Alessandro Aldini
email: alessandro.aldini@uniurb.it

Vincenzo Fano
email: vincenzo.fano@uniurb.it

Pierluigi Graziani
email: pierluigi.graziani@unich.it

Theory of Knowing Machines: Revisiting Gödel and the Mechanistic Thesis

Keywords: Church-Turing Thesis, Mechanism, Epistemic Arithmetic, Gödelian Arguments, Intuitive provability, Knowledge operator

Church-Turing Thesis, mechanistic project, and Gödelian Arguments offer different perspectives of informal intuitions behind the relationship existing between the notion of intuitively provable and the definition of decidability by some Turing machine. One of the most formal lines of research in this setting is represented by the theory of knowing machines, based on an extension of Peano Arithmetic, encompassing an epistemic notion of knowledge formalized through a modal operator denoting intuitive provability. In this framework, variants of the Church-Turing Thesis can be constructed and interpreted to characterize the knowledge that can be acquired by machines. In this paper, we survey such a theory of knowing machines and extend some recent results proving that a machine can know its own code exactly but cannot know its own correctness (despite actually being sound). In particular, we define a machine that, for (at least) a specific case, knows its own code and knows to be sound.

Introduction

After the seminal paper by Turing in 1950 [START_REF] Turing | Computing machinery and intelligence[END_REF], the launch of the mechanistic project, intended to establish whether minds can be explained, either extensionally or intensionally, in purely mechanist terms, was contrasted by the socalled Gödelian Arguments. These represent the effort done by several scholars to interpret Gödel's Incompleteness Theorems [START_REF] Gödel | Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme[END_REF] with the purpose of refuting mechanism. In particular, several speculative ideas, like the anti-mechanist arguments by Lucas [START_REF] Lucas | Minds, machine and Gödel[END_REF][START_REF] Lucas | Satan stultified: a rejoinder to Paul Benacerraf[END_REF] and Penrose [START_REF] Penrose | Beyond the doubting shadow[END_REF], contributed to animate the debate. On the other hand, authors like Benacerraf [START_REF] Benacerraf | God, the devil and Gödel[END_REF], Chihara [START_REF] Chihara | On alleged refutations of mechanism using Gödel's incompleteness results[END_REF], and Shapiro [START_REF] Shapiro | Incompleteness, mechanism, and optimism[END_REF] tried to follow more sophisticated lines of reasoning for the analysis of the relation between human mind and machines [START_REF] Fano | Mechanical intelligence and Gödelian arguments[END_REF]. Most of these approaches preserve intensional elements on properties of human mind that make severely informal the argumentation and, more confusing, make hard even to define precisely what the (anti-)mechanistic thesis claims.

In this setting, a series of results presented by Reinhardt [START_REF] Reinhardt | Epistemic theories and the interpretation of Gödel's incompleteness theorems[END_REF], Carlson [START_REF] Carlson | Knowledge, machines, and the consistency of Reinhardt's strong mechanistic thesis[END_REF], and Alexander [START_REF] Alexander | A machine that knows its own code[END_REF], clarify some typically informal aspects of the Gödelian Arguments. This is done in the framework of a theory, called Epistemic Arithmetic (EA), developed independently by both Reinhardt and Shapiro [START_REF] Reinhardt | The consistency of a variant of Church's thesis with an axiomatic theory of an epistemic notation[END_REF][START_REF] Shapiro | Epistemic and intuitionistic arithmetic[END_REF][START_REF] Reinhardt | Absolute versions of incompleteness theorems[END_REF], and encompassing an epistemic notion of intuitive provability. In particular, Reinhardt uses such an axiomatic framework to study both ramifications of the Church-Turing Thesis [START_REF] Kleene | Mathematical Logic[END_REF]: every effectively calculable function is computable by a Turing Machine and the consequences of Gödel's Incompleteness Theorems, with the aim of strengthening the formal elements behind the philosophical debate on the knowledge that can be acquired by machines.

EA is the language of Peano Arithmetic enriched by a modal operator K for knowledge, which is the notation used by Shapiro and from Carlson on (see, e.g., [START_REF] Shapiro | Epistemic and intuitionistic arithmetic[END_REF][START_REF] Carlson | Knowledge, machines, and the consistency of Reinhardt's strong mechanistic thesis[END_REF][START_REF] Alexander | A machine that knows its own code[END_REF]). According to Shapiro, K means "ideally, or potentially, knowable", while Carlson, analogously, says "can eventually come to be known". On the other hand, Reinhardt prefers the more specific interpretation "it is intuitively provable that" [START_REF] Reinhardt | The consistency of a variant of Church's thesis with an axiomatic theory of an epistemic notation[END_REF] and, to strengthen such an idea, uses the epistemic operator B for beweisbar (meaning provable in German). By following the same motivation, we adopt the operator B, which is closer to the intended interpretation of its role, and use the two intuitions -provability and knowledge -interchangeably, by assuming that the notion of knowability [START_REF] Shapiro | Epistemic and intuitionistic arithmetic[END_REF] is actually limited to intuitive provability. Hence, the idea behind such a modal operator is to express a definition of decidability by human mind (humanly provable) that occurs in many forms both in the mechanistic thesis and in the Gödelian Arguments.

The formal interpretation of B passes through the definition of the properties at the base of an epistemic notion of knowledge. For instance, it is expected that humanly provable statements are closed under logic consequence, meaning that if we can intuitively prove φ → ψ and we have an intuitive proof of φ, then by combining these we derive an intuitive proof of ψ. In other words, we are representing modus ponens as a rule for intuitive provability. Analogously, we also expect the soundness of intuitive provability (what can be proved is true, thus stating the infallibilism of knowledge) and that what is humanly provable includes all tautologies. Using this latter rule in the setting of intuitive provability amounts to establish an introspection principle: if φ can be proved then such a knowledge can be proved. Such a principle may be seen as an instance of the general, largely discussed and controversial, KK (knowing that one knows) law (see, e.g., [START_REF] Hawthorne | Knowledge and Lotteries[END_REF][START_REF] Hintikka | Knowing that one knows[END_REF]). However, intuitive provability ("it can be proved") is weaker than the general notion of knowledge ("I know that") and relies on idealizations abstracting from space, time, and complexity constraints, thus making the introspection rule less debatable with respect to the classical contrast between internalist and externalist theories of knowledge (see, e.g., [START_REF] Bonjour | The Structure of Empirical Knowledge[END_REF][START_REF] Goldman | What is justified belief?[END_REF]).

As we will see, all the laws informally expressed above are formalized through very common axioms of epistemic modal systems, see, e.g., [START_REF] Hintikka | Knowledge and Belief: An Introduction to the Logic of the Two Notions[END_REF] for a comprehensive discussion. Therefore, in essence, knowability is treated in axioms form, while any attempt to specify precisely its meaning by following model theoretic approaches is intentionally avoided. In fact, representing knowledge as a predicate easily leads to contradictions forcing all true propositions to be provable [START_REF] Löb | Solution of a problem of Henkin[END_REF][START_REF] Myhill | Some remarks on the notion of proof[END_REF][START_REF] Montague | Syntactical treatments of modality[END_REF][START_REF] Reinhardt | The consistency of a variant of Church's thesis with an axiomatic theory of an epistemic notation[END_REF][START_REF] Reinhardt | Epistemic theories and the interpretation of Gödel's incompleteness theorems[END_REF].

Then, the aim of an axioms system based on the operator B is to apply deductive reasoning to prove the (in)consistency of statements specified in an appropriately formulated logic and representing conjectures related to Church's Thesis and the mechanist project, with a specific interest towards the following: the property of being humanly provable is equivalent to decidability by some Turing Machine.

In the rest of the paper, we first illustrate such a theory of knowledge (Section 2). Then, we survey the main results obtained by recasting Gödel Incompleteness Theorems and by analyzing variants of the Church-Turing Thesis in this setting (Section 3). We also show how to extend a tradeoff result obtained by Alexander [START_REF] Alexander | A machine that knows its own code[END_REF] about the relation between knowledge of soundness and knowledge of own code for (knowing) machines. Some conclusions terminate the paper (Section 4). This paper is a full and revised version of an extended abstract presented at HaPoC 2015 [START_REF] Aldini | A note on knowing machines[END_REF].

