Propositional Abduction in Modal Logic - CiteSeerX
Propositional Abduction in Modal Logic Marta Cialdea Mayer2
Fiora Pirri1
1 Dipartimento di Informatica e Sistemistica, Universita di Roma \La Sapienza" via Salaria 113, 00198 Roma, Italia 2 Dipartimento D.S. Informatica, Terza Universita degli Studi di Roma via Segre 2, 00146 Roma, Italia e-mail: fcialdea,[email protected]
Abstract
In this work, the problem of performing abduction in modal logics is addressed, along the lines of [3], where a proof theoretical abduction method for full rst order classical logic is de ned, based on tableaux and Gentzen-type systems. This work applies the same methodology to face modal abduction. The non-classical context enforces the value of analytical proof systems as tools to face the meta-logical and proof-theoretical questions involved in abductive reasoning. The similarities and di erences between quanti ers and modal operators are investigated and proof theoretical abduction methods for the modal systems K , D, T and S 4 are de ned, that are sound and complete. The construction of the abductive explanations is in strict relation with the expansion rules for the modal logics, in a modular manner that makes local modi cations possible. The method given in this paper is general, in the sense that it can be adapted to any propositional modal logic for which analytic tableaux are provided. Moreover, the way towards an extension to rst order modal logic is straightforward. This
work has been partially supported by Basic Research Action, Medlar II
1
1 Abductive reasoning and modal logics Many forms of commonsense reasoning are not deductive in nature. Reasonable inferences may be performed even if leading to conclusions that are not certain but only plausible. Abduction is one of such inference schemata that re ect unsound forms of inference. 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. A precise de nition of abductive explanations is obviously dependent on the logic underlying the theory : some fact could explain ' in classical logic, but not in intuitionistic or linear logic. So, let us assume that a logic L is given, with a corresponding relation of logical consequence j=L . Then an abduction problem in L consists of a background theory and a formula ' such that 6j=L '. A solution of the problem given by the pair h; 'i is to be chosen among the formulae such that [ f g j=L '. In general, it is not even required that j=L satis es the tarskian properties for a consequence relation. For example, it makes sense to think of (and it would be worth exploring) non-monotonic explanations, i.e. cases where explains ' but ^ does not [2], so that the notion of explanation is related to a non-monotonic consequence relation. In this perspective, abduction in the logic programming context, with its tricks (like constraints), can be seen as the form of abduction that is carried out in the logic induced by the semantics of the arrow and negation-as-failure operators. Recently, much attention has been paid to abductive reasoning in di erent elds of Arti cial Intelligence, such as diagnosis [4, 5, 13, 12], interpretation of natural language sentences [7, 15], plan recognition [1], scene interpretation [14]. This work is a contribution to the characterization of abduction as a form of inference whose context is only the logic it depends upon. Abstraction from any particular problem helps to focus on general theoretical questions, while results can be obtained that bene t the applicative elds. In this work, the problem of performing abduction in modal logics is addressed. The meaningfulness of such a problem can be recognized by considering two simple examples: the rst is from Arti cial Intelligence, the second from program veri cation. Let us assume that an agent is reasoning about her own knowledge, represented by a modal theory , and she observes that ' is consistent with her knowledge, i.e. 3' is true. However, 3' is not derivable in . Abduction would produce what is needed to sanction 3'. Of course, the importance of epistemic abduction is clearer when considering multiple agent contexts: for example, both natural language understanding and user modeling require the ability to abduce the goals of the speaker, and student modeling includes the abduction of the student's beliefs from the observation of her answers to questions and problems. As a second example, let us assume that a program is described by a 2
theory in a temporal or dynamic logic L and that the termination property for is represented by the formula '. Then, determining the class of input data for which terminates can be seen as abducing a precondition such that [ f g j=L '. In [3] analytical proof systems are proposed as suitable tools to face the metalogical and proof-theoretical questions involved in abductive reasoning and a proof theoretical abduction method for full rst order classical logic is de ned. The underlying idea is to solve the abduction problem h; 'i by building a formula that closes the open branches of a tableau for [ f:'g. The method does not require any preliminary reduction of formulae into normal forms, but dynamic skolemization is used. In the rst order case, quanti ers are to be reintroduced following a sort of reverse skolemization. This work applies the same methodology to face modal abduction. There are obvious similarities between quanti ers and modal operators (see, among others, [11] for a functional translation method of modal formulae into rst order formulae). As far as the tableau based abduction method is concerned, however, a main di erence lies in that the structure of a quanti ed formula is essentially rebuildable from its Skolem forms, while the trace of modal operators, once they are dropped, is lost. This fact induces a recursive reformulation of the construction of the abduced formulae, thus shifting the view from a meta-logical level to a logical one. The resulting construction is in strict relation with the expansion rules for the modal logic, in a modular manner that makes local modi cations possible, so as to t the method to di erent modal systems. In fact, the abductive method given in this paper is general, in the sense that it can be adapted to any propositional modal logic for which analytic tableaux are provided. Moreover, the way towards an extension to rst order modal logic is straightforward.
1.1 Basic notions about modal logics
The logical language of modal systems consists of all the classical logical symbols and the modal operator 2 (necessity). Modal formulae are de ned consequently. Formulae of the form 2 are called boxed formulae. The modal operator 3 is de ned as follows: 3 :2: . In the sequel we consider modal systems characterizable by a subset of the following schemata:
(K) 2( ! ) ! (2 ! 2 ) (T) 2 ! (D) 2 ! 3 (4) 2 ! 22 and the necessitation rule: 2 . In particular, the systems K (with axiom K), D (with axioms K and D), T (with axioms K and T) and S 4 (with axioms K, T and 4) are considered. 3
The semantics of modal logics is de ned as usual in terms of L-modal structures M = hW ; RL ; Vi, where W is a non empty set (of possible worlds), RL is a binary relation on W (accessibility relation), whose properties depend on the logic L, and V is a truth assignment, that maps W P into ftrue; falseg, where P is the set of propositional symbols of the language. The notion of truth of a formula in a world w 2 W , (M; w) j= , is de ned as usual. The notion of logical consequence in a modal logic L is not always given the same de nition. Two versions are commonly used: (F ) j=FL i for every L-modal structure M: (for all w 2 W and for all 2 , (M; w) j= ) implies (for all w 2 W , (M; w) j= ). (K) j=KL i for every L-modal structure M: for all w 2 W (for all 2 , (M; w) j= implies (M; w) j= ). Clearly, if is not empty, j=KL i j=FL (
^ ) !
