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 true. Two disjunctions (conjunctions) are considered equal and the equality sign '=' will be used if they disjoin (conjoin) sets of equal elements. 2.1 Propositional semantic tableaux and sequent calculus The propositional tableaux method [6] and Gentzen deduction system for classical logic [8] are essentially the same calculus considered from two di erent standpoints. We shall de ne propositional semantic tableaux as introduced by 5 Fitting [6], while the version of signed tableaux will be considered in order to highlight the substantial identity of such a calculus with the sequent system. 2.1.1. Semantic tableaux are used as refutation systems. They are built by means of a set of rules that preserve satis ability. The tableau expansion rules are the following::-rules) ::' ' :false true -rules) 1 ^ 2 1 :( 1! 2) 1 :( 1 _ 2) : 1 2 : 2 : 2 -rules) 1 _ 2 1j 2 1! 2 : 1j 2 :( 1 ^ 2) : 1j: 2 If ' is a formula, a tableau for ' is a tree whose root is labelled by ' and every non-root node is obtained from a preceding node in the same branch by means of the application of an expansion rule. The following de nition generalizes this notion to tableaux for sets of formulae. De nition 3 (Tableaux) Let f'1; : : : ; 'sg be a nite set of formulae of L0. 1. The following one branch tree is a tableau for f'1; : : : ; 'sg: '1 '2 :: 's 2. If T is a tableau for f'1; : : : ; 'sg and T * results from T by the application of a tableau expansion rule, then T * is a tableau for f'1; : : : ; 'sg. Each branch of a tableau can be thought of as the conjunction of the formulae appearing in it and the whole tableau as the disjunction of its branches. A tableau branch is satis able if the conjunction of all the formulae labelling the branch is satis able. A tableau is satis able if one of its branches is satis able. The tableau system preserves satis ability, that is, if T is a satis able tableau for , then any application of a tableau expansion rule yields another satis able tableau. A branch B of a tableau is called closed if both ' and :' occur in B, for some formula ', or if the atom false occurs in B; otherwise it is called open. A tableau is closed i all its branches are closed. 6 De nition 4 (Refutations) A tableau refutation of ' is a closed tableau for '. De nition 5 (Tableaux proofs) A tableau proof of ' is a closed tableau for :'. ' is a theorem of the tableau system if ' has a tableau proof. The tableau system is sound and complete, i.e. ' is a tautology i ' has a tableau proof. 2.1.2. A Gentzen-type calculus is a proof system given by a set of validity preserving rules. Proofs are trees labelled by sequents, i.e. constructs of the form ) , where and are nite sets of formulae and ) is a new symbol. A sequent ) is true in an interpretationM if either some formula in is false in M or some formula in is true in M. In other terms, the sequent f 1; : : : ; ng ) f 1; : : : ; mg is interpreted as the implication ( 1^ : : :^ n)!( 1 _ : : : _ m). The following rules are the propositional inference rules of the classical sequent calculus. In order to simplify the notation, here and in the following we shall write ; 1; :::; n instead of [f 1; :::; ng in antecedents or consequents of sequents. (: ) ) ) ; ;: ) ( ) :) ; ) ) : ; (^ ) ) ; ; ) ; ^ ) ( ) ^) ) ; ; ) ; ) ^ ; (_ ) ) ; ) ; ; ) ; _ ) ( ) _) ) ; ; ) _ ; (! ) ) ) ; ; ; ) ; ! ) ( ) !) ; ) ; ) ! ; An axiom of the calculus is a sequent ) such that \ 6= ;. The above rules can be read either downwards, so that they allow us to perform deductions and construct proofs of valid sequents starting from axioms, or upwards. In the latter case they reduce the question of the validity of a sequent to the question of the validity of one or two simpler sequents. As we shall be concerned mainly by this second reading of the rules, the calculus will be called the reduction calculus. De nition 6 (Reduction tree (R-tree)) Let be a sequent. A reduction tree T for is a nite tree whose root (endsequent) is and every node i in T is either a leaf or it is the conclusion of an inference rule of the reduction calculus, where the premisse is (the premisses are) the node(s) in T immediately above i. 7 De nition 7 (Sequent proof) An R-tree T over the sequent is a sequent proof for if all its leaves are axioms. The sequent calculus is sound and complete: a propositional sequent is valid if and only if there exists a proof tree for it. 2.1.3. In order to recognize the fundamental identity of the two calculi, it may be useful to give a brief presentation of the original notion of tableau, dealing with signed formulae, i.e. expressions of the form T' or F', where ' is a formula [21]. The signed version of the rules given in section 2.1.1 is obtained by pre xing the formulae with T . Dual rules are immediately de ned for F formulae. For example, the following are rules of the signed tableaux system: -rules) T ( 1 ^ 2) T 1 F ( 1! 