Epistemic Arithmetic

We start by introducing some notation used to describe the language of Peano Arithmetic (PA)3 : φ, ψ denote well formed formulas (wff, for short), which, if not specified, are considered to include only one free variable x. 4 A sentence, usually denoted by σ, is a wff without free variables, and any set of sentences closed under logical consequence is called theory. Terms and substitution principle are defined as usual: if t is a term, then x | t denotes the substitution of x by t. An assignment is a function s : V → U from the domain of variables to the reference universe, such that s(x | t) denotes the function assigning t to x and s(y) to every variable y = x.

Epistemic Arithmetic extends Peano Arithmetic with the modal operator B. The language of EA contains every wff of the language of PA and the additional formulas of the form Bφ whenever φ is a wff of EA. In the following, given a set Φ of wff we also use the notation BΦ to denote the set {Bφ | φ ∈ Φ}. It is worth noticing that Bφ is treated as an atomic formula. Then, in EA a structure M with respect to a universe U is defined in the classical way for the first order part related to PA and includes also a boolean interpretation function for B. More precisely, for each assignment s, M |= Bφ[s] means that φ can be intuitively proved when the free variables of φ are interpreted according to s. Obviously, the interpretation of Bφ[s] does not depend on s(x) whenever x does not occur free in φ.

Moreover, we write M |= φ if M |= φ[s] for each assignment s and we say that φ is valid if M |= φ for each structure M; φ is logical consequence of a set of sentences Σ (denoted Σ |= φ) if for each structure M it holds that:

∀σ ∈ Σ : M |= σ ⇒ M |= φ.
We observe that, in such an epistemic extension of PA, compactness and completeness results can be proved. 5As far as the axiomatization of EA is concerned, we first recall the Peano axioms:

1. ∀x(S(x) = 0) 2. ∀x∀y((S(x) = S(y)) → (x = y)) 3. ∀x(x + 0 = x) 4. ∀x∀y(x + S(y) = S(x + y)) 5. ∀x(x • 0 = 0) 6. ∀x∀y(x • S(y) = x • y + x) 7. ∀y 1 . . . ∀y n ((φ(x | 0) ∧ ∀x(φ → φ(x | S(x)))) → ∀xφ) for each wff φ
establishing that 0 is not in the codomain of S (see 1), S is injective (see 2), while 3 to 6 define the rules for sum and product, and 7 expresses the induction schema, where φ is a formula with free variables x, y 1 , . . . , y n .

To the standard Peano axioms we add the universal closure of the following basic axioms of knowledge:

-B1. B∀xφ → ∀xBφ -B2. B(φ → ψ) → (Bφ → Bψ) -B3. Bφ → φ -B4. Bφ → BBφ
where B2-B4 formalize the intuitive properties illustrated in Section 1 and are essentially taken from the modal system S4 [START_REF] Lewis | Symbolic Logic[END_REF][START_REF] Hintikka | Knowledge and Belief: An Introduction to the Logic of the Two Notions[END_REF]. To complete the axioms system, the first order condition B1 establishes that the statement "φ can be proved to be valid" implies the knowledge of each element that can be assigned to the free variable x of φ and the provability of the formula under each such assignment. This condition represents a sort of accessibility principle ensuring that the elements assigned to the free variable of φ should be accessible to the knower in some way, and from which we notice that the following desirable condition can be derived:

M |= B∀xφ[s] ⇒ M |= Bφ[s(x|u)] ∀u ∈ U
which states that if φ can be proved under the interpretation provided by M, then it is possible to prove φ(x) for each assignment of the variable x in the universe U. Now, given a wff φ, we define the B-closure of φ as φ ∪ Bφ, which extends easily to sets of formulas. Then, the standard axiomatization of the theory of knowledge for EA is given by the B-closure of B1-B4 and of the axioms of PA, with the modus ponens being the unique rule of this formal system. The theory of knowledge axiomatized in such a way extends conservatively the classical interpretation of PA.

In particular, let us characterize the following property emphasizing the role of each element of the theory of knowledge. If T is a theory with an axiomatization of the form Σ ∪ BΣ extending the theory of knowledge, then:

σ ∈ T ⇔ Bσ ∈ T.
Hence, we have a correspondence between the valid sentences of T and the sentences that can be intuitively proved in T . On one hand, if σ ∈ Σ ∪ BΣ then, by B4, BBσ ∈ T and, by B3, Bσ ∈ T . Now, assume that σ 2 derives by modus ponens from σ 1 , σ 1 → σ 2 ∈ T . By induction hypothesis, both Bσ 1 and B(σ 1 → σ 2 ) are in T and, by applying B2, we derive Bσ 2 ∈ T . On the other hand, by B3 we have that Bσ ∈ T implies σ ∈ T .

Finally, giving for granted the accessibility principle, it is possible to reason about a simple theory of knowledge [START_REF] Carlson | Knowledge, machines, and the consistency of Reinhardt's strong mechanistic thesis[END_REF][START_REF] Alexander | A machine that knows its own code[END_REF] in which B1 is replaced by the rule of necessitation: if φ, then Bφ. Hence, it can be shown that the theory of knowledge corresponds to the set of sentences that can be proved in the modal system S4.

Knowing Entities and Machines

The statements presented in the next section are considered with respect to a knower reasoning about basic arithmetic. Therefore, we refer to the standard interpretation of the model of arithmetic, extended formally in the following way. Given a set of sentences Σ in EA, the structure N Σ over the set of naturals N is such that, for any wff φ with one free variable x, it holds that:

N Σ |= Bφ[s] iff Σ |= φ(x | s(x))
where n is the numeral associated to the natural number n (i.e., the term of the language of PA corresponding to n). Informally, Bφ is satisfied by N Σ under assignment s if and only if φ is logical consequence of Σ whenever replacing each free occurrence of x in φ by the numeral associated to s(x). Then, we say that a theory T in EA is a (knowing) entity if N T is a model of the theory of knowledge. Finally, by using Carlson's notation [START_REF] Carlson | Knowledge, machines, and the consistency of Reinhardt's strong mechanistic thesis[END_REF], a recursively enumerable entity is said to be a (knowing) machine.

Properties of knowing machines

While Shapiro was more involved with the description in EA of the grounds of constructive mathematics [START_REF] Shapiro | Epistemic and intuitionistic arithmetic[END_REF], Reinhardt's main intent was to use the theory of knowledge illustrated above for studying variants of the Church-Turing Thesis. The effective version under analysis is based on a notion of weak decidability, according to which a property φ of natural numbers is weakly decidable if there exists a theorem proving procedure by an idealized human mathematician that can eventually produce each n satisfying the property, i.e., such that φ(n) is true. Such a notion is expressed in EA in terms of the modal operator B, thus leading to a definition of weak B-decidability: ∀n(φ(n) → Bφ(n)), i.e., given a formula φ with one free variable, it holds that the assignments of the variable satisfying φ are known. The objective is then to analyze the relationship between properties that are weakly B-decidable and the Turing Machines (TMs) that formalize the decision algorithm for these properties. Therefore, weak B-decidability expresses in the framework of EA a notion of humanly provable as discussed in Section 1, and its equivalence with respect to Turing computability is represented by the following formula (which we simply call Turing Thesis):

∃e∀x(Bφ ↔ x ∈ W e ) (1)
where we assume that W e is the recursively enumerable set with Gödel number represented by the PA language term e. Notice that such a statement can be seen as a constructive, effective version of Church's Thesis: a recursively enumerable set exists (and, therefore, a TM enumerates its elements) that contains all and only the assignments of x making φ intuitively provable, that is to say, Turing computability coincides with weak B-decidability.

As we will show later, we have that ( 1) is consistent in EA [START_REF] Reinhardt | The consistency of a variant of Church's thesis with an axiomatic theory of an epistemic notation[END_REF]. However, validity of the Turing Thesis implies that there exists an absolutely undecidable statement. Formally, the following theorem holds.

Theorem 1 (Incompleteness of B). For every theory T in which (1) holds, then there exists φ with one free variable such that:

T ∃x(φ(x) ∧ ¬Bφ(x)).