2
If is empty, the two notions collapse, so that in particular the notion of validity they induce is the same (this is the reason why any of the two versions can be often used without discrepancies). The version (F ) corresponds to the notion of derivability in axiomatic systems (with necessitation), while (K) corresponds to derivability of the corresponding sequent in sequent calculi. Consequently, the main di erence between the two versions is that the following properties: (i) if j= ' then [ f:'g is inconsistent (ii) if [ f g j= ' then j= ! ' hold for (K) but not for (F ) (their converses are true in both cases). Thus, (K) is closer to the classical notion of logical consequence. In recent literature (F ) is often preferred because of its greater generality. However, in this work, we use (K) just because it de nes a stronger relation and, consequently, it corresponds to a more general notion of abduction: if [ f g j=KL ' then [ f g j=FL ', but the converse is not true. In the context of abduction, the use of (K) is more reasonable (there does not seem to be much sense in assuming that is a good explanation for 2 ) and it allows the duality between deduction and abduction to be preserved. Of course, the same concepts could be de ned by use of (F ) and implication, but the notation would be far more cumbersome. In the following we shall simply use the symbol j=L instead of j=KL . 4
1.2 Basic notions about modal abduction
Abductive explanations are in general required to respect some fundamental conditions, in order to be accepted as \interesting". If well settled consequence relations j=L are considered, such as the classical, modal or intuitionistic ones, the following restrictions are to be reasonably imposed on explanations for an abduction problem h; 'i: (i) is consistent with , i.e. 6j=L : . (ii) is a minimal explanation for the abduction problem h; 'i, i.e. for any formula , if [ f g j=L ' and j=L , then j=L . Moreover, the syntactical form of explanations is usually restricted; for example, in rst order classical logic, an explanation is a prenex formula whose matrix is a conjunction of literals. A logical reason for this has been pointed out by [9]: if = f1 ; : : : ; n g is nite, then any minimal explanation for h; 'i is equivalent to 1 ^ : : : ^ n ! '. The following de nition is a straightforward adaptation to the modal language of the classical syntactical restriction on explanations. De nition 1 (modal C-formulae) A formula is a modal C-formula if it is built up from literals using only the modal operators 2 and 3 and conjunction. Consequently: De nition 2 (L-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=L '. When an explanation is required to be either minimal or consistent with the theory, it will be explicitly stated. L-equivalent explanations are considered as identical, so that L-explanations are in fact equivalence classes of formulae (the terms L-equivalence, L-derivability, etc. are used in the obvious sense).
2 Modal abduction by use of analytic tableaux Semantic tableaux are used as refutation systems. They are built by means of a set of rules that preserve satis ability of sets of formulae. In this work we make use of the unsigned version of modal tableaux [6]. Modal expansion rules are stated as follows, where set union is brie y denoted by the comma, in the obvious sense; for example, ; ; 0 is a short-cut for f g [ [ 0 . Moreover, 2 is a set of boxed formulae and 0 is a set of non boxed formulae. :2 ; 2; 0 2; 0 -rule) K -rule) :2 ; 4 : ; : ; ; 2
D -rule)
2; 0
T -rule) 22 ; ; ;
5
The systems K , D, T and S 4 are characterized by the addition of modal rules to the classical ones, as the following table shows:
K D T S4
K -rule K -rule K -rule 4 -rule In the sequel we shall generically speak of -rule and -rule; the version {
D -rule T -rule T -rule
actually referred to depends on the context. If is a set of formulae, an L-tableau for is a tree whose root is labeled 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 of the system L. A branch B of a tableau is closed if both ' and :' occur in some node of 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. A L-tableau refutation of is a closed L-tableau for . The L-tableau systems are sound and complete, i.e. is L-unsatis able i has an L-tableau refutation [6]. The standard formulation of analytic tableau systems does not make explicit reference to structural rules. However, we assume that the following form of weakening is among the rules of the systems: 2; 0 2 where 0 contains no boxed formulae. This rule is certainly redundant for the aims of theorem proving, but it is needed to guarantee the completeness of the abduction procedure for the systems K , T and S 4 (the reason can be found in the proof given in the Appendix). In the classical case, the construction of an explanation can be done only on the basis of the set of the open leaves of the tableau, without need to store and work on the structure of the whole tableau: a literal is chosen for each open leaf, that closes the corresponding branch; the set of literals is conjoined and quanti ers are reintroduced. In fact, the elimination of quanti ers by means of skolemization preserves the information about their mutual dependencies that is needed to reintroduce them. In contrast to these, modal operators cannot be correctly reintroduced without considering the structure of the tableau, where they may disappear without leaving any trace. This fact is intuitively explained by the following simple example. Let us consider the more general problem of nding a formula that is L-inconsistent with a set . Let L = D, = f2' _ 2 g and 0 = f2(' _ )g; the following trees are D-tableaux for and 0 , respectively. 2(' _ ) 2' _ 2
2' '
'_ '
2
6
Both tableaux have the same open leaves. Nevertheless, while the rst one closes with the addition of = 3:' ^ 3: , the closure of the latter requires the stronger 0 = 3(:' ^ : ).
2.1 The set of abductive explanations
Let h; 'i be an abduction problem in L. Then, as 6j= ', there is no L-tableau refutation for [ f:'g. A solution for the abduction problem h; 'i can be found among the formulae that force the closure of a tableau for [ f:'g. Of course, it would be meaningless to close an unexpanded tableaux by adding the atom false or the formula '. So, if only non trivial explanations are looked for, the tableaux should in principle be developed as far as possible. However, also tableaux that can still be expanded are to be considered, because (i) -rules may be reapplicable, and (ii) -rules may cancel some formulae. The following de nitions cope with the determination of the point where the construction of a tableau can be interrupted in order to perform abduction. Here and in the following, tableaux for [ f:'g will be called tableaux for the abduction problem h; 'i. De nition 3 (Compulsory expansion of a branch) The expansion of a branch B with leaf is compulsory if one or more of the following conditions hold: (i) contains no literals and can be expanded. (ii) contains some formula that can be expanded by means of a classical logical rule; (iii) contains some 2 that can be expanded but has not yet been expanded in B; In other words, the expansion of B is compulsory whenever it can be performed without the loss of information that is useful for abduction and without cycling. Note that the application of a -rule is not always compulsory. De nition 4 (Fundamental tableaux) Let T be a tableau. A branch B of T is fundamental if its expansion is not compulsory. A node is fundamental if it is the leaf of a fundamental branch. T is fundamental if all its branches are fundamental. Note that a fundamental node always contains some literals, unless the logic considered is K . In this case, a node that contains only boxed formulae is fundamental. We next turn to the characterization of the set of abductive explanations in L built up from a fundamental L-tableau T for an abductive problem. It will be denoted by E L (T ). The following de nition is relative to the logics T and S 4. Its variants for K and D are stated later on. 7
De nition 5 (Construction of explanations in T and S 4) If L is either T or S 4 and T is a fundamental L-tableau, the set E L (T ) is de ned recursively
as follows. 1. If T consists of the single fundamental node , then: 1a. If contains either false or a pair of complementary literals, then E L (T ) = ftrueg. 1b. Otherwise E L (T ) = f: j 2 and is a literalg. Note that, as is fundamental, E L (T ) is not empty. 2. Otherwise, di erent cases are to be considered, according to the form of the rst expansion rule applied in T . 2a. T has the form (=) , where = is an application of a classical non-
T0
branching rule (weakening included). In this case E L (T ) = E L (T 0 ). 2b. T has the form (=) , where = is an application of a -rule. T 0 T1 In this case E L (T ) = f'0 ^ '1 j '0 2 E L (T0 ); '1 2 E L (T1 )g. 2 ; , where = is an application of the -rule 2c. T has the form (=) T0 to 2 . We distinguish two cases: (i) If either is empty or it contains only boxed formulae, then E L (T ) = f3' j ' 2 E L (T0 )g; (ii) otherwise E L (T ) = E L (T 0 ). 0 2d. T has the form (=) :2 ; 2; , where = is an application of the T0 -rule to :2 . Then E L (T ) = f2' j ' 2 E L (T0 )g. As an example, let = f2(p ^ 2q ! r)g and ' = 23r. The following (leftmost) tree, where 2 = f2(p ^ 2q ! r); 2:rg, is a fundamental S 4-tableau for [ f:'g. The tree at its right side is an intuitive view of the construction of a particular explanation, 23(p ^ 2q) (the tree is to be read bottom up). 23(p ^ 2q) 2(p ^ 2q ! r); 32:r 2; p ^ 2q ! r 3(p ^ 2q) 2; p ^ 2q ! r; :r p ^ 2q 2; :(p ^ 2q); :r 2; r; :r p ^ 2q true closed 2; :p; :r 2; :2q; :r p 2q open 2; ; :q q
open
Suitable modi cation to the de nition of the set E L (T ) are easy to state for other logics. As far as the system D is considered, clauses 1, 2a, 2b and 2d remain the same, while clause 2c is replaced by the following: 8
0 2c0 . T has the form (=) 2T ; , where = is an application of the -rule. Then 0 E D (T ) = f3' j ' 2 E (T0 )g.