2) T 1 F ( 1 _ 2) F 1 T 2 F 2 F 2 -rules) T ( 1 _ 2) T 1jT 2 T ( 1! 2) F 1jT 2 F ( 1 ^ 2) F 1jF 2 The signed systems has also rules that deals with negation: :-rules) T:' F' F:' T' A branch is closed when both T' and F' occur in it. When a rule is applied to a formula, that formula is marked as used. A tableau corresponds to a set of sequents. Every branch of the tableau corresponds to a sequent, a top sequent of an R-tree: a non-marked occurrence of a formula T' on the branch means that ' occurs on the left of the sequent arrow, a non-marked occurrence of F' means that ' occurs on the right of the sequent arrow. The tableaux de ned in section 2.1.1 correspond to signed tableaux where formulae are all signed with T , i.e. their branches correspond to sequents where the right hand side is empty. The fundamental identity of the rules of the tableau and sequent systems can be easily recognized. Given such a correspondence between the two calculi, in the sequel we shall often use ambigous expressions, such as trees to denote both R-trees and tableaux, and notations, such as to denote both sequents and sets of sentences. Tableaux and R-trees have properties that make them interesting from the standpoint of proof theory. They enjoy the subformula property: every formula added to a branch (in a sequent) is a subformula of some formula already on that branch (occurring in a lower sequent). This is essentially what gives tableau and R-trees their analytic character. Propositional tableaux and R-trees have another property, that we call the decomposition property. We de ne a mapping from tableaux to formulae as 8 follows: F(T ) is the disjunction of the open branches of T , where each branch B is identi ed with the conjunction of the unexpanded formulae (formulae to which no expansion rule has been applied). Then the formula labelling the root of a tableaux T for f g is equivalent to F(T ); in particular, if T is acceptable (see next subsection), F(T ) = DNF ( ), where DNF ( ) is the reduced disjunctive normal form of '. Analogously, if T is an R-tree, F(T ) is the conjunction of the non-axiom leaves of T , where each sequent f 1; : : : ; ng ) f 1; : : : ; mg is identi ed with the disjunction : 1_: : :_: n_ 1_: : :_ m. Then, the endsequent of an R-tree T is equivalent to F(T ); in particular, if T is acceptable, F(T ) = CNF ( ), where CNF ( ) is the reduced conjunctive normal form of the formula corresponding to .1 Tableaux can obviously be used also to transform a formula into CNF : the conjunction of the negation of the branches of a tableau T for f: g is CNF ( ). The same duality holds for R-trees. In fact, \tableaux are, above all, the best way known to logicians to nd a clause form (or a CNF) to a given sentence or sequent" [1]. In propositional logic, the order of application of the rules in a tree is interchangeable (permutability property) and a set of formulae can have di erent tableaux (a sequent di erent R-trees) that di er only for the order of application of the rules. However, if T 1 and T 2 are di erent tableaux (R-trees) for the same (set of) formulae, then F(T 1) F(T 2). 2.2 The set of abductive explanations Let h ; 'i be an abduction problem. Then, as 6j= ', there is no tableau refutation for [ f:'g and there is no sequent proof for ) '. The basic idea underlying this work is that a solution for the abduction problem h ; 'i can be found among the formulae that force the closure of a tableau for [ f:'g; equivalently, among the formulae that, when added to the left hand side of the top sequents of an R-tree for ) ', make them all valid. Of course, it would be meaningless to close an unexpanded tableaux by adding the atom false or the formula '. So, the trees have to be developed as far as possible and a careful choice of the literals that close each branch has to be performed. In the sequel, we de ne some notions that are useful in this respect. Here and in the following, tableaux for [f:'g and R-trees above ) ' will be generically called trees for the abduction problem h ; 'i. De nition 8 (Acceptable trees) Let T be a tableau. A branch B of T is fundamental if each non literal formula occurring in B has been expanded with the appropriate expansion rule. T is acceptable if all its branches are fundamental. Let T be an R-tree. A sequent of T is fundamental i it contains only atoms. T is acceptable if all its leaves are fundamental. 1The decomposition and subformula properties hold for the above formulations of the calculi, that make no use of structural rules. 