Reasoning by symmetry in non-monotonic inference Bela¨ıd Benhamou, Tarek Nabhani and Pierre Siegel Laboratoire des Sciences de l’Information et des Systmes (LSIS) Centre de Mathmatiques et d’Informatique 39, rue Joliot Curie - 13453 Marseille cedex 13, France Email: {benhamou, nabhani, siegel}@cmi.univ-mrs.fr

Abstract—Symmetry had been well studied in classical logics and constraint programming since a decade. Early, Krishnamurthy showed that some tricky formulas admit short proofs when augmenting the propositional logic resolution proof system by the symmetry rule. However, in Artificial Intelligence, we usually manipulate incomplete information and need to include uncertainty to reason on knowledge with exceptions and nonmonotonicity. Several non classic logics are introduced for that purpose, but as far as we know, symmetry for these frameworks had not been studied yet. Here, we are interested to extend the notion of symmetry to that non classical logics such as preferential logics, X-logics and default logics, then give a new symmetry inference rule for the X-logics and the default logics. Finally, we show how symmetry reasoning is profitable for these logics and how they handle some symmetries that do not exist in classical logics.

I. I NTRODUCTION Symmetry is by definition a multidisciplinary concept. It appears in many fields ranging from mathematics to artificial intelligence, chemistry and physics. It reveals different forms and uses, even inside the same field. In general, it returns to a transformation, which leaves invariant (does not modify its fundamental structure and/or its properties) an object (a figure, a molecule, a physical system, a formula or a constraints network...). For instance, rotating a chessboard up to 180 degrees gives a board that is indistinguishable from the original one. Symmetry is a fundamental property that can be used to study these various objects, to finely analyze these complex systems or to reduce the computational complexity when dealing with combinatorial problems. As far as we know the principle of symmetry in AI has been first introduced by Krishnamurthy [1] to improve resolution in propositional logic. Symmetries for Boolean constraints are studied in depth in [2], [3], [4]. The authors showed how to detect them and proved that their exploitation is a real improvement for several automated deduction algorithms efficiency. Since that, many research works on symmetry appeared. For instance, the static approach used by James Crawford et al. in [5] for propositional logic theories consists in adding constraints expressing global symmetry of the problem. This technique has been improved in [6] and extended to 0-1 Integer Logic Programming in [7]. The notion of interchangeability in CSPs is introduced in [8] and symmetry for CSPs is studied earlier in [9], [10]. Since a great number of constraints could be added in the static approach, some researchers proposed to add the

constraints during the search. In [11], [12], [13], authors post some conditional constraints which remove the symmetric 1 of the partial interpretation in case of backtracking. In [14], [15], [16], [17], authors proposed to use each subtree as a nogood to avoid exploration of some symmetric interpretations and the group equivalence tree conceptual for value symmetry elimination is introduced in [18]. More recently Walsh in [19] studied various new propagators to break various symmetries among them the one acting simultaneously on both variables and values. As stated by Krishnamurthy in his seminal work [1], symmetry is one of the most promising approach for deriving short proofs for many tricky formulas. He proposed a resolution proof system augmented with a local symmetry rule (LS-Res) that Stephan Szeider in a more recent work [20] showed to be stronger than resolution with the global symmetry rule which is itself stronger than the classical resolution proof system. Urquhart mentioned in his recent paper [21] that even random instances might contain many local symmetries. This means that, when solving combinatorial problems, one can encounter symmetrical sub-problems at different nodes of the search tree. This hypothesis is clearly verified on many application domains such as circuit design and planning. A given circuit without global symmetries, might handle a symmetrical equivalent sub-circuit up to the renaming of inputs and outputs. Similarly, in planning problems, two partial plans (or sets of actions) might also lead to symmetrical states. On the other hand, within the framework of the Artificial intelligence, an important paradigm is to take into account incomplete information (uncertain information, revisable information...). An essential component of the intelligence (that is human, animal or artificial) is indeed to be related to a certain capacity of adaptation of the reasoning. Contrary to the mode of reasoning formalized by a conventional or a classical logic, a result deducible from information (from a knowledge, or from beliefs) is not true but only probable in the sense that it can be invalidated further, and can be revised when adding new information. For example, it is admitted that a normal bird flies. Thus, if it is known that Tweety is a bird, then one will conclude from it that naturally Tweety flies. If it is learned thereafter that Tweety is a Penguin, this conclusion will have to be revised. This is impossible in a classical logic having the monotony property: an information deducible from 1 By