Considering system K , it must be noted that K -tableaux may have fundamental nodes containing only boxed formulae and literals, so, besides deleting clause 2c, item 1b has to be replaced by: 1b0 . Otherwise, has the form 20 ; , where is a (possibly empty) set of literals; then E K (T ) = f: j 2 g 0 [f3' j ' 2 E (T ) for some fundamental K -tableau T 0 for 0 g. The methods are sound and complete, in the sense stated by the following theorems, that hold for L = K; D; T; S 4.
Theorem 1 (Soundness) Let T be a fundamental L-tableau for and 2 E L (T ). Then there exists a closed L-tableau T 0 for [ f g. Theorem 2 (Completeness) Let be any set of modal formulae and an L-consistent C-formula such that [ f g is L-unsatis able. Then there exists a fundamental L-tableau T * for and a C-formula ' 2 E L (T ) such that j=L '. The proof of Theorem 1 is a straightforward induction on T . The signi cant steps in proving completeness can be found in the Appendix.
3 Existence of minimal explanations
If h; 'i is an abductive problem, let (; ') denote the set of all the fundamental L-tableaux for [ f:'g. By the completeness theorem, the set of minimal explanations for h; 'i is included in the set
E L (; ') =
[
T 2(;')
E L (T )
The question whether min(E L (; ')) may be empty is addressed here. In the case of rst order logic, there are abduction problems that admit no minimal explanation [9], because there exist in nite descending chains of formulae. This is not the case for modal logic. For the non transitive logics K , D and T , the existence of minimal explanations follows from the fact that the number of fundamental tableaux for a given problem h; 'i is nite, therefore E L (; ') is nite. In fact, at any step in the expansion of a tableau there is only a nite number of choices and the construction of any tableau terminates, because each application of a rule of K 9
and D decreases the number of logical operators; in T the number of logical operators may not be reduced by a -rule, but it is as soon as a -rule is applied (if a node contains no -formulae, reapplications of -rules can be avoided). The case of the logic S 4 is not so simple. In fact, there exist in nite tableaux, i.e. there may be an in nite number of fundamental tableaux for a given abductive problem. The fact that in nite tableaux cycle does not help, because every cycle could generate weaker explanations and in S 4 there is an in nite number of distinct modal functions, even of a single propositional variable. So, in principle, a given problem may lack minimal explanations. However, the following theorem solves the problem of the existence of minimal explanations for S 4 too.
Theorem 3 Let h; 'i be an abductive problem in a propositional modal logic L that enjoys the nite model property. Then: (i) there exists a minimal explanation of h; 'i, and (ii) for any explanation of h; 'i there exists a minimal explanation min such that j= min . A proof of this theorem is given in the Appendix.
4 Concluding remarks Abduction in a logic may be approached in several ways, at least one for every deductive method de ned for the logic. However abduction in modal logic has never been addressed so far. Obviously, the results of this work are only a rst step towards a good characterization of modal abduction and leave a number of problems to be solved. An improvement of the method can be obtained by use of pre xed tableaux, that in modal deduction reduces the number of tableaux to be considered. Apart from a gain in eciency, pre xes could help in storing important information on the \history" of the modal operators in the same way as the use of Skolem functions code the position and quality of quanti ers that are eliminated. However, commonly used pre xes [6] do not allow the abductive construction to be based only on the leaves of the tree, like in classical logic. In fact, those pre xes keep track of the application of modal rules whatsoever, without storing any information about which rules have been applied. A careful rede nition of pre xes to be used for abduction is therefore to be done, following the intuition given in [10, 11, 8].
Acknowledgments The authors are indebted to the anonymous referees for useful suggestions, which helped to improve the preliminary version. 10
References [1] D. E. Appelt and M. E. Pollack. Weighted abduction for plan ascription. Technical Report 491, SRI International, 1990. [2] C. Boutilier and V. Becher. Abduction as belief revision inference rules. Technical Report 93-23, University of British Columbia, 1993. [3] M. Cialdea Mayer and F. Pirri. First order abduction via tableau and sequent calculi. Bulletin of the IGPL, 1:99{117, 1993. [4] R. Davis. Diagnostic reasoning based on structure and behaviour. Arti cial Intelligence, 24:347{410, 1984. [5] J. de Kleer and C. B. Williams. Diagnosing multiple faults. Arti cial Intelligence, 32:97{130, 1987. [6] M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Reidel Publishing Company, Dordrecht, 1983. [7] J. R. Hobbs, M. Stickel, D. Appelt, and P. Martin. Interpretation as abduction. Arti cial Intelligence, 63:301{353, 1993. [8] P. Jackson and H. Reichgelt. A general proof method for rst order modal logic. In Proceedings of the Eleventh International Joint Conference on Arti cial Intelligence (IJCAI-89), pages 942{944, 1989. [9] P. Marquis. Extending abduction from propositional to rst order logic. In Proceedings of the First International Workshop on Fundamentals of Arti cial Intelligence Research (FAIR '91), pages 141{155, 1991. SpringerVerlag, LNAI 535. [10] H.J. Ohlbach. A resolution calculus for modal logic. In Proceedings of the Ninth Conference on Automated Deduction (CADE-88), pages 500{516. Springer Verlag, 1988. [11] H.J. Ohlbach. Translations methods for non-classical logics: an overview. Bulletin of the IGPL, 1:69{89, 1993. [12] D. Poole. Representing knowledge for logic-based diagnosis. In Proceedings of the International Conference on Fifth Generation Computer Systems, pages 1282{1289, 1988. [13] R. Reiter. A theory of diagnosis from rst principles. Arti cial Intelligence, 32:57{96, 1987. [14] R. Reiter and A. K. Mackworth. A logical framework for depiction and image interpretation. Arti cial Intelligence, 41:125{155, 1989. 11
[15] M. E. Stickel. Rationale and methods for abductive reasoning in naturallanguage interpretations. In Proceedings Natural language and logic: International Computer Science Conference, pages 233{252, 1989.