9 De nition 9 (Threads and closing sets) Let B be a fundamental branch of a tableau. The thread associated to B, (B) is the set of the literals labeling nodes of B. The closing set for B is (B) = f: j 2 (B)g, where : is the complement of . If T is an acceptable tableau, the set S(T ) of the minimal closing sets of (the open branches of) T is f (B) j B is an open branch in T and there is no branch B0 in T such that (B0) (B)g. Let = ) be a fundamental sequent. The thread associated to , ( ) is [f: j 2 g. The closing set for is ( ) = f: j 2 ( )g = f: j 2 g [ . If T is an acceptable R-tree, the set S(T ) of the minimal closing sets of (the non axiom leaves of) T is f ( ) j is a non axiom leaf in T and there is no leaf 0 in T such that ( 0) ( )g. S(T ) will sometimes be brie y called the set of the closing sets for T , but it has to be noted that it collects the subset-minimal closing sets of the open branches (non-axiom leaves) of T . As a consequence of the decomposition property, if T 1 e T 2 are acceptable trees for the same set of formulae or sequent, then S(T 1) = S(T 2); therefore, if is a set of propostional formulae (a propositional sequent), the set S( ) of the minimal closing sets of the open branches (non axiom leaves) of any tableau (R-tree) for is well de ned. De nition 10 (Closures) Let T be an acceptable tableau (an acceptable Rtree) for the abduction problem h ; i and S(T ) = f 1; : : : ; ng the set of the closing sets of T . Let g be any choice function for the elements of S(T ), i.e. g( i) 2 i. (i) If S(T ) = ;, T has a single closure, the atom true. (This is the case where j= ). (ii) If S(T ) 6= ; and for any choice function g for the elements of S(T ) the set fg( 1); : : : ; g( n)g contains a pair of complementary literals, then the only closure for T is the atom false. (This is the case where j= : ). (iii) Otherwise = g( 1)^ : : :^ g( n) is a closure for T i does not contain a pair of complementary literals. In the above de nition, it is intended that is the conjunction of the literals in the set fg( 1) ^ : : : ^ g( n)g, i.e. literals are not repeated. If T is any acceptable tableau or R-tree, we denote by E(T ) the set of all the closures of T . Clearly, di erent acceptable trees (tableaux or R-trees) for the same abduction problem h ; 'i have the same closures. Let then: E( ; ') = E(T ) for any tree T for h ; 'i 10 The following theorem establishes completeness and a form of soundness of the tableau/sequent based procedure that solves abduction problems by generation of the set E( ; '). Theorem 1 Let h ; 'i be an abduction problem. Then every element of E( ; ') is a non-contradictory explanation for h ; 'i (soundness) and any minimal and non-contradictory explanation for h ; 'i is an element of E( ; ') (completeness). Clearly, the set of all but only the minimal and -consistent explanations for h ; 'i can be obtained by ltering the set E( ; ') and retaining only its minimal elements that are consistent with . In the next subsection we show how to perform the consistency test, fundamentally without any extra e ort, and give a characterization of minimal explanations that corresponds to a nondeterministic algorithm for the construction of a single minimal and -consistent explanation for h ; 'i. 2.3 Minimality and consistency with the theory 2.3.1. A straightforward consequence of Theorem 1 is that the set of the minimal explanations for h ; 'i is equal to the set min(E( ; ')) of the minimal closures for any tree T for h ; 'i, i.e. min(f j for all 2 S(T ), ( \ ) 6= ;g). where, if is a set of formulae, min( ) denotes its subset containing only minimal elements w.r.t. j=. This result leads to the construction of the set of minimal explanations of h ; 'i, but it does not help to build minimal explanations one by one, because in order to test their minimality each of them has to be compared with all the other elements of E( ; '). We are now going to show how this can be done, using only the set of the closing sets of any tree for h ; 'i. De nition 11 Let h 1; : : : ; mi be any ordered and non-empty set of sets of literals and a conjunction of literals. Then is minimally determined by h 1; : : : ; mi i one of the following conditions holds: (a) m = 1 and 2 1; (b) m > 1, is minimally determined by h 1; : : : ; m 1i and \ m 6= ;; (c) m > 1, condition (b) does not hold and there exists 2 \ ( m ( 1 [ : : : [ m 1)) such that, if ^ , then : 62 and is minimally determined by h 1; : : : ; m 1i. The condition (c) of De nition 11 ensures the soundness of the procedure (i.e. when T is a tree for an abductive problem and 1; : : : ; m are the closing set in S(T ), then only minimal explanation are generated) but completeness is 11 lost, unless any permutation of the sets 1; : : : ; m is considered. Next de nition copes with this fact. De nition 12 Let 1; : : : ; m be sets of literals and a conjunction of literals. Then is minimally determined by f 1; : : : ; mg i there exists a permutation h p1 ; : : : ; pmi of 1; : : : ; m, such that is minimally determined by h p1 ; : : : ; pmi. Lemma 1 Let T be a tree with non-empty S(T ) and such that false is not a minimal closure of T . Then, a C-formula is a minimal closure for T i is minimally determined by S(T ). 