symmetric, we mean global symmetric

a knowledge C, will remain deducible if C is increased. To manage the problem of exceptions, several logical approaches in Artificial intelligence had been introduced. Many non-monotonic formalisms were presented since about thirty years, but the problem of symmetry within this framework was not studied. Symmetry reasoning is however relevant for knowledge representation. For instance, in the previous example, it is interesting to consider that the normal birds belong to the same class with respect to some basic properties, and then they are all symmetrical in this sense. In this work we investigate symmetry in two non-classical logics: first, we will extend semantic symmetry2 to the preferential logic, second, we extend syntactic symmetry3 to the X-logic formalism and try to study the relationship between both kind of symmetries. The rest of the paper is organized as follows: Section 2 gives the main definitions of symmetry in both propositional logic and constraint programming and shows that they are equivalent. In section 3, we study symmetry in preferential logics. Section 4 extends symmetry to the X-logic formalism and Section 5 concludes the work and gives some perspectives. II. S YMMETRY IN CLASSICAL LOGICS AND CONSTRAINT PROGRAMMING

Since Krishnamurthy’s [1] symmetry definition in propositional logic, several other definitions are given in the CP community. The most known among them always consider two families of symmetry: syntactic symmetry and semantic symmetry. Freuder in his useful work [8], introduced the notions of full (semantic) and neighborhood (syntactic) interchangeabilities, where two domain values are interchangeable in a CSP, if they can be substituted for each other without any effects to the CSP. In the other hand Benhamou in [10] defined two levels of semantic symmetry and a notion of syntactic symmetry. He also showed that the Full interchangeability of Freuder is a particular case of semantic symmetry and Neighborhood interchangeability is a particular case of syntactic symmetry. More recently a work of Cohen et al [22] that won the prize of the best paper in CP’2005, discussed most of the known definitions and gathered them in two definitions: symmetry of solutions and symmetry of constraints. We will summarize in the following both semantic and syntactic symmetry in propositional logic and show their relationship with the solution and constraint symmetries in CSPs. A. Symmetry in propositional logic We recall here briefly the symmetry definition given in [1], [2], [3]. Definition 1 (Semantic symmetry): Let F be a propositional formula given in CNF and LF its complete set 4 of literals. A semantic symmetry of F is a permutation σ defined on LF such that F |=| σ(F ) 2 Variable

permutations that preserve the models of the formula permutations that preserves the syntax of the formula 4 The set of literals of F containing each variable of F and its negation 3 Variable

In other words a semantic symmetry of a formula is a variable permutation that conserves the set of the models of the formula. It also conserves the set of no-goods. Definition 2 (Syntactic symmetry): Let F be a propositional formula given in CNF and LF its complete set of literals. A syntactic symmetry of F is a permutation σ defined on LF such that the following conditions hold: 1) ∀l ∈ LF , σ(¬l) = ¬σ(l), 2) σ(F ) = F In other words, a syntactical symmetry of a formula is a variable permutation that leaves the formula invariant. Definition 3: Two literals l and l0 of a formula F are symmetrical if there exists a symmetry σ of F such that σ(l) = l0 . Theorem 1: Each syntactical symmetry of a formula F is a semantic symmetry of F . Proof: It is trivial to see that syntactic symmetry is a sufficient condition to semantic symmetry. Indeed, if σ is syntactic symmetry of F , then σ(F ) = F , thus it results that F and σ(F ) have the same set of models. Each syntactic symmetry is a semantic symmetry and the converse is in general not true. On other hand, Krishnamurthy introduced the following symmetry rule to augment the resolution proof system. Definition 4: If L is the propositional logic, A a set of formulas of L, B a formula of L and σ a symmetry of A (a permutation of propositional variables of A such that A = σ(A)), then the symmetry rule can defined as follows: A`B A ` σ(B) Many hard problems for resolution have been shown to be polynomial when using symmetry in resolution. For instance, finding some of the Ramsey’s numbers or solving the pigeonhole problem are known to be exponential for classical resolution, while short proofs can be made for both of them when adding the symmetry rule to the resolution proof system. We will see in Section 4 how to extend this rule to non-monotonic logics. B. Symmetry in CSPs We summarize briefly the CSP symmetry definitions given in [22]. We start by the solution symmetry. Definition 5: Let P = (V, D, C) be a finite discrete CSP where V is its set of variables, D its set of domains, and C its set of constraints. A solution symmetry of P is a permutation of the set V × D that preserves the set of solutions of P . On the other hand constraint symmetry is defined. Before introducing the constraint symmetry notion we define the microstructure of a CSP. Definition 6: Let P = (V, D, C) be a binary CSP. The microstructure of P is a graph with the set of vertices V × D where each edge corresponds either to a tuple permitted by a specific constraint, or to a tuple allowed because there is no constraint between the associated variables.