Appendix This appendix contains the proofs of Theorem 2 and Theorem 3. For the completeness proof, the following obvious result is used.
Lemma 1 If L is K , D or T , and if ; : j=L :' then 2; :2 j=L :2'. If 2; : j=S4 :2' then 2; :2 j=S4 :2'. Proof of Theorem 2 Let us assume that is a set of modal formulae and an L-consistent C-formula such that [ f g is L-unsatis able. Then there exists a closed tableau T for [ f g. We may assume T is fundamental, because every tableau can be further expanded to a fundamental one. Every node in T has the form 0 [ , where 0 contains descendants of formulae in and
contains descendants of . As is satis able and it is a C-formula, every is satis able. The proof shows how a tableau T * for is generated, such that E L (T ) contains a formula that is a logical consequence of . Essentially, T * is obtained from T by elimination of all the 's from the nodes and (some of) the nodes generated by the application of expansion rules to formulae in some 's. T * is clearly fundamental. T * is de ned inductively, with respect to a tableau T for [ , where is a satis able set of C-formulae; the same induction on T that builds T * shows that there exists ' 2 E L (T ) such that j= '. The base of the induction is straightforward and the only non trivial case of the induction step is when T begins with an application of a -rule to a formula in . The corresponding cases for the four systems are treated as follows. (K ) If the logic is K , D or T , then T has the form 0
0
(=) 2; ; ;2 ; ;: ; :2 [T 0 ] where = is an application of the -rule to :2 . The case of the logic K is treated as follows. By the induction hypothesis, there exists '0 2 E K (T 0 ) such that ; : j= '0 . T * is then de ned as the tree: 12
2; 0 (weakening) 2 Note that 2 is not empty, otherwise [ f: g would be unsatis able. By de nition, E K (T ) = f3' j ' 2 E (T 0 ) for some fundamental K tableau T 0 for g. Clearly, T 0 is a fundamental K -tableau for , so 3'0 2 E K (T ). 2 ; :2 j= 3'0 follows by Lemma 1. (D) In the case of system D, the tableau T * is de ned as: 2; 0 ( -rule) D [T 0 ] By the induction hypothesis, there exists '0 2 E D (T 0 ) such that ; : j= '0 . By de nition, ' = 3'0 2 E D (T ) and 2 ; :2 j= ' by Lemma 1. (T ) The case of sistem T is similar, but for the fact that T is obtained by applying the weakening rule to 2; 0 , yielding 2, then a series of T rules are applied to formulae in 2. The tableau is subsequently expanded like T 0 above. We note that the addition of boxed formulae to all the nodes of a tableau does not preclude the generation of any old explanation. (S 4) If L is S 4, then T has the form 0
0
(=) 2;; 2; ;2 ; ;2 ;; ::2 [T 0 ] Let T 0 be the fundamental tableau for ; 2, given by the inductive construction, such that there is a formula '0 2 E S4 (T 0 ), with ; 2 ; : j= '0 . If = f 0 ; : : : ; n g then T has the following form:
2; 0 2 0 ; 2 .. .
(weakening) (=0 )
0 ; : : : ; n?1 ; 2 (=n ) ; 2 [T0 ]
where =0 , : : :, =n are applications of -rules. By de nition, '0 is among the explanations built for ; 2 and 2; 0 ; : : : ; n?1 and : : : 13
and 2; 0 . In other words, '0 percolates upwards in correspondence of the expansion rules =n , : : :, =1 . The abductive construction at the level =0 gives 3'0 among the explanations of 2, so that 3'0 2 E S4 (T ). Since ; 2 ; : j= '0 then also 2 ; : j= '0 hence by Lemma 1, 2 ; :2 j= 3'0 .
Proof of Theorem 3 The abduction problem h; 'i has clearly at least one explanation, therefore a proof of (ii) proves (i) too. Let us assume that (ii) is false, i.e. that there exists an in nite set ? = f' = '0 ; '1 ; '2 ; : : : ; 'i ; : : :g in the language of h; 'i, such that for all i, j= 'i ! 'i+1 but 6j= 'i+1 ! 'i . We show rst that there exists an in nite subset ?0 of ?, ?0 = f 0 ; 1 ; 2 ; : : :g and a nite M=fW ,R,Vg with a world w 2 W such that for all i, (M; w) 6j= i+1 ! i . As the nite model property holds, for all i, 6j= 'i+1 ! 'i i there exists a nite interpretation Mi and a world wi such that (Mi ; wi ) 6j= 'i+1 ! 'i . The number of nite interpretations of a xed nite language is nite. Therefore there exists at least an interpretation M and a world w such that for an in nite number of i, (M; w) 6j= 'i+1 ! 'i . The subset ?0 = f 0 = 'k ; 1 ; 2 ; : : :g is de ned as follows: 0 is the rst 'k in ? such that (M; w) 6j= 'k+1 ! 'k . For all i k, 'i+1 2 ?0 i (M; w) 6j= 'i+1 ! 'i . Next step is to show that for all i, (M; w) 6j= i+1 ! i , i.e. that for all i and p > 0, if 'i 2 ?0 and 'i+p 2 ?0 , then (M; w) 6j= 'i+p ! 'i . If p = 1, it follows immediately from the construction of ?0 . Otherwise, as the elements 'i+1 ; : : : 'i+p?1 are not in ?0 , the following relations hold: (M; w) j= 'i+1 ! 'i (M; w) j= 'i+2 ! 'i+1
:::
(M; w) j= 'i+p?1 ! 'i+p?2 Moreover, as j= 'i ! 'i+1 , (M; w) j= 'i 'i+p?1 . Since 'i+p 2 ?0 , (M; w) 6j= 'i+p ! 'i+p?1 . Consequently, (M; w) 6j= 'i+p ! 'i , i.e. (M; w) 6j= i+1 ! i . Finally, let us assume that for some i, (M; w) j= i . Then, (M; w) j= i+1 ! i , but this is false by the construction of ?0 . Therefore, for all i, (M; w) 6j= i . But then (M; w) j= i ! i?1 , again contradicting the construction of ?0 .
