Jonathan Ben-Naim
email: jbennaim@lif.univ-mrs.fr

Lif

Pivotal and Pivotal-discriminative Consequence Relations *

In the present paper, we investigate consequence relations that are both paraconsistent and plausible (but still monotonic). More precisely, we put the focus on pivotal consequence relations, i.e. those relations that can be defined by a pivot (in the style of e.g. D. Makinson). A pivot is a fixed subset of valuations which are considered to be the important ones in the absolute sense. We worked with a general notion of valuation that covers e.g. the classical valuations as well as certain kinds of many-valued valuations. In the many-valued cases, pivotal consequence relations are paraconsistant (in addition to be plausible), i.e. they are capable of drawing reasonable conclusions which contain contradictions. We will provide in our general framework syntactic characterizations of several families of pivotal relations. In addition, we will provide, again in our general framework, characterizations of several families of pivotal-discriminative consequence relations. The latter are defined exactly as the plain version, but contradictory conclusions are rejected. We will also answer negatively a representation problem that was left open by Makinson. Finally, we will put in evidence a connexion with X-logics from Forget, Risch, and Siegel. The motivations and the framework of the present paper are very close to those of a previous paper of the author which is about preferential consequence relations. * This is an updated version of the paper of the same title published in The Journal of Logic and Computation. This version just contains a better presentation (so the numbering of definitions and propositions is different).

Introduction

One of the main motivations of this paper is to combine some tools used in Paraconsistent Reasoning on the one hand and Plausible Reasoning on the other hand to deal with both incomplete and inconsistent information. Actually, the motivations of the present paper are very close to those of a previous paper of the author which is about preferential consequence relations [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF].

Many-valued consequence relations have been developed with the aim of dealing with inconsistent information. These relations are defined in frameworks where valuations can assign more than two different truth values to formulas. In fact, they tolerate contradictions within the conclusions, but reject the principle of explosion according to which a single contradiction entails the deduction of every formula.

Independently, plausible (generally non-monotonic) consequence relations have been developed with the aim of dealing with incomplete information. Choice functions are central tools to define plausible relations. Indeed, suppose we have at our disposal a function µ, called a choice function, which chooses in any set of valuations V , those elements which are preferred, not necessarily in the absolute sense, but when the valuations in V are the only ones under consideration. Then, we can define a plausible consequence relation in the following natural way: a formula α follows from a set of formulas Γ iff every model for Γ chosen by µ is a model for α.

In the present paper, we put the focus on a particular family of choice functions. Let us present it. Suppose some valuations are considered to be the important ones in the absolute sense and collect them in a set I, called a pivot. This defines naturally a choice function. Indeed, simply choose in any set of valuations, those elements that belong to I. Those choice functions which can be defined in this manner constitute the aforementioned family. The consequence relations defined by this family are called pivotal consequence relations. Their importance has been put in evidence by D. Makinson in [START_REF] Makinson | Bridges between classical and nonmonotonic logic[END_REF][START_REF] Makinson | Bridges from Classical to Nonmonotonic Logic[END_REF] where it is shown that they constitute an easy conceptual passage between classical and plausible non-monotonic relations. Indeed, they are perfectly monotonic but already display some of the distinctive features (i.e. the choice functions) of plausible non-monotonic relations.

For a long time, research efforts on paraconsistent relations and plausible relations were separated. However, in many applications, the information is both incomplete and inconsistent. For instance, the semantic web or big databases inevitably contain inconsistencies. This can be due to human or material imperfections as well as contradictory sources of information. On the other hand, neither the web nor big databases can contain "all" information. Indeed, there are rules of which the exceptions cannot be enumerated. Also, some information might be left voluntarily vague or in concise form. Consequently, consequence relations that are both paraconsistent and plausible are useful to reason in such applications.

Such relations first appear in e.g. [Pri91, Bat98, KL92, AA00, KM02]. The idea begins by taking a many-valued framework to get paraconsistency. Then, only those models that are most preferred according to some particular binary preference relation on valuations (in the style of [START_REF] Shoham | Reasoning About Change[END_REF][START_REF] Shoham | A semantical approach to nonmonotonic logics[END_REF]) are relevant for making inference, which provides plausibility (and in fact also non-monotonicity). In [START_REF] Avron | Formula-preferential systems for paraconsistent non-monotonic reasoning[END_REF][START_REF] Avron | A formula-preferential base for paraconsistent and plausible reasoning systems[END_REF], A. Avron and I. Lev generalized the study to families of binary preference relations which compare two valuations using, for each of them, this part of a certain set of formulas it satisfies. The present paper follows this line of research by combining many-valued frameworks and choice functions.

More explicitly, we will investigate pivotal consequence relations in a general framework. According to the different assumptions which will be made about the latter, it will cover various kinds of frameworks, including e.g. the classical propositional one as well as some many-valued ones. Moreover, in the many-valued frameworks, pivotal relations lead to non-trivial conclusions is spite of the presence of contradictions and are thus useful to deal with both incomplete and inconsistent information. However, they will not satisfy the Disjunctive Syllogism (from α and ¬α ∨ β we can conclude β), whilst they satisfy it in classical frameworks.

In addition, we will investigate pivotal-discriminative consequence relations. They are defined exactly as the plain version, but any conclusion such that its negation is also a conclusion is rejected. In the classical framework, they do not bring something really new. Indeed, instead of concluding everything in the face of inconsistent information, we will simply conclude nothing. On the other hand, in the many-valued frameworks, where the conclusions are rational even from inconsistent information, the discriminative version will reject the contradictions among them, rendering them all the more rational.

As a first contribution, we will characterize, in our general framework, several families of pivotal(-discriminative) consequence relations. To do so, we will use techniques very similar to those of a previous paper of the author [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF]. The latter is about another family of choice functions. Let us present it. Suppose we are given a binary preference relation ≺ on states labelled by valuations (in the style of e.g. [START_REF] Kraus | Nonmonotonic reasoning, preferential models, and cumulative logics[END_REF][START_REF] Lehmann | What does a conditional knowledge base entail?[END_REF][START_REF] Schlechta | Coherent systems[END_REF]). This defines naturally a choice function. Indeed, choose in any set of valuations V , each element which labels a state which is ≺-preferred among those states which are labelled by the elements of V . Those choice functions which can be defined in this manner constitute the aforementioned family. The consequence relations defined by this family of choice functions are called preferential(-discriminative) consequence relations. In fact, the present paper provides an example of how the techniques developed in [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF] (especially, in the discriminative case) can be adapted to new families of choice functions. Note that, in the non-discriminative case, the techniques of [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF] are themselves strongly inspired by the work of K. Schlechta [START_REF] Schlechta | Coherent systems[END_REF].

In many cases, our characterizations will be purely syntactic. This has a lot of advantages, let us quote some important ones. Take a set of syntactic conditions that characterizes a family of pivotal consequence relations. This gives a syntactic point of view on this family defined semantically, which enables us to compare it to conditions known on the "market", and thus to other consequence relations. This can also give rise to questions like: if we modified the conditions in such and such a natural-looking way, what would happen on the semantic side? More generally, this can open the door to questions that would not easily come to mind otherwise or to techniques of proof that could not have been employed in the semantic approach. Some characterizations of pivotal consequence relations, valid in classical frameworks, can be found in the literature, e.g. [START_REF] Rott | Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning[END_REF][START_REF] Makinson | Bridges between classical and nonmonotonic logic[END_REF][START_REF] Makinson | Bridges from Classical to Nonmonotonic Logic[END_REF]. But, to the author knowledge, the present paper contains the first systematic work of characterization for them in non-classical frameworks. Similarly, it seems that the author is the first to investigate pivotal-discriminative consequence relations.

As a second contribution, we will answer negatively a representation problem that was left open by Makinson, namely, in an infinite classical framework, there does not exist a "normal" characterization for the family of all pivotal consequence relations. Approximatively, a characterization is called normal iff it contains only conditions universally quantified and of limited size. This constitutes the more innovative part of the paper. A last contribution is that a certain family of pivotal consequence relations will be shown to be precisely a certain family of X-logics, which were introduced by Forget, Risch, and Siegel [START_REF] Forget | Preferential logics are X-logics[END_REF].

The rest of the paper is organized as follows. In Section 2.1, we introduce our general framework and the different assumptions which will sometimes be made about it. We will see that it covers in particular the many-valued frameworks of the well-known paraconsistent logics F OUR and J 3 . In Section 2.2, we present choice functions and some of their well-known properties. We will see which properties characterize those choice functions that can be defined by a pivot. In Section 2.3, we define pivotal(-discriminative) consequence relations and give examples of them in both the classical and the many-valued frameworks. In Section 3, we provide our characterizations. In Section 4, we answer negatively the problem that was left open by Makinson. In Section 5, we put in evidence a connexion with X-logics. Finally, we conclude in Section 6.

Background

Semantic structures

Definitions and properties

The framework is exactly the one presented in [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF]. We will work with general formulas, valuations, and satisfaction. A similar approach has been taken in two well-known papers [START_REF] Makinson | Bridges from Classical to Nonmonotonic Logic[END_REF][START_REF] Lehmann | Nonmonotonic Logics and Semantics[END_REF].

Definition 1 We say that S is a semantic structure iff S = F, V, |= where F is a set, V is a set, and |= is a relation on V × F.
Intuitively, F is a set of formulas, V a set of valuations for these formulas, and |= a satisfaction relation for these objects (i.e. v |= α means the formula α is satisfied in the valuation v, i.e. v is a model for α).