The complement of this graph will define the microstructure complement and the microstructure can be extended to nonbinary CSPs resulting in a Hyper-graphs microstructure. Definition 7: If P = (V, D, C) is a CSP, then a constraint symmetry of P is an automorphism of its microstructure (or, equivalently, of the microstructure complement) C. The relationship between symmetry in logics and symmetry in CSPs The symmetry definitions introduced in CSPs are related to the former ones introduced in propositional logic. We will show in the following that the definitions in both frameworks are equivalent. Proposition 1: Solution symmetry, respectively, constraint symmetry in CSPs are equivalent to semantic symmetry, respectively, to syntactic symmetry in propositional logic. Proof: Consider for instance the direct SAT encoding F of a CSP P = (V, D, C) [23] where a boolean variable is introduced for each CSP variable-value pair, and where a clause forbidding each tuple disallowed by a specific constraint is added as well as another clause ensuring that a value is chosen for each variable in its domain. It is then trivial that the solution symmetry of the CSP P is equivalent to the semantic symmetry of its SAT encoding F and the constraint symmetry of P is equivalent to the syntactic symmetry of F . III. S YMMETRY IN P REFERENTIAL L OGICS In Artificial intelligence, an important paradigm is to take into account incomplete information (uncertain information, revisable information...). Unlike to classical logic, a deduction made from some knowledge, or from some beliefs is not true but only probable in the sense that it can be invalidated further, and can be revised when adding new information. For example, it is admitted that a normal student is young. Thus, if it is known that John is a student, one will conclude from it rather naturally that John is young. If it is learned thereafter that John is fifty years old, then the conclusion saying that John is young will be revised. This is impossible in classical logic having the monotony property: an information that we deduce from a knowledge C, still always deducible when the knowledge C grows. Several non-monotonic formalisms were presented for that purpose since about thirty years, but the problem of symmetry within these frameworks was not studied. The problem is however relevant for knowledge representation. For instance, in the previous example, it is interesting to consider that the normal students belong to the same class with respect to some properties, and that they are all symmetrical in this sense. Initially, simplest is to start from a preferential approach, such as it was initiated by Bossu-Siegel [24], [25], taken again by Shoam [26], and Besnard-Siegel [27] , then by Kraus, Lehmann and Magidor in [28]. All these approaches are built on a classical logic (propositional calculus, predicate calculus, modal logic) where the semantic of the inference is given by “A formula A implies a formula B if each model of A is a model of B”. However, a preferential approach, in its