14
Fiora Pirri1
1 Dipartimento di Informatica e Sistemistica, Universita di Roma \La Sapienza" via Salaria 113, 00198 Roma, Italia 2 Dipartimento D.S. Informatica, Terza Universita degli Studi di Roma via Segre 2, 00146 Roma, Italia e-mail: fcialdea,[email protected]
Abstract
In this work, the problem of performing abduction in modal logics is addressed, along the lines of [3], where a proof theoretical abduction method for full rst order classical logic is de ned, based on tableaux and Gentzen-type systems. This work applies the same methodology to face modal abduction. The non-classical context enforces the value of analytical proof systems as tools to face the meta-logical and proof-theoretical questions involved in abductive reasoning. The similarities and di erences between quanti ers and modal operators are investigated and proof theoretical abduction methods for the modal systems K , D, T and S 4 are de ned, that are sound and complete. The construction of the abductive explanations is in strict relation with the expansion rules for the modal logics, in a modular manner that makes local modi cations possible. The method given in this paper is general, in the sense that it can be adapted to any propositional modal logic for which analytic tableaux are provided. Moreover, the way towards an extension to rst order modal logic is straightforward. This
work has been partially supported by Basic Research Action, Medlar II
1
1 Abductive reasoning and modal logics Many forms of commonsense reasoning are not deductive in nature. Reasonable inferences may be performed even if leading to conclusions that are not certain but only plausible. Abduction is one of such inference schemata that re ect unsound forms of inference. 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. A precise de nition of abductive explanations is obviously dependent on the logic underlying the theory : some fact could explain ' in classical logic, but not in intuitionistic or linear logic. So, let us assume that a logic L is given, with a corresponding relation of logical consequence j=L . Then an abduction problem in L consists of a background theory and a formula ' such that 6j=L '. A solution of the problem given by the pair h; 'i is to be chosen among the formulae such that [ f g j=L '. In general, it is not even required that j=L satis es the tarskian properties for a consequence relation. For example, it makes sense to think of (and it would be worth exploring) non-monotonic explanations, i.e. cases where explains ' but ^ does not [2], so that the notion of explanation is related to a non-monotonic consequence relation. In this perspective, abduction in the logic programming context, with its tricks (like constraints), can be seen as the form of abduction that is carried out in the logic induced by the semantics of the arrow and negation-as-failure operators. Recently, much attention has been paid to abductive reasoning in di erent elds of Arti cial Intelligence, such as diagnosis [4, 5, 13, 12], interpretation of natural language sentences [7, 15], plan recognition [1], scene interpretation [14]. This work is a contribution to the characterization of abduction as a form of inference whose context is only the logic it depends upon. Abstraction from any particular problem helps to focus on general theoretical questions, while results can be obtained that bene t the applicative elds. In this work, the problem of performing abduction in modal logics is addressed. The meaningfulness of such a problem can be recognized by considering two simple examples: the rst is from Arti cial Intelligence, the second from program veri cation. Let us assume that an agent is reasoning about her own knowledge, represented by a modal theory , and she observes that ' is consistent with her knowledge, i.e. 3' is true. However, 3' is not derivable in . Abduction would produce what is needed to sanction 3'. Of course, the importance of epistemic abduction is clearer when considering multiple agent contexts: for example, both natural language understanding and user modeling require the ability to abduce the goals of the speaker, and student modeling includes the abduction of the student's beliefs from the observation of her answers to questions and problems. As a second example, let us assume that a program is described by a 2
theory in a temporal or dynamic logic L and that the termination property for is represented by the formula '. Then, determining the class of input data for which terminates can be seen as abducing a precondition such that [ f g j=L '. In [3] analytical proof systems are proposed as suitable tools to face the metalogical and proof-theoretical questions involved in abductive reasoning and a proof theoretical abduction method for full rst order classical logic is de ned. The underlying idea is to solve the abduction problem h; 'i by building a formula that closes the open branches of a tableau for [ f:'g. The method does not require any preliminary reduction of formulae into normal forms, but dynamic skolemization is used. In the rst order case, quanti ers are to be reintroduced following a sort of reverse skolemization. This work applies the same methodology to face modal abduction. There are obvious similarities between quanti ers and modal operators (see, among others, [11] for a functional translation method of modal formulae into rst order formulae). As far as the tableau based abduction method is concerned, however, a main di erence lies in that the structure of a quanti ed formula is essentially rebuildable from its Skolem forms, while the trace of modal operators, once they are dropped, is lost. This fact induces a recursive reformulation of the construction of the abduced formulae, thus shifting the view from a meta-logical level to a logical one. The resulting construction is in strict relation with the expansion rules for the modal logic, in a modular manner that makes local modi cations possible, so as to t the method to di erent modal systems. In fact, the abductive method given in this paper is general, in the sense that it can be adapted to any propositional modal logic for which analytic tableaux are provided. Moreover, the way towards an extension to rst order modal logic is straightforward.
1.1 Basic notions about modal logics
The logical language of modal systems consists of all the classical logical symbols and the modal operator 2 (necessity). Modal formulae are de ned consequently. Formulae of the form 2 are called boxed formulae. The modal operator 3 is de ned as follows: 3 :2: . In the sequel we consider modal systems characterizable by a subset of the following schemata:
(K) 2( ! ) ! (2 ! 2 ) (T) 2 ! (D) 2 ! 3 (4) 2 ! 22 and the necessitation rule: 2 . In particular, the systems K (with axiom K), D (with axioms K and D), T (with axioms K and T) and S 4 (with axioms K, T and 4) are considered. 3
The semantics of modal logics is de ned as usual in terms of L-modal structures M = hW ; RL ; Vi, where W is a non empty set (of possible worlds), RL is a binary relation on W (accessibility relation), whose properties depend on the logic L, and V is a truth assignment, that maps W P into ftrue; falseg, where P is the set of propositional symbols of the language. The notion of truth of a formula in a world w 2 W , (M; w) j= , is de ned as usual. The notion of logical consequence in a modal logic L is not always given the same de nition. Two versions are commonly used: (F ) j=FL i for every L-modal structure M: (for all w 2 W and for all 2 , (M; w) j= ) implies (for all w 2 W , (M; w) j= ). (K) j=KL i for every L-modal structure M: for all w 2 W (for all 2 , (M; w) j= implies (M; w) j= ). Clearly, if is not empty, j=KL i j=FL (
^ ) !