Notation 2 Let F, V, |= be a semantic structure, Γ ⊆ F, and V ⊆ V. Then,

M Γ := {v ∈ V : ∀ α ∈ Γ, v |= α}, T (V ) := {α ∈ F : V ⊆ M α }, D := {V ⊆ V : ∃ Γ ⊆ F, M Γ = V }.
Suppose L is a language, ¬ a unary connective of L, and F the set of all wffs of L. Then,

T d (V ) := {α ∈ F : V ⊆ M α and V ⊆ M ¬α }, C := {V ⊆ V : ∀ α ∈ F, V ⊆ M α or V ⊆ M ¬α }.
Intuitively, M Γ is the set of all models for Γ and T (V ) the set of all formulas satisfied in V . Roughly speaking, T d (V ) is this part of T (V ) that is not contradictory. D is the set of all those sets of valuations that are definable by a set of formulas and C the set of all those sets of valuations that do not satisfy both a formula and its negation. As usual, M Γ,α , T (V, v) stand for respectively M Γ∪{α} , T (V ∪ {v}), etc.

Remark 3

The notations M Γ , T (V ), etc. should contain the semantic structure on which they are based. To increase readability, we will omit it. There will never be any ambiguity. We will omit similar things with other notations in the sequel, for the same reason.

A semantic structure defines a basic consequence relation: Notation 4 We denote by P the power set operator. Let F, V, |= be a semantic structure. We denote by ⊢ the relation on

P(F ) × F such that ∀ Γ ⊆ F, ∀ α ∈ F, Γ ⊢ α iff M Γ ⊆ M α .
Let |∼ be a relation on

P(F ) × F. Then, |∼(Γ) := {α ∈ F : Γ |∼ α}.
Suppose L is a language, ¬ a unary connective of L, F the set of all wffs of L, and Γ ⊆ F. Then, we say that

Γ is consistent iff ∀ α ∈ F, Γ ⊢ α or Γ ⊢ ¬α.
The following trivial facts hold, we will use them implicitly in the sequel:

Remark 5 Let F, V, |= be a semantic structure and Γ, ∆ ⊆ F. Then:

M Γ,∆ = M Γ ∩ M ∆ ; ⊢(Γ) = T (M Γ ); M Γ = M ⊢(Γ) ; Γ ⊆ ⊢(∆) iff ⊢(Γ) ⊆ ⊢(∆) iff M ∆ ⊆ M Γ .
Sometimes, we will need to make some of the following assumptions about a semantic structure: Definition 6 Let F, V, |= be a semantic structure. The define the following assumptions:

(A0) M F = ∅; (A1) V is finite.
Suppose L is a language, ¬ a unary connective of L, and F the set of all wffs of L. Then, define:

(A2) ∀ Γ ⊆ F, ∀ α ∈ F, if α ∈ T (M Γ ) and ¬α ∈ T (M Γ ), then M Γ ∩ M α ⊆ M ¬α .
Suppose ∨ and ∧ are binary connectives of L. Then, define:

(A3) ∀ α, β ∈ F, we have: M α∨β = M α ∪ M β ; M α∧β = M α ∩ M β ; M ¬¬α = M α ; M ¬(α∨β) = M ¬α∧¬β ; M ¬(α∧β) = M ¬α∨¬β .
Clearly, those assumptions are satisfied by classical semantic structures, i.e. structures where F , V, and |= are classical. In addition, we will see, in Sections 2.1.2 and 2.1.3, that they are also satisfied by certain many-valued semantic structures.

The semantic structure defined by F OUR

The logic F OUR was introduced by N. Belnap in [START_REF] Belnap | How computers should think[END_REF][START_REF] Belnap | A useful four-valued logic[END_REF]. This logic is useful to deal with inconsistent information. Several presentations are possible, depending on the language under consideration. For the needs of the present paper, a classical propositional language will be sufficient. The logic has been investigated intensively in e.g. [AA94, AA96, AA98], where richer languages, containing an implication connective ⊃ (first introduced by A. Avron [START_REF] Avron | Natural 3-valued logics : characterization and proof theory[END_REF]), were considered.

Notation 7

We denote by A a set of propositional symbols (or atoms). We denote by L c the classical propositional language containing A, the usual constants f alse and true, and the usual connectives ¬, ∨, and ∧. We denote by F c the set of all wffs of L c .

We briefly recall a meaning for the logic F OUR (more details can be found in [CLM99, [START_REF] Belnap | How computers should think[END_REF][START_REF] Belnap | A useful four-valued logic[END_REF]). Consider a system in which there are, on the one hand, sources of information and, on the other hand, a processor that listens to them. The sources provide information about the atoms only, not about the compound formulas. For each atom p, there are exactly four possibilities: either the processor is informed (by the sources, taken as a whole) that p is true; or he is informed that p is false; or he is informed of both; or he has no information about p.

Notation 8 Denote by 0 and 1 the classical truth values and define:

f := {0}; t := {1}; ⊤ := {0, 1}; ⊥ := ∅.
The global information given by the sources to the processor can be modelled by a function s from A to {f , t, ⊤, ⊥}. Intuitively, 1 ∈ s(p) means the processor is informed that p is true, whilst 0 ∈ s(p) means he is informed that p is false. Then, the processor naturally builds information about the compound formulas from s. Before he starts to do so, the situation can be be modelled by a function v from F c to {f , t, ⊤, ⊥} which agrees with s about the atoms and which assigns ⊥ to all compound formulas. Now, take p and q in A and suppose 1 ∈ v(p) or 1 ∈ v(q). Then, the processor naturally adds 1 to v(p ∨ q). Similarly, if 0 ∈ v(p) and 0 ∈ v(q), then he adds 0 in v(p ∨ q). Of course, such rules hold for ¬ and ∧ too.

Suppose all those rules are applied recursively to all compound formulas. Then, v represents the "full" (or developed) information given by the sources to the processor. Now, the valuations of the logic F OUR can be defined as exactly those functions that can be built in this manner (i.e. like v) from some information sources. More formally,

Definition 9 We say that v is a four-valued valuation iff v is a function from F c to {f , t, ⊤, ⊥} such that v(true) = t, v(f alse) = f and ∀ α, β ∈ F c , 1 ∈ v(¬α) iff 0 ∈ v(α); 0 ∈ v(¬α) iff 1 ∈ v(α); 1 ∈ v(α ∨ β) iff 1 ∈ v(α) or 1 ∈ v(β); 0 ∈ v(α ∨ β) iff 0 ∈ v(α) and 0 ∈ v(β); 1 ∈ v(α ∧ β) iff 1 ∈ v(α) and 1 ∈ v(β); 0 ∈ v(α ∧ β) iff 0 ∈ v(α) or 0 ∈ v(β).
We denote by V 4 the set of all four-valued valuations.

The definition may become more accessible if we see the four-valued valuations as those functions that satisfy Tables 1, 2, and 3 below:

v(α) v(¬α) f t t f ⊤ ⊤ ⊥ ⊥ Table 1. v(β) f t ⊤ ⊥ v(α) f f t ⊤ ⊥ t t t t t ⊤ ⊤ t ⊤ t ⊥ ⊥ t t ⊥ v(α ∨ β) Table 2. v(β) f t ⊤ ⊥ v(α) f f f f f t f t ⊤ ⊥ ⊤ f ⊤ ⊤ f ⊥ f ⊥ f ⊥ v(α ∧ β) Table 3.
In the logic F OUR, a formula α is considered to be satisfied iff the processor is informed that it is true (it does not matter whether he is also informed that α is false).

Notation 10 We denote by |=

4 the relation on V 4 × F c such that ∀ v ∈ V 4 , ∀ α ∈ F c , we have v |= 4 α iff 1 ∈ v(α).
Proof systems for the consequence relation ⊢ based on the semantic structure F c , V 4 , |= 4 (i.e. the semantic structure defined by F OUR) can be found in e.g. [START_REF] Arieli | Logical bilattices and inconsistent data[END_REF][START_REF] Arieli | Reasoning with logical bilattices[END_REF][START_REF] Arieli | The value of the four values[END_REF].

Note that the F OUR semantic structure satisfies (A0) and (A3). In addition, if A is finite, then (A1) is also satisfied. However, (A2) is not satisfied by this structure. In Section 2.1.3, we turn to a many-valued semantic structure which satisfies (A2).

The semantic structure defined by J 3

The logic J 3 was introduced in [START_REF] Schewe | Sur un problème de Jaśkowski[END_REF] to answer a question posed in 1948 by S. Jaśkowski, who was interested in systematizing theories capable of containing contradictions, especially if they occur in dialectical reasoning. The step from informal reasoning under contradictions and formal reasoning with databases and information was done in [START_REF] Carnielli | Formal inconsistency and evolutionary databases[END_REF] (also specialized for real database models in [START_REF] De Amo | A logical framework for integrating inconsistent information in multiple databases[END_REF]), where another formulation of J 3 called LFI1 was introduced, and its first-order version, semantics and proof theory were studied in detail. Investigations of J 3 have also been made in e.g. [START_REF] Avron | Natural 3-valued logics : characterization and proof theory[END_REF], where richer languages than our L c were considered.

The valuations of the logic J 3 can be given the same meaning as those of the logic F OUR, except that the consideration is restricted to those sources which always give some information about an atom. More formally,

Definition 11 We say that v is a three-valued valuation iff v is a function from F c to {f , t, ⊤} such that v(true) = t, v(f alse) = f and ∀ α, β ∈ F c , 1 ∈ v(¬α) iff 0 ∈ v(α); 0 ∈ v(¬α) iff 1 ∈ v(α); 1 ∈ v(α ∨ β) iff 1 ∈ v(α) or 1 ∈ v(β); 0 ∈ v(α ∨ β) iff 0 ∈ v(α) and 0 ∈ v(β); 1 ∈ v(α ∧ β) iff 1 ∈ v(α) and 1 ∈ v(β); 0 ∈ v(α ∧ β) iff 0 ∈ v(α) or 0 ∈ v(β).
We denote by V 3 the set of all three-valued valuations.