most general form, says ”A implies B if all the preferred models of A are models of B“. The preferred models of A are models that have relevant properties for the management of the exceptions. This concept of preference can be defined by a relation of pre-order (a transitive and reflexive relation) between interpretations, the preferred models being the minimal models for this relation. For our elementary example, if I and J are interpretations and the pertinent information is ”Young”, then the pre-order relation can be defined by: I ≺ J iff any young individual in J is young in I Definition 8: Let L be a classical logic, and F the set of all formulas of L. If A is a subset of formulas (or a formula) of F , then A¯ is the set is of the formulas logically implied by ¯ A. The set of formula A is deductively closed if A = A. Definition 9: A preferential relation ≺ is any pre-order relation between interpretations (a pre-order relation ≺ is transitive and reflexive). Besides, if the relation ≺ is antisymmetric, then ≺ becomes an order. Intuitively, we can consider the students who are not young as exceptions (Abnormal students). Therefore I ≺ J if the set of exceptions of I is included in the set of exceptions of J. Definition 10: If A is a set of formulas, a minimal model M of A is an interpretation which satisfies A (M ` A) and which is minimal for the relation ≺ defined on the set of models of A. That is, if M 0 is a model of A such that M 0 ≺ M , then M ≺ M 0 (or, equivalently, M 0 = M if ≺ is antisymmetric). Definition 11: In a classical manner, if ≺ is a preferential relation, we define the preferential-model logic inference `≺ as follows: A `≺ B iff each minimal model of A is a model of B. Proposition 2: If a language L has a finite set of variables, then each of its consistent formulas F has at least a minimal model, and for each model M of F there exists a minimal model M 0 such that M 0 ≺ M . Remark 1: The propositional logic satisfies the conditions of Proposition 2. Each consistent propositional formula F admits at least a minimal model, and fore each model M of F there exists a model M 0 of F such that M 0 ≺ M . Example 1: To represent the sentences “Generally students are young” and “Lea is a student ”, we can use a preferential approach, that is close to the circumscription [29]. An additional predicate “Abnormal” is added and our first sentence is translated to “A student, which is not abnormal is young”. Now, if the predicates St, Ab, and Y o, respectively, denotes “Student”, “Abnormal” and “Young”, then in first order logic, we obtain the following set of formulas. A = {St(Lea), ∀x(St(x) ∧ ¬Ab(x)) → Y o(x)} By instantiating the constant Lea to the variable x, we translate the set of formulas into propositional logic and obtain the following set of formulas: A ≡ {St(Lea), (St(Lea) ∧ ¬Ab(Lea) → Y o(Lea)} Thus: A ≡ {St(Lea), Ab(Lea) ∨ Y o(Lea)}

The set A has an amount of eight interpretations, among them the three following which are the models: M1

= {St(Lea), Ab(Lea), Y o(Lea)}

M2

= {St(Lea), ¬Ab(Lea), Y o(Lea)}

M3

= {St(Lea), Ab(Lea), ¬Y o(Lea)}

In a classical logic, it is impossible to infer from A that Lea is young. Indeed, A has both models were Lea is young (M1 and M2 ), and models were Lea is not young, particularly those where Lea is abnormal (M3 ). To obtain the result “Lea is young”, we will prefer the models that have fewer abnormal students. Thus, in a preferential model approach, the relation ≺ can be defined as: M ≺ M 0 iff “each individual which is abnormal in M is abnormal in M 0 ”. According to this relation, we obtain the following preferences between the models of A: M1 ≺ M3 , M3 ≺ M1 , M2 ≺ M1 , M2 ≺ M3 . Therefore A has only one minimal model which is M2 , and Lea is young in this model. We can then infer that Lea is young in this preferential approach. It is then important to infer all the symmetrical of the literal Y o(Lea). A. Symmetry We recall that a semantic symmetry of A is a permutation σ of the propositional variables of A such that A and σ(A) have the same models. Now, we extend the definition of semantic symmetry to preferential-model logics and show how literals could be symmetrical in this non-classic logic but not symmetrical in a classic logic. Definition 12 (Semantic preferential symmetry): If `≺ is a preferential-model inference, A a set of formulas, and σ a permutation defined on the variables of A, then σ is a symmetry of A, iff A and σ(A) have the same set of minimal models. Definition 13: Two literals l and l0 are symmetrical in A iff there exists a semantic preferential symmetry σ of A such that σ(l) = l0 . Example 2: If we take the previous example and add the fact that “John is a student”, then we obtain the following set of formulas: A0 = {St(Lea), St(John), Ab(Lea) ∨ Y o(Lea), Ab(John) ∨ Y o(John)} which admits nine models. It is easy to see that the individuals Lea and John are symmetrical in both classical and preferential logics in the sense that each predicate literals where Lea appears is symmetrical to the predicate literals where we substitute John to Lea. If we consider the “abnormal” like the pertinent information on which it is based the preference and the permutation σ = (St(Lea), St(John))(Ab(Lea), Ab(John)) (Y o(Lea), Y o(John)) then we can easily see that σ is a semantic preferential symmetry of the formula A0 . Indeed, there is one minimal model that is M ={St(Lea), St(John), ¬Ab(Lea), ¬Ab(John), Y o(Lea), Y o(John)} which is preserved by σ. We can see that the literals Y o(Lea) and Y o(John) are symmetrical as well as Ab(Lea) and Ab(John). Now, if we add to A0 the information “John is not