2
If is empty, the two notions collapse, so that in particular the notion of validity they induce is the same (this is the reason why any of the two versions can be often used without discrepancies). The version (F ) corresponds to the notion of derivability in axiomatic systems (with necessitation), while (K) corresponds to derivability of the corresponding sequent in sequent calculi. Consequently, the main di erence between the two versions is that the following properties: (i) if j= ' then [ f:'g is inconsistent (ii) if [ f g j= ' then j= ! ' hold for (K) but not for (F ) (their converses are true in both cases). Thus, (K) is closer to the classical notion of logical consequence. In recent literature (F ) is often preferred because of its greater generality. However, in this work, we use (K) just because it de nes a stronger relation and, consequently, it corresponds to a more general notion of abduction: if [ f g j=KL ' then [ f g j=FL ', but the converse is not true. In the context of abduction, the use of (K) is more reasonable (there does not seem to be much sense in assuming that is a good explanation for 2 ) and it allows the duality between deduction and abduction to be preserved. Of course, the same concepts could be de ned by use of (F ) and implication, but the notation would be far more cumbersome. In the following we shall simply use the symbol j=L instead of j=KL . 4
1.2 Basic notions about modal abduction
Abductive explanations are in general required to respect some fundamental conditions, in order to be accepted as \interesting". If well settled consequence relations j=L are considered, such as the classical, modal or intuitionistic ones, the following restrictions are to be reasonably imposed on explanations for an abduction problem h; 'i: (i) is consistent with , i.e. 6j=L : . (ii) is a minimal explanation for the abduction problem h; 'i, i.e. for any formula , if [ f g j=L ' and j=L , then j=L . Moreover, the syntactical form of explanations is usually restricted; for example, in rst order classical logic, an explanation is a prenex formula whose matrix is a conjunction of literals. A logical reason for this has been pointed out by [9]: if = f1 ; : : : ; n g is nite, then any minimal explanation for h; 'i is equivalent to 1 ^ : : : ^ n ! '. The following de nition is a straightforward adaptation to the modal language of the classical syntactical restriction on explanations. De nition 1 (modal C-formulae) A formula is a modal C-formula if it is built up from literals using only the modal operators 2 and 3 and conjunction. Consequently: De nition 2 (L-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=L '. When an explanation is required to be either minimal or consistent with the theory, it will be explicitly stated. L-equivalent explanations are considered as identical, so that L-explanations are in fact equivalence classes of formulae (the terms L-equivalence, L-derivability, etc. are used in the obvious sense).
2 Modal abduction by use of analytic tableaux Semantic tableaux are used as refutation systems. They are built by means of a set of rules that preserve satis ability of sets of formulae. In this work we make use of the unsigned version of modal tableaux [6]. Modal expansion rules are stated as follows, where set union is brie y denoted by the comma, in the obvious sense; for example, ; ; 0 is a short-cut for f g [ [ 0 . Moreover, 2 is a set of boxed formulae and 0 is a set of non boxed formulae. :2 ; 2; 0 2; 0 -rule) K -rule) :2 ; 4 : ; : ; ; 2
D -rule)
2; 0
T -rule) 22 ; ; ;
5
The systems K , D, T and S 4 are characterized by the addition of modal rules to the classical ones, as the following table shows:
K D T S4
K -rule K -rule K -rule 4 -rule In the sequel we shall generically speak of -rule and -rule; the version {
D -rule T -rule T -rule
actually referred to depends on the context. If is a set of formulae, an L-tableau for is a tree whose root is labeled 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 of the system L. A branch B of a tableau is closed if both ' and :' occur in some node of 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. A L-tableau refutation of is a closed L-tableau for . The L-tableau systems are sound and complete, i.e. is L-unsatis able i has an L-tableau refutation [6]. The standard formulation of analytic tableau systems does not make explicit reference to structural rules. However, we assume that the following form of weakening is among the rules of the systems: 2; 0 2 where 0 contains no boxed formulae. This rule is certainly redundant for the aims of theorem proving, but it is needed to guarantee the completeness of the abduction procedure for the systems K , T and S 4 (the reason can be found in the proof given in the Appendix). In the classical case, the construction of an explanation can be done only on the basis of the set of the open leaves of the tableau, without need to store and work on the structure of the whole tableau: a literal is chosen for each open leaf, that closes the corresponding branch; the set of literals is conjoined and quanti ers are reintroduced. In fact, the elimination of quanti ers by means of skolemization preserves the information about their mutual dependencies that is needed to reintroduce them. In contrast to these, modal operators cannot be correctly reintroduced without considering the structure of the tableau, where they may disappear without leaving any trace. This fact is intuitively explained by the following simple example. Let us consider the more general problem of nding a formula that is L-inconsistent with a set . Let L = D, = f2' _ 2 g and 0 = f2(' _ )g; the following trees are D-tableaux for and 0 , respectively. 2(' _ ) 2' _ 2
2' '
'_ '
2
6
Both tableaux have the same open leaves. Nevertheless, while the rst one closes with the addition of = 3:' ^ 3: , the closure of the latter requires the stronger 0 = 3(:' ^ : ).
2.1 The set of abductive explanations
Let h; 'i be an abduction problem in L. Then, as 6j= ', there is no L-tableau refutation for [ f:'g. A solution for the abduction problem h; 'i can be found among the formulae that force the closure of a tableau for [ f:'g. Of course, it would be meaningless to close an unexpanded tableaux by adding the atom false or the formula '. So, if only non trivial explanations are looked for, the tableaux should in principle be developed as far as possible. However, also tableaux that can still be expanded are to be considered, because (i) -rules may be reapplicable, and (ii) -rules may cancel some formulae. The following de nitions cope with the determination of the point where the construction of a tableau can be interrupted in order to perform abduction. Here and in the following, tableaux for [ f:'g will be called tableaux for the abduction problem h; 'i. De nition 3 (Compulsory expansion of a branch) The expansion of a branch B with leaf is compulsory if one or more of the following conditions hold: (i) contains no literals and can be expanded. (ii) contains some formula that can be expanded by means of a classical logical rule; (iii) contains some 2 that can be expanded but has not yet been expanded in B; In other words, the expansion of B is compulsory whenever it can be performed without the loss of information that is useful for abduction and without cycling. Note that the application of a -rule is not always compulsory. De nition 4 (Fundamental tableaux) Let T be a tableau. A branch B of T is fundamental if its expansion is not compulsory. A node is fundamental if it is the leaf of a fundamental branch. T is fundamental if all its branches are fundamental. Note that a fundamental node always contains some literals, unless the logic considered is K . In this case, a node that contains only boxed formulae is fundamental. We next turn to the characterization of the set of abductive explanations in L built up from a fundamental L-tableau T for an abductive problem. It will be denoted by E L (T ). The following de nition is relative to the logics T and S 4. Its variants for K and D are stated later on. 7
De nition 5 (Construction of explanations in T and S 4) If L is either T or S 4 and T is a fundamental L-tableau, the set E L (T ) is de ned recursively
as follows. 1. If T consists of the single fundamental node , then: 1a. If contains either false or a pair of complementary literals, then E L (T ) = ftrueg. 1b. Otherwise E L (T ) = f: j 2 and is a literalg. Note that, as is fundamental, E L (T ) is not empty. 2. Otherwise, di erent cases are to be considered, according to the form of the rst expansion rule applied in T . 2a. T has the form (=) , where = is an application of a classical non-
T0
branching rule (weakening included). In this case E L (T ) = E L (T 0 ). 2b. T has the form (=) , where = is an application of a -rule. T 0 T1 In this case E L (T ) = f'0 ^ '1 j '0 2 E L (T0 ); '1 2 E L (T1 )g. 2 ; , where = is an application of the -rule 2c. T has the form (=) T0 to 2 . We distinguish two cases: (i) If either is empty or it contains only boxed formulae, then E L (T ) = f3' j ' 2 E L (T0 )g; (ii) otherwise E L (T ) = E L (T 0 ). 0 2d. T has the form (=) :2 ; 2; , where = is an application of the T0 -rule to :2 . Then E L (T ) = f2' j ' 2 E L (T0 )g. As an example, let = f2(p ^ 2q ! r)g and ' = 23r. The following (leftmost) tree, where 2 = f2(p ^ 2q ! r); 2:rg, is a fundamental S 4-tableau for [ f:'g. The tree at its right side is an intuitive view of the construction of a particular explanation, 23(p ^ 2q) (the tree is to be read bottom up). 23(p ^ 2q) 2(p ^ 2q ! r); 32:r 2; p ^ 2q ! r 3(p ^ 2q) 2; p ^ 2q ! r; :r p ^ 2q 2; :(p ^ 2q); :r 2; r; :r p ^ 2q true closed 2; :p; :r 2; :2q; :r p 2q open 2; ; :q q
open
Suitable modi cation to the de nition of the set E L (T ) are easy to state for other logics. As far as the system D is considered, clauses 1, 2a, 2b and 2d remain the same, while clause 2c is replaced by the following: 8
0 2c0 . T has the form (=) 2T ; , where = is an application of the -rule. Then 0 E D (T ) = f3' j ' 2 E (T0 )g.