As previously, the definition may become more accessible if we see the three-valued valuations as those functions that satisfy Tables 4, 5, and 6 below:

v(α) v(¬α) f t t f ⊤ ⊤ Table 4. v(β) f t ⊤ v(α) f f t ⊤ t t t t ⊤ ⊤ t ⊤ v(α ∨ β) Table 5. v(β) f t ⊤ v(α) f f f f t f t ⊤ ⊤ f ⊤ ⊤ v(α ∧ β) Table 6.
We turn to the satisfaction relation.

Notation 12

We denote by |= 3 the relation on

V 3 × F c such that ∀ v ∈ V 3 , ∀ α ∈ F c , we have v |= 3 α iff 1 ∈ v(α).
Proof systems for the consequence relation ⊢ based on the semantic structure F c , V 3 , |= 3 (i.e. the semantic structure defined by J 3 ) have been provided in e.g. [START_REF] Avron | Natural 3-valued logics : characterization and proof theory[END_REF][START_REF] Schewe | Sur un problème de Jaśkowski[END_REF] and chapter IX of [START_REF] Epstein | The Semantic foundation of logic[END_REF]. The J 3 structure satisfies (A0), (A3) and (A2). In addition, if A is finite, then it satisfies (A1) too.

Choice functions

Definitions and properties

In many situations, an agent has some way to choose in any set of valuations V , those elements that are preferred (the bests, the more normal, etc.), not necessarily in the absolute sense, but when the valuations in V are the only ones under consideration. In Social Choice, this is modelled by choice functions [Che54, Arr59, Sen70, AM81, Leh02, Leh01].

Definition 13 Let V be a set, V ⊆ P(V), W ⊆ P(V), and µ a function from V to W. We say that µ is a choice function iff ∀ V ∈ V, µ(V ) ⊆ V .

Several properties for choice functions have been put in evidence by researchers in Social Choice. For the sake of completeness, we present two important ones though we will not investigate them in the present paper (a better presentation can be found in [START_REF] Lehmann | Nonmonotonic Logics and Semantics[END_REF]).

Suppose W is a set of valuations, V is a subset of W , and v ∈ V is a preferred valuation of W . Then, a natural requirement is that v is a preferred valuation of V . Indeed, in many situations, the larger a set is, the harder it is to be a preferred element of it, and he who can do the most can do the least. This property appears in [START_REF] Chernoff | Rational selection of decision functions[END_REF] and has been given the name Coherence in [START_REF] Moulin | Choices functions over a finite set: A summary[END_REF].

We turn to the second property. Suppose W is a set of valuations, V is a subset of W , and suppose all the preferred valuations of W belong to V . Then, they are expected to include all the preferred valuations of V . The importance of this property has been put in evidence by [Aiz85, AM81] and has been given the name Local Monotonicity in e.g. [START_REF] Lehmann | Nonmonotonic Logics and Semantics[END_REF].

In [START_REF] Schlechta | New techniques and completeness results for preferential structures[END_REF], Schlechta showed that Coherence and Local Monotonicity characterize those choice functions that can be defined by a binary preference relation on states labelled by valuations (in the style of e.g. [START_REF] Kraus | Nonmonotonic reasoning, preferential models, and cumulative logics[END_REF]). Now, we turn to properties relevant for the paper, i.e. properties which characterize those choice functions that can be defined by a pivot (in the style of e.g. D. Makinson [START_REF] Makinson | Bridges between classical and nonmonotonic logic[END_REF][START_REF] Makinson | Bridges from Classical to Nonmonotonic Logic[END_REF]). A pivot is a fixed subset of valuations which are considered to be the important ones in the absolute sense. Details will be given in Section 2.2.2. Definition 14 Let V be a set, V ⊆ P(V), W ⊆ P(V), and µ a choice function from V to W. We say that µ is strongly coherent

(SC) iff ∀ V, W ∈ V, µ(W ) ∩ V ⊆ µ(V ).
Suppose F, V, |= is a semantic structure. We say that µ is definability preserving (DP) iff

∀ V ∈ V ∩ D, µ(V ) ∈ D.
In addition, suppose V ∈ V. We say that µ is universe-codefinable (UC) iff

V \ µ(V) ∈ D.
Definability Preservation has been put in evidence first in [START_REF] Schlechta | Some results on classical preferential models[END_REF]. One of its advantages is that when the choice functions under consideration satisfy it, we will provide characterizations with purely syntactic conditions. To the author knowledge, Strong Coherence and Universe-codefinability are first introduced in the present paper. An advantage of Universe-codefinability is that it provides a link with X-logics [START_REF] Forget | Preferential logics are X-logics[END_REF]. We will see it in Section 5. Now, we turn to a last property:

Definition 15 Suppose L is a language, ¬ a unary connective of L, F the set of all wffs of L, F, V, |= a semantic structure, V ⊆ P(V), W ⊆ P(V), and µ be a choice function from V to W.

We say that µ is coherency preserving (CP) iff

∀ V ∈ V ∩ C, µ(V ) ∈ C.
Proof of (2). Direction: "→". Take the same I as for (0). Then, by verbatim the same proof,

∀ V ∈ V, µ(V ) = V ∩ I. It remains to show V \ I ∈ D. As µ is UC, V \ µ(V) ∈ D. But, V \ µ(V) = V \ (V ∩ I) = V \ I.
Direction: "←". Verbatim the proof of (0), except that in addition V \ I ∈ D. We show that µ is UC:

V \ µ(V) = V \ (V ∩ I) = V \ I ∈ D.

Pivotal(-discriminative) consequence relations 2.3.1 Definitions

Suppose we are given a semantic structure and a choice function µ on the valuations. Then, it is natural to conclude a formula α from a set of formulas Γ iff every model for Γ chosen by µ is a model for α. More formally: Definition 18 Let F, V, |= be a semantic structure and |∼ a relation on P(F ) × F. We say that |∼ is a pivotal consequence relation iff there exists a SC choice function µ from D to

P(V) such that ∀ Γ ⊆ F, ∀ α ∈ F, Γ |∼ α iff µ(M Γ ) ⊆ M α
In addition, if µ is DP, CP, etc., then so is |∼.

We called these relations "pivotal" because, in the light of Proposition 17, they can be defined equivalently with pivots, instead of SC choice functions. Their importance has been put in evidence by D. Makinson in e.g. [START_REF] Makinson | Bridges between classical and nonmonotonic logic[END_REF][START_REF] Makinson | Bridges from Classical to Nonmonotonic Logic[END_REF], where he showed that they constitute easy conceptual passage from basic to plausible non-monotonic consequence relations. Indeed, they are perfectly monotonic but already display some of the distinctive features (i.e. the choice functions) of plausible non-monotonic relations. Note that pivotal (resp. DP pivotal) consequence relations correspond to Makinson's pivotal-valuation (resp. pivotal-assumption) relations. We will give an example of how they can be used to draw plausible conclusions from incomplete information in Section 2.3.2.

Moreover, if a many-valued semantic structure is considered, they lead to rational and non-trivial conclusions is spite of the presence of contradictions and are thus useful to treat both incomplete and inconsistent information. However, they will not satisfy the Disjunctive Syllogism. We will give an example with the F OUR semantic structure in Section 2.3.3.

Characterizations of pivotal consequence relations, valid in classical frameworks, can be found in the literature. For instance, the following result appears to be part of folklore for decades: DP pivotal consequence relations correspond precisely to those supraclassical closure operations that are compact and satisfy Disjunction in the premisses. For more details see e.g. [START_REF] Rott | Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning[END_REF][START_REF] Makinson | Bridges between classical and nonmonotonic logic[END_REF][START_REF] Makinson | Bridges from Classical to Nonmonotonic Logic[END_REF]. Now, we turn to a qualified version of pivotal consequence. It captures the idea that the contradictions in the conclusions should be rejected.

Definition 19

Let L be a language, ¬ a unary connective of L, F the set of all wffs of L, F, V, |= a semantic structure, and |∼ a relation on P(F ) × F. We say that |∼ is a pivotal-discriminative consequence relation iff there exists a SC choice function

µ from D to P(V) such that ∀ Γ ⊆ F, ∀ α ∈ F, Γ |∼ α iff µ(M Γ ) ⊆ M α and µ(M Γ ) ⊆ M ¬α
In addition, if µ is DP, CP, etc., then so is |∼.

If a classical semantic structure is considered, the discriminative version does not bring something really new. Indeed, the only difference will be to conclude nothing instead of everything in the face of inconsistent information. On the other hand, with a many-valued structure, the conclusions are rational even from inconsistent information. The discriminative version will then reject the contradictions in the conclusions, rendering the latter all the more rational.

Example in the classical framework

Let L be a classical propositional language of which the atoms are r, q, and p. Intuitively, r means Nixon is a republican, q means Nixon is a quaker, and p means Nixon is a pacifist. Let F be the set of all wffs of L, V the set of all classical two-valued valuations of L, and |= the classical satisfaction relation for these objects. Then, V is the set of the 8 following valuations: v 0 , v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , and v 7 , which are defined in the obvious way by the following table:

r q p v 0 0 0 0 v 1 0 0 1 v 2 0 1 0 v 3 0 1 1 v 4 1 0 0 v 5 1 0 1 v 6 1 1 0 v 7 1 1 1
Now, consider the class of all republicans and the class of all quakers. Consider that a republican is normal iff he is not a pacifist and that a quaker is normal iff he is a pacifist. And, consider that a valuation is negligible iff (in it) Nixon is a non-normal individual of some class. Then, collect the non-negligible valuations in a pivot I. More formally:

I = {v ∈ V : if v |= r, then v |= ¬p; and if v |= q, then v |= p}.
Finally, let |∼ be the pivotal consequence relation defined by the SC choice function µ I .