abnormal”, then both Y o(Lea) and Y o(John) as well as ¬Ab(Lea) and ¬Ab(John) remains symmetrical two by two in the preferential logic since M remains the single minimal model of the theory where all the literals are true. However, the literals Y o(Lea) and Y o(John) are not symmetrical in a classical logic. Indeed, the new formula contains the following three extended models: M1

=

{St(Lea), St(John), Ab(Lea), Y o(Lea), ¬Ab(John), Y o(John)} {St(Lea), St(John), ¬Ab(Lea), Y o(Lea),

M2

=

M3

= {St(Lea), St(John), Ab(Lea), ¬Y o(Lea), ¬Ab(John), Y o(John)}

¬Ab(John), Y o(John)}

where both literals are not symmetrical. We can see for instance that the model M3 does not remain a model if we permute Y o(Lea) and Y o(John). It is then important to see that some literals could be symmetrical in a preferential approach but not symmetrical in a classical logic. This is due to the relaxation of the symmetry conditions in the preference approaches that considers only the minimal models. This information is new and is promising for symmetry reasoning in non classical logics. We extended the semantic symmetry notion to preferential logics, now we try to extend the notion of syntactic symmetry to non-classical logics. To do that we chose the X-logics [30] as a baseline framework. IV. S YMMETRY IN X-L OGICS We have seen that it is easy to extend the notion of semantic symmetry to preferential-model logics, but the syntactic symmetry definition in such logics seems to be not trivial. For this purpose, we will use the X-logic formalism that looks to have some syntactic important properties that we will use to extend the notion of syntactic symmetry to non-classical logics. Definition 14 (X-logics definition): Let X be a set of formulas of the propositional logic L (X is not necessarily deductively closed). The non-monotonic inference relation `X is defined by A `X B iff (A ∪ B) ∩ X ⊆ A¯ ∩ X Remark 2: • In other words, A `X B if every theorem ¯ That of (A ∪ B), which is in X is also a theorem of A. is, by adding the knowledge B to A the set of theorems which are in X does not grow. As the classical logic inference ` is monotonic, then we also have A¯ ∩ X ⊆ (A ∪ B)∩X, and therefore, it is possible to define the X¯ logic inference `X by A `X B iff (A ∪ B)∩X = A∩X. • For the particular case where X = F (the set of all possible formulas of the logic L), the inference `X is identical to the classical inference `. On other hand if X is empty, then every formula B can be inferred from A. The set X can be seen as a potentiometer that regulates the inference. Intuitively, if a formula A encodes an information (a knowledge, or some beliefs..), X can be considered as the