2.3.2. If is any set of consistent formulae and T is any tableau for or T is an R-tree for ( ) ), then T will be generically called an analytic tree for and E( ) will denote E(T ), i.e. E( ; false). In order to obtain from E( ; ') the explanations for h ; 'i that are consistent with , we note that the set of all the minimal and -consistent explanations for h ; 'i is min(E( ; ')) min(E( )). Thus, we can rst generate all the minimal explanations for the negation of the theory (i.e. the set min(E( )) and then remove them from min(E( ; ')). As remarked in [12], this is reasonable, if we assume that the theory is not frequently modi ed, so that the set min(E( )) can be stored once and for all and used each time an abduction problem has to be solved. However, the subformula property enjoyed by the tableau/sequent calculi can help do something more. The formulation of the systems can be modi ed so that the origin of every formula as a subformula of or a subformula of ' can be recognized. In the case of the sequent calculus, for example, the rules can be modi ed so that, in the sequents of a deduction, formulae never change sides. The rules (: ) ), ( ) :), (! ) ) and ( ) !) are to be dropped and the following rules are added: (:^ )) ;: ) ; ;: ) ;:( ^ ) ) () :^) ) : ;: ; ) :( ^ ); (:_ )) ;: ;: ) ;:( _ ) ) () :_) ) : ; ; ) : ; ) :( _ ); (! )) ;: ) ; ; ) ; ! ) () !) ) : ; ; ) ! ; (:! )) ; ;: ) ;:( ! ) ) () :!) ) ; ; ) : ; ) :( ! ); A sequent ) in this calculus is an axiom if and only if either \ 6= ;, or contains a pair of complementary formulae, or contains a pair of complementary formulae. 12 The leaves i ) i of an acceptable R-tree for a sequent ) ' have the property that every literal in i is an ancestor of a subformula in and every literal in i is an ancestor of a subformula in '. Such a feature, i.e. analicity of R-trees (and tableaux), has no correspondence in resolution proofs. If ( 1 ) 1); : : : ; ( m ) m) are all the axiom leaves in an R-tree for ( ) ') and ( 1 ) 1); : : : ; ( n ) n) are the non-axiom leaves, it follows that the leaves of an R-tree T for ( ) ) are exactly ( 1 ) ); : : : ; ( m ) ); ( 1 ) ); : : : ; ( n ) ). The non-axiom leaves of T include certainly ( 1 ) ); : : : ; ( n ) ), but possibly also some of the ( i ) ), let us say ( 1 ) ); : : : ; ( k ) ) for some k m. To simplify the notation, let us rename ( i ) ) as ( n+i ) ) for i = 1; :::; k. If a C-formula is an element of E( ) then is a closure for T , and for all 2 S(T ), \ 6= ;. Therefore, an element of min(E( ; ')) is consistent with i for some 2 S(T ), \ = ;. Note that the sequents ( 1 ) ) : : : ; ( n+k ) ), and consequently the sets in S(T ), can be built after the construction of the R-tree for ) ' with a minimal e ort. The method does not require the explicit construction of min(E( )). The same mechanism can be adapted to the tableau-based method. For example, by the permutability of the rules in the propositional case, the tableau can be built by rst applying the expansion rules to as far as possible, collecting the threads deriving from and nally expanding :'. Alternatively, mimicking the sequent-based method, threads can be thought of as pairs h ; :'i, where the elements of are subformulae of and the elements of :' are subformulae of :'. 2.3.3. Basing on the above observations and the results in 2.3.1, here follows a characterization of minimal and -consistent explanations, that does not refer to the set of all the explanations for the abduction problem, but makes use of the collection of closing sets for any analytic tree for it. Such a characterization is equivalent to the usual de nition. On its basis, an algorithm can easily be de ned, that non deterministically builds a single explanation for h ; 'i that is minimal and consistent with . If T is any tree for h ; 'i, let T 0 be the subtree of T where no rule is applied to :' ('). The characterization of -consistent explanations for h ; 'i refers to the two sets of closing sets S(T ) and S(T 0) and is justi ed by the following lemma. Lemma 2 Let T be any analytic tree for . Then a C-formula is consistent with i there exists 2 S(T ) such that \ = ;. In fact, if is consistent with , then there is some branch in any tree for that is not closed by adding (and expanding) . Then there is also a minimal branch that is not closed by . So, for some 2 S(T ) it must be that \ = ;. Lemmas 1 and 2 merge in the following: 13 Theorem 2 Let T be any tree for h ; 'i and T 0 the subtree of T where no rule is applied to :' ('). Then a C-formula is a minimal and -consistent explanation for h ; 'i i is minimally determined by S(T ) and there exists 2 S(T 0) such that \ = ;. The above theorem clearly gives a way to de ne a sound and complete nondeterministic algorithm for the construction of a single explanation for an abduction problem. 