Then, |∼ leads to "jump" to plausible conclusions from incomplete information. For instance, r |∼ ¬p and q |∼ p. But, we fall into triviality if we face new information that contradict previous "hasty" conclusions. For instance, {r, p} |∼ α, ∀ α ∈ L, and {q, ¬p} |∼ α, ∀ α ∈ L. This is the price to pay for being monotonic, whereas conclusions that are only plausible are accepted.

In addition, |∼ is not paraconsistent and some sets of formulas are rendered useless because there is no model in the pivot for them, though there are models for them. For instance, {q, r} |∼ α, ∀ α ∈ L.

Example in the F OUR framework

Consider the F OUR semantic structure F c , V 4 , |= 4 and suppose A = {r, q, p} (these objects have been defined in Section 2.1.2). In addition, make the same considerations about Nixon, the classes, normality, etc., as in Section 2.3.2, except that this time a valuation is considered to be negligible iff (in it) the processor is informed that Nixon is an individual of some class, but he is not informed that Nixon is a normal individual of that class. See Section 2.1.2 for recalls about the sources-processor systems. Again, collect the non-negligible valuations in a pivot I. More formally:

I = {v ∈ V 4 : if v |= r, then v |= ¬p; and if v |= q, then v |= p}.
Let |∼ be the pivotal consequence relation defined by the SC choice function µ I .

Then, again |∼ leads to "jump" to plausible conclusions from incomplete information. For instance, r |∼ ¬p and q |∼ p. Moreover, though "hasty" conclusions are never withdrawn, we do not fall into triviality when we face new information that contradict them. For instance, {r, p} |∼ p and {r, p} |∼ ¬p and {r, p} |∼ r and {r, p} |∼ ¬r.

In addition, |∼ is paraconsistent. For instance, {p, ¬p, q} |∼ p and {p, ¬p, q} |∼ ¬p and {p, ¬p, q} |∼ q and {p, ¬p, q} |∼ ¬q. And, less sets of formulas are rendered useless because there is no model in the pivot for them, though there are models for them. For instance, this time, {q, r} |∼ p and {q, r} |∼ ¬p and {q, r} |∼ q and {q, r} |∼ ¬q and {q, r} |∼ r and {q, r} |∼ ¬r.

However, |∼ does not satisfy the Disjunctive Syllogism. Indeed, for instance, {¬r, r ∨ q} |∼ q.

Characterizations

The first contributions of the paper are characterizations (in many cases, with purely syntactic conditions) of several families of pivotal and pivotal-discriminative consequence relations. Sometimes, we will need to make some assumptions about the semantic structure under consideration. However, no assumption will be needed for the two following families:

• the pivotal consequence relations (Section 3.2);

• the DP pivotal consequence relations (Section 3.1).

We will assume (A0) for:

• the UC pivotal consequence relations (Section 3.2).

We will need (A3) and (A1) for:

• the CP pivotal-discriminative consequence relations (Section 3.4);

• the CP DP pivotal-discriminative consequence relations (Section 3.3).

We will assume (A3), (A1), and (A2) for:

• the pivotal-discriminative consequence relations (Section 3.4);

• the DP pivotal-discriminative consequence relations (Section 3.3).

We will assume (A0), (A3), and (A1) for:

• the CP UC pivotal-discriminative consequence relations (Section 3.4).

We will need (A0), (A3), (A1), and (A2) for:

• the UC pivotal-discriminative consequence relations (Section 3.4).

The non-discriminative and definability preserving case

In the present section, we provide, in a general framework, a characterization for the family of all DP pivotal consequence relations. We will use techniques very similar to those of [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF] (see the DP and non-discriminative case). The latter are themselves strongly inspired by the work, in a classical propositional framework, of K. Schlechta (see Proposition 3.1 of [START_REF] Schlechta | New techniques and completeness results for preferential structures[END_REF]). The idea is to get to the remarkable equality: µ(M Γ ) = M |∼(Γ) . Thanks to it, properties like Strong Coherence can be easily translated in syntactic terms (i.e. using only the language, ⊢, |∼, etc.).

We show that (|∼4) holds. Let Γ ⊆ F.

Then, |∼(Γ) = T (µ(M Γ )) = T (M T (µ(MΓ)) ) = T (f (M Γ )) = T (M T (µ f (MΓ)) ) = T (µ f (M Γ )) = T ({v ∈ M Γ : ∀ W ∈ D, if v ∈ W , then v ∈ f (W )}) = T ({v ∈ M Γ : ∀ ∆ ⊆ F, if v ∈ M ∆ , then v ∈ f (M ∆ )}) = T ({v ∈ M Γ : ∀ ∆ ⊆ F, if v ∈ M ∆ , then v ∈ M |∼(∆) }). Direction: "←". Suppose |∼ satisfies (|∼4). Let f be the function from D to D such that ∀ Γ ⊆ F, we have f (M Γ ) = M |∼(Γ) . Note that f is well-defined. Indeed, if Γ, ∆ ⊆ F and M Γ = M ∆ , then, by (|∼4), |∼(Γ) = |∼(∆). In addition, by (|∼4), we clearly have ∀ Γ ⊆ F, |∼(Γ) = T (µ f (M Γ )).
And finally, by Lemma 23, µ f is a SC choice function.

Proof of (1). Direction: "→". Verbatim the proof of (0), except that in addition (A0) holds and µ is UC. We show that |∼ satisfies

(|∼5). As µ is UC, V \ µ(V) ∈ D. But, by Lemma 24, µ(V) = µ f (V) = {v ∈ V : ∀ W ∈ D, if v ∈ W , then v ∈ f (W )} = {v ∈ V : ∀ ∆ ⊆ F, if v ∈ M ∆ , then v ∈ f (M ∆ )} = {v ∈ V : ∀ ∆ ⊆ F, if v ∈ M ∆ , then v ∈ M |∼(∆) }.
Direction: "←". Verbatim the proof of (0), except that in addition (A0) holds and |∼ satisfies (|∼5). But, because of (|∼5), V \ µ f (V) ∈ D. Therefore µ f is UC. Note that in this direction (A0) is not used.

The discriminative and definability preserving case

In the present section, we will characterize certain families of DP pivotal-discriminative consequence relations. We need an inductive construction introduced in We turn to the representation result:

Definition 29 Suppose L is a language, ¬ a unary connective of L, ∨ a binary connective of L, F the set of all wffs of L, F, V, |= a semantic structure, and |∼ a relation on P(F ) × F. Then, consider the following conditions: ∀ Γ, ∆ ⊆ F, ∀ α, β ∈ F,

M Γ ⊆ M α ∪ M E iff M Γ ⊆ M Γ∪{α} ∪ M E iff M Γ ⊆ M (Γ∪{α})∨E iff T (M (Γ∪{α})∨E ) ⊆ T (M Γ ) iff T (M Γ∪{α} ∪ M E ) ⊆ T (M Γ ) iff T (M Γ∪{α} ) ∩ T (M E ) ⊆ T (M Γ ) iff ⊢(Γ, α) ∩ ⊢(E) ⊆ ⊢(Γ) iff ⊢(Γ, α) ∩ E ⊆ ⊢(Γ).
Proof of (1). Let Γ ⊆ F and α ∈ F. Then:

Γ |∼ α iff ⊢(Γ, α) ∩ E ⊆ ⊢(Γ) iff ⊢(Γ, α) ∩ ⊢(E) ⊆ ⊢(Γ) iff T (M Γ∪{α} ) ∩ T (M E ) ⊆ T (M Γ ) iff T (M Γ∪{α} ∪ M E ) ⊆ T (M Γ ) iff T (M (Γ∪{α})∨E) ) ⊆ T (M Γ ) iff M Γ ⊆ M (Γ∪{α})∨E) iff M Γ ⊆ M Γ∪{α} ∪ M E iff M Γ ⊆ M α ∪ M E iff M Γ ∩ (V \ M E ) ⊆ M α iff M Γ ∩ I ⊆ M α .

Conclusion

We provided, in a general framework, characterizations for families of pivotal(-discriminative) consequence relations. We showed, in an infinite classical framework, that there is no normal characterization for the family of all pivotal consequence relations. And, we showed that UC pivotal consequence relations are precisely those X-logics such that X is closed under the basic entailment.

Beyond the contributions, an interest of the present paper is to give an example of how the techniques developed in [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF] (in particular in the discriminative case) can be adapted to new properties (here Strong Coherence in the place of Coherence). So naturally, we turn now to conclusions similar to those of [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF]. In many cases, our conditions are purely syntactic. In fact, when the choice functions under consideration are not necessarily definability preserving, we provided solutions with semi-syntactic conditions. We managed to do so thanks to Lemmas 23 and 24. An interesting thing is that we used them both in the plain and the discriminative versions. This suggests that they can be used in yet other versions. In addition, Lemmas 31 and 32 have been applied both here and previously in [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF] to characterize families of consequence relations defined in the discriminative manner by DP choice functions. But, [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF] is about coherent choice functions, whilst the present paper is about strongly coherent choice functions. This suggests that these lemmas can be applied with yet other properties.

