First order abduction via tableau and sequent calculi

The formalization of abductive reasoning is still an open question: there is no general agreement on the boundary of some basic concepts, such as preference criteria for explanations, and the extension to rst order logic has not been settled. Investigating the nature of abduction outside the context of resolution based logic programming still deserves attention, in order to characterize abductive explanations without tailoring them to any xed method of computation. In fact, resolution is surely not the best tool for facing meta-logical and proof-theoretical questions. In this work the analysis of the concepts involved in abductive reasoning is based on analytical proof systems, i.e. tableaux and Gentzen-type systems. A proof theoretical abduction method for rst order classical logic is de ned, based on the sequent calculus and a dual one, based on semantic tableaux. The methods are sound and complete and work for full rst order logic, without requiring any preliminary reduction of formulae into normal forms. In the propositional case, two di erent characterizations are given for abductive explanations, each of them being the declarative counterpart of a di erent algorithm for the generation of explanations. The rst one corresponds to the generation of the whole set of minimal and consistent explanations, where minimality is checked by comparison with the other elements of the set. The second characterization corresponds to a (non-deterministic) algorithm for generating a single minimal explanation that is consistent with the theory. The rst order versions of the abductive systems make use of uni cation and dynamic herbrandization/skolemization of formulae. The construction of the abduced formula is pursued by means of de-skolemization. The rst order methods are very loose in discarding explanations that are not minimal. In fact, the question of minimality in rst order abduction is a main issue. As usually de ned, minimality is undecidable, for two di erent reasons: (i) determining whether an explanation is better than another one is in general undecidable; (ii) the set of explanations may be in nite. Moreover, because of (ii), a minimal element may not exist. Such problems suggest that the minimality requirement should be relaxed, possibly de ning it w.r.t. a stronger relation than logical consequence. 2 1 Abductive reasoning 1.1 Preliminaries Abduction is a form of reasoning that infers premises from a conclusion. Its characteristic logical schema is the inference of ' from '! and . It is an unsound form of inference, that re ects some forms of commonsense reasoning [14, 15, 18], where causes for events are to be hypothesized, and diagnostic reasoning [16]. More generally, abductive reasoning is a way to solve problems where an observed event ' is not explained by the presently adopted theory and an explanation for ' has to be looked for. Precisely, an abduction problem is given by a background theory (a set of formulae) and a formula ' such that: 1) 6j= ' 2) 6j= :' A solution of the problem given by the pair h ; 'i is to be looked for among the formulae such that [ f g j= '. In abductive reasoning explanations are required to respect some fundamental conditions, in order to be accepted as \interesting". Although there is no general agreement on the exact boundary between interesting and noninteresting explanations, the following three reasonable restrictions are usually imposed on explanations for an abduction problem h ; 'i: (i) is consistent with (or -consistent), i.e. 6j= : . (ii) is a minimal explanation for the abduction problem h ; 'i, i.e. for any formula , if [ f g j= ' and j= , then j= . (iii) has some restricted syntactical form; for example, it is a prenex formula whose matrix is a conjunction of literals. The syntactical restriction imposed on explanations in this work is stated in the following de nitions. De nition 1 (C-formulae) A formula is a ( rst-order) C-formula if it is built up from literals using only quanti ers and conjunction. De nition 2 (Explanations for abduction problems) is an explanation for the abduction problem h ; 'i if it is a C-formula in the language of [ f'g and [ f g j= '. The reason why explanations are not required to be prenex is that this fact simpli es both de nitions and proofs. However, equivalent explanations are considered as identical, so that explanations are in fact equivalence classes of formulae. When an explanation is required to be either minimal or consistent with the theory, it will be explicitly stated. In what follows, whenever we speak of minimality in a set of formulae, it is intended minimality with respect to j=.3 1.2 Proof theoretical methods for performing abduction The development of modern logic highlighted the fundamental duality between abduction and deduction, due to the fact that, if is a logical theory and ' an observed fact, then for any (in any logic where sentences can \cross" the logical entailment symbol