) →Bel(ϕ → ¬ψ)∨Background(ϕ)∨Background(¬ϕ) 24
According to principle (NecT rig1) an agent cannot update its beliefs with sentence ϕ unless ϕ is something that the agent has perceived (ϕ is a perceptual datum of the agent). According to principle (NecT rig2) if the agent is focused on the expectation that ψ then the agent cannot update is beliefs with ϕ unless either the agent recognizes a contradiction between ϕ and its scrutinized expectation that ψ or ϕ (or ¬ϕ) is the content of an available background expectation. Both principles formally express the following postulate. An agent can reconsider its previous knowledge with some piece of information ϕ only if: 1) ϕ is some piece information that agent has perceived and which is collected as a perceptual datum and 2) either the agent recognizes (is aware of) the contradiction and incompatibility between the perceptual datum ϕ (the object of its perception) and its scrutinized expectation or the probabilistic expectation that ϕ (or the expectation that ¬ϕ) is (mentally) available at a background level. Thus according to the previous postulate if an agent is not aware of the inconsistency between the perceptual datum ϕ and its actual scrutinized expectation that ψ and does not have access to the information concerning the plausibility of ϕ then the agent cannot revise its knowledge base on the basis of the perceptual datum. The following example is given in order to defend the plausibility of the present postulate.

Example 4. Mary goes shopping downtown. She is looking for a nice pair of shoes for New Year's Eve party. She remembers having heart from Bill that a well-stocked shoe shop has been opened at the main square of the town. Mary trusts Bill since she thinks that Bill gives always good advice. Thus she decides to reach the main square of the town in order to find the shoe shop. Now Mary expects that ϕ 1 = she will 23 This postulate corresponds to the following semantic constraint: for every w

∈ W if R ϕ! (M, w) = ∅ then ϕ = DAT A(w).
24 Given the definition Background(ϕ) = def retrieve(ϕ) and property (P erc 1 ) of perceptual actions (see section 2.3) we can express the semantics corresponding to this principle by the following first order formula:

for every w ∈ W if R ϕ! (M, w) = ∅ and T EST (w) = ψ and there is a w such that w ∈ B(w) and R

observe(ϕ) 1 (w ) = ∅ and R observe(ψ) 1 (w ) = ∅ then R retrieve(ϕ) 2 (w ) = ∅ or R retrieve(¬ϕ) 1 (w ) = ∅.
find a shop selling a nice pair of shoes at the main square with high probability and focuses her attention on this. When walking toward the shop Mary observes ϕ 2 = there is a Japanese restaurant in the corner of the street. Nevertheless Mary does not care about ϕ 2 . Indeed: 1) ϕ 2 is not evaluated to be incompatible with ϕ 1 and 2) Mary does not have a background available expectation that ϕ 2 nor a background available expectation that ¬ϕ 2 which makes her able to recognize the implausibility of the perceived fact ϕ 2 . Thus Mary does not reconsider her knowledge base according to what she has perceived since both a recognition of implausibility of the perceived fact and a recognition of incompatibility between the perceived fact and the scrutinized expectation that ϕ 1 are lacking. Finally Mary arrives at the main square of the town where she expects to find the shoe shop and to buy a nice pair of shoes. But Mary sees that no shop is there. Mary recognizes the inconsistency between her scrutinized expectation (ϕ 1 ) and what is being perceived. Indeed there is not a shoe shop at the place where she expected to find a shoe shop. Since Mary is aware of the incompatibility between the perceived fact and her actual scrutinized expectation she can reconsider her belief base according to the perceptual datum.

Given the definitions of astonishment and mismatch-based surprise (section 3.1) and supposing that the previous two principles (NecT rig1) and (NecT rig2) are added to our logic SURPRISE+ the following becomes a provable theorem.

Proposition 3. Surp+ ϕ! ∧ T est(ψ) → M ismatchSurprise(ϕ, ψ) ∨ Astonishment(ϕ)
According to Proposition 3 if the agent is focused on the expectation that ψ then the agent cannot revise its knowledge base with the perceived fact ϕ unless either the agent gets surprised by the mismatch between the perceptual datum ϕ and the scrutinized expectation that ψ or the agent gets astonished by the recognized implausibility of ϕ. This proposition expresses a general cognitive principle: belief update with a perceived fact ϕ is triggered only if the agent is surprised or astonished by the perception of ϕ, that is Some form of surprise is a necessary precondition for belief update. This is for us a crucial principle for designing resource bounded cognitive agents which are focused on a small portion of their complete informational state and which need some mechanism for "signaling" that beliefs must be updated.

After having characterized two "necessary" preconditions for triggering belief update we move toward a brief investigation of the "necessary and sufficient conditions". We only provide here some general intuitions about this issue.

