Explanatory relations in arbitrary logics based on satisfaction systems, cutting and retraction

03/05/2018 ∙ by Marc Aiguier, et al. ∙ University of the Andes Télécom ParisTech Université Paris-Dauphine 0

The aim of this paper is to introduce a new framework for defining abductive reasoning operators based on a notion of retraction in arbitrary logics defined as satisfaction systems. We show how this framework leads to the design of explanatory relations satisfying properties of abductive reasoning, and discuss its application to several logics. This extends previous work on propositional logics where retraction was defined as a morphological erosion. Here weaker properties are required for retraction, leading to a larger set of suitable operators for abduction for different logics.

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1 Introduction

Since its introduction by Charles Peirce in [peirce1958collected]

, abduction has motivated a large body of research in several scientific fields, e.g. philosophy of science, logics, law, artificial intelligence, to mention a few. Abduction, whatever the adopted view on its treatment, involves a background theory (

), an observation also called explanandum (), and an explanation (). The observation may be seen as a surprising phenomenon that is inconsistent with the background theory. It may also be consistent with the background theory but not directly entailed by this theory, which is the case considered in this paper. Several constraints can be imposed on the explanations and on the process of their production. One can allow changing the background theory, or not, consider as non relevant explanations those that entail the observation on their own without engaging the background knowledge. Hence, several forms of abduction can be defined depending on the chosen criteria. Despite their divergence, most of these models agree to define abduction as an explanatory reasoning allowing us to infer the best explanation of an observation. This contributes to the field of explainable artificial intelligence. Explanatory relations, trying to model common sense and everyday reasoning, find applications in many domains, such as diagnosis [console1991, eiter1995], forensics [han2011], argumentation [booth2014a, booth2014b], language understanding [hobbs2004], image understanding [JA:SMC-14, YY:KI-15], etc. (it is out of the scope of this paper to describe applications exhaustively). Then, as a form of inference, several works have studied rationality postulates that are more appropriate to govern the process of selecting the best explanations, e.g. [Flach96, PPU99]. From a computational point of view, a very large number of papers has tackled the definition of abductive procedures, mainly in propositional logics. An attractive approach, governed by what is called the AKM model, is based on semantic tableaux tailored for particular logics (e.g. propositional logics [Aliseda97], first order and modal logics [MP93, MP95]), which was the basis for several extensions (e.g. [bienvenu2008, Britz17, eiter1995, halland2012]). In these works, the explanatory reasoning process is split into two stages: (i) generating a set of hypotheses from the formulas that allow closing the open branches in the tableau constructed from , and (ii) selecting the preferred solutions from this plain set by considering some of the criteria discussed above.

Our aim in this paper is to introduce a new framework for defining abductive reasoning operators in arbitrary logics in the framework of satisfaction systems. To this end, we propose on a new notion of cutting, from which operators of retraction are derived. We show that this framework leads to the design of explanatory relations satisfying the rationality postulates of abductive reasoning introduced in [PPU99] and adapted here to the proposed more general framework, and present applications in several logics. This extends previous work on abduction in propositional logics where retraction was defined as a morphological erosion [BL02, IB:arXiv-18, BPU04], as well as abduction in description logics for image understanding [JA:SMC-14]. Here weaker properties are required for retraction, that allow defining a larger set of suitable operators for abduction for different logics. This approach is similar to the one proposed for revision in [AABH18], where revision operators were defined from relaxation in satisfaction systems, and then instantiated in various logics. An important feature of the proposed explanations based on retraction is that generation and selection steps are merged, or at least the set of generated hypotheses is reduced, thus facilitating the selection step.

The paper is organized as follows. In Section 2, we recall the useful definitions and properties of satisfaction systems, and provide examples in propositional logic, Horn logic, first order logic, modal propositional logic and description logic. In Section 3 we introduce our first contribution, by defining a notion of cutting, from which explanations are then defined. In Section 4, we propose to define particular cuttings, based on retractions of formulas. Then in Section 5, we instantiate the proposed general framework in various logics.