Considering system K , it must be noted that K -tableaux may have fundamental nodes containing only boxed formulae and literals, so, besides deleting clause 2c, item 1b has to be replaced by: 1b0 . Otherwise, has the form 20 ; , where is a (possibly empty) set of literals; then E K (T ) = f: j 2 g 0 [f3' j ' 2 E (T ) for some fundamental K -tableau T 0 for 0 g. The methods are sound and complete, in the sense stated by the following theorems, that hold for L = K; D; T; S 4.
Theorem 1 (Soundness) Let T be a fundamental L-tableau for and 2 E L (T ). Then there exists a closed L-tableau T 0 for [ f g. Theorem 2 (Completeness) Let be any set of modal formulae and an L-consistent C-formula such that [ f g is L-unsatis able. Then there exists a fundamental L-tableau T * for and a C-formula ' 2 E L (T ) such that j=L '. The proof of Theorem 1 is a straightforward induction on T . The signi cant steps in proving completeness can be found in the Appendix.
3 Existence of minimal explanations
If h; 'i is an abductive problem, let (; ') denote the set of all the fundamental L-tableaux for [ f:'g. By the completeness theorem, the set of minimal explanations for h; 'i is included in the set
E L (; ') =
[
T 2(;')
E L (T )
The question whether min(E L (; ')) may be empty is addressed here. In the case of rst order logic, there are abduction problems that admit no minimal explanation [9], because there exist in nite descending chains of formulae. This is not the case for modal logic. For the non transitive logics K , D and T , the existence of minimal explanations follows from the fact that the number of fundamental tableaux for a given problem h; 'i is nite, therefore E L (; ') is nite. In fact, at any step in the expansion of a tableau there is only a nite number of choices and the construction of any tableau terminates, because each application of a rule of K 9
and D decreases the number of logical operators; in T the number of logical operators may not be reduced by a -rule, but it is as soon as a -rule is applied (if a node contains no -formulae, reapplications of -rules can be avoided). The case of the logic S 4 is not so simple. In fact, there exist in nite tableaux, i.e. there may be an in nite number of fundamental tableaux for a given abductive problem. The fact that in nite tableaux cycle does not help, because every cycle could generate weaker explanations and in S 4 there is an in nite number of distinct modal functions, even of a single propositional variable. So, in principle, a given problem may lack minimal explanations. However, the following theorem solves the problem of the existence of minimal explanations for S 4 too.
Theorem 3 Let h; 'i be an abductive problem in a propositional modal logic L that enjoys the nite model property. Then: (i) there exists a minimal explanation of h; 'i, and (ii) for any explanation of h; 'i there exists a minimal explanation min such that j= min . A proof of this theorem is given in the Appendix.
4 Concluding remarks Abduction in a logic may be approached in several ways, at least one for every deductive method de ned for the logic. However abduction in modal logic has never been addressed so far. Obviously, the results of this work are only a rst step towards a good characterization of modal abduction and leave a number of problems to be solved. An improvement of the method can be obtained by use of pre xed tableaux, that in modal deduction reduces the number of tableaux to be considered. Apart from a gain in eciency, pre xes could help in storing important information on the \history" of the modal operators in the same way as the use of Skolem functions code the position and quality of quanti ers that are eliminated. However, commonly used pre xes [6] do not allow the abductive construction to be based only on the leaves of the tree, like in classical logic. In fact, those pre xes keep track of the application of modal rules whatsoever, without storing any information about which rules have been applied. A careful rede nition of pre xes to be used for abduction is therefore to be done, following the intuition given in [10, 11, 8].
Acknowledgments The authors are indebted to the anonymous referees for useful suggestions, which helped to improve the preliminary version. 10
References [1] D. E. Appelt and M. E. Pollack. Weighted abduction for plan ascription. Technical Report 491, SRI International, 1990. [2] C. Boutilier and V. Becher. Abduction as belief revision inference rules. Technical Report 93-23, University of British Columbia, 1993. [3] M. Cialdea Mayer and F. Pirri. First order abduction via tableau and sequent calculi. Bulletin of the IGPL, 1:99{117, 1993. [4] R. Davis. Diagnostic reasoning based on structure and behaviour. Arti cial Intelligence, 24:347{410, 1984. [5] J. de Kleer and C. B. Williams. Diagnosing multiple faults. Arti cial Intelligence, 32:97{130, 1987. [6] M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Reidel Publishing Company, Dordrecht, 1983. [7] J. R. Hobbs, M. Stickel, D. Appelt, and P. Martin. Interpretation as abduction. Arti cial Intelligence, 63:301{353, 1993. [8] P. Jackson and H. Reichgelt. A general proof method for rst order modal logic. In Proceedings of the Eleventh International Joint Conference on Arti cial Intelligence (IJCAI-89), pages 942{944, 1989. [9] P. Marquis. Extending abduction from propositional to rst order logic. In Proceedings of the First International Workshop on Fundamentals of Arti cial Intelligence Research (FAIR '91), pages 141{155, 1991. SpringerVerlag, LNAI 535. [10] H.J. Ohlbach. A resolution calculus for modal logic. In Proceedings of the Ninth Conference on Automated Deduction (CADE-88), pages 500{516. Springer Verlag, 1988. [11] H.J. Ohlbach. Translations methods for non-classical logics: an overview. Bulletin of the IGPL, 1:69{89, 1993. [12] D. Poole. Representing knowledge for logic-based diagnosis. In Proceedings of the International Conference on Fifth Generation Computer Systems, pages 1282{1289, 1988. [13] R. Reiter. A theory of diagnosis from rst principles. Arti cial Intelligence, 32:57{96, 1987. [14] R. Reiter and A. K. Mackworth. A logical framework for depiction and image interpretation. Arti cial Intelligence, 41:125{155, 1989. 11
[15] M. E. Stickel. Rationale and methods for abductive reasoning in naturallanguage interpretations. In Proceedings Natural language and logic: International Computer Science Conference, pages 233{252, 1989.