It has been noticed by some psychologists [START_REF] Meyer | Towards a process analysis of emotions: The case of surprise[END_REF] that the trigger of a belief update process depends on the intensity of surprise associated with the perception of some fact ϕ: the higher the intensity of surprise relative to the perception of ϕ, the higher the probability that the agent will revise its knowledge with ψ.

In our view a first rough approximation of the necessary and sufficient preconditions for belief update is obtained by introducing the previous dimension: the intensity of surprise associated with the perception of ϕ.

We suggest the following as a plausible solution to the identification of the "necessary and sufficient" preconditions for belief update. We establish that if the agent is scrutinizing the expectation that ϕ then it updates its belief base with ψ if and only if:

• ψ is a perceptual datum and

• either the agent gets surprised by the mismatch between the perceptual datum ψ and the scrutinized expectation that ϕ and the intensity of mismatch-based surprise exceeds a given threshold Δ or

• the agent gets astonished by the recognition of implausibility of ψ and the intensity of astonishment exceeds threshold Δ.

We can express formally the previous principle.

(NecSuffT rig) ϕ! ∧T est(ψ) ←→T est(ψ)∧(IntensityM ismatchSurprise(ϕ, ψ) > Δ ∨ IntensityAstonishment(ϕ) > Δ)

Let us note two relevant facts. On one hand we want to emphasize that both personality factors and motivational factors can affect the value of the threshold Δ and that the value of Δ changes due to the evolution and dynamics of goals and intentions. Since Δ has an intrinsic dynamic nature, its value is not in principle the same for all possible worlds w in a model. Nevertheless it seems plausible to state that the higher is the motivational relevance of the perceived fact (more important is ϕ given actual goals and intentions of the agent) and the lower is the value of Δ. This implies that I am more prone to revise my knowledge when I perceive something which is relevant with respect to my actual motivations than when I perceive something which is completely irrelevant with respect to my actual motivations.

On the other hand we want to emphasize that the previous characterization (NecSuffT rig) of "necessary and sufficient preconditions" for belief update is somehow still unsatisfactory. It must be stressed that a more articulated model of the process would require a distinction between belief change vs. belief rejection. Indeed after having been surprised by the perceived fact ϕ, not necessarily the agent "decides" to update its beliefs.

The agent may simply decide to reject ϕ if the source of information is evaluated to be unreliable [START_REF] Castelfranchi | Representation and integration of multiple knowledge sources: Issues and questions[END_REF]. This means that once the agent has been surprised, the possibility of updating beliefs with a perceived fact ϕ also depends on the reliability assigned to the source of information (reliability of the sensors or reliability of the communicative source etc...). Indeed after being surprised by the perception of ϕ, I am more prone to revise my knowledge with ϕ (instead of rejecting the perceptual datum ϕ) when I consider my sensors to be reliable ("so it is not a hallucination!") than when I consider my sensors to be unreliable.

Conclusion

We have provided in this paper a conceptual and formal clarification of the notion of surprise thanks to the elaboration of the ontology developed in section 3. Each kind of surprise has been associated with a particular phase of the cognitive processing and involves particular kinds of epistemic representations (representation and expectation under scrutiny, perceptual data, presupposed frame, background expectations and beliefs).

We have identified two main kinds of surprise: mismatch-based surprise and astonishment. The first has been defined as the surprise due to a recognized inconsistency between an expectation under scrutiny and a perceived fact. The second has been defined as the surprise due to the recognition of implausibility of the perceived fact where this recognition is based either on the retrieval of a background expectation or on some inferential process (classical deduction or probabilistic inference). We have compared our model with existing psychological models of surprise and shown that an analytic investigation of the concept is still missing and that in these models some important aspects of this cognitive phenomenon are ignored.

In the second part of the paper (section 4) we have investigated the role of surprise in triggering belief update. We think in fact that the notion of surprise should be exploited by current formal models of information update. We have provided several justifications of our theoretical position. Indeed on one hand we think that in designing cognitive agents we must relax the assumption that in principle any perception produces a reconsideration of pre-existent beliefs and expectations. Since realistic agents are by definition resource-bounded [START_REF] Wasserman | Resource-bounded Belief Revision[END_REF][START_REF] Cherniak | Minimal rationality[END_REF], they should not waste time and energy in reasoning out and reconsider their knowledge on the basis of every piece of information they get. To relax the previous assumption seems indeed a necessary desideratum to bridge the existing gap between formal models of belief change and cognitive theories of belief dynamics. On the other hand we think that after having relaxed the previous assumption we must look for the cognitive precursors of belief change. We have stressed that surprise is perhaps the most important causal precursor of belief change. We have presented a method to integrate surprise in a formal model of belief update and to investigate its functional role. More work must be done in order to improve the present model. From a strictly formal point view. We have not yet completeness results for our modal logic of surprise. From a more theoretical point view. We have characterized several kinds of informational mental states such as scrutinized expectations (expectations on which the agent focuses its attention) and background expectations (expectations which are available at a mere automatic and effortless level). Moreover we have characterized mental processes which are responsible for modifying those scrutinized expectations (we have called them retrieve mental operations) by transforming one background expectation into a scrutinized one. We still miss a systematic explanation and formal account of why certain expectations rather other ones go in background and become accessible. Moreover we have not explained why certain expectations rather than other ones get scrutinized by the agent.