Proof. Let us assume φ(x) = ¬(x ∈ W x ). Then, from (1) we derive:

∃e∀x(B¬(x ∈ W x ) ↔ x ∈ W e )
and, by taking x = e, it follows:

∃e(B¬(e ∈ W e ) ↔ e ∈ W e ). (2)
Since, by applying B3, we have:

∀e(B¬(e ∈ W e ) → ¬(e ∈ W e )) (3)
from ( 2) and (3) we obtain:

∃e((Bφ(e) ↔ ¬φ(e)) ∧ (Bφ(e) → φ(e))).

Now, recalling that, by applying tautologies, Bφ(e) → ¬φ(e) and Bφ(e) → φ(e) imply ¬Bφ(e), and that ¬φ(e) → Bφ(e) and Bφ(e) → φ(e) imply φ(e), then it is immediate to observe that a tautological consequence of ( 4) is the following:

∃e(φ(e) ∧ ¬Bφ(e)) (5)
which corresponds to the statement of the theorem.

Notice that B3 and (5) imply:

∃e(¬Bφ(e) ∧ ¬B¬φ(e))

because (B¬φ(e) → ¬φ(e)) and φ(e) hold, and therefore it must be ¬B¬φ(e), thus stating the absolute undecidability of φ(e). Such an incompleteness result can be viewed as a version of Gödel's first incompleteness theorem, that is, if T is sound, then it is also incomplete [START_REF] Reinhardt | Absolute versions of incompleteness theorems[END_REF].

Analogously, even Gödel's second incompleteness theorem can be recast in this setting to show that the consistency of T is absolutely unprovable. More precisely, a generalized version of Gödel's second incompleteness theorem defined in EA establishes that for any intuitively definable upper bound for intuitive provability we have an absolute impossibility of a consistency proof [START_REF] Reinhardt | Absolute versions of incompleteness theorems[END_REF]. Formally, the following second incompleteness of B theorem holds, where we use the notation φ to represent the symbolic expression n, with n the Gödel number of the formula φ.

Theorem 2 (Incompleteness of B). Assume the existence of a formula ψ with one free variable such that for every sentence σ of T it holds that:

T B(Bσ → ψ(σ)) (6)
then it also holds that: T B¬BConψ where the predicate Con expressing consistency of its argument is defined as ∀x¬(ψ(x) ∧ ¬ψ(x)).

Before proving this theorem, it is worth observing that ψ represents an intuitive upper bound for provability: it is provable that if σ is known, then ψ holds when applied to the symbolic representation of the Gödel number of σ. In fact, notice that the formula above is an instance of the B-closure of B3 stating the knowledge of the bounds of intuitive provability, which are established by ψ. Then, the result states that the consistency of ψ is absolutely unprovable or, in other words, no interesting upper bounds for intuitive provability can be defined. The following proof relies on a preliminary result by Reinhardt [START_REF] Reinhardt | Absolute versions of incompleteness theorems[END_REF] stating the applicability of Gödel Fixed Point Lemma6 in the setting of EA.

Proof. By Gödel Fixed Point Lemma, we can choose a sentence σ such that:

T B(σ ↔ B¬ψ(σ)). (7)
We now construct the derivation sequence:

σ → B¬ψ(σ) by ( 7) and B3 → BB¬ψ(σ) by B4 → Bσ by ( 7) and B2 → ψ(σ) by ( 6) and B3.

At the same time, we also have the following derivation sequence:

σ → B¬ψ(σ) by ( 7) and B3 → ¬ψ(σ) by B3.

Hence, from σ → ψ(σ) and σ → ¬ψ(σ) we obtain σ → ¬Conψ by definition of Con. Since it is possible to prefix with B every step of the proof above, it also holds B(σ → ¬Conψ) and, equivalently, B(Conψ → ¬σ), which, by applying B2, implies:

BConψ → B¬σ. (8)
Now:

BConψ → B¬σ → ¬ψ(σ) by ( 6) and B3.

Since it is possible to prefix with B every step, we have B(BConψ → ¬ψ(σ)) from which, by B2, it follows BBConψ → B¬ψ(σ). From this and by virtue of B4 we derive: BConψ → B¬ψ(σ) → σ by ( 7) and B3.

Then, by applying B3, from (8) we derive BConψ → ¬σ, which, in conjunction with BConψ → σ proved above, imply ¬BConψ. Again, by prefixing B to each previous step, we get: B¬BConψ.

This result completes the part concerning the revisiting of the Gödel incompleteness theorems in EA starting from the Turing Thesis. In the following, it is our interest to consider stronger versions of the Turing Thesis involving the knowledge that can be acquired in the theory about such a thesis. The strongest claim, which we call Reinhardt's schema, establishes that the index of the TM deciding Bφ in the Turing Thesis is known. This claim can be refuted in EA, as stated by the following theorem.

Theorem 3 (Reinhardt's schema [START_REF] Reinhardt | Epistemic theories and the interpretation of Gödel's incompleteness theorems[END_REF]). ∃eB∀x(Bφ ↔ x ∈ W e ) is not consistent in EA.

Notice that Reinhardt's schema states that a TM exists for which it is known that it enumerates all (and only) the elements (for which it is known) that make φ true. By citing Carlson's intuition, I know to be a TM and I know which one. The inconsistency of this schema can be viewed as an alternative characterization of Gödel's first incompleteness theorem.

Proof. By B1, from Reinhardt's schema we derive:

∃e∀xB(Bφ ↔ x ∈ W e )
Now, assume φ(x) = ¬(x ∈ W x ) and x = e, hence we have:

B(Bφ ↔ ¬φ)
while by the B-closure of B3:

B(Bφ → φ).
From the conjunction of the two above, by applying tautologies and distributivity, we derive B(φ ∧ ¬Bφ) and then Bφ ∧ B¬Bφ, and applying B3:

Bφ ∧ ¬Bφ.
A weaker version of Reinhardt's schema consists of moving the outermost B operator to prefix the whole formula, thus obtaining the so-called Strong Mechanistic Thesis, which we call Carlson's schema from the author who proved its consistency in EA.

Theorem 4 (Carlson's schema [START_REF] Carlson | Knowledge, machines, and the consistency of Reinhardt's strong mechanistic thesis[END_REF]). B∃e∀x(Bφ ↔ x ∈ W e ) is consistent in EA.

By citing Carlson, I know that the set of x for which I know φ(x) is recursively enumerable or, by rephrasing an hypothesis studied by Benacerraf independently, I know to be a TM but I do not know which one.

To prove Theorem 4, in [START_REF] Carlson | Knowledge, machines, and the consistency of Reinhardt's strong mechanistic thesis[END_REF] it is shown that the theory of knowledge plus Carlson's schema is a (knowing) machine. In order to establish the induction proving the result, Carlson generalizes the modal operator B by introducing a collection of operators B t , where t belongs to a linearly ordered set and can be interpreted as the amount of steps (either logical or temporal) needed to prove the formula guarded by B t . In such a stratified version of the theory of knowledge, each formula is such that any occurrence of B t in the scope of another occurrence B t satisfies the condition t < t . Then, the key parts of the proof rely on showing that the standard axiomatization of the stratified theory of knowledge for EA plus Carlson's schema is recursively enumerable and that such a result is inherited by the non-stratified theory.

Theorem 4 implies, as a corollary deriving from the application of B3, the validity of the Turing Thesis. Another interesting corollary of the theorem is related to the first incompleteness of B theorem.

Corollary 1. Given T and φ as in Theorem 1, then:

T B∃e(φ(e) ∧ ¬Bφ(e)).

Proof. By Theorem 1, (5) holds and if (1) holds whenever prefixed by B, then so does [START_REF] Benacerraf | God, the devil and Gödel[END_REF], because under such a hypothesis it is possible to prefix with B every step of the proof of Theorem 1. Now, it is sufficient to notice that (1) prefixed by B is exactly Carlson's schema.