  [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF]: Notation 27 N denotes the natural numbers: {0, 1, 2, . . .} and N + the positive natural numbers: {1, 2, . . .}.Definition 28 Let L be a language, ¬ a unary connective of L, F the set of all wffs of L, F, V, |= a semantic structure, |∼ a relation on P(F ) × F, and Γ ⊆ F. Then,H 1 (Γ) := {¬β ∈ F : β ∈ ⊢(Γ, |∼(Γ)) \ |∼(Γ) and ¬β ∈ ⊢(Γ, |∼(Γ))}. Let i ∈ N with i ≥ 2. Then, H i (Γ) := {¬β ∈ F : β ∈ ⊢(Γ, |∼(Γ), H 1 (Γ), . . . , H i-1 (Γ)) \ |∼(Γ)and ¬β ∈ ⊢(Γ, |∼(Γ), H 1 (Γ), . . . , H i-1 (Γ))

A link with X-logicsIn this section, we investigate a link between pivotal consequence relations and pertinence consequence relations (alias X-logics) which were first introduced by Forget, Risch, and Siegel[START_REF] Forget | Preferential logics are X-logics[END_REF]. Suppose some formulas are considered to be the pertinent ones in the absolute sense and collect them in a set E. Then, it is natural to conclude a formula α from a set of formulas Γ iff every pertinent basic consequence of Γ ∪ {α} is a basic consequence of Γ (i.e. the addition of α to Γ does not yield more pertinent formulas than with Γ alone). This constitutes a pertinence consequence relation. More formally,

Acknowledgments

I acknowledge Karl Schlechta, David Makinson, and Nicolas Ollinger for useful comments and suggestions.

Proof of (0). Let Γ ⊆ F. As µ is DP, µ(M Γ ) ∈ D. Thus,

Proof of (1). Let Γ, ∆ ⊆ F and suppose ⊢(Γ) = ⊢(∆). Then, M Γ = M ∆ . Thus, |∼(Γ) = T (µ(M Γ )) = T (µ(M ∆ )) = |∼(∆).

Proof of (2). Let

Proof of (4). Let Γ, ∆ ⊆ F. Then, by (0) and SC,

Consequently, µ is a choice function. In addition, µ is obviously DP.

Proof of (7). Let Γ, ∆ ⊆ F. Then, by (|∼3), we get

Proof of (1). Direction: "⊆". Suppose the contrary, i.e. suppose

But, we will show:

Proof of (0.1). Let Γ ⊆ F and suppose Γ is consistent.

Proof of (0.2). Let Γ, ∆ ⊆ F and suppose M Γ = M ∆ . Then, ⊢(Γ) = ⊢(∆). Thus, by (|∼0),

Proof of (1). Direction: "→". Verbatim the proof of (0), except that µ is no longer CP, whilst (A2) now holds. Note that, in (0), CP was used only to show (|∼9) and (|∼10). But, (|∼10) is no longer required to hold and we are going to get (|∼9) by another mean. Indeed, by Lemma 32 (1) and Strong Coherence of µ, (|∼9) holds.

Direction: "←". Verbatim the proof of (0), except that (|∼10) does no longer hold, whilst (A2) now holds. However, in (0), (|∼10) was used only to show that µ is CP, which is no longer required. Note that we do not need to use (A2) in this direction.

Therefore, by definition of |∼, we have |∼(Γ) = T (∅) = F c . Thus, p ∈ |∼(Γ), which is impossible. Case 2: µ(V) ⊆ M p . Then, by (0),

Proof of (4). µ is clearly a choice function. We show that µ satisfies SC. Let V, W ⊆ V. Then, µ(W

Proof of (5). By (3), ∀ Γ ∈ {Γ 1 , . . . , Γ λ }, |∼ ′ (Γ) = T (µ(M Γ )) = T (M Γ \ X ) = |∼(Γ). But, (Γ 1 , . . . , Γ λ , |∼(Γ 1 ), . . . , |∼(Γ λ )) ∈ Φ. Therefore, (Γ 1 , . . . , Γ λ , |∼ ′ (Γ 1 ), . . . , |∼ ′ (Γ λ )) ∈ Φ. Consequently, as λ, Φ is a normal characterization, |∼ ′ is not a pivotal consequence relation.

To the author knowledge, Coherency Preservation has been first introduced in [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF]. An advantage of it is that when the choice functions under consideration satisfy it, we will not need to assume (A2) to show our characterizations (in the discriminative case).

Pivots

Suppose some valuations are considered to be the important ones in the absolute sense and collect them in a set I, called a pivot. Then, I defines naturally a choice function µ I which chooses in any set of valuations, simply those elements which belong to I. More formally, Definition 16 Let V be a set. We say that I is a pivot on V iff I ⊆ V.

Let I be a pivot on V. We denote by µ I the function from P(V) to P(V) such that ∀ V ⊆ V,

Pivots have been investigated extensively by D. Makinson in [START_REF] Makinson | Bridges between classical and nonmonotonic logic[END_REF][START_REF] Makinson | Bridges from Classical to Nonmonotonic Logic[END_REF]. In the present section, we show that the properties of Strong Coherence, Definability Preservation, and Universecodefinability characterize those choice functions that can be defined by a pivot. More precisely:

Proposition 17 Let V be a set, V, W ⊆ P(V), and µ a choice function from V to W. Then:

(0) µ is SC iff there exists a pivot I on V such that ∀ V ∈ V, µ(V ) = µ I (V ).

Suppose F, V, |= is a semantic structure and V ∈ V. Then:

(1) µ is SC and DP iff there exists a pivot I on V such that I ∈ D and ∀ V ∈ V, µ(V ) = µ I (V );

(2) µ is SC and UC iff there exists a pivot I on V such that V \I ∈ D and ∀V ∈ V, µ(V ) = µ I (V ).

Proof Proof of (0). Direction: "→". Let I = {v ∈ V : ∃ V ∈ V, v ∈ µ(V )} and suppose V ∈ V. If v ∈ µ(V ), then v ∈ V and, by definition of I, v ∈ I. Consequently, µ(V ) ⊆ V ∩ I.

Consequently, V ∩ I ⊆ µ(V ).

Direction: "←". There exists I ⊆ V such that ∀ V ∈ V, µ(V ) = V ∩ I. We show that µ satisfies SC. Let V, W ∈ V. Then, µ(W

Proof of (1). Direction: "→". Take the same I as for (0). Then, by verbatim the same proof, ∀ V ∈ V, µ(V ) = V ∩ I. It remains to show that I ∈ D. As M ∅ = V, V ∈ D. Thus, as µ is DP, µ(V) ∈ D. But, µ(V) = V ∩ I = I.

Direction: "←". Verbatim the proof of (0), except that in addition

Definition 20 Let F, V, |= be a semantic structure and |∼ be a relation on P(F ) × F. Then, consider the following conditions: ∀ Γ, ∆ ⊆ F,

Note that those conditions are purely syntactic when there is a proof system available for ⊢ (which is the case with e.g. the classical, F OUR, and J 3 semantic structures).

Proof Direction: "→".

There exists an DP SC choice function µ from D to P(V) such that ∀ Γ ⊆ F, |∼(Γ) = T (µ(M Γ )). We will show:

Direction: "←". Suppose |∼ satisfies (|∼0), (|∼1), (|∼2), and (|∼3). Let µ be the function from D to P(V) such that ∀ Γ ⊆ F, µ(M Γ ) = M |∼(Γ) . We will show:

(5) µ is well-defined; (6) µ is a DP choice function;

The non-discriminative and not necessarily definability preserving case

In the present section, we will investigate in particular the family of all pivotal consequence relations. Unlike in Section 3.1, the choice functions considered here are not necessarily definability preserving. As a consequence, we will no longer have at our disposal the remarkable equality:

. Therefore, we cannot translate properties like Strong Coherence in syntactic terms. Moreover, we will put in evidence, in Section 4, some limits of what can be done in this area. Approximatively, we will show, in an infinite classical framework, that there does not exist a characterization (of the aforementioned family) made of conditions which are universally quantified and of limited size.

We provide a solution with semi-syntactic conditions. To do so, we will use techniques very similar to those of [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF] (see the non-DP and non-discriminative case). The latter are themselves strongly inspired by the work of K. Schlechta (see Proposition 5.2.5 of [START_REF] Schlechta | Coherent systems[END_REF]). Technically, the idea begins by building from any function f , a SC choice function µ f such that whenever f "covers" some SC choice function, it necessarily covers µ f . Definition 22 Let V be a set, V ⊆ P(V), W ⊆ P(V) and f a function from V to W. We denote by µ f the function from V to P(V) such that ∀ V ∈ V,

Lemma 23 Let V be a set, V ⊆ P(V), W ⊆ P(V) and f a function from V to W. Then, µ f is a SC choice function.

Proof µ f is obviously a choice function. We show that it satisfies Strong Coherence. Suppose the contrary, i.e. suppose ∃ V, W ∈ V and

Therefore, simply by definition of µ f , v ∈ µ f (W ), which is impossible.

Lemma 24 Let V be a set, V, W, and X subsets of P(V), f a function from V to W, and µ a SC choice function from

Suppose F, V, |= is a semantic structure satisfying (A0), D ⊆ V, and µ is UC. Then:

Definition 25 Let F, V, |= be a semantic structure and |∼ be a relation on P(F ) × F. Then, consider the following conditions: ∀ Γ ⊆ F,

Proposition 26 Let F, V, |= be a semantic structure and |∼ a relation on P(F ) × F. Then:

Suppose F, V, |= satisfies (A0). Then:

(1) |∼ is a UC pivotal consequence relation iff |∼ satisfies (|∼4) and (|∼5).