For the moment we leave unsolved these formal and theoretical problems and we postpone them to future work.

  (a) for all w ∈ W : B ϕ (w) = {w |w ∈ B(w) and M, w |= ϕ}. (b) for all w ∈ W : W ϕ w = {w ∈ W w |M, w |= ϕ}.

(

  Upd 10 ) Surp+ [ϕ!] Belϕ (Upd 11 ) Surp+ Bel m o 1 → [ϕ!] Bel m o 1 for each m > 0 (Upd 12 ) Surp+ P (ϕ|ψ) = c → [ψ!] P (ϕ) = c where P (ϕ|ψ) = P (ϕ∧ψ) P (ψ)

The necessity for a distinction between a mere activity of seeing, hearing, smelling something and a complex cognitive activity of perceptual recognition of an object or event has also been stressed by[START_REF] Dretske | Knowledge and the flow of information[END_REF].

In the extended version of this paper(Lorini and Castelfranchi, 2006a) we investigate also those forms of surprise due to the invalidation of the agent's presupposed frame. We provide a general definition of frame (or script) as agglomerate of conditional beliefs and argue that a special kind of surprise (called disorientation) arises from the invalidation and revision of the conditional beliefs which are part of a given presupposed frame of the agent.

We use the term "informational mental state" in order to distinguish it from a "motivational mental state" (a desire, intention, wish, goal and so on).

Notice that "to feel surprised" should not be confused with "having awareness of our own surprise" (for a distinction between phenomenal consciousness and access consciousness see[START_REF] Bloch | On a confusion about a function of consciousness[END_REF][START_REF] Chalmers | Facing up to the problem of consciousness[END_REF].

Our test function is comparable to the awareness function defined in[START_REF] Fagin | Belief, awareness, and limited reasoning[END_REF].

Several empirical evidences exist showing that in being active and available at an automatic and effortless level background (passive) expectations can affect subject's performances and judgments and can conflict with conscious (scrutinized) expectations (on this see[START_REF] Matt | Differential effects of voluntary expectancies on reaction times and event-related potentials: evidence for automatic and controlled expectancies[END_REF][START_REF] Sommer | The expectancies that govern the P300 amplitude are mostly automatic and unconscious[END_REF].

Note that this definition is equivalent to the following definition:Astonishment(ψ) = def Datum(ψ) ∧ ( retrieve(¬ψ)∨ retrieve(ψ) ).

Formal proofs of theorems, lemmas and propositions are given in an extended version of this paper(Lorini and Castelfranchi, 2006a).

See for instance[START_REF] Hansson | A Textbook of Belief Dynamics: Theory Change and Database Updating[END_REF] for a complete account of belief revision applied to belief bases.

Obviously we assume that the expectation under scrutiny is a special kind of explicit belief (see Figure1).

As noticed in the previous section it could also be a probabilistic inference (a post-hoc reconstruction of the probability of the perceived fact).

This assumption is made in order to prevent from dividing by zero (condition 2c in definition 6) when redefining the probability function after an update with a sentence with probability zero. More general approaches to updating with sentences with probability zero are discussed in[START_REF] Halpern | Lexicographic probability, conditional probability, and nonstandard probability[END_REF][START_REF] Halpern | Reasoning about uncertainty[END_REF].

For belief revision with inconsistent information see for instance[START_REF] Herzig | Sensing and revision in a modal logic of belief and action[END_REF]; van[START_REF] Van Benthem | Dynamic logic for belief revision[END_REF].

The same result is obtained in[START_REF] Kooi | Probabilistic dynamic epistemic logic[END_REF].

Acknowledgment

We are very grateful to Johan van Benthem and to the anonymous referees of this paper for their helpful comments on the content of our work. Our research has been supported by the European Project "MindRACES: from Reactive to Anticipatory Cognitive Embodied Systems" (IST-511931).