Informally, this result states that knowledge of the Turing Thesis implies knowledge that there exists an absolutely undecidable sentence in EA. In particular, the corollary requires B prefixed to B3, which is the only axiom for B used in the proof of (5) in Theorem 1. In general, all the proofs related to variants of the Church-Turing Thesis discussed so far rely on the validity of B(Bφ → φ), stating that in the formal system the soundness (factivity, as called by Alexander) of knowledge can be proved. Under such a condition, we have Theorem 4 stating that for each wff φ, an unspecified TM enumerates recursively the set of values for which one knows that φ is satisfiable (which can be read as "I know to have some code related to φ"). On the other hand, we have Theorem 3 stating that the knowledge of the identity of such a TM cannot be acquired (which can be read as "I cannot know my own code related to φ"). In between the limiting results stated by Reinhardt and Carlson, Alexander has recently proved a dichotomy revealing the relation between knowledge of factivity and knowledge of own code. Either a machine can know to be sound (that is, B(Bφ → φ) is valid) as well as that it has some code (without knowing which, as proved by Carlson), or it can know its own code exactly (thus proving the consistency of Reinhardt's schema) but in such a case cannot know its own soundness (that is, B(Bφ → φ) is not valid anymore). As a consequence, we emphasize that renouncing knowledge of soundness implies that the machine loses also knowledge of incompleteness.

Providing that the axioms of EA mod factivity consist of the axioms of EA except for the universal closure of B3 prefixed by B (i.e., B(Bφ → φ) is not valid), it is possible to prove that: Theorem 5 (Alexander [START_REF] Alexander | A machine that knows its own code[END_REF]). Reinhardt's schema is consistent in EA mod factivity.

and then to construct the previous dichotomy.

In order to sketch the main intuition behind the proof (see [START_REF] Alexander | The Theory of Several Knowing Machines[END_REF] for details), it is worth considering the family of axioms Σ(n), for n ∈ N, which essentially consists of the axioms of EA mod factivity minus B3 and plus the additional schemes: ∀x(Bφ ↔ b(x, φ) ∈ W n) for any wff φ with one free variable x where b is a canonical computable bijection -hence definable in the language of PA -mapping pairs (represented by the symbolic Gödel number of φ and its input x) to numerals. Since by completeness and compactness of EA, Σ(n) is recursively enumerable, then, by the Church-Turing Thesis, chosen φ as above, for every n ∈ N there exists a total computable function f : N → N such that:

W f (n) = {b(m, φ) | Σ(n) |= φ(x | m)}
meaning that the set of elements that, assigned to x in φ, make φ a logical consequence of Σ(n), is recursively enumerable by a set of index f (n). Now, by Kleene's Recursion Theorem, it holds that there exists e ∈ N such that W e = W f (e) .

Hence, N Σ(e) satisfies ∀x(Bφ ↔ b(x, φ) ∈ W ē), because for each s with s(x) = m it holds:

N Σ(e) |= Bφ[s] ⇔ Σ(e) |= φ(x | s(x)) ⇔ Σ(e) |= φ(x | m) ⇔ b(m, φ) ∈ W e ⇔ N Σ(e) |= b(m, φ) ∈ W ē ⇔ N Σ(e) |= (b(x, φ) ∈ W ē)[s].
In addition, by construction, N Σ(e) satisfies also all the instances of the other axioms of Σ(e). Then, for each axiom φ of Σ(e), it holds that Bφ is satisfied by N Σ(e) because Σ(e) |= φ and φ is a sentence, so that, by definition of N Σ(e) , it follows N Σ(e) |= Bφ. Notice that, by virtue of this condition, N Σ(e) satisfies Reinhardt's schema. To complete the proof, it remains to show that N Σ(e) satisfies B3, which is not included in Σ(e), i.e., N Σ(e) |= Bφ → φ. Supposing N Σ(e) |= Bφ[s] for some s, then Σ(e) |= φ(x | s(x)) and, since N Σ(e) |= Σ(e), we derive N Σ(e) |= φ(x | s(x)).

Summarizing, we have described a recursively enumerable set of sentences (including EA mod factivity and Reinhardt's schema) that turns out to be an entity, which represents the result expressed by Theorem 5.

In the framework of such a machine, we show a result related to a specific case. Consider φ(x) = (x ∈ W x ) and x = e. Thus from: which expresses a limited form of knowledge of soundness allowed in EA mod factivity. More precisely, we have a specific function φ for which the related machine knows its own code and knows to be sound with respect to a specific input x. We notice that, by definition, φ turns out to represent an instance of an interpreter function 7 and the specific input x is the index of the machine itself. Therefore, by following the same intuitions derived from Reinhardt and Carlson theorems, "If I am a universal TM knowing my own code, then I know the soundness of the computation provided when I am fed with my own code". By virtue of the previous results and according to Alexander's dichotomy, knowledge about other TMs is much more limited. As a consequence, in general a universal TM knowing its own code cannot know the soundness of what it proves when interpreting other TMs. On the other hand, if a (universal) TM can prove the soundness of its knowledge, then it cannot know its own code. In our opinion, this is an interesting enhancement of the tradeoff result provided by Alexander representing an additional formal element for the analysis of the Gödelian Arguments.

∃eB∀x(Bφ ↔ x ∈ W e )

Conclusion

Historically, the idea of a provability logic is first discussed by Gödel [START_REF] Gödel | Zum intuitionistischen Aussagenkalkül[END_REF][START_REF] Gödel | Eine Interpretation des intuitionistischen Aussagenkalküls[END_REF] with the aim of defining a formal semantics for intuitionistic truth [START_REF] Troelstra | Constructivism in Mathematics[END_REF]. Gödel's calculus is based on propositional logic and on the modal operator 2, and is basically equivalent to S4 [START_REF] Lewis | Symbolic Logic[END_REF]. The resulting notion of provability shall not be confused with formal deducibility for theories including PA, as such a correspondence leads to a contradiction [START_REF] Gödel | Eine Interpretation des intuitionistischen Aussagenkalküls[END_REF]. Following Gödel's attempt, several approaches have been proposed either to define axiom systems for formal deducibility, or to find an exact provability semantics for S4-like modal systems. For a comprehensive survey, we refer, e.g., to [START_REF] Artemov | Provability logic[END_REF].

In this setting, the theory of knowing machines offers a proof-theoretic framework to reason about the notions of intuitive provability and consistency of TMs. In practice, the extended results show precisely the relation between selfawareness of soundness and of own code in the setting of knowing machines, and show some compatibility with philosophical arguments like the following suggestions due to Gödel himself [START_REF] Wang | From Mathematics to Philosophy[END_REF]:

On the other hand, on the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory.

The results provided in EA are obtained by following formal lines of reasoning even if starting from the definition of an epistemic notion of intuitive provability. In fact, on one hand, it is worth noticing that the modal operator B necessitates an atomic treatment of formulas of the form Bφ and allowing only for an axiomatic representation of its properties. Hence, the lack of a precise modeltheoretic semantics represents a weakness. However, on the other hand, it is also worth observing that all the proofs of the results provided by Reinhardt, Carlson, and ourselves derive from the formal application of deductive reasoning and proof theoretic techniques. The unique exception is given by Theorem 5, which relies on the application of the Church-Turing Thesis (see, as an example, [START_REF] Mauro | The informal side of computability: Church-Turing thesis, in practice[END_REF] for the informal aspects of practical uses of the Church-Turing Thesis).

Even in the setting of EA as defined by Shapiro [START_REF] Shapiro | Epistemic and intuitionistic arithmetic[END_REF], the consistency of epistemic variants of Myhill's version of the Church-Turing Thesis [START_REF] Myhill | Intensional set theory[END_REF] can be demonstrated [START_REF] Flagg | Church's thesis is consistent with epistemic arithmetic[END_REF][START_REF] Goodman | Flagg realizability in arithmetic[END_REF][START_REF] Horsten | In defense of epistemic arithmetic[END_REF]. Such variants are captured by the following informal statement: if it is intuitively provable that for each x there exists y such that it holds that φ(x, y) can be proved, then φ determines a total recursive function. This is analogous to the Turing Thesis discussed in this paper and is independent from the formalization of Carlson's schema, which represents the strongest mechanistic position demonstrated in the framework defined by Reinhardt.