2 Satisfaction systems

We recall here the basic notions of satisfaction systems needed in this paper. The presentation follows the one in [AABH18], where we give a more complete presentation of satisfaction systems, including the properties and their proofs, that are omitted here.

2.1 Definition and examples

Definition 1 (Satisfaction system).

A satisfaction system consists of

  • a set of sentences,

  • a class of models, and

  • a satisfaction relation .

Let us note that the non-logical vocabulary, so-called signature, over which sentences and models are built, is not specified in Definition 1111The set of logical symbols is defined in each particular logic and does not depend on a theory.. Actually, it is left implicit. Hence, as we will see in the examples developed in the paper, a satisfaction system always depends on a signature.

Example 1.

The following examples of satisfaction systems are of particular importance in computer science and in the remainder of this paper.

Propositional Logic (PL)

Given a set of propositional variables , we can define the satisfaction system where is the least set of sentences finitely built over propositional variables in , the symbols and (denoting tautologies and antilogies, respectively), and Boolean connectives in , contains all the mappings ( and are the usual truth values), and the satisfaction relation is the usual propositional satisfaction.

Horn Logic (HCL)

A Horn clause is a sentence of the form where is a finite (possibly empty) conjunction of propositional variables and is a propositional variable. The satisfaction system of Horn clause logic is then defined as for PL except that sentences are restricted to be conjunctions of Horn clauses.

Modal Propositional Logic (MPL)

Given a set of propositional variables , we can define the satisfaction system where

  • is the least set of sentences finitely built over propositional variables in , the symbols and , Boolean connectives in , and modalities in ;

  • contains all the Kripke models where is an index set, is a family of functions from to , and is an accesibility relation;

  • the satisfaction of sentences by Kripke models, , is defined by for every where is defined by structural induction on sentences as follows:

    • iff for every ,

    • Boolean connectives are handled as usual,

    • iff for every such that , and

    • is the same as .

First Order Logic (FOL) and Many-sorted First Order Logic

We detail here only the many-sorted variant of FOL, FOL being a particular case. Signatures are triplets where is a set of sorts, and and are sets of function and predicate names respectively, both with arities in and respectively ( is the set of all non-empty sequences of elements in and where denotes the empty sequence). In the following, to indicate that a function name (respectively a predicate name ) has for arity (respectively ), we will note (respectively ).
Given a signature , we can define the satisfaction system where:

  • is the least set of sentences built over atoms of the form where and for every , ( is the term algebra of sort built over with sorted variables in a given set ) by finitely applying Boolean connectives in and quantifiers in .

  • is the class of models defined by a family of non-empty sets (one for every ), each one equipped with a function for every and with an n-ary relation for every .

  • Finally, the satisfaction relation is the usual first-order satisfaction.

As for PL, we can consider the logic FHCL of first-order Horn Logic whose models are those of FOL and sentences are restricted to be conjunctions of universally quantified Horn sentences (i.e. sentences of the form where is a finite conjunction of atoms and is an atom).

Description logic (DL)

Signatures are triplets where , and are nonempty pairwise disjoint sets where elements in , and are called concept names, role names and individuals, respectively.
Given a signature , we can define the satisfaction system where:

  • contains 222The description logic defined here is better known under the acronym . all the sentences of the form , and where , and is a concept inductively defined from and binary and unary operators in and in , respectively.

  • is the class of models defined by a set equipped for every concept name with a set , for every relation name with a binary relation , and for every individual with a value .

  • The satisfaction relation is then defined as:

    • iff ,

    • iff ,

    • iff ,

    where is the evaluation of in inductively defined on the structure of as follows:

    • if with , then ;

    • if then ;

    • if (resp. ), then (resp. );

    • if , then ;

    • if , then ;

    • if , then .

2.2 Knowledge bases and theories

Let us now consider a fixed but arbitrary satisfaction system (since the signature is supposed fixed, the subscript will be omitted from now on).