3 Full rst order abduction 3.1 Automated theorem proving in rst order sequent calculus The rst order rules for the universal quanti er in sequent calculus are usually given the following formulation, where t is any term and a a free variable not occurring in ;8x (x) ) : (8 ) ) ; (t) ) ;8x (x) ) ( ) 8) ) (a); ) 8x (x); These rules preserve validity. However, if we need falsity preserving rules, so that validity is preserved in the calculus also when the rules are read upwards, a di erent version of (8 ) ) has to be adopted: (8 ) ) ; (t);8x (x) ) ;8x (x) ) The main problem in automatizing proof search in this calculus is the choice of the term t in such a rule. Sometimes the rule has been stated as follows [11, 7]: (8 ) ) ; (t1); : : : ; (tk);8x (x) ) ;8x (x) ) where t1; : : : ; tk are terms from the language up to a given depth. This allows the mechanization of the calculus to perform validity checking. In [10, 2] it is proposed to delay the choice of the terms until an attempt to unify a pair of formulae occurring in the opposite sides of the same sequent succeeds. The notion of metavariable or dummy is introduced and the following version of the rule is given, that we shall also adopt: (8 ) ) ; (di);8x (x) ) ;8x (x) ) where di is a new metavariable When the rule is used in a reduction calculus, it is meant to build not a proper derivation or proof, but just a skeleton that can be changed into a derivation or 14 a proof by application of a substitution of terms for dummies. The problem is that the application of substitutions does not preserve the correcness of ( ) 8) inferences, because the eigenvariable conditions may be violated. Along the lines of Fitting [6], that de nes a tableau calculus handling also dynamic skolemization (see next section), it has been proposed in [20] to modify the formulation of the ( ) 8) rule by using Herbrand (or Skolem) functions, so that application of substitutions preserves correctness: ( ) 8) ) (hi(d0; :::; dn)); ) 8x (x); where hi is a new function and d0; :::; dn are all the metavariables occurring in ;8x (x) ) . The calculus de ned in [20] that deals with intuitionistic logic is actually more sophisticated, in that a list of "dominating" dummies is attached to every formula in a sequent, that augments with every application of (8 ) ) and determines the set of variables that have to occur in an h-term introduced by an application of ( ) 8). However, in order to simplify notation, exposition and proofs, we shall not deal with such an improvement in this work. 3.2 Free variable semantic tableaux and R-trees In the following the main formal de nitions, both for the case of free-variable semantic tableaux [6] and for the sequent calculus, are introduced. Let L be a rst order language extending L0. Terms, formulae, literals, free and bound occurences of variables in a formula are de ned as usual. Lsko is the extension of L, obtained by adding a distinct (countable) set of new symbols, the metavariables (or dummy variables) d0; d1; d2 : : : and, for any n, a (countable) set of new n-place function symbols, called Skolem functions or h-functions2 hn0 ; hn1 ; : : : (the superscript will often be omitted). The rst order tableau rules include the propositional expansion rules and the following rules, where d is a new metavariable that does not occur elsewhere in the tableau, h is a new Skolem function, and d1; : : : ; dn are all the metavariables occurring in the branch: ( rules) 8x (x) (d) :9x (x) : (d) ( rules) 9x (x) (h(d1; : : : ; dn)) :8x (x) : (h(d1; : : : ; dn)) The reduction calculus we are adopting contains the following quanti er rules, where d is a new metavariable that does not occur elsewhere in the R-tree, 2The reason for this double naming is that these functional symbols are used as Skolem functions in the tableau (refutation) system, while they rather appear as Herbrand functions (h-functions) in the sequent (validation) calculus. 15 h is a new Skolem function, and d1; : : : ; dn are all the metavariables occurring in ;8x (x); . (8 ) ) ; (d);8x (x) ) ;8x (x) ) ( ) 8) ) (h(d0; :::; dn)); ) 8x (x); (9 ) ) ; (h(d0; :::; dn)) ) ; 9x (x) ) ( ) 9) ) (d); 9x (x); ) 9x (x); The unrestricted rules, i.e. (8 ) ) and ( ) 9) correspond to the -rules for tableaux, while (9 ) ) and ( ) 8) correspond to -rules. Note that, because of the -rules, the decomposition property does not hold. Clearly, as -rules may be applied in nitely many times, rst order trees may be in nite. However, the de nitions of R-trees and tableaux are extended to rst order so that only nite trees are considered. De nition 13 (Uninstantiated tableau and reduction tree (U-tree)) Let be a set of formulae. An uninstantiated tableau T for is a nite tree that is either a one-branch tableau for