The abstractions behind the notion of intuitive provability, which cover all the aspects related to space and time constraints [START_REF] Reinhardt | Absolute versions of incompleteness theorems[END_REF], do not allow for reasoning about the relation between knowability and complexity. As future work, it would be interesting to investigate such a relation, in order to formalize what informally stated, e.g., by Benacerraf [START_REF] Benacerraf | God, the devil and Gödel[END_REF]:

It seems to be consistent with all this that I am indeed a Turing machine, but one with such a complex machine table (program) that I ascertain what it is.

  we derive: ∃eB(Bφ(e) ↔ e ∈ W e ) and: B(Bφ(e) → φ(e))

We employ the standard nonlogical symbols of PA syntax 0, S, +, •, and the usual application order on the operators.

A variable x occurs free in φ if it is not in the scope of any quantifier of φ defined over x.

In particular, the set of valid wff is recursively enumerable and if Σ is recursively enumerable then so is {φ | Σ |= φ}.

Let ψ(x) be any formula in the language of PA with one free variable x. Then, there exists a sentence σ of which it can be proved that σ ↔ ψ(σ).

Given any gödelization of functions, an interpreter fu is a function mimicking the behavior of any other function, i.e., fu(x, y) = fx(y). As an example, the universal TM is an interpreter. We recall that interpreters represent a classical tool in computability theory and play a fundamental role for programming languages.


Marie-Jeanne Lesot

Adrien Revault D'allonnes

Information quality and uncertainty

Keywords: information scoring, information processing, uncertainty type, competence, reliability, plausibility, credibility, linguistic uncertainty

The quality of a piece of information depends, among others, on the certainty that can be attached to it, which relates to the degree of confidence that can be put in it. This paper discusses various components to be considered when assessing this certainty level. It shows that they cover a wide range of different types of uncertainty and provide a highly relevant application domain for theoretical questioning about uncertainty modelling. It also describes several frameworks that have been considered for this task.

Introduction

Information quality (see e.g. [START_REF] Berti-Equille | Assessment and analysis of information quality: a multidimensional model and case studies[END_REF]) and its implementation in the domain of information evaluation (see e.g. [START_REF]Information Evaluation[END_REF]) aim at providing guidance and help to users in the drowning quantity of information they are nowadays overwhelmed with, in particular due to the dramatic increase of Web usage, e.g. through blogs and social networks, such as Facebook and Twitter. One specificity of these new media is that everyone can participate in the information spread and be a source of information, making the question of a relevance measure of the available information crucial. As a consequence, it is necessary to dispose of tools for automatically assessing their quality: there is an acute need for automatic methods to identify the "best", e.g. understood as the most useful, pieces of information.

Numerous criteria and properties have been proposed and considered to to that aim [START_REF] Berti-Equille | Assessment and analysis of information quality: a multidimensional model and case studies[END_REF][START_REF]Information Evaluation[END_REF]. This paper1 focuses on the certainty dimension, numerically evaluated as a degree of certainty that can be attached to any piece of information. In a schematic view, it exploits the argument according to which a certain piece of information is worthier than a doubtful one. Insofar, it is related to the task that aims at assessing the trust that can be put in a piece of information. It can be underlined that such a degree of trust can mean either evaluating the reality of the fact the piece of information reports [START_REF]OTAN: Annex to STANAG[END_REF][START_REF] Demolombe | Reasoning about trust: a formal logical framework[END_REF][START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF] or the extent to which the rater is convinced, based on the process with which he forms an opinion about this piece of information [START_REF] Besombes | An extension of STANAG2022 for information scoring[END_REF][START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF][START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF].

Even if uncertainty is only one of its components, information quality appears as a highly relevant application framework for the theoretical domain of uncertainty modelling. Indeed, it turns out to be a very challenging one, raising critical requirements that lead to question existing models and possibly to develop new ones. As discussed in this paper, information processing involves several types of uncertainty that must be distinguished, appropriately modelled and possibly combined: information-related uncertainty covers a wide spread spanning over several dimensions. As detailed in the following, one can mention distinctions between objective and subjective uncertainty, as well as between general vs contextual uncertainty.

This paper first discusses various kinds of uncertainty that can be attached to a piece of information in Section 2, organising them according to their cause, i.e. the characteristic of the considered piece of information that triggers them. Section 3 discusses the two axes objective-subjective and general-contextual. Section 4 briefly describes some theoretical frameworks that have been proposed to model uncertainty for information evaluation.

Sources of uncertainty in the information processing framework

This section discusses 5 sources of uncertainty that can be considered in the framework of information processing, structuring them according to their cause: it distinguishes the uncertainties respectively triggered by the content of a piece of information, its source, its context, its formulation and its automatic extraction.

In order to illustrate these types, it considers the following fictitious piece of information together with two basic meta-data, namely author and publication date:

On February 15th 2015, the International Olympic Committee declared "In 2048, the Summer Olympic Games will probably take place in November"

Content-related uncertainty: what is said?

The degree of uncertainty attached to a piece of information obviously depends on its content, i.e. the answer to the question "what does it say?": for the running example, it for instance relates to the assertion that can be schematically written as "2048 Summer Olympic Games dates = November".

More precisely, the piece of knowledge provided by the considered information can trigger a surprise effect that influences its uncertainty level: an unexpected fact can, at least at first, appear as more uncertain than a known one. The surprise effect can be measured with respect to two types of background, leading to distinguish between the notions of plausibility and credibility.

Knowledge context: plausibility Surprise can be defined as compared to the personal background of the information rater, i.e. as the compatibility of the considered piece of information with his/her knowledge, which is defined as plausibility [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF].

For instance for the running example, the asserted date may appear highly atypical, in particular to people living in the North hemisphere, who usually do not associate November with summer. As a consequence, they may receive the information with more caution and consider it as more uncertain than people living in the South hemisphere. Along the same lines, for someone with knowledge about the history of the Olympic Games, for instance knowing that the situation where the summer games take place in November already occurred (in 1956, for the Melbourne Games), the fact may appear as less uncertain.

Plausibility can be considered as the first component in the conviction establishing process [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF], that determines an a priori confidence level attached to a considered piece of information.

Other information context: credibility Surprise can also be defined with respect to other available pieces of information, e.g. other assertions provided in the same period regarding the location and dates of the Olympic Games: in this case, the considered piece of information is compared to other statements, building the credibility component [START_REF]OTAN: Annex to STANAG[END_REF][START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF].

More precisely, the assessment of credibility relies on the identification of corroboration or invalidation of the considered piece of information, defining another type of background for the evaluation of its attached uncertainty. This dimension both depends on the content of the information and the context of its assertion, it is more detailed in the section discussing the latter (Section 2.3).

Source-related uncertainty: who says it?

The uncertainty attached to an assertion also depends on its source, i.e. the answer to the question "who says it?": for the running example, it for instance relates to the fact that the International Olympic Committee provides it, who can be considered as a qualified source. The question is then to define the characteristics that make a source "qualified", this section discusses some of them, a more complete discussion can be found in [START_REF] Pichon | Multidimensional approach to reliability evaluation of information sources[END_REF] for instance.

It must be underlined that, altogether, the qualification of a source is contextual: it may not be the same for all pieces of information and may for instance depend on their topics, i.e. on their contents. However, some of its components remain topic-independent and general.

Source trustworthiness: reliability The reliability of the source corresponds to an a priori assessment of its quality, independently of the considered piece of information: it indicates whether, in general, the assertions it provides can be trusted or should be considered with caution.

In the seminal model for information evaluation [START_REF]OTAN: Annex to STANAG[END_REF], reliability plays a major role: this model represents the information score as a bi-gram defined as the concatenation of two symbols measured on two discrete graded scales associated to linguistic labels. The first one is called reliability, although it may depend on several distinct dimensions [START_REF] Capet | Information evaluation in the military domain: Doctrines, practices and shortcomings[END_REF]: its explicit and direct presence in the final score underlines its crucial role.

This subjective dimension, that may take different values for different raters, is difficult to define formally and thus to measure. It can be related to the concept of source reputation although the latter may be as difficult to model and quantify. In the case of Twitter sources, it has for instance been proposed to establish it from measurable quantities such as the number of followers or the source social status [START_REF] Ulicny | Toward formal reasoning with epistemic policies bout information quality in the twittersphere[END_REF].