Joël K Kazadi
email: joelkazadi21@gmail.com

Ishara Musimwa
email: musimwa.ishara@etu.univ-paris1.fr

General formula for the p-derivative of a univariate polynomial function of degree n

Keywords: Derivative, Polynomial function, Induction process. JEL Classification : C02

This paper proposes a general formula of the p-order derivate for an univariate polynomial function, i.e. a polynomial of only one variable, of degree n. Based on a simple algebraic manipulation of the general formula, we show that it's now easier to deduce the derivative of any order of any polynomial without passing through the (p-1) successive derivations. Finally, we consider an appropriate illustration in order to show that the successive derivations of the polynomial function and our general formula yield same results.

Introduction

A function or mapping from the set A to the set B, denoted by : f : A → B, is a triplet (A, B, graph(f )) where A and B are two sets and graph(f ) is a subset of A × B such that for every a ∈ A, there is one and only one b ∈ B sucht that (a, b) ∈ graph(f ) :

[∀x ∈ A, ∃y ∈ B : (x, y) ∈ graph(f )]∧{[(a, b) ∈ graph(f )∧(a, c) ∈ graph(f )] ⇒ (b = c)}.
In other words, for every a ∈ A there is exactely one element denoted f (a) ∈ B such that the ordered pair (a, f (a)) ∈ graph(f ).

• The set A is called the Domain of f ;

• The set B is called the codomain of f ;

• The set graph(f ) is called the graph of f (:= {(a, b) ∈ A × B|b = f (a)});

• The unique element f (a) such that (a, f (a)) ∈ graph(f ) is called the image of a by f .

Propositions

• Proposition 1 :

A 1 ⊂ A 2 ⇒ f (A 1 ) ⊂ f (A 2 ) Proof : Consider y ∈ f (A 2 ) ⇒ ∃x ∈ A 1 : y = f (x) Since A 1 ⊂ A 2 ⇒ x ∈ A 1 ⇒ x ∈ A 2 ⇒ f (x) ∈ f (A 2 ) ⇒ y ∈ f (A 2 ) • Proposition 2 : f (A 1 ∪ A 2 ) = f (A 1 ) ∪ f (A 2 ) Proof : f (A 1 ∪ A 2 ) = f (A 1 ) ∪ f (A 2 ) ⇐⇒ f (A 1 ∪ A 2 ) ⊂ f (A 1 ) ∪ f (A 2 ) ∧ f (A 1 ∪ A 2 ) ⊃ f (A 1 ) ∪ f (A 2 ) Let's consider first f (A 1 ∪ A 2 ) ⊂ f (A 1 ) ∪ f (A 2 ) Now let's take y ∈ f (A 1 ∪ A 2 ) ⇒ ∃x ∈ A 1 ∪ A 2 : y = f (x) Therefore x ∈ A 1 ∪ A 2 ⇔ x ∈ A 1 ∨ x ∈ A 2 ⇔ f (x) ∈ f (A 1 ) ∨ f (x) ∈ f (A 2 ) ⇔ y ∈ f (A 1 ) ∨ y ∈ f (A 2 ) ⇔ y ∈ f (A 1 ) ∪ f (A 2 ) Let's consider secondly f (A 1 ∪ A 2 ) ⊃ f (A 1 ) ∪ f (A 2 ) Now let's take y ∈ f (A 1 ) ∪ f (A 2 ) ⇔ y ∈ f (A 1 ) ∨ y ∈ f (A 2 ) ⇒ ∃x ∈ A 1 ∨ x ∈ A 2 : y = f (x) Therefore x ∈ A 1 ∨ x ∈ A 2 ⇔ x ∈ A 1 ∪ A 2 ⇔ f (x) ∈ f (A 1 ∪ A 2 ) ⇔ y ∈ f (A 1 ∪ A 2 ) • Proposition 3 : f (A 1 ∩ A 1 ) ⊂ f (A 1 ) ∩ f (A 2 ) Proof : Consider y ∈ f (A 1 ∩ A 2 ) ⇒ ∃x ∈ A 1 ∩ A 2 : y = f (x) Since x ∈ A 1 ∩ A 2 ⇔ x ∈ A 1 ∧ x ∈ A 2 ⇔ f (x) ∈ f (A 1 ) ∧ f (x) ∈ f (A 2 ) ⇔ y ∈ f (A 1 ) ∧ y ∈ f (A 2 ) ⇔ y ∈ f (A 1 ) ∩ f (A 2 )

Exemples of functions

The identity mapping on A, denoted by id A , is the mapping from A to A defined by id A (a) = a for every a ∈ A.