or it is obtained from a tableau for by application of rst order expansion rules. Let be a sequent. An uninstantiated reduction tree T over is a nite tree whose root (endsequent) is and every node i in T is either a leaf or it is the conclusion of an inference rule of the reduction calculus, where the premisse is (the premisses are) the node(s) in T immediately above i. Substitutions are de ned as usual. They are intended to a ect only metavariables, that are all distinct from bound variables, so that substitutions are always free. De nition 14 (Instantiated tree (I-tree)) is an instantiated tree (either a tableau or an R-tree) if it is obtained by applying a substitution of terms for metavariables to every formula of a U-tree. I.e., for every U-tree T and substitution , T is an I-tree. De nition 15 (Refutations and proofs) An instantiated tableau for is a refutation of if it is closed. is a proof of ' if is a refutation of :'. An instantiated reduction tree for is a proof of if all its leaves are axioms. The free variable semantic tableau system is sound and complete [6], and so is the free variable sequent system. Let be a set of formulae (a sequent). In the general rst order case, a possibly in nite collection of U-trees can be constructed for : T 1; T 2; : : :. Two of them may di er either on the order of application of the rules of the calculus, 16 or because one is the extension of the other, or both (they are extensions of two other U-trees that are permutations one of the other). In fact, in spite of the reusability of -rules, and expansion rules are not interchangeable. Moreover, every U-tree T i for may correspond to di erent I-trees, one for each substitution applicable to T i. For the purposes of both proof search and abduction, the number of the trees that have to be considered can be reduced. For example, completeness is not lost if we admit only substitutions that are generated as most general uni ers of formulas occurring both positive and negative in the same branch (sequent). Although this kind of results is surely important, we shall not deal with this question in this work. The conceptual point is in fact that in nity cannot be ruled out of rst order logic. However, the problems that arise with the undecidability of the predicate calculus mainly re ect upon the preference criteria of consistency with the theory (obviously) and minimality. In fact, not only is implication between rst-order C-formulae undecidable [19], but also a rst order abduction problem can have an in nite number of explanations, so, in general, we cannot determine whether a given explanation is minimal just by comparison with the others. There are even cases where no explanation is minimal [12]. These observations suggest that both criteria of consistency with the theory and minimality should be somehow relaxed. This point is not developed further here, but we de ne an abductive method where only a small part of the non-minimal explanations are recognized and rejected. W.r.t. propositional abduction, a new font of non-determinism stems from the choice of a single nite U-tree for the abduction problem (i.e. we stop at any stage of the development of a possibly in nite tree) and the choice of a substitution for it. On the basis of the resulting I-tree, a set of explanations for the original problem is built. As a nal remark, we note that in rst order logic it is undecidable even to detemine whether a pair h ; 'i is a genuine abduction problem. Consequently, any method for performing abduction should not rely on the assumptions that j= ' and 6j= :'. The following de nitions determine when a tree is acceptable for abduction, i.e. when its construction can be interrupted in order to generate a set of explanations. De nition 16 (Acceptable U-tree (AU-tree)) A branch in a tableau is fundamental if every non literal occurrence in the branch has been applied the appropriate expansion rule at least once. A sequent is fundamental i it only contains either atoms or formulas of the form 9x in the consequent or 8x in the antecedent. An U-tree is acceptable if all its branches (leaves) are fundamental. De nition 17 (Acceptable I-tree (AI-tree)) An I-tree = T is acceptable i T is an AU-tree. 17 For each of the ( nite) AI-trees for an abduction problem h ; 'i, we show how to build a nite set of explanations EFOL( ) for h ; 'i. Soundness will amount to saying that if a given formula is an element of EFOL( ), then is not contradictory and [ f g j= '. However, is not necessarily a minimal explanation. In particular, it may be the case that j= ' even if non valid explanations are generated. Completeness will guarantee that, for any non-contradictory formula such that [ f g j= ', there exists an AI-tree for h ; 'i such that EFOL( ) contains a logical consequence of . 