Reliability can also be assessed by comparing previous source assertions with the ground truth when the occurrence of events has make it possible to establish whether the source was right or wrong [START_REF] Ellouedi | Assessing sensor reliability for mutisensor data fusion with the transferable belief model[END_REF][START_REF] Destercke | Methods for the evaluation and synthesis of multiple sources of information applied to nuclear computer codes[END_REF]. This approach highlights the fact that reliability is a dynamic concept whose measure should evolve with time. It also relates this dimension to validity [START_REF] Demolombe | Reasoning about trust: a formal logical framework[END_REF], according to which if the source produces a piece of information, then the latter is true2 .

Another component of reliability can be derived from the formulation used by the source: the number of citations that are contained in its publications allows to evaluate the extent to which it cites its own sources [START_REF] Castillo | Information credibility on twitter[END_REF][START_REF] Morris | Tweeting is believing? understanding microblog credibility perceptions[END_REF][START_REF] Pichon | Multidimensional approach to reliability evaluation of information sources[END_REF]. Now, offering a possibility to track back the origin of the provided information contributes to its reliability. Another indication can be derived from the amount of grammatical and spelling errors [START_REF] Castillo | Information credibility on twitter[END_REF][START_REF] Morris | Tweeting is believing? understanding microblog credibility perceptions[END_REF][START_REF] Pichon | Multidimensional approach to reliability evaluation of information sources[END_REF]: it is argued that a grammatically mistake-free text advocates for analysis capacity and critical way of thinking, which are desirable qualities for reliable sources. Although these quantities are related to the question "how is it said", discussed in Section 2.4, they capture an uncertainty originated from the source, allowing to infer some of the source characteristics, whereas the components described in Section 2.4 measure uncertainty originated from the expression itself.

Source expertise level: competence A distinct component of source-related uncertainty comes from its competence, that measures the extent to which it is entitled to provide the information, i.e. whether it is legitimated to give it [START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF][START_REF] Pichon | Multidimensional approach to reliability evaluation of information sources[END_REF]. In the considered example, it can for instance be considered that the IOC is much more competent than a taxi driver would be, leading to a lower uncertainty regarding the date of the 2048 Olympic Games than would occur if the latter provided the information.

Competence relates to the source expertise and appears to be a topic-dependent component: the IOC would be significantly less competent to provide information about the World Football Champions' cup; likewise, the taxi driver would be a legitimate source about efficient routes or traffic jams for instance, leading to less uncertain pieces of information regarding these topics.

It is worth noticing that two types of competence can be distinguished, an essential one and a more accidental one, that respectively apply to experts and witnesses [START_REF] Pichon | Multidimensional approach to reliability evaluation of information sources[END_REF]. Indeed, an essential competence can be established from the source fields of study and possibly diplomas, or from official roles: they provide a theoretical expertise and indeed entitle a source to make assertions about a given topic. On the other hand, a geographical or temporal proximity to an event provides an empical competence, granting witnesses a local expertise level.

Source intention

The assessment of the certainty degree attached to a piece of information, or the degree of trust put in it, can also depend on source characteristics even more difficult to establish, related to its intention: indeed, a source may for instance pursue an aim of desinformation, with the intention to lure the information rater. The certainty degree should obviously be reduced in such a communication paradigm, if it can be recognised as such.

This dimension is related to a sincerity feature, which captures the tendency of the source to tell the truth or not (see also [START_REF] Demolombe | Reasoning about trust: a formal logical framework[END_REF]): it can be considered that sincerity is a general characteristic of the source, describing its global tendency, whereas its intention is more contextual and varies for each piece of information. Sincerity can be considered as being related to the source reliability, as they both depend on the truth of the source assertions. The notion of sincerity may be seen as integrating a judgment component, that takes into account the source intention when interpreting the reason why it is wrong.

Source implication Another source characteristic is captured by its commitment degree, i.e. the extent to which it is involved in the propagation of the information it produces. Commitment depends on what the source may loose if it produces erroneous information, and, insofar, can be seen as related to its reputation.

It has for instance been proposed, in the case of Twitter sources, to measure the commitment degree as a function of the energy they put in their accounts [START_REF] Morris | Tweeting is believing? understanding microblog credibility perceptions[END_REF][START_REF] Pichon | Multidimensional approach to reliability evaluation of information sources[END_REF], in turn quantified by the richness of their profile, e.g. the number of filled fields or the presence of a picture, or the number of publications.

The source commitment also influences the uncertainty that can be attached to its assertions, under the interpretation that a highly committed source should be less prone to produce erroneous content and may be trusted.

Successive sources: hearsay A specific case for the evaluation of the source of an information occurs when the piece of information is not directly obtained, i.e. when it results from a series of successive sources, following a scheme of the form "S 1 says that S 2 says that ... S n says that F " where F is the fact and S i , i = 1..n the sources. Dedicated models have been proposed to process such cases, see e.g. [START_REF] Cholvy | Evaluation of information reported: a model in the theory of evidence[END_REF][START_REF] Cholvy | Collecting information reported by imperfect agents[END_REF].

Indeed, for such pieces of information, all previous source-uncertainty related components are measured not with respect to F (except for S n ) but, for S i , with respect to "S i+1 says that... S n says that F ": competence then for instance measures whether S i is entitled to report the assertions of S i+1 .

Context-related uncertainty: when is it said?

Another meta-data that influences the certainty attached to a considered piece of information relates to the context of its assertion, undestood as the global answer to the question "when is it said?". Different components can be considered, a purely temporal one as well as a more global one that depends on other available assertions.

Temporal context The date associated to an assertion contributes to the certainty level that can be attached to it, both in comparison with the date of the reported event and with the current date.

Indeed, the gap between the reported event and the assertion influences the uncertainty: information provided too much in advance may be considered with caution, decreasing their certainty level. For instance for the running example, if the assertion is about the Olympic Games in 2084, it may be interpreted as less certain.

On the other hand, a comparison with the current date can influence the importance that should be granted to a considered piece of information: when faced with an information stream, it can be useful to take into account older, and possibly out-of-date, pieces of information to a lesser degree than the more recent ones. It has for instance been proposed to associate each assertion with a currentness score [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF], so as to weight down the pieces of information according to their possible obsolescence. It can be underlined that such a model makes the evaluation sensitive to the information order, possibly leading to different results if a piece of information I 1 is published before I 2 or reciprocally. Such a behaviour can be considered as a realistic approach to model the uncertainty evolution when faced with an information stream.

It can be noted that beside these relative date comparisons, with respect to the information content and the current date, an absolute effect can be considered: some dates do bear meaning and influence the evaluation of their content. This component depends on a cultural dimension that makes difficult its general implementation. For instance, one can consider that information produced on April 1st os less certain than others; announcements contained election campaigns may also require a specific processing.

Other assertion context: credibility The evaluation of the uncertainty attached to a piece of information classicaly includes a cross-checking step, aiming at identifying complementary information backing up or undermining it: confirmations and invalidations respectively increase and decrease its certainty level.

The credibility dimension can be understood as a degree of confirmation resulting from comparison of the piece of information to be rated with the available information [START_REF]OTAN: Annex to STANAG[END_REF][START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF][START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF][START_REF] Revault D'allonnes | Dynamics of trust bulding: models of information cross-checking in a multivalued logic framework[END_REF].

In the seminal model [START_REF]OTAN: Annex to STANAG[END_REF], the second symbol of the bigram measures this confirmation degre, as indicated by the description it are accompanied by. It can be underlined that its inguistic labels mainly describe the information certainty, across the scale improbable, doubtful, possibly true, probably true, confirmed by other sources, showing the relation with this underlying essential component.

The principle of credibility evaluation [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF][START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF] consists in aggregating several assertions, said to be homologous, that refer to the same content. It thus depends on the choice of a similarity measure that measures the degree of confirmation by assessing the extent to which an homologous piece of information corroborates the information to be rated (see e.g. [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF] for a discussion on such eligible measures and their components).