Polynomial function: Definitions and properties

In this paper, the focus will be on polynomial functions. In algebra, a one-indeterminate polynomial y is defined as an expression of the form:

y = a 0 X 0 + a 1 X 1 + a 2 X 2 + • • • + a n-2 X n-2 + a n-1 X n-1 + a n X n , ( [1]
)

where n ∈ N, (a 0 , a 1 , a 2 , . . . , a n-2 , a n-1 , a n ) ∈ R n+1 and X is a formal symbol called the "indeterminate of the polynomial"1 . As shown in equation [START_REF] Tomasz | Solving Problems in Mathematical Analysis[END_REF], the polynomial is a null sequence of elements from a certain rank, i.e. a near-zero sequence because there are a finite number of coefficients a i [START_REF] Stetter | Numerical Polynomial Algebra[END_REF].

On the other hand, a polynomial function has the following general form:

y = a 0 x 0 + a 1 x 1 + a 2 x 2 + • • • + a n-2 x n-2 + a n-1 x n-1 + a n x n . ([2])
Concretely, in this document, we define a real polynomial function as following :

f : R -→ R x -→ y = f (x) = n i=0 a i x i
Now it is established above that a polynomial function is defined for any real. As main properties, it is continuous and derivable.

Contuinity

By definition, a function is continuous on its domaine D,

if ∀x 0 ∈ D, lim x→x 0 f (x) = f (x 0 )
Let's show by induction that the real polynomial function of degree n is continuous in R

Initial conditions :

For n = 0, we have a constant function define as f (x) = k with k ∈ R By definition a limit, it is established that lim

x→x 0 k = k
For n = 1, we have an identity function define as f (x) = x with k ∈ R By definition a limit, it is established that lim

x→x 0 x = x 0
Induction hypothesis : Now we assume that any polynomial h of degree n -1 is continuous in R and let f be a polynomial of degree n. Let x 0 be any arbitrary real number.

We have :

h(x) = n-1 i=0 a i x i is continuous by assumptions i.e. lim x→x 0 h(x) = h(x 0 ). Let's show that f (x) = n i=0 a i x i is also continous. lim x→x 0 f (x) = lim x→x 0 n i=0 a i x i = lim x→x 0 n-1 i=1 a i x i + a n x n = lim x→x 0 h(x) + a n x.x n-1 = lim x→x 0 h(x) + lim x→x 0 a n × lim x→x 0 x × lim x→x 0 x n-1 = h(x 0 ) + a n × x 0 × x n-1 0 = n-1 i=0 a i x i 0 + a n x n 0 = n i=0 a i x i 0 = f (x 0 )
Hence the real polynomial function f is continuous in R since this proof is true ∀x 0 ∈ R. By this principle of induction, every polynomial with real coefficients is continuous in R .

Derivability

In this paper, we study the derivatives of a polynomial function. Before doing so, it is useful to propose a formal definition of the derivative of a function f, both from an analytical and a geometrical point of view.

Geometrical definition

Let y = f (x) be a continuous function in x 0 . The derivative of y with respect to its explanatory variable x -noted f (x) -measures the slope of y at a point M of coordinates (x 0 , y 0 ) on the representative curve of y noted C f .

Figure 1: Geometrical interpretation of the derivative

If f is derivable in x 0 , then its representative curve C f admits a tangent T M at the point M (x 0 , y 0 ) whose angular coefficient is f (x 0 ). In other words, the derivative number is the angular coefficient of the tangent2 to the representative curve of a function at each point equal to the corresponding value of the derivative.

Analytical definition

Let y = f (x) be a continuous function in x 0 . We say that f is derivable in x 0 if its increasing rate: ∆y ∆x = y -y 0 x -x 0 admits a finite limit in x 0 . This limit, when it exists, is the derivative number of f. It is denoted

dy dx = f (x).
The derivative of y with respect to x thus corresponds to the limit of the increasing rate of y in x 0 as x tends to x 0 . This can be written:

dy dx = lim x→x 0 y -y 0 x -x 0 .
Given that ∆x = x -x 0 , the derivative can be rewritten such as:

dy dx = lim ∆x→0 f (x + ∆x) -f (x) ∆x . ([3])
The notion of derivative is very important in Economics because it allows us to analyze the marginal variations of economic quantities on the one hand, and to determine their extrema after optimization on the other hand (Sydsaeter et al., 2016).

Using this analytical definition [START_REF] Knut | Essential Mathematics for Economic Analysis[END_REF], we can prove easilly that real polynomial functions are derivable. Considering the polynomial function :