3.3 First order explanations The notions of thread, closing set and closure for acceptable I-trees are immediate extensions of the corresponding propositional ones. Note however that, in the case of sequents, only literals are collected in threads and closing sets. If is an acceptable I-tree, then E( ) denotes, as before, the set of all the closures of . Now, an element of E( ) is in the language Lsko L, so it may not be an explanation for h ; 'i. In order to obtain explanations, all the metavariables and the skolem terms introduced in the tree for h ; 'i are to be replaced by suitably quanti ed variables. De nition 18 (Reverse skolemization) 3 Let be a formula in Lsko and let st( ) = ft1; : : : ; tkg be the set of the terms occurring in that are not in L. Let htp1 ; : : : ; tpki be any total ordering of the elements in st( ) such that for all i and j, if tpi properly occurs in tpj then i < j. Then Qx1 : : :Qxk 0 is obtained from by reverse skolemization on the basis of htp1 ; : : : ; tpki i 0 is obtained from by replacing each term tpi with the (new) variable xi; for all i, if tpi is a metavariable, then xi is existentially quanti ed, otherwise, if tpi is a skolem term, then xi is universally quanti ed. The set of all the formulae obtainable by reverse skolemization from will be denoted by desk( ). The fact that desk( ) is not necessarily a singleton is illustrated by the sample case where = p(d1; h1(d1); d2; h2(d2)); in fact the two elements of desk( ) = f9x18x29x38x4p(x1; x2; x3; x4); 9x18x29x38x4p(x3; x4; x1; x2)g are not equivalent. An algorithm for reverse skolemization can be found in [4]. 3Although we are using the term skolemization, that is more familiar to the AI community, what is actually de ned is reverse herbrandization. 18 De nition 19 (First order closures) Let be any I-tree for the abduction problem h ; 'i and E( ) the set of all the closures of . Then the set of the rst order closures of is EFOL( ) = f j 2 min(desk( )) for some 2 min(E( ))g: If h ; 'i is an abduction problem and < = f j is an acceptable I-tree for h ; 'ig, then EFOL( ; ') = [ 2< EFOL( ) The de nition of EFOL( ; ') is quite naive. Investigating the relations that hold between the sets EFOL( 1) and EFOL( 2) when 1 is an expansion of 2 would lead to a better characterization of EFOL( ; '). The following theorem establishes that the abduction calculus that generates elements of EFOL( ; ') is sound and complete. Theorem 3 Let h ; 'i be an abduction problem. Then for any element of EFOL( ; '), [f g j= ' (soundness). If is a consistent C-formula such that [ f g j= ', then there exists an element of EFOL( ; ') such that j= (completeness). Example 1 Let contain the single formula 8x8y8z(p(x; y)^p(y; z)!p(x; z)) and ' = 9xp(x; x). This is the exampled used in [12] to show that in the rst order case there may be no minimal explanations. An unistantiated tableau for [ f:'g is the following tree T : 19 (1) 8x8y8z(p(x; y)^ p(y; z)! p(x; z)) (2) :9xp(x; x) ... (3) p(d1; d2) ^ p(d2; d3)! p(d1; d3) (from 1) (4) :p(d4; d4) (from 2) ! ! ! ! ! ! ! ! ! ! ! aaaaaaaaaaa (5) :(p(d1; d2) ^ p(d2; d3) (from 3) QQQQQQQQ (7) :p(d1; d2) (8) :p(d2; d3) (6) p(d1; d3) The three threads of T are 1 = f:p(d4; d4);:p(d1; d2)g, 2 = f:p(d4; d4); :p(d2; d3)g and 3 = f:p(d4; d4); p(d1; d3)g. If the substitution = fd1=d4; d1=d3g is applied to T , then 3 closes and the set S(T ) contains the two following closing sets: 1 = fp(d1; d1); p(d1; d2)g, 2 = fp(d1; d1); p(d2; d1)g. The corresponding minimal closures are p(d1; d1) and p(d1; d2)^ p(d2; d1), that, by deskolemization, generate the explanations 9xp(x; x) (trivial) and 9x9y(p(x; y)^ p(y; x)). Clearly, the tree T can be further expanded and new explanations generated. Example 2 The following simple example shows the use of herbrand functions in the sequent case. Let = f8x(8yp(x; y)!q(x))g and ' = 8xq(x). The following derivation tree T is an U-tree for ) '. 20 8x(8yp(x; y)!q(x)) ) q(h0); p(d; h1(d))8x(8yp(x; y)!q(x)) ) q(h0);8yp(d; y) ; 8x(8yp(x; y)!q(x)); q(d) ) q(h0)8x(8yp(x; y)!q(x));8yp(d; y)!q(d) ) q(h0)8x(8yp(x; y)!q(x)) ) q(h0)8x(8yp(x; y)!q(x)) ) 8xq(x)Applying the substitution = fh0=dg makes the leaf8x(8yp(x; y)!q(x)); q(d) ) q(h0)an axiom. So, the only closing set of T is = fq(h0); p(h0; h1(h0))g. Conse-quently, EFOL(T ) = f8xq(x);8x8yp(x; y)g.4 Concluding remarksThe analysis of some of the still open problems in the formalization of abductivereasoning by means of sequent/tableau calculi has led to:(a) characterize a single minimal and -consistent explanation for a proposi-tional abduction problem without referring to the set of all the explana-tions; prove the soundness and completeness of such a characterization;(b) de ne a method for performing abduction in full rst order logic, thatdoes not require any preliminary transformation of formulae into normalforms; prove the soundness and completeness of such a method;(c) single out as a central point the necessity of a relaxation of the minimalityrequirement in rst order abduction; minimality should be de ned in termsof a reasonable ordering relation v between C-formulae, that is decidableand such that for any and , if v then j= .Much work still remains to be done. Here follow some of the main points tobe addressed, excluding the question of minimality.