j=): ; j= ' iff ;:' j= : The di erence lies in that, while deductive problems usually consist in verifying whether a given formula follows from a theory (possibly instantiating some variables), abduction problems are generative. Thus, as pointed out since the earliest papers on modern abduction [13, 17, 5], any deductive system that can be used not only to test, but also to generate consequences can be used to perform abduction. Most proof theoretical methods for performing abduction are based on resolution (see for example [12, 13, 5, 9]). As a prerequisite for the use of resolution based methods, the theory and the negation of the observation must rst of all be transformed into clause form. Moreover, works on abduction in the context of logic programming are often in uenced by the linear resolution view, even in the de nition of the basic concepts involved. The above sketchy observations suggest that investigating the nature of abduction outside the context of resolution and logic programming still deserves attention, in order to characterize abductive explanations without tailoring them to any xed method of computation. Moreover, the fact that many of the questions that should be addressed are meta-logical and proof-theoretical in nature suggests the use of proof systems that are, in that respect, more suited than resolution. This work proposes to base the analysis of the concepts invoved in abductive reasoning on non-resolution logical systems, i.e. tableaux and Gentzen systems, whose importance in automated reasoning has been often neglected in the computer science community (the relation between these methods and resolution has been clearly analysed in [1]). Classical tableau and sequent calculi enjoy of analicity, a feature that resolution lacks and that makes proof theoretical investigations clearer. They are especially promising tools to deal with abduction: the interpretation represented by a given branch of a tableau or by a sequent is clearly a partial interpretation, and abduction can be framed in the context of three valued semantics [3]. In the rest of the work a proof theoretical abduction method for rst order classical logic is de ned, based on the sequent calculus and a dual one, based on semantic tableaux. The methods work for full rst order logic, without requiring any preliminary reduction of formulae into normal forms. Soundness and completeness are established. In the propositional case, where the generation of the set of explanations is bound to terminate, two di erent characterizations are given for abductive explanations. Both identify explanations on the basis of 4 a given set of tableaux branches (leaves of a sequent derivation tree) and each of them is the declarative counterpart of a di erent algorithm for the generation of explanations. The rst one corresponds to the generation of the whole set of minimal and -consistent explanations, built by an incremental method that uses the branches (leaves) one by one and discards them as they are used. Minimal explanations are singled out by comparision with the other elements of the whole set. The second characterization corresponds to a (non-deterministic) algorithm for generating a single minimal explanation that is consistent with the theory. Such a method requires that a given set of tableaux branches (leaves of a sequent derivation tree) is stored and used till the algorithm terminates. The rst order versions of the abductive systems make use of uni cation and dynamic herbrandization/skolemization of formulae. The construction of the abduced formula is pursued by means of de-skolemization. The undecidability of rst order logic re ects on the fact that it may be impossible to terminate the construction of a tableau (derivation tree) and, therefore, the process of generation of explanations may not terminate. Consequently, the set of explanations may be in nite. Therefore, it is obvious that determining whether a given explanation is minimal is in general undecidable (even w.r.t. subsumption), and also that a minimal element may not exist. The method proposed here constructs the set of explanations in an incremental manner and the minimality check is very loose. In fact, the problem of identifying a reasonable relaxation of minimality w.r.t.j=, that ensures decidability, deserves a di erent work. 2 Abduction via sequent calculi and tableaux: the propositional case In this section we are going to de ne a sound and complete propositional method for computing minimal and -consistent explanations for an abduction problem. We consider a propositional language L0, containing two distinct propositional letters, true and false. Formulae, clauses and literals are de ned as usual. Clauses and conjunctions of literals will sometimes be identi ed with the set of their literals, so that set operations on clauses or conjunctions of literals are allowed. We make the convention that the empty disjunction is equivalent to the atom false and the empty conjunction to the atom