f (x) = n i=0 a i x i dy dx = lim ∆x→0 n i=0 a i (x + ∆x) i - n i=0 a i x i ∆x = lim ∆x→0 1 ∆x n i=0 a i i k=0 i k x k (∆x) i-k - n i=0 a i x i = lim ∆x→0 1 ∆x n i=0 a i i-1 k=0 i k x k (∆x) i-k + i i x i (∆x) i-i - n i=0 a i x i = lim ∆x→0 1 ∆x n i=0 a i i-1 k=0 i k x k (∆x) i-k + x i - n i=0 a i x i = lim ∆x→0 1 ∆x n i=0 a i i-1 k=0 i k x k (∆x) i-k + n i=0 a i x i - n i=0 a i x i = lim ∆x→0 1 ∆x n i=0 a i i-1 k=0 i k x k (∆x) i-k = lim ∆x→0 n i=0 a i i-1 k=0 i k x k (∆x) i-k-1 = lim ∆x→0 n i=0 a i i-2 k=0 i k x k (∆x) i-k-1 + i i -1 x i-1 = lim ∆x→0 n i=0 a i i-2 k=0 i k x k (∆x) i-k-1 + ix i-1 = lim ∆x→0 n i=0 a i i-2 k=0 i k x k (∆x) i-k-1 + lim ∆x→0 n i=1 a i (ix i-1 ) = n i=0 i × a i x i-1 = 0 + a 1 + 2a 2 x + 3a 3 x 2 + • • • + (n -2)a n-2 x n-3 + (n -1)a n-1 x n-2 + na n x n-1
This result shows that the derivative of a real polynominial function of degree n is a new polynomial of degree n -1. It is possible to proceed by the same way to find the derivative of this new polynomial which would be of degree n -2. The aim of this paper is to find a general formula to find the p-derivative of the polynomial function of degree n.

Development of the formula

In this article, we try to find a general formula for the derivative of any order of a univariate polynomial function. The latter is specified as follows:

y = f (x) = n i=0 a i x i , ([4])
where n is the order of the polynomial and a i the coefficients associated to the independent variable. Disaggregating the expression of equation 3 above, we obtain the following exploded form:

y = f (x) = a 0 + a 1 x + a 2 x 2 + • • • + a n-2 x n-2 + a n-1 x n-1 + a n x n .
Let us derivate this function successively. Using the definition of derivative etablished in equation [START_REF] Knut | Essential Mathematics for Economic Analysis[END_REF], we obtained the following results:

dy dx = a 1 + 2a 2 x + 3a 3 x 2 + • • • + (n -2)a n-2 x n-3 + (n -1)a n-1 x n-2 + na n x n-1
Proceeding in the same way, it is possible to find the remaining (n -1) successive derivatives. The final results are given as follows:

d 2 y dx 2 = 2a 2 +6a 3 x+• • •+(n-2)(n-3)a n-2 x n-4 +(n-1)(n-2)a n-1 x n-3 +n(n-1)a n x n-2 . . . . . . . . . d n-2 y dx n-2 = (n-2)(n-3) . . . [n-(n-1)]a n-2 x n-n +(n-1)(n-2) . . . [n-(n-2)]a n-1 x n-(n-1) +n(n-1) . . . [n-(n-3)]anx n-(n-2) d n-1 y dx n-1 = (n-1)(n-2) . . . [n-(n-2)][n-(n-1)]a n-1 x n-n +n(n-1) . . . [n-(n-3)][n-(n-2)]a n x n-(n-1) d n y dx n = n(n -1) . . . [n -(n -3)][n -(n -2)][n -(n -1)]a n x n-n .
If we take a closer look at these successive derivatives, we can see that it is possible to rewrite them in the following forms:

dy dx = 1! 0! a 1 + 2! 1! a 2 x+ 3! 2! a 3 x 2 +• • •+ (n -2)! (n -3)! a n-2 x n-3 + (n -1)! (n -2)! a n-1 x n-2 + n! (n -1)! a n x n-1 d 2 y dx 2 = 2! 0! a 2 + 3! 1! a 3 x + • • • + (n -2)! (n -4)! a n-2 x n-4 + (n -1)! (n -3)! a n-1 x n-3 + n! (n -2)! a n x n-2 . . . . . . . . . d n-2 y dx n-2 = (n -2)! (n -n)! a n-2 x n-n + (n -1)! [n -(n -1)]! a n-1 x n-(n-1) + n! [n -(n -2)]! a n x n-(n-2) d n-1 y dx n-1 = (n -1)! (n -n)! a n-1 x n-n + n! [n -(n -1)]! a n x n-(n-1) d n y dx n = n! (n -n)! a n x n-n .
From the logic of these n successive derivatives, we can therefore give a general formulation that allows us to easily find one of them without having to know those that preceded it:

d p y dx p = n i=p i! (i -p)! a i x i-p , with p ∈ [0, n]. ([5])
Equation [5] gives the general formula for the p-order derivative of a univariate polynomial function of degree n. This formula allows us to compute one of the successive derivatives of a polynomial function without having to compute all the other derivatives that came before the one we want to compute. It saves us from the tedious work of computing the (p -1) previous derivatives, especially when p is very close to n, the degree of the polynomial.