(1) Proving interesting properties of rst order trees, such as \expanding an U-tree further on does not loose information" and \if two trees di er for theorder of application of two rules, then any of them can be further expandedso that possibly lost information is recovered". As a consequence of theseproperties, the di erent trees that have to be generated to perform rstorder abduction are all instances of nite subtrees of the same tree.(2) Formulating the rst order free variable calculi on the style of [20], withthe aim of reducing the dependencies of skolem terms from dummies asmuch as possible. In fact, a list of "dominating" dummies can be attachedto every formula, that augments with every application of a -rule to thatformula and determines the set of variables that have to occur in a skolem21 term introduced by an application of a -rule. This mechanism reducesthe failures in attempts to unify pairs of formulae, without loosing thesoundness of the procedure, therefore improving e ciency.(3) Characterizing and comparing the set of explanations that is generatedby di erent methods (the resolution based ones and the tableau/sequentmethod), before ltering them w.r.t. minimality and consistency with thetheory. If di erent methods do not produce the same un ltered set, theset generated by a given method can be a measure of its e ciency.(4) Studying abduction in modal, intuitionistic and linear logics, by means ofthe same proof-theoretical tools.References[1] A. Avron. Gentzen-type systems, resolution and tableaux. Journal ofAutomated Reasoning, 10:265{281, 1993.[2] K. A. Bowen. Programming with full rst-order logic. In J. E. Hayes,D. Michie, and Y.-H. Pao, editors, Machine Intelligence, volume 10, pages421{440. Halsted Press, 1982.[3] M. Cialdea Mayer, F. Pirri, and C. Pizzuti. Natural properties of abductivehypotheses in three valued logic. In International Conference on Logic Pro-gramming (ICLP '93) Postconference Workshop on Abduction, Budapest,Hungary, 1993.[4] P. T. Cox and T. Pietrzykowski. A complete nonredundant algorithm forreversed skolemization. Theoretical Computer Science, 28:239{261, 1984.[5] P. T. Cox and T. Pietrzykowski. Causes for events: their computationand applications. In Proceedings of the Eighth Conference on AutomatedDeduction (CADE-86), pages 608{621, 1986.[6] M. Fitting. First Order Logic and Automated Theorem Proving. Springer-Verlag, New York, 1989.[7] J. H. Gallier. Logic and Computer Science Foundations of AutomaticTheorem Proving. Harper & Row, New York, 1986.[8] G. Gentzen. Untersuchungen uber das logische schliessen. MathematischeZeitschrift, 39:176{210, 405{431, 1935.[9] K. Inoue. Linear resolution for consequence nding. Arti cial Intelligence,56:301{353, 1992.22 [10] S. Kanger. A simpli ed proof method for elementary logic. In P. Bra ortand D. Hirshberg, editors, Computer Programming and Formal Systems,pages 87{94. North Holland, 1963.[11] S.C. Kleene. Mathematical Logic. John Wiley & Sons, New York, 1966.[12] P. Marquis. Extending abduction from propositional to rst-order logic.In Proceedings of the First International Workshop on Fundamentals ofArti cial Intelligence Research (FAIR '91), pages 141{155, 1991. Springer-Verlag, LNAI 535.[13] C. G. Morgan. Hypothesis generation by machine. Arti cial Intelligence,2:179{187, 1971.[14] D. Poole. Explanation and prediction: an architecture for default andabductive reasoning. Computational Intelligence, 5:97{110, 1989.[15] D. Poole. Compiling a default reasoning system into prolog. New Genera-tion Computing, 9:3{38, 1991.[16] D. Poole, R. Goebel, and R. Aleliunas. Theorist: a logical reasoning systemfor default and diagnosis. In N. Cercone and G. Mc Calla, editors, TheKnowledge Frontier, pages 331{352. Springer Verlag, 1987.[17] H. E. Jr. Pople. On the mechanization of abductive logic. In Proceedings ofthe Third International Joint Conference on Arti cial Intelligence (IJCAI-73), pages 147{152, 1973.[18] R. Reiter and J. de Kleer. Foundations of assumption-based truth mainte-nance systems: Preliminary report. In Proceedings of the National Confer-ence on Arti cial Intelligence, pages 183{188, 1987.[19] M. Schmidt-Schauss. Implication of clauses is undecidable. TheoreticalComputer Science, 59:287{296, 1988.[20] N. Shankar. Proof search in intuitionistic sequent calculus. In Proceedingsof the Tenth Conference on Automated Deduction (CADE-90), pages 522{536, 1992.[21] R.M. Smullyan. First-Order Logic. Springer-Verlag, Berlin, 1968.23