Proof Proof of (0). Direction: "→" There exists a SC choice function µ from

Note that those conditions are purely syntactic when there is a proof system available for ⊢ Proposition 30 Suppose L is a language, ¬ a unary connective of L, ∨ and ∧ binary connectives of L, F the set of all wffs of L, F, V, |= a semantic structure satisfying (A3) and (A1), and |∼ a relation on P(F ) × F. Then: Suppose F, V, |= satisfies (A2). Then:

(1) |∼ is a DP pivotal-discriminative consequence relation iff |∼ satisfies (|∼0), (|∼6), (|∼7), ( |∼8) and (|∼9).

Before we show Proposition 30, we need to introduce Lemmas 31 and 32 below, taken from [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF]:

Suppose L is a language, ¬ a unary connective of L, ∨ and ∧ binary connectives of L, F the set of all wffs of L, F, V, |= a semantic structure satisfying (A3) and (A1), and |∼ a relation on P(F ) × F satisfying (|∼6), (|∼7), and (|∼8).

Lemma 32 From [START_REF] Ben-Naim | Preferential and Preferential-discriminative Consequence Relations[END_REF]. Suppose L is a language, ¬ a unary connective of L, ∨ and ∧ binary connectives of L, F the set of all wffs of L, F, V, |= a semantic structure satisfying (A3) and (A1), V ⊆ P(V), µ a DP choice function from D to V, |∼ the relation on

, and Γ ⊆ F. Then:

(0) |∼ satisfies (|∼6), (|∼7), and (|∼8);

(1

We come to the proof of Proposition 30.

Proof Proof of (0). Direction: "→". There exists a CP DP SC choice function µ from D to P(V) such that

We will show: Then, let µ be the function from

We will show: (0.2) µ is well-defined. Clearly, µ is a DP choice function. In addition, as |∼ satisfies (|∼9), µ is strongly coherent. We will show:

The discriminative and not necessarily definability preserving case

Unlike in Section 3.3, the conditions of the present section will not be purely syntactic. The translation of properties like Strong Coherence in syntactic terms is blocked because we do no longer have the following useful equality: µ(M Γ ) = M Γ,|∼(Γ),H(Γ) , which hold when the choice functions under consideration are definability preserving (but this is not the case here). Thanks to Lemmas 23 and 24 (stated in Section 3.2), we will provide a solution with semi-syntactic conditions.

Definition 33 Let L be a language, ¬ a unary connective of L, F the set of all wffs of L, F, V, |= a semantic structure, and |∼ a relation on P(F ) × F. Then, consider the following conditions:

Proposition 34 Suppose L is a language, ¬ a unary connective of L, ∨ and ∧ binary connectives of L, F the set of all wffs of L, F, V, |= a semantic structure satisfying (A3) and (A1), and |∼ a relation on P(F ) × F. Then: Proof Proof of (2). Direction: "→". There exists a SC choice function µ from

Then, |∼ satisfies obviously (|∼0).

Let f be the function from

Consequently, by Lemma 32 (0), |∼ satisfies (|∼6), (|∼7), and (|∼8).

In addition, by Lemma 32 (1),

Direction: "←". Suppose (|∼0), (|∼6), (|∼7), (|∼8), and (|∼11) hold. Let f be the function from

But, by Lemma 23, µ f is a SC choice function.

Proof of (3). Direction: "→". Verbatim the proof of (2), except that in addition (A0) holds and µ is UC. We show that (|∼12) holds. As µ is UC,

Direction: "←". Verbatim the proof of (2), except that in addition (A0) holds and |∼ satisfies (|∼12). But, because of (|∼12), V \ µ f (V) ∈ D. Therefore µ f is UC. Note that (A0) is not used in this direction.

Proof of (0). Direction: "→". Verbatim the proof of (2), except that (A2) does no longer hold, whilst µ is now CP. Note that (A2) was used, in (2), only to apply Lemma 32 (1) to get ∀Γ ⊆ F, f (M Γ ) = M Γ,|∼(Γ),H(Γ) . But, we will get this equality by another mean.

In addition, by verbatim the proof of (0.1) of Proposition 30, |∼ satisfies (|∼10).

Direction: "←". Verbatim the proof of (2), except that (A2) does no longer hold, whilst |∼ satisfies now (|∼10). But, in this direction, (A2) was not used in (2). It remains to show that µ f is CP. By verbatim the proof of (0.3) of Proposition 30, we get that f is CP.

Proof of (1). Direction: "→". Verbatim the proof of (2), except that (A2) does no longer hold, whilst (A0) now holds and µ is now UC and CP. Note that (A2) was used, in (2), only to apply Lemma 32 (1) to get ∀Γ ⊆ F, f (M Γ ) = M Γ,|∼(Γ),H(Γ) . But, by verbatim the proof of (0), we get anyway ∀ Γ ⊆ F, f (M Γ ) = M Γ,|∼(Γ),H(Γ) . In addition, by verbatim the proof of (0.1) of Proposition 30, |∼ satisfies (|∼10). And, by verbatim the proof of (3), |∼ satisfies (|∼12).

Direction: "←".

Verbatim the proof of (2), except that (A2) does no longer hold, whilst (A0) now holds and |∼ satisfies now (|∼10) and (|∼12). But, in this direction, (A2) was not used in (2).

In addition, by verbatim the proof of (0), µ f is CP. And, because of (|∼12), V \ µ f (V) ∈ D. Therefore µ f is UC. Note that (A0) is not used in this direction.

Nonexistence of normal characterizations

Now, suppose there is no normal characterization of R.

Here are examples (i.e. (C1), (C2), and (C3) below) that will give the reader (we hope) a good idea which conditions cannot characterize R. This will thus make clearer the range of our impossibility result (Proposition 37 below). To begin, consider the following condition:

Then, (C1) cannot characterize R. Indeed, suppose the contrary, i.e.

suppose

Then, take λ = 3 and the relation

Then, 3, Φ is a normal characterization of R. We give the easy proof of this, so that the reader can check that a convenient relation Φ can be found quickly for all simple conditions like (C1).

But actually, we are not limited to simple operations (like e.g. ∪, ∩, \). More complex conditions than (C1) are also excluded. For instance, let f be any function from P(F ) to P(F ) and consider the following condition:

Then, (C2) cannot characterize R. Indeed, suppose it characterizes R. Then, take the relation

It can be checked that 3, Φ is a normal characterization of R. We leave the easy proof to the reader.

We can even go further combining universal (not existential) quantifiers and functions like f . For instance, let G be a set of functions from P(F ) to P(F ) and consider the following condition:

Then, (C3) cannot characterize R. Indeed, suppose it characterizes R. Then, take the relation

It can be checked that 3, Φ is a normal characterization of R. The easy proof is left to the reader. Finally, a good example of a condition which is not excluded is (|∼4). We have seen in Proposition 26 that it characterizes the family of all pivotal consequence relations.

Impossibility results

In the present section, we will show, in an infinite classical framework, that there is no normal characterization for the family of all pivotal consequence relations (in other words, (|∼4) cannot be replaced by a simpler condition in Proposition 26). In the same vein, in Proposition 5.2.15 of [START_REF] Schlechta | Coherent systems[END_REF], K. Schlechta showed that there does not exist a normal characterization for the family of all preferential consequence relations.

Note that he used the word "normal" in a more restrictive sense (see Section 1.6.2.1 of [START_REF] Schlechta | Coherent systems[END_REF]). Approximatively, a characterization of R is called normal by Schlechta iff it contains only conditions like (C1), i.e. conditions which are universally quantified, "apply" |∼ at most |F | times, and use only elementary operations like e.g. ∪, ∩, \ (complex structures, functions, etc are not allowed). We have been inspired by the techniques of Schlechta. We will need Lemma 5.2.14 of [START_REF] Schlechta | Coherent systems[END_REF]:

Recall that A and F c have been introduced in Section 2.1.2. Note that the subscript in V Γ is written just to keep in mind that V Γ depends on Γ.

Proposition 37 Suppose A is infinite and F c , V, |= is a classical propositional semantic structure. Then, there doesn't exist a normal characterization for the family of all pivotal consequence relations.

Proof

We will show:

Now, by lemma 36, we get:

Then, define:

Then, we will show:

Let µ be the function from D to P(V) such that ∀ V ∈ D, µ(V ) = V \ X . We will show: (4) µ is a SC choice function. Let |∼ ′ be the pivotal consequence relation defined by µ. We will show the following, which entails a contradiction:

(5) |∼ ′ is not a pivotal consequence relation.

Proof of (0). Let V ⊆ V and suppose |V | ≤ |A|. Obviously, T (V) ⊆ T (V \ V ). We show T (V \ V ) ⊆ T (V). Suppose the contrary, i.e. suppose ∃ α ∈ T (V \ V ), α ∈ T (V). Then, ∃ v ∈ V, v ∈ M α . Now, define: W := {w ∈ V : for all atom q occurring in α, w(q) = v(q)}. Then, ∀ w ∈ W , we have w(α) = v(α) and thus w ∈ M α . As the number of atoms occurring in α is finite and A is infinite, we get

Proof of (1). It suffices to show that |∼ is not a pivotal consequence relation. Suppose the contrary, i.e. suppose there exists a SC choice function µ from

Definition 38 Let F, V, |= be a semantic structure and |∼ a relation on P(F ) × F. We say that |∼ is a pertinence consequence relation (alias X-logic) iff there exists E ⊆ F such that

In addition, if ⊢(E) = E, we say that |∼ is closed.