Two extreme cases can be observed when using this calculation formula:

• Case 1 : p = 0
If the order of the derivative is zero, then we will find the original polynomial function.

d 0 y dx 0 = n i=0 i! (i -0)! a i x i-0 ⇒ y = n i=0 a i x i .
"The derivative of order zero of a polynomial function is equal to the function itself ".

• Case 2 : p = n

If the order of the derivative is equal to the degree of the polynomial function, then we will find a constant function.

d n y dx n = n i=n i! (i -n)! a i x i-n ⇒ d n y dx n = n!a n .
"The n-derivative of a polynomial function of degree n is always equal to a constant".

Corollary

Note that if p > n, then d p y dx p = 0. For a polynomial function of degree n, its derivative of order greater than n will always be zero.

Proof of the general formula by induction

The general formula is :

d p y dx p = n i=p i! (i -p)! a i x i-p
Initial condition : let's replace p by some integers 1, 2 and 3. We get following results :

• p = 1 dy dx = n i=1 i! (i -1)! a i x i-1 dy dx = 1! 0! a 1 + 2! 1! a 2 x + • • • + (n -1)! (n -2)! a n-1 x n-2 + n! (n -1)! a n x n-1 • p = 2 d 2 y dx 2 = n i=2 i! (i -2)! a i x i-2 d 2 y dx 2 = 2! 0! a 2 + 3! 1! a 3 x + • • • + (n -1)! (n -3)! a n-1 x n-3 + n! (n -2)! a n x n-2 • p = 3 d 3 y dx 3 = n i=3 i! (i -3)! a i x i-3 d 3 y dx 3 = 3! 0! a 3 + 4! 1! a 4 x + • • • + (n -1)! (n -4)! a n-1 x n-4 + n! (n -3)! a n x n-3
Induction hypothesis : Since with p = 1, ..., p = 3, the general formula is true; let's suppose that it is true for every p ∈ N, in other words :

d p y dx p = n i=p i! (i -p)! a i x i-p
Induction process : Let's prove that this formula is true for every p + 1, i.e.

d p+1 y dx p+1 = n i=p+1 i! (i -p -1)! a i x i-p-1 d p+1 y dx p+1 = d d p y dx p dx = d n i=p i! (i -p)! a i x i-p dx = n i=p+1 (i -p)i! (i -p)! a i x i-p-1 = n i=p+1 (i -p)i! (i -p)(i -p -1)! a i x i-p-1 = n i=p+1 i! (i -p -1)! a i x i-p-1
It is clear that, both by successive derivatives and by the application of the general formula, the final results are not contradictory. The two methods are good for derivating a polynomial. The advantage with the general formula is that it is more appropriate for polynomial functions of very high degree, i.e. where n is very large.

Conclusion

In this paper, we proposed a general formula to calculate the p-derivative of a univariate polynomial function of degree n. Many Mathematics books mention the successive derivation to find the derivative of any order of a polynomial function. This can be inefficient when derivating polynomial functions of very high order.

Thus, the general formula has the advantage of making this task easy in the sense that it is not forced to go through the previous derivatives to calculate the one we are looking for. This is a saving of effort and a possibility to find the derivative of any order whatever the degree of the polynomial function one is dealing with.

It would have been more interesting to also find such a formula with integrals, the inverse operation of the derivation being the integration. We think that future research efforts along these lines will be very commendable. For our part, we have limited ourselves to suggest a general formula for the derivation of a polynomial function in order to facilitate the calculation of one of the successive derivatives of this univariate polynomial function.

X is not a variable that can take real values.

The equation of the tangent to C f is written: y = f (x 0 ) + f (x 0 )(x -x 0 ).


Frank O Wagner

SOME REMARKS ON SHARPLY 2-TRANSITIVE GROUPS AND NEAR-DOMAINS

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Recall that a permutation group G acting on a set X is sharply 2-transitive if for any two pairs (x, y) and (x ′ , y ′ ) of distinct elements of X there is a unique g ∈ G with gx = x ′ and gy = y ′ . Then G has involutions, and either involutions have fixed points and G is of permutation characteristic 2, or the action on X is equivalent to the conjugation action on the set I of involutions. In that case, all translations, i.e. products of two distinct involutions, are also conjugate and have the same order p, which is either an odd prime or ∞; the number p (or 0 if p = ∞) is the permutation characteristic of G. We say that G splits if it has a regular normal subgroup N; in that case G = N ⋊ C G (i) for any involution i ∈ I. Note that Tent, Rips and Segev have constructed non-split sharply 2-transitive permutation groups of characteristic 0 and 2. V. D. Mazurov asked in the Kourovka Notebook (question 12.48): Let G be a sharply 2-transitive permutation group.