Notation 1.

Let be a set of sentences.

  • is the sub-class of whose elements are models of , i.e. for every and every , . When is restricted to a formula (i.e. ), we will denote the class of model of by , rather than .

  • is the set of semantic consequences of . In the following, we will also denote to mean that .

  • iff .

  • Let . Let us note . When is restricted to one model , will be equivalently noted .

  • Let us note , i.e. the set of models in which all formulas are satisfied. In PL, MPL and FOL, is empty because the negation is considered. Similarly, the negation is involved in the DL , hence is empty. In HCL, only contains the unique model where all propositional variables have a truth value equal to 1. In FHCL, contains all models where for every predicate name , .

Definition 2 (Knowledge base and theory).

A knowledge base (KB) is a finite set of sentences (i.e. and the cardinality of belongs to ). A set of sentences is said to be a theory if and only if .
A theory is
finitely representable if there exists a KB such that .
A class of models is
finitely axiomatizable if there exists a finite KB such that . A satisfaction system is finitely axiomatizable if each of its classes of models is finitely axiomatizable.

Note that in DL, a knowledge base consists classically of a set of axioms (in the form ), called TBox, and a set of assertions (in the form or ), called ABox.

Classically, the consistency of a theory is defined as . The problem of such a definition of consistency is that its significance depends on the considered logic. Hence, this consistency is significant for FOL, while in FHCL it is a trivial property since each set of sentences is consistent because always contains which is non empty. Here, for the notion of consistency to be more appropriate for our purpose of defining abduction for the largest family of logics, we propose a more general definition of consistency, the meaning of which is that given a theory , is not restricted to trivial models.

Definition 3 (Consistency).

is consistent if .

Proposition 1 ([Aabh18]).

For every , is consistent if and only if .

Hence, for every , is inconsistent is equivalent to .

2.3 Internal logic

Following [Dia08, GB92], the satisfaction system-independent definition of Boolean connectives is straightforward. This will be useful when we give general results of preserving explanatory relation along Boolean connectives. Let be a satisfaction system. A sentence is a

  • semantic negation of when ;

  • semantic conjunction of and when ;

  • semantic disjunction of and when ;

  • semantic implication of and when .

has (semantic) negation when each sentence has a negation. It has (semantic) conjunction (respectively disjunction and implication) when any two sentences have conjunction (respectively disjunction and implication). As usual, we note negation, conjunction, disjunction and implication by , , and .

Example 2.

PL has all semantic Boolean connectives. FOL has all semantic Boolean connectives when sentences are restricted to closed formulas, otherwise (i.e. sentences can be open formulas) it only has semantic conjunction. Finally, MPL has only semantic conjunction.

3 Explanation in satisfaction systems

The process of inferring the best explanation of an observation is usually known as abduction. In a logic-based approach, the background of abduction is given by a knowledge base (KB) and a formula (the observation) such that is consistent. Besides this fact, which can be expressed equivalently as , some works further require that . We do not impose this last requirement here.

Let us start by introducing the notion of explanation of with respect to .

Definition 4 (Set of explanations).

Let be a KB. Let be a formula consistent with T. The set of explanations of over is the set defined as:

Note that this definition does not impose that . In some cases a preferred explanation of with respect to the background knowledge base could be a formula such that .

Since abduction aims to infer the best explanations, the notion of explanation given in Definition 4 only captures candidate explanations of with respect to . Some additional properties are needed to define the key notion of “preferred explanations". Following the works in [Aliseda97, Flach96, Flach00a, Flach00b, PPU99, PPU03], we will study some preference criteria and give their logical properties when abduction is regarded as a form of inference.

Definition 5 (Explanatory relation).

Let be a KB. An explanatory relation for is a binary relation such that:

Now, we define an (abstract) explanatory relation, the behavior of which will consist in cutting in as much as possible but still under the constraint that it remains consistent (i.e. it is not equal to ). A cutting will then generate a sequence of subsets of that we can order by inclusion. Moreover, this sequence cannot be extended by inverse inclusion. This gives rise to the notion of a cutting for a KB and a formula .