We introduce a new assumption about semantic structures (in fact, simply a weak version of (A3)):

Definition 39 Suppose L is a language, ∨ a binary connective of L, F the set of all wffs of L, and F, V, |= a semantic structure. Then, define the following condition:

We will show that when (A4) is assumed, then UC pivotal consequence relations are precisely closed pertinence consequence relations. We need before Notation 40 and the very easy Proposition 41 (which we will use implicitly in the sequel).

Notation 40 Suppose L is a language, ∨ a binary connective of L, F the set of all wffs of L, Γ ⊆ F and ∆ ⊆ F. Then: Γ ∨ ∆ := {α ∨ β : α ∈ Γ and β ∈ ∆}.

Proposition 41 Suppose L is a language, ∨ a binary connective of L, F the set of all wffs of L, F, V, |= a semantic structure satisfying (A4), Γ ⊆ F, and ∆ ⊆ F. Proof of (0). Let Γ ⊆ F and α ∈ F. Then:


Fabien Teytaud
email: teytaud@lisic.univ-littoral.frfonlupt@lisic.univ-littoral.fr

Cyril Fonlupt

A CRITICAL REASSESSMENT OF EVOLUTIONARY ALGORITHMS ON THE CRYPTANALYSIS OF THE SIMPLIFIED DATA ENCRYPTION STANDARD ALGORITHM

Keywords: SIMPLIFIED DATA ENCRYPTION STANDARD, META-HEURISTICS, GENETIC ALGORITHM

In this paper we analyze the cryptanalysis of the simplified data encryption standard algorithm using meta-heuristics and in particular genetic algorithms. The classic fitness function when using such an algorithm is to compare n-gram statistics of a the decrypted message with those of the target message. We show that using such a function is irrelevant in case of Genetic Algorithm, simply because there is no correlation between the distance to the real key (the optimum) and the value of the fitness, in other words, there is no hidden gradient. In order to emphasize this assumption we experimentally show that a genetic algorithm perform worse than a random search on the cryptanalysis of the simplified data encryption standard algorithm.

However as the brute force algorithms are inadequate for cryptanalysis for standard encryption schemes (DES requires a 56-bit length key for instance), meta-heuristics and evolutionary methods have drown a fair amount of attention from the cryptanalysis field.

As early as 1993, Spillman [START_REF] Spillman | Cryptanalysis of Knapsack Cipher using Genetic Algorithms[END_REF] was the first to introduce an evolutionary approach based on genetic algorithm for cryptanalysis to discover a simple substitution cipher. Mathhews [5] used genetic algorithms for transposition ciphers. In this work, GAs were used to seek the accurate permutation. In the same way, Jacobsen [START_REF] Thomas Jakobsen | A Fast Method for Cryptanalysis of Substituion Ciphers[END_REF] proposed a hill-climbing approach to attack simple and polyalphabetical substitution ciphers.

More recently, several interesting studies were carried out for cryptanalysis of SDES via meta-heuristics and evolutionary approaches. Rao et al [7] were the first ones to study how several optimization heuristics (tabu search algorithms, simulated annealing and genetic algorithm) could match 9 to 10 bits of the target in about 15 to 20 minutes. No algorithms performed better than the others. This work was enhanced by Garg in several papers [START_REF] Garg | Genetic Algorithm Attack on SImplified Data Encryption Standard Algorithm[END_REF] [START_REF] Garg | A Comparison between Memetic algorithm and Genetic Algorithm for the cryptanalysis of Simplified Data Encryption Standard algorithm[END_REF] [10] [11] where she studied how to use memetic and genetic algorithms to break the SDES key. Other works regarding cryptography using evolutionary tools can be found in [12][13][14].

All the work described in the previous section provides a test-bed for evaluating the performance of memetic and evolutionary approaches and furthermore showed that unlike classic methods for breaking encryption schemes that require mathematical knowledge, these new approaches were attractive as they need little cryptological knowledge. However, we strongly believe that these approaches are biased and that on average they perform no better than random search and that the use of evolutionary schemes to break state-of-the-art encryption process require a better insight understanding of the encryption process.

The paper is organized as follows: Section 2 presents the SDES algorithm. Section 3 presents the random search while section 4 briefly describes the genetic algorithms. Experiments and discussions about the possible flaws of previous papers are explained in Section 5. Section 6 draws some conclusions and presents some hints to efficiently use memetic and evolutionary algorithms in cryptanalysis.

2.THE SIMPLIFIED DATA ENCRYPTION STANDARD

The SDES algorithm [START_REF] Edward | A simplified Data Encryption Standard Algorithm[END_REF] is a simple encryption algorithm, it was devised for pedagogical purposes. It is a symmetric-key algorithm which means that the sender and the recipient share the same key.

It uses a 10-bit key and takes an 8-bit block of plain-text to produce an 8-bit block of cipher text. The decryption process works similarly, except the cipher is the input and the produced plain-text is the output.

The algorithm consists in five steps: an initial permutation (IP), a complex function fk, another permutation function (SW), another application of the fk function and eventually a final permutation (IP -1 ). This final permutation is the inverse of the initial permutation. These different steps are now detailed.

Key Generation

SDES is based on the use of a 10-bit key. From this key, two 8 bit sub-keys are produced respectively called K1 and K2. These two sub-keys are produced through different and easy binary operations: circular left shift and permutations.

Initial And Final Permutations

As previously said, the algorithm takes as input an 8-bit block. The first operation is an initial permutation called IP. The initial permutation is (1,5,2,0,3,7,4,6) which means that the 5th bit will be the 2nd bit after this initial permutation.

For instance, if we define the 8-bit block as (m0, m1, m2, m3, m4, m5, m6, m7). Then, IP(m0, m1, m2, m3, m4, m5, m6, m7) = (m1,m5,m2,m0,m3,m7,m4,m6).

The F k Function

The fk function can be seen as the heart of the simplified data encryption standard SDES scheme, it is a complex function that involves a combination of permutation and non linear functions called Sboxes.

The f k function can be summed up as:

f k ( L , R)=( L⊕F ( R , SK ), R)
Where F is a mapping from a 4-bit strings to a 4-bit strings, L is the leftmost 4 bits, R the rightmost 4 bits and SK the subkey. The mapping F is the trickiest part of the scheme as it involves non linear functions.

The first step is known as an expansion/permutation operation. It basically consists in mapping from a 4-bit string to an 8-bit strings.

For instance if the 4-bit input is (m 0 , m 1 , m 2 , m 3 , m 4 ), the expansion/permutation operation consists in: (m 4 , m 1 , m 2 , m 3 , m 2 , m 3 , m 4 , m 1 ).

An exclusive or (xor) is performed on the output of the expansion/permutation operation with the first sub-key K 1 :

(m 4 ⊕ k 1,1 , m 1 ⊕k 1,2 , m 2 ⊕k 1,3 , m 3 ⊕k 1,4 , m 2 ⊕k 1,5 , m 3 ⊕k 1,6 , m 4 ⊕k 1,7 , m 1 ⊕k 1,8 )
where k 1,i indicates the ith bit of the sub-key.

This output is usually depicted for clarity reasons in a matrix form:

M = | m 4 ⊕k 1,1 m 2 ⊕k 1,5 m 1 ⊕k 1,2 m 3 ⊕k 1,6 m 2 ⊕k 1,3 m 4 ⊕k 1,7 m 3 ⊕k 1,4 m 1 ⊕k 1,8 |
At this step, the first row of this matrix M is used as input to the S0 box to produce a 2-bit output while the second row is used to produce another 2-bit output from the S1 box.

S 0 = [ 1 3 0 3 0 2 2 1 3 1 1 1 2 0 3 2 ] S 1 = [ 0 2 3 2 1 0 0 1 2 1 1 0 3 3 0 3 ]
The process is quite easy. The 1st and 4th bits are turned into an integer to specify the row of the S0 while the 2nd and 3rd bits are used to specify the column. For instance, if the first row of M is (1,0,1,0), the output of the S0 box will be 2 (row 2 (10), column 1 (01)). This value is eventually turned into a binary number and delivers 2 bits. Additionally, the last row of M is used in the same way that S0 as an input for S1 to produce two more bits that are merged (the two bits of S0 on the left, two bits of S1 on the right).

These 4 bits undergo a last permutation called P4 (m2, m4, m3, m1) to provide the output the F function.

A final xor operation is performed on the output of the F function with the 4 leftmost bits of the input, the 4 rightmost bits being untouched.

As it can be seen, the output of the fk function only alters the leftmost 4 bits of the input. The purpose of the Switch function is to invert the input to iterate the fk function with the rightmost 4 bits.

The Switch Function

The switch function is a relatively simple function. It simply permutes the first four bits with the last four bits: SW(m0, m1, m2, m3, m4, m5, m6, m7) = (m4,m5,m6,m7,m0,m1,m2,m3).

After this step, another iteration of the fk function is performed allowing to encrypt the 4 rightmost bits. However during this step, a slight modification is realized. Instead of xoring the output of the expansion/permutation operation with the sub-key K1, the sub-key K2 is used instead. A summary of the SDES algorithm is presented in Figure 1.

3.RANDOM SEARCH

Random Search (RS) belongs to the family of numerical optimization. It does not require the gradient of the function to be optimized and hence, can be performed on functions which are not continuous or differentiable. Such optimization can be applied to black-box optimization. The principle of this method is relatively simple : at each iteration a random solution from the search space is generated. If this solution is better than the current best solution, this solution replaces the current best solution. At the end, the best solution found by the algorithm is returned.

Needless to say, this method is a basic method which is studied here only to enhance the problematic of this paper. This method can be seen as a lower bound of the quality of a search algorithm.