The aggregation step can take into account various dimensions, among which the previous degree of confirmation, the individual uncertainty attached to the homologous information [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF][START_REF] Revault D'allonnes | An architecture for the evolution of trust: Definition and impact of the necessary dimensions of opinion making[END_REF][START_REF] Revault D'allonnes | Dynamics of trust bulding: models of information cross-checking in a multivalued logic framework[END_REF], but also the relations between the sources [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF]: one can consider a refined notion of confirmation and invalidation, weighting them according to affinity or hostility relations between sources. Indeed, a confirmation provided by sources known to be in affinity relation should have a lower influence than a confirmation by independent, not to say hostile, sources: friendly sources are expected to be in agreement and to produce somehow redundant information.

As the temporal component, the credibility dimension makes uncertainty evaluation sensitive to the order of the pieces of information in a stream, taking into account more subtle relations than their publication dates only. This dynamical behaviour, source of many a theory of argumentation, considers that two confirmations followed by an invalidation may lead to a different level of uncertainty than a confirmation followed by a contradiction and another confirmation might [START_REF] Revault D'allonnes | Dynamics of trust bulding: models of information cross-checking in a multivalued logic framework[END_REF].

Formulation-related uncertainty: how is it said?

The words used in a piece of information play a major role on the attached uncertainty level, both because of the imprecision they convey and the uncertainty they intrinsically convey. The additional role of linguistic quality, that influences the assesment of the source reliability, has been discussed in Section 2.2.

Natural language is often imprecise (see e.g. [START_REF] Zadeh | Precisiated natural language (PNL)[END_REF]), allowing for fuzziness of the conveyed message, which can lead to uncertainty: if, for instance, the IOC asserts that the 2048 Games will take place "around the end of the year", some uncertainty is attached to the fact that the games will take place in November. In this case, uncertainty arises from the approximate compatibility between the rated piece of information and the query (e.g. regarding the Games date): only a partial answer is available. Such imprecision also plays a role in the identification of homologous information involved in the cross-checking step of credibility assessment discussed in Section 2.3.

Beside imprecision, the used words also convey uncertainty: they give indication regarding the source own level of uncertainty and influence the overall evaluation of the uncertainty [START_REF] Auger | Expression of uncertainty in linguistic data[END_REF]. In the case of the considered example for instance, the linguistic expression contains the adverb "probably" whose presence increases the uncertainty of the final evaluation.

Linguistic works (see e.g. [21,[START_REF] Ba | Uncertainty of an event and its markers in natural language processing[END_REF]) propose classification of uncertainty bearing terms, making it possible to assess the global expressed uncertainty. Such terms include adjectives (such as certain, likely or improbable), modal verbs (e.g. may, might, could, should), adverbs (such as certainly, possibly or undeniably) or complex idiomatic structures. Modifiers such as "very" can be used to reinforce or weaken the previous linguistic tags.

Automatic processing-related uncertainty: how is it extracted?

A fifth level of uncertainty comes from the fact that the available pieces of information are automatically processed, which can introduce errors in the content identification and thus for many of the components mentioned in the previous sections.

Indeed, the evaluation of the uncertainty attached to a piece of information according to the previously cited dimensions for instance include the use of tools for named entity detection, event and relationship identification and date extraction [START_REF] Ba | Uncertainty of an event and its markers in natural language processing[END_REF]. They also require to solve difficult linguistic tasks, as negation handling and anaphora resolution, that still are challenges for automatic text processing systems. These uncertainties can be measured automatically, for instance through performance rates of the corresponding methods, i.e. using recognition rate, recall or precision.

Among the examples of the encountered difficulties, one can for instance mention possible errors in the text topic identification, possibly leading to erroneous assessment of the source competence (see Section 2.2). Similarly, the identification of the date in the processed document may result in mistakes in the evaluation of the temporal content (see Section 2.3). The most impacted dimension is probably credibility (Section 2.3), that relies on the extraction of homologous pieces of information, and therefore both on all the documents processing and the computation of their similarities. It can be noticed that this task is sometimes performed semi-automatically, in order to guarantee its quality, crucial for the whole system [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF].

Uncertainty types for information

Form a formal point of view, the various uncertainty types discussed in the previous section can be classified according to two axes, opposing objective s subjective uncertainties as well as general vs contextual ones.

It can be underlined that the considered uncertainties also differ in their very nature: for instance, some express structural doubts about the phenomena, as content plausibility or recognition rate for instance, whereas the linguistically triggered uncertainty on the other hand captures an imprecision level.

Objective vs subjective uncertainty A first axis discriminating the listed uncertainty types refers to the position of the rater and his/her implication in the evaluation: some of them actually do not depend on the rater and constitutes objective dimensions, whereas others are subjective.

Indeed, the evaluation of the uncertainty triggerend by the automatic processing step for instance is objective and can be automatically measured. Similarly, the evaluation of the degree of confirmation between two pieces of information, i.e. the credibility dimension, does not depend on the rater and is identical for all users.

On the other hand, the plausibility dimension is subjective: it is measured by comparison to the rater's background knowledge and therefore varies from one rater to another. Likewise, most source evaluation criteria can be considered as subjective: for instance, not all users may agree on the competence fields of a given source, nor on its intention.

General vs contextual uncertainty Another discriminating axis refers to the dependence of the dimension to the rated piece of information: some criteria are evaluated generally, a priori, i.e. independently of any information, whereas other characterise the considered one.

As an example, the source reliability does not depend on the rated piece of information and similarly applies to all the source assertions. The category of general criteria also involve the evaluation of the uncertainty triggered by automatic processing step, which is measured globally, for all types of information. Similarly, the measure of the formulation-related uncertainty relies on a linguistic modelling of uncertainty expression: the latter is built generally, not for a specific piece of information.

On the other hand, the source competence for instance is topic-dependent and thus varies from one piece of information to the other. In that sense, it is considered to be contextual. Obviously, the content credibility, as well as the temporal dimension, are contextual too.

Formal frameworks for information scoring

As discussed in the previous sections, the uncertainty to be considered in the domain of information quality covers different types. As a consequence, distinct formal frameworks have been considered to represent it or some of its components. A central issue is to dispose of aggregation operators to combine the individual uncertainty scores obtained for each considered component. It can be observed that some propositions focus on this aggregation issue, in a multicriteria aggregation approach, using for instance Choquet integrals [START_REF] Pichon | Multidimensional approach to reliability evaluation of information sources[END_REF]. This section briefly discusses the main existing uncertainty modelling frameworks applied to the case of information evaluation, distinguishing them depending on whether they model symbolic, ordered or numerical uncertainties.

Symbolic framework Symbolic approaches in the domain of information evaluation include logical representation, in particular in the framework of modal logics [START_REF] Demolombe | Reasoning about trust: a formal logical framework[END_REF][START_REF] Demolombe | A logical account of trust in information sources[END_REF][START_REF] Herzig | A logic of trust and reputation[END_REF][START_REF] Cholvy | When reported information is second hand[END_REF], that allow to perform logical inferences to characterise the sources and the pieces of information. However, they usually do not model the attached uncertainty.

The first formal framework for information evaluation considering uncertainty has been proposed in the seminal model [START_REF]OTAN: Annex to STANAG[END_REF]: it represents the information score as a bi-gram defined as the concatenation of two symbols measured on two discrete graded scales associated to linguistic labels: according to the descriptions they are accompanied by, the first one captures the source reliability and the second one the information credibility. However it has been shown [START_REF] Cholvy | Information evaluation in fusion: a case study[END_REF][START_REF] Besombes | An extension of STANAG2022 for information scoring[END_REF][START_REF] Capet | Information evaluation in the military domain: Doctrines, practices and shortcomings[END_REF] that this symbolic approach raises some difficulties, among others regarding the manipulation and comparison of the obtained scores.

Ordered framework: extended multivalued logic In order to ease the manipulation of uncertainty scores, it has been proposed to exploit an extended multivalued logic framework [START_REF] Revault D'allonnes | Trust-moderated informationlikelihood. a multi-valued logics approach[END_REF][START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] to model the process of trust building: trust can be defined on a single discrete graded scale, clarified with linguistic labels, improving the legibility of a unique degree with a semantic interpretation. Moreover, this framework is equipped with formal tools to combine the truth degrees through logical operations that generalise conjunction, disjunction or implication, as well as arithmetical ones [START_REF] Seridi | Approximate reasoning for processing uncertainty[END_REF].