Definition 6 (Cutting).

Let be a KB and let be a formula. A cutting for and is any such that for every , , is closed under set-theoretical union and contains , and the poset is well-founded 333Let us recall that a poset is well-founded if every non-empty subset has a minimal element with respect to , or equivalently there does not exist any infinite descending chain..
Let us denote the set of minimal elements for in .
In the following, given a KB and a formula , a cutting for and will be denoted 444To simplify the notations, does not index cuttings because as we will see, will be often constant.

Note that in Definition 6, we do not impose that is consistent (“who can do more, can do less”). However, the case where it is not would not be very interesting since would then be empty.

Remark 1.

If then there exists a trivial cutting for , namely .

As is closed under set-theoretical union and then it is inductive, by the Hausdorff maximal principle, every chain is contained in any maximal chain (and then maximal chains exist). Moreover, as is well-founded, every maximal chain has a least element which belongs to .

Definition 7 (Explanatory relation based on cuttings).

Let be a KB, and let us define a set of cuttings by choosing a cutting for every in Sen: . Let us define the binary relation as follows:

By Remark 1, is well defined. Obviously, is an explanatory relation. We will later add some stability properties to to ensure good properties of this explanatory relation.

Remark 2.

If is a cutting for and , then we can define a relation based on cuttings such that satisfies the equivalence of Definition 7 (i.e. is precisely the cutting chosen for in the set ).

The next example shows how our general definition via cuttings can capture some explanatory relations defined in the literature.

Example 3.

Abduction via semantic tableau [MP93] and resolution [SNA06] generates a cutting, and then an explanatory relation. We illustrate this fact for abduction via semantic tableau in the framework of the propositional logic 555Note that semantic tableau methods have been extended to modal logic [Bienvenu09, MP95], first-order logic [Marquis91, RAN06], DL [halland2012], etc., and in the same way we would be able to generate a cutting from them..

Semantic tableaux are used as refutation systems. Let be a set of propositional formulas The tableau expansion rules are as follows:

A tableau is then a sequence of sets of sets of formulas such that, for every , is obtained from by the application of a tableau expansion rule on a formula of a set in . At each step , every set in which contains both and for some propositional variable is removed from .
A formula is a theorem of a KB if there exists a finite sequence such that and . As an example, let us show that is a theorem of . The tableau method provides the finite sequence , using -rules. The set contains a unique set, with both and , which is then removed, and becomes empty.

Let us observe that the tableau expansion rules break propositional formulas on their main Boolean connectives. Hence, tableaux are necessarily finite, and then two cases can occur:

  1. the last set of the sequence is empty, and then we have that ; or

  2. every in only contains literals but no literal has its negation in .

Following [MP93], if is any consistent choice function for the elements of , i.e. for , , then if is consistent with , then is an explanation of for ( is even the minimal one according to the definition of minimality given in [MP93]).

We now show that the way the tableau is generated in [MP93] defines a cutting . Before defining the cutting , let us introduce some useful notions. Let be a tableau for such that . For every , , let us denote the disjunction of the negation of all the literals , i.e. . Then, let us set . We can define the cutting as follows:

Obviously we have and . Moreover, for any , , hence . It is not difficult to show that for every , , . Moreover, the tableau is finite, which completes the proof that is a cutting.

Let us illustrate this construction on an example. Let be the KB and let be the observation. The tableau method applied to generates four sets where:

  • ;

  • ;

  • ;

  • ;

This leads to the following formulas :

  • ;

  • ;

  • ;

  • .

A consistent choice satisfying minimality is for instance .

It is interesting to note that there is an alternative way of looking at . The descending chains to obtain the minimal element provide a method to order the models of .

Definition 8 (Relation on models).