4.GENETIC ALGORITHMS

Evolutionary algorithms belong to the family of stochastic optimization algorithms. These methods are bio-inspired techniques that crudely mimic reproduction, mutations, crossovers and selection. They are modeled according to Darwin's evolution theory. In these algorithms, individuals represent candidate solutions of the optimization problem. These algorithms do not require the gradient of the problem to be optimized. Genetic Algorithms (GA) [START_REF] Holland | Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence[END_REF] [2] belong to the family of evolutionary algorithms. They are mainly used with a discrete search space, meaning that they are used for combinatorial optimization problems. A population of candidate solutions evolves, and generations after generations, individuals are biased towards the best ones (according to the fitness function). Crossovers and mutations are random operators used for exploring the search space. Algorithm 1 illustrates this method. This algorithm is used in Section 5 as it is praised in [START_REF] Garg | A Comparison between Memetic algorithm and Genetic Algorithm for the cryptanalysis of Simplified Data Encryption Standard algorithm[END_REF] as a good scheme to hack the key of SDES.

Algorithm 1.

A generic Genetic Algorithm.

5.EXPERIMENTS

Checking The Encryption Process Performance

Using bio-inspired algorithms for the cryptanalysis of the simplified DES is based on a very strong assumption: a fitness function can guide the search towards the perfect encryption key.

The technique that is currently used for any language (in this paper, English is used without loss of generality) is to compare n-gram statistics of the decrypted message with those of the target message.

To evaluate the suitability of a proposed key K, the encrypted text is decrypted using this test key K, and the statistics of the decrypted text are compared to the statistics of the target language. Usually, only unigrams and bigrams are used since trigrams require more computing This fitness function is defined as follows: (1) with α and β being the weights given to unigram and digram, where Ei is the percentage of the ith letter in the English language and E(i,j) is the frequency of a letter i by a letter j in English (for instance the frequency of "th" is 1.52 while the frequency of "ld" is 0.02). Di is the frequency of a letter i in the decrypted text and D(i,j) is the frequency of the digram ij in the decrypted message. This fitness function is 0 if the correct key K is used in the decryption process and the greater the value of Fk, the worst the solution.

F k =α ∑ i |E i -D i |+β∑ i | E (i , j) -D (i, j ) |
It is quite obvious that this fitness function can be used in a minimization process and when this function tends towards 0 we can assume that the text is close to the plain-text. It is a good practice to evaluate this fitness functions over several texts and the longer the text is, the more accurate the statistics of unigram and digram are. In the case of genetic or memetic algorithms, this fitness function is used to evaluate candidate key and to guide the evolutionary process towards the good solutions (keys).

Figure 2. Relevance of the fitness function

For methods like genetic algorithms, memetic algorithms or hill-climbing strategies, a strong assumption is made: there exists an implicit gradient to better solutions to the best solution. In other words this means that the more bits a solution shares with the real key ( i.e. the target key), the greater the fitness value, thus guiding the research to share more bits with the real key. In [START_REF] Garg | A Comparison between Memetic algorithm and Genetic Algorithm for the cryptanalysis of Simplified Data Encryption Standard algorithm[END_REF] [13], Garg shows that using memetic algorithm, several bits of the key are correctly recovered after several minutes for cipher text ranging from 100 to 1000 characters. However, we think that there is some serious flaw in the reasoning: the fact that this fitness function is fruitful to the search landscape.

In order to show that using the fitness function Fk to guide the evolutionary process is more or less like trying to find a needle in a haystack, we devise the following experiment: starting with a given key of 10 bits, we perturb one of the 10 bits of this original key to get a new key and we evaluate Fk over this new key. This experiment is iterated 9 times, one for each possible key with a distance 1 from the original key, then we go on with all the possible keys with a distance of 2 from the original key and so on until we evaluate the fitness function for all possible keys with a distance of 9 (note that for all cases, the number of possible keys with a distance n is equal to the number of keys with a distance 10-n). 1024 different keys. From these figures the first point to note is that there is absolutely no correlation between any two perturbed keys whatever its distance from the original key is! Furthermore, it seems that solutions at one bit distance from the target key are attributed a misleading fitness value [START_REF] Goldberg | Messy genetic algorithms: motivation, analysis, and first results[END_REF] When comparing these figures with results from [START_REF] Garg | A Comparison between Memetic algorithm and Genetic Algorithm for the cryptanalysis of Simplified Data Encryption Standard algorithm[END_REF][10][13], we can understand why the memetic approach seems to remain stuck to no more than 9 bits matched (out of 10), the solutions at a distance of one bit of the local optimum act as a deterrent and hopping from these solutions to converge to the global optimum is rather intractable.

Comparison Between A Random Search And A Genetic Algorithm.

In the previous section, the fitness landscape of the SDES is examined and we show that is is quite deceptive. In this section, we study the number of bits matching with the target key for two methods: a GA-based approach and the classic random search scheme. To assess our approach and to show that using a GA with the fitness function presented is Section 5.1 and used in [11] for cryptanalysis is not an efficient tool for cryptanalysis, we carry out the following experiments.

As in [START_REF] Garg | A Comparison between Memetic algorithm and Genetic Algorithm for the cryptanalysis of Simplified Data Encryption Standard algorithm[END_REF], a simple GA is programmed whose fitness function is equation 1 and is compared to a random search method. The purpose of these experiments is to show whether or not a GA performs better or not than a random search for the cryptanalysis of SDES.

The GA is tuned as in [START_REF] Garg | A Comparison between Memetic algorithm and Genetic Algorithm for the cryptanalysis of Simplified Data Encryption Standard algorithm[END_REF], the population size was set to 10, the probability for crossover is set to 0,95 while the probability for mutation is 0.05. 10 generations are performed, so that the total number of evaluations is the same than the experiment using random search. Table 2 sums up the results for the GA approach for texts of various lengths (from 100 to +100k characters) while table 1 shows the results for the RS scheme. There is no specific parameters for the RS scheme: a solution is simply randomly generated. For each text size, 100 runs of the GA and RS are performed, the standard deviation and the average fitness of the number of matched bits are computed. The best of the 100 runs is recorded.

According to these results it is quite obvious that the GA does not outperform the RS. Moreover is seems that the GA heuristic seems to add some kind of deleterious mechanism to the search.

From all these results it seems clear that we can conclude that using a GA with this fitness function is irrelevant and that a basic random search, usually considered as a lower bound, is more efficient.

We can also say that this kind of problems is not suited to meta-heuristics like GA as the SDES like all cryptographic schemes is essentially devised as a misleading problem with no gradient. If some kind of gradient was available to the problem, SDES, like DES, would be much more easily hackable. To conclude, using an evolutionary approach for the cryptanalysis of SDES is not an adequate response. Moreover, it seems that it in some cases it performs worse than the random search.

Figure 1 .

 1 Figure 1. The SDES algorithm 1 .

Figures 2

 2 Figures 2 and 3 represent the fitness values of the perturbation of 8 random keys (4 per figure). Due to the lack of space, only 8 different keys are displayed but are fully representative of the

Figure 3 .

 3 Figure 3. Relevance of the fitness function using only unigrams

Table 1 .

 1 Results of the random search.

	Text size	Fitness Mean	Standard deviation Best fitness found
	100	5,4	2,27	10
	200	6,4	1,57	10
	400	6,9	1,52	10
	800	6,7	1,49	10
	1600	7,4	2,01	10
	3200	6,3	1,34	8
	6400	6,9	1,97	10
	12800	6,7	2	10
	25600	5,8	2,74	10
	51200	6,4	2,72	10
	102400	5,2	1,48	7

Table 2 .

 2 Results of the Genetic Algorithm.

	Text size	Fitness Mean	Standard deviation Best fitness found
	100	5,4	2,27	10
	200	5,6	1,51	8
	400	5,7	1,57	7
	800	5,8	2,1	10
	1600	4,9	1,66	8
	3200	6,1	1,91	10
	6400	6,4	1,90	9
	12800	5,4	1,90	8
	25600	6,3	2,16	10
	51200	5,8	1,40	8
	102400	5	1,89	8

Figure from http://homepage.smc.edu/morgan_david/vpn/des.htm

CONCLUSIONS

In this paper we emphasize the importance of having a relevant fitness function when meta-heuristic methods, and in particular when genetic algorithms are used to the cryptanalysis of the SDES . We first study the landscape of the fitness function in order to show that there is no correlation between the fitness function and the distance to the correct key. This is an important fact because this means that having a good score (according to the fitness) does not mean that we are close to the optimum. Then we compare the efficiency of the genetic algorithm with a random search. From these experiments, we can conclude that the genetic algorithm does not converge correctly to the optimum. The SDES algorithm uses a 10-bit key, meaning the search space equals to 1024. This allows us to perform a random search. In fact the results is quite interesting: in average the random search is better than the genetic algorithm. The number of correct bits for all text sizes is 6.37 for the random search against 5.67 for the genetic algorithm. Moreover 8 times (over 10) the optimum (i.e. the correct key) is found with the random search against 4 times over 10 for the genetic algorithm. The main explication comes from the bad fitness function used. Figures 2 and3 show in particular that being at a distance of 1 to the correct key corresponds generally to a fitness trap. There is no doubt that the genetic algorithm converges to a local optimum.

We strongly believe that using evolutionary computing techniques might be of help for cryptanalysis but it should be used not as a silver bullet to magically retrieve the key but more as the tool for cryptanalysis. The most complex part of SDES consists of a combination of the non-linear functions (called Sbox); these Sboxes might be built using evolutionary computation approaches and may also be hacked with this approach.