The extension [START_REF] Revault D'allonnes | Trust-moderated informationlikelihood. a multi-valued logics approach[END_REF][START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF] of classical multivalued logic consists in introducing an additional degree that allows to distinguish between facts that are 'neither true nor false', i.e. that have a neutral truth value, and facts whose truth values cannot be evaluated: it makes it possible to distinguish between ignorance and neutral knowledge, which is for instance required to distinguish between a source whose reliability is unknown from a source with intermediate reliability.

Numerical frameworks: probability, possibility and evidence Probability theory is one of the most frequent framework used to model uncertainties. In the case of information evaluation, it can for instance naturally be used to quantify the uncertainty related to the extraction process, e.g. to measure error recognition rates of the applied automatic tools. However, many components of information evaluation uncertainty cannot be considered as having a probabilistic nature. Moreover, they need to distinguish between ignorance and uniform distribution, as sketched above, which cannot be implemented in the probabilistic framework. Furthermore, probabilities impose strong axiomatic constraints, restricting the choice of aggregation operators. Finally, probability theory often requires to set a priori distributions, which may be a difficult task in the case of information evaluation.

Possibility theory [START_REF] Zadeh | Fuzzy sets as the basis for a theory of possibility[END_REF] allows to represent the ignorance case separately from the neutral one and offers a wide variety of aggregation operators allowing to model many different behaviours for the combination of the considered uncertainty dimensions. It has for instance be applied to assess the uncertainty that can be attached to an event e, to answer the question "did e take place?", based on a set of pieces of information, enriching the binary answer yes/no with a confidence level [START_REF] Lesot | Proposition of a semi-automatic possibilistic information scoring process[END_REF].

The theory of belief functions [START_REF] Shafer | A mathematical theory of evidence[END_REF] generalises the probability and the possibility theories, offering a very rich expression power. It has been applied to information evaluation in particular to the issues of reported information [START_REF] Cholvy | Information evaluation in fusion: a case study[END_REF][START_REF] Cholvy | Evaluation of information reported: a model in the theory of evidence[END_REF] and source reliability measures [START_REF] Pichon | Relevance and truthfulness in information correction and fusion[END_REF][START_REF] Cholvy | When reported information is second hand[END_REF].

Conclusion

This chapter considered the issue of uncertainty in the domain of information evaluation, discussing the various types of uncertainty that can be attached to a piece of information, describing either the event it reports or its intrinsically attached trust. Many components can be distinguished, whose combination build to a complex notion for which several theoretical frameworks have been considered, so as to capture its diverse facets.

Among other topics related to uncertainty in the context of information evaluation, dynamics and validation offer challenging issues opening the way to research directions. The need for modelling the temporal evolution of uncertainty comes from the availability of information streams, beyond the individual pieces of information, as briefly mentioned previously. It also comes from the possible evolution of the general components of the source characteristics: if, for instance, the reliability of a source proves to change over time, it may require to re-evaluate the uncertainty attached to previously assessed pieces of information this source had provided, and, consequently, also to the information they are analogous to.

The issue of validation aims at assessing the quality of the proposed uncertainty models, both regarding the considered components and the chosen formal framework. Now its difficulty comes from the lack of data allowing to perform empirical studies: in the case of real data, it is difficult to dispose of expected scores to which the computed ones can be compared. The use of artificial data raises the challenge of their realistic generation controlling their relevance.

This paper is based on part of the panel which has been organised by Prof. Kovalerchuk at IPMU2012 on the general topic "Uncertainty Modelling"

Conversely, a source is said to be complete if, when a piece of information is true, the source provides it[START_REF] Demolombe | Reasoning about trust: a formal logical framework[END_REF]. This useful characterisation, related both to the source omniscience and "sharing communication type", is however less central for the assessment of the information uncertainty.


La 0.7 Sr 0.3 Mn0 3 (LSMO) suspended bolometers of various geometries have been fabricated using silicon micromachining techniques. Thanks to adequate geometry design and to the operating conditions (temperature and bias current), these uncooled bolometers could be limited by phonon noise and achieve NEP values below 1 pW Hz -1/2 around 300 K.

I. INTRODUCTION

HE La 0.7 Sr 0.3 MnO 3 (LSMO) material attracts great attention due to its high metal-insulator temperature, thus creating expectations for the realization of infrared bolometers operating at room temperature [1,2]. It shows a large electrical resistance variation as a function of temperature at room temperature (Temperature Coefficient of the Resistance around 0.02 K -1 in the 300-330 K range). It exhibits also a very low noise level compared to other resistive materials such as semiconductors (a-Si, a-Si:H, a-Ge, poly SiGe) and other oxide materials (semiconducting YBCO, VOx, etc.) [3,4]. We successfully fabricated LSMO/SrTiO 3 and LSMO/CaTiO 3 suspended bridges to form single line structures or square areas constituted of parallel lines (of width 2 or 4 µm). Very low thermal conductance and small time constant of few ms were measured.

The optical sensitivity versus frequency and the low frequency noise of the devices were measured at temperatures ranging from 290 K to 320 K and at various bias current. A discussion will be made about the proper choice of the optimized operating conditions. II. FABRICATION LSMO films were grown on buffer layers on Si (001) substrates by a reactive MBE (Molecular Beam Epitaxy) system. The fabrication process of the free-standing bridges needs only two photolithographic steps (Fig. 1). The electrode pads are defined by standard UV photolithography using a KI etching. The bridge geometry was defined by standard UV photolithography, followed by Ar ion beam milling down to the silicon substrate. The photoresist was not removed for protecting LSMO thin film and Au contact pads. RIE in SF6 gas was applied from the front side in order to etch the silicon under what would become suspended microbridges [5].

III. RESULTS

The electrical resistivity of suspended bridges measured as a function of temperature was very close to the characteristics of the non-suspended LSMO thin film, thus demonstrating that the fabrication process did not degrade the quality of LSMO. The temperature coefficient of resistance R (defined as 1/R × dR/dT) ranged from 0.012 to 0.021 K -1 . The optical sensitivity of such devices is superimposed to the temperature derivative of the resistance, thus showing the bolometric nature of the response (Fig. 2).

The thermal conductances of suspended strips were measured and are very close to the calculated ones. The measured values (of the order of 10 -7 W.K -1 ) are more than 4 orders of magnitude lower than the thermal conductance of non-suspended strips [6]. The resulting sensitivity is thus increased by more than 4 orders of magnitude compared to non-suspended bolometers. Low frequency noise measurements performed on the suspended structures show very low 1/f noise values as well as low white noise values. These results are consistent with the previous measurements on non suspended LSMO films [3,4]. Noise spectra typically consist of a low frequency noise and a white noise. No lorentzian component was observed.

In case of the suspended bridges, 1/f noise is also very low (  /n~7×10 -31 m 3 ). The white noise is dependent of current bias. It is not explained by the Johnson noise which has to be independent of the current value. The dependence is linked to the phonon noise indeed. The noise power spectral density due to phonon noise is quadratic dependent of the current bias and is inversely proportional to G. As the thermal conductance G is very low, phonon noise becomes dominant in the middle frequency range. This behavior could be observed since the excess noise is very low for LSMO compared to other thermoresistive materials used at 300 K [4]. Very low noise equivalent power NEP values have been measured (Fig. 3). A discussion about optimization of the design and the choice of operating conditions (temperature and current bias) of uncooled LSMO suspended bolometers will be given to achieve phonons noise limited uncooled bolometers.

B. Guillet 1 ,Fig. 1 . 2 .

 112 Fig. 1. Scanning Electrical Microscope photograph of a suspended bridge Fig. 2. Optical sensitivity and dR/dT of a suspended bridge versus temperature

Fig. 3 .

 3 Fig. 3. NEP measurements of various bolometers at 300 K

Kavli, Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA Uncooled phonon noise limited La 0.7 Sr 0.3 MnO 3 suspended bolometers