(1) Does G possess a regular normal subgroup if a point stabilizer is locally finite? (2) Does G possess a regular normal subgroup if a point stabilizer has an abelian subgroup of finite index?

We shall answer question (b) affirmatively in permutation charateristic 0. In fact, we shall show a more general result for near-domains.

Date: 28/2/2022. 2010 Mathematics Subject Classification. 20B22.

Near-domains and near-fields.

Instead of working with sharply 2-transitive groups, we shall work in the equivalent setting of near-domains.

Definition 1. (K, 0, 1, +, •) is a near-domain if for all a, b, c ∈ K (1) (K, 0, +) is a loop, i.e. a + x = b and y + a = b have unique solutions, with a + 0 = 0 + a = a; (2) (K \ {0}, 1, •) is a group, and 0 • a = a • 0 = 0; (3) left distributivity holds: a • (b + c) = a • b + a • c; (4) for all a, b ∈ K there is d a,b ∈ K such that a + (b + x) = (a + b) + d a,b • x for all x. A near-domain is a near-field if addition is associative.
Hence a near-field is a skew field iff right distributivity holds.

Fact 2 (Tits, Karzel). A sharply 2-transitive permutation group G is isomorphic to the group of affine transformations of some near-domain K, i.e. of the set of permutations {x → a + bx : a, b ∈ K, b = 0}; the centraliser of any involution is isomorphic to the multiplicative group K × . It is split iff K is a near-field.

Let E be the set {d ∈ K : 1 + d = d + 1}. Since the additive loop of K is powerassociative, it is easy to see that 1 generates a subfield of K contained in E, which is either Q or F p . Thus K has a characteristic, which is easily seen to be equal to the permutation characteristic of G. Note that in characteristic > 2 there is a unique maximal sub-near-field, which is equal to E.

Fact 3 ([1]

). For all a, b, c ∈ K we have:

(1) d a,a = 1.

(

2) d a,b (b + a) = a + b. (3) cd a,b c -1 = d ca,cb . (4) d a,b = d a,c d c+a,-c+b d -c,b . (5) If a, b ∈ E then (a + b) 2 ∈ E. (6) |K × : C K × (d a,b )| = ∞ if d a,b = 1.
Let now A be any subgroup of finite index in K × which avoids all non-trivial coefficients d a,b for a, b ∈ K. Kerby [START_REF] Kerby | On infinite sharply multiply transitive groups, Hamburger Mathematische Einzelschriften[END_REF]Theorem 8.26] has shown that K must be a near-field in the following cases:

(1) charK = 0 and |K × : A| = 2, (2) charK = 2, |K × : A| = 2 and |E| > 2,

(3) charK = p > 2 and |K × : A| < |E|. We shall adapt the proof of (3) to characteristic 0. Lemma 4. Suppose d a,1/k = 1. Then d a,n/k = 1 for all n ∈ N.

Proof. By induction on n. This is clear for n = 0 and n = 1. So suppose it holds for n, and consider

n + 1 k + a = n k + 1 k + a = n k + 1 k + a = n k + a + 1 k = n k + a + 1 k = a + n k + 1 k = a + n k + 1 k = a + n + 1 k . Proposition 5. If A ≤ K × is
(i-j) -1 (j+a),1 = d j+a,-j+i = 1, so (i-j) -1 (j+a) ∈ E. Since -(i-j) -1 j ∈ Q ⊆ E, we have [-(i -j) -1 j + (i -j) -1 (j + a)] 2 = (i -j) -1 a 2 ∈ E, and d a2,i-j = 1. But 0 < (i -j) n! ≤ n is integer,
and there is an integer k > 0 with ij = 1 k . By Lemma 4 we obtain d a2,1 = 1 and a 2 ∈ E, a contradiction. Corollary 6. Let G be a sharply doubly transitive permutation group of characteristic 0 whose point stabilizer is virtually abelian. Then G is split.

Proof. If K is the associated near-domain, K × has an abelian subgroup A of finite index. Now any non-trivial d a,b has a centralizer of infinite index in K × , so d a,b / ∈ A. We finish by Proposition 5.

  a subgroup of finite index avoiding all nontrivial d a,b and char(K) = 0, then K is a near-field.Proof. Recall that Q ⊆ E. If K = E, then d a,b = 1 for all a, b ∈ K and K is a near-field. So assume E K × ,and take a ∈ K \ E 2 -1 . Let n = |K × : A|. Then there are distinct i > j in {0, 1, 2, . . . , n}/n! with d a,i A = d a,j A; since d -j,i = 1 we obtain d a,i = d a,j d j+a,-j+i d -j,i = d a,j d j+a,-j+i . Hence d j+a,-j+i ∈ A, and d j+a,-j+i = 1 by assumption. Now d