set of “pertinent” informations. The set A implies a set B of information, for the X-logic inference, if the addition of B to A does not produce more pertinent formulas than those produced by A alone. The set X is any set of formula, not necessary closed. It is possible to deduce some properties on the X-logic by adding properties on the set X. For example, if the complement of X, is deductively closed, then the corresponding X-logic will have very interesting properties such as that one of “Cumulativity”. If X is the set of positive formulas (formulas that do not contain the logical operators → and ¬), then we obtain the Closed World Assumption (CWA). A. Symmetry Now we will deal with symmetry in X-logics. We extend the definition of syntactic symmetry to the framework of Xlogic and give an extended rule of symmetry that can be used to make short proofs by using symmetrical formulas within this framework. Definition 15: Let A be a set of formulas of the propositional logic, X the subset of pertinent formulas on which the X-logic inference `X is built and σ is a variable permutation. The permutation σ is a syntactic symmetry of A in the considered X-logic if the following conditions hold: 1) σ(A) = A, 2) σ(X) = X Now we extend the symmetry rule of Krishnamurthy to the X-logic framework. Proposition 3: Let A and B two formulas or two sets of formulas of the classical logic L and σ a syntactic symmetry of the considered X-logic. We have the following rule: A `X B A `X σ(B) Proof: To prove that A `X σ(B) we shall prove that A ∪ σ(B) ∩ X = A¯ ∩ X. We have by the hypothesis A `X B, that is (A ∪ B) ∩ X = A¯ ∩ X. Thus, we have σ(A ∪ B ∩ X) = σ(A¯ ∩ X) which is equivalent to ¯ ∩ σ(X). Since σ is a syntactic σ(A ∪ B) ∩ σ(X) = σ(A) symmetry of the X-logics, then it preserve the propositional logic theorems. This gives σ(A ∪ B) ∩ σ(X) = σ(A) ∩ σ(X) which is equivalent to σ(A) ∪ σ(B) ∩ σ(X) = σ(A) ∩ σ(X). Since both A and X are invariant under σ, then we deduce A ∪ σ(B) ∩ X = A¯ ∩ σ(X). Therefore, A `X σ(B). Example 3: Take for instance the student example encoded by the following set of formulas: A0 = {St(Lea), St(John), Ab(Lea) ∨ Y o(Lea), Ab(John) ∨ Y o(John)} Consider the set X = {¬Ab(Lea), ¬Ab(John)} and a permutation σ = (St(Lea), St(John))(Ab(Lea), Ab(John)) (Y o(Lea), Y o(John)). The permutation σ is a X-logic symmetry of A0 . We have A0 `X Y o(Lea) and by symmetry we have A0 `X Y o(John) since σ(Y o(Lea)) = Y o(John). This rule could be used to infer all the symmetrical formulas of an inferred formula, its implementation in a theorem prover will help to shorten proof of a theorem.

B. Relationship between symmetry in X-logic and in preferential logics It is important to study the relation between symmetry in preferential logics and symmetry in X-logics. An important result on the relationship between the preferential logics and the X-logics is given in [30]. The main theorem is the following: Theorem 2: Every preferential logic is a X-logic. That is, for each preferential-model inference `≺ there exists an equivalent X-logic inference `X . From this theorem, one can then define a syntactic symmetry form for a preferential logic. The approach consists in translating the given preferential logic into its equivalent X-logic and then apply the syntactic symmetry definition given in the previous section on the resulting X-logic. On the other hand one can ask the converse question: how can we define a semantic symmetry for a X-logic? this question is more difficult than the first one, since the definition of X-logics is completely syntactic and no semantic is known for the X-logics. Besides, it is shown in [30] that in general, a X-logic is not necessarily a preferential logic. It is then difficult to define a semantic symmetry for the X-logics in the general case, but when the complement of X (denoted by Xc ) is deductively closed, that is X¯c = Xc , the corresponding X-logic satisfies the cumulativity property, we believe (we conjecture) that in this particular case that the X-logic is a preferential logic. It is then important in future to prove this conjecture and try to define a semantic symmetry for these particular X-logics where X¯c = Xc . V. C ONCLUSION AND PERSPECTIVES The main purpose behind this work is to extend the notion of symmetry to non-classical logics formalisms. We defined the notion of semantic symmetry in preferential-model logics and showed how some information can be inferred by using this symmetry extension in preferential logics while such deductions could not be done in a classical logic. The other point studied here is the extension of syntactic symmetry definition to the framework of X-logics where a new symmetry inference rule was given. As a future work, one can try to study in depth the relationship between the semantic symmetry in preferential logic and the syntactic symmetry defined in the X-logics framework that remains an important point. An other point that we want to investigate is to study symmetry for default logics and provide a proof procedure that will take advantage of the new symmetry rule. R EFERENCES [1] B. Krishnamurty, “Short proofs for tricky formulas,” Acta informatica, no. 22, pp. 253–275, 1985. [2] B. Benhamou and L. Sais, “Theoretical study of symmetries in propositional calculus and application,” Eleventh International Conference on Automated Deduction, Saratoga Springs,NY, USA, 1992. [3] B.Benhamou and L.Sais, “Tractability through symmetries in propositional calculus,” Journal of Automated Reasoning (JAR),, vol. 12, pp. 89–102, 1994. [4] B. Benhamou, L. Sais, and P. Siegel, “Two proof procedures for a cardinality based language,” in proceedings of STACS’94, Caen France, pp. 71–82, 1994.

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