Appendix This appendix contains the proofs of Theorem 2 and Theorem 3. For the completeness proof, the following obvious result is used.
Lemma 1 If L is K , D or T , and if ; : j=L :' then 2; :2 j=L :2'. If 2; : j=S4 :2' then 2; :2 j=S4 :2'. Proof of Theorem 2 Let us assume that is a set of modal formulae and an L-consistent C-formula such that [ f g is L-unsatis able. Then there exists a closed tableau T for [ f g. We may assume T is fundamental, because every tableau can be further expanded to a fundamental one. Every node in T has the form 0 [ , where 0 contains descendants of formulae in and
contains descendants of . As is satis able and it is a C-formula, every is satis able. The proof shows how a tableau T * for is generated, such that E L (T ) contains a formula that is a logical consequence of . Essentially, T * is obtained from T by elimination of all the 's from the nodes and (some of) the nodes generated by the application of expansion rules to formulae in some 's. T * is clearly fundamental. T * is de ned inductively, with respect to a tableau T for [ , where is a satis able set of C-formulae; the same induction on T that builds T * shows that there exists ' 2 E L (T ) such that j= '. The base of the induction is straightforward and the only non trivial case of the induction step is when T begins with an application of a -rule to a formula in . The corresponding cases for the four systems are treated as follows. (K ) If the logic is K , D or T , then T has the form 0
0
(=) 2; ; ;2 ; ;: ; :2 [T 0 ] where = is an application of the -rule to :2 . The case of the logic K is treated as follows. By the induction hypothesis, there exists '0 2 E K (T 0 ) such that ; : j= '0 . T * is then de ned as the tree: 12
2; 0 (weakening) 2 Note that 2 is not empty, otherwise [ f: g would be unsatis able. By de nition, E K (T ) = f3' j ' 2 E (T 0 ) for some fundamental K tableau T 0 for g. Clearly, T 0 is a fundamental K -tableau for , so 3'0 2 E K (T ). 2 ; :2 j= 3'0 follows by Lemma 1. (D) In the case of system D, the tableau T * is de ned as: 2; 0 ( -rule) D [T 0 ] By the induction hypothesis, there exists '0 2 E D (T 0 ) such that ; : j= '0 . By de nition, ' = 3'0 2 E D (T ) and 2 ; :2 j= ' by Lemma 1. (T ) The case of sistem T is similar, but for the fact that T is obtained by applying the weakening rule to 2; 0 , yielding 2, then a series of T rules are applied to formulae in 2. The tableau is subsequently expanded like T 0 above. We note that the addition of boxed formulae to all the nodes of a tableau does not preclude the generation of any old explanation. (S 4) If L is S 4, then T has the form 0
0
(=) 2;; 2; ;2 ; ;2 ;; ::2 [T 0 ] Let T 0 be the fundamental tableau for ; 2, given by the inductive construction, such that there is a formula '0 2 E S4 (T 0 ), with ; 2 ; : j= '0 . If = f 0 ; : : : ; n g then T has the following form:
2; 0 2 0 ; 2 .. .
(weakening) (=0 )
0 ; : : : ; n?1 ; 2 (=n ) ; 2 [T0 ]
where =0 , : : :, =n are applications of -rules. By de nition, '0 is among the explanations built for ; 2 and 2; 0 ; : : : ; n?1 and : : : 13
and 2; 0 . In other words, '0 percolates upwards in correspondence of the expansion rules =n , : : :, =1 . The abductive construction at the level =0 gives 3'0 among the explanations of 2, so that 3'0 2 E S4 (T ). Since ; 2 ; : j= '0 then also 2 ; : j= '0 hence by Lemma 1, 2 ; :2 j= 3'0 .
Proof of Theorem 3 The abduction problem h; 'i has clearly at least one explanation, therefore a proof of (ii) proves (i) too. Let us assume that (ii) is false, i.e. that there exists an in nite set ? = f' = '0 ; '1 ; '2 ; : : : ; 'i ; : : :g in the language of h; 'i, such that for all i, j= 'i ! 'i+1 but 6j= 'i+1 ! 'i . We show rst that there exists an in nite subset ?0 of ?, ?0 = f 0 ; 1 ; 2 ; : : :g and a nite M=fW ,R,Vg with a world w 2 W such that for all i, (M; w) 6j= i+1 ! i . As the nite model property holds, for all i, 6j= 'i+1 ! 'i i there exists a nite interpretation Mi and a world wi such that (Mi ; wi ) 6j= 'i+1 ! 'i . The number of nite interpretations of a xed nite language is nite. Therefore there exists at least an interpretation M and a world w such that for an in nite number of i, (M; w) 6j= 'i+1 ! 'i . The subset ?0 = f 0 = 'k ; 1 ; 2 ; : : :g is de ned as follows: 0 is the rst 'k in ? such that (M; w) 6j= 'k+1 ! 'k . For all i k, 'i+1 2 ?0 i (M; w) 6j= 'i+1 ! 'i . Next step is to show that for all i, (M; w) 6j= i+1 ! i , i.e. that for all i and p > 0, if 'i 2 ?0 and 'i+p 2 ?0 , then (M; w) 6j= 'i+p ! 'i . If p = 1, it follows immediately from the construction of ?0 . Otherwise, as the elements 'i+1 ; : : : 'i+p?1 are not in ?0 , the following relations hold: (M; w) j= 'i+1 ! 'i (M; w) j= 'i+2 ! 'i+1
:::
(M; w) j= 'i+p?1 ! 'i+p?2 Moreover, as j= 'i ! 'i+1 , (M; w) j= 'i 'i+p?1 . Since 'i+p 2 ?0 , (M; w) 6j= 'i+p ! 'i+p?1 . Consequently, (M; w) 6j= 'i+p ! 'i , i.e. (M; w) 6j= i+1 ! i . Finally, let us assume that for some i, (M; w) j= i . Then, (M; w) j= i+1 ! i , but this is false by the construction of ?0 . Therefore, for all i, (M; w) 6j= i . But then (M; w) j= i ! i?1 , again contradicting the construction of ?0 .
14