Let be a KB and let be a formula such that . Let be a cutting for and . Let us define as follows:

(1)

Let and be a binary relation over . We define as if and only if and . We also define . Note that the relation is reflexive, but not necessarily transitive (hence it is not a pre-order).

Theorem 1.

Let be the cutting for a KB and a formula used in the definition of . For any , the following equivalence holds:

Proof.

() By definition of , we have . Let us suppose . By the definition of , the statement means that there exists such that . As satisfies the Hausdorff maximal principle, there exists a maximal chain , the least element of which is . Hence, by the definition of , for every , we have that:

  1. for every , and , and

  2. for every , .

This proves that .

() Let us suppose that . This means that either and in this case the conclusion is obvious, or there does not exist a minimal element such that . Let be the least element (for inclusion) of such that . This least element exists because contains and is well-founded. As satisfies the Hausdorff maximal principle, there exists a maximal chain which contains , and then cannot be the least element of . Therefore, there exist some models which belong to some elements in such that whence we can deduce that for some models , we have . ∎

The explanatory relation satisfies a number of logical properties. Most of these properties are (rationality) postulates defined in [PPU99] up to some adaptations. Let us recall them, adapted to the satisfaction system context, for any KB , explanatory relation for and formulas :

Now, we will show that, with an appropriate structure on the set of cuttings , adding a limited set of rather intuitive stability and monotony requirements, we can get strong results on the explanatory relation , according to the above postulates. Recall that is defined by choosing a cutting for each in . A first requirement is that for every we have:

(2)

This will be directly used in Property (1) of the following Theorem.

Theorem 2.

Let be a satisfaction system, a KB, a set of cuttings and the explanatory relation based on cuttings of Definition 7. The following properties are satisfied, for every :

  1. Assume that satisfies Equation 2. If , and , then .

  2. If and with , then .

  3. iff there exists a relation based on cuttings such that .

  4. If is finitely axiomatizable for every and has conjunction, then for every cutting , we have that and , where is a relation based on cuttings such that the cutting associated with is .

Proof.
  1. The first property is obviously satisfied because and mean that and and, by assumption, .

  2. means that there exists such that , and . As (hence ), and , we can deduce that .

  3. The “if part” is obvious. To prove the “only if part”, let us notice that for every , we can build in a saturated chain starting at . As , this saturated chain satisfies all the conditions of Definition 6, and then it is a cutting for and . By Remark 2, we can define the relation based on cuttings such that the cutting corresponding to is precisely . By construction of , it is clear that .

  4. Again by Remark 2, it makes sense to consider as the explanatory relation defined by a family of cuttings in which the one associated with is . Let . As is finitely axiomatizable, there exists a finite KB such that . Let us set . We obviously have that is consistent, hence , and then we can conclude that .

It is interesting to note that property (1) generalizes to satisfaction systems the properties LLE and RLE of [PPU99]. Similarly, Property (2) corresponds to RS, and Properties (3) and (4) to E-Con.

If also has Boolean connectives in , the explanatory relation satisfies additional logical properties.

Lemma 1.

If is a cutting for and , and , then is a cutting for and .

Proof.

For every , we have that . It is clear that if is not well-founded so it is . Moreover, is closed by union of sets. Thus all the conditions for defining a cutting are satisfied.∎

Lemma 2.

If is a cutting for and and , then is a cutting for and .

Proof.

Similar to the one of the previous lemma. ∎

In the following results, we also assume an additional structure on , according to Lemma 1 and Lemma 2, by imposing the following constraints:

(3)
(4)
(5)
(6)
Theorem 3.

Let be a satisfaction system with conjunction, disjunction and implication. Let be a KB, and a set of cuttings satisfying Equation 36. The following properties are satisfied, for every :

  1. If and , then .

  2. If and , then .

  3. If and , then .

  4. If (and ), then .

  5. If , then .

  6. If and , then .

  7. If and , then .

  8. For every cutting such that is total, if and , then .

  9. If and , then .

  10. If , then , for .

Proof.
  1. By hypothesis, there exists