Natural properties of abductive hypotheses inthree-valued logic�Marta Cialdea Mayer(*), Fiora Pirri(*), Clara Pizzuti(**)(*) Dipartimento di Informatica e SistemisticaUniversit�a di Roma \La Sapienza"via Salaria 113, 00198 Roma, Italiae-mail:fcialdea,pirrig@assi.ing.uniroma1.it(**) CRAILocalit�a S.Stefano, 87036 Rende (CS), Italiae-mail: clara@icsvmhpo.bitnet clara@crai.itFebruary 20, 1997SummaryThis paper shows some interesting properties of Kleene's three-valued logic in relation to abductive reasoning. A semanticalcharacterization of abductive explanations is proposed, based onthe notion of minimal three-valued model. This establishes a re-lation between the minimization problem in abductive reasoningand three-valued semantics, in the same sense as non-monotonicreasoning deals with minimization in two-valued semantics.1 IntroductionAbductive reasoning is that form of unsound reasoning that infers a conclu-sion � from premises � � and . Its importance in Arti�cial Intelligencelies both on the fact that it re ects some forms of commonsense reasoning[15, 16, 19], where causes for events are to be hypothesized, and on its linkwith diagnostic reasoning [14]. In fact, in the AI literature, abductive rea-soning is a way to solve problems where an observed event � is not explained�This work has been partially supported by Basic Research Action, Medlar II1

by the presently adopted theory � and an explanation for � has to be lookedfor. So, assuming that � 6j= �, if is an explanation for � in the context of�, then � is a logical consequence of � [ f g.The �rst explicit studies on abductive reasoning are due to Pierce, al-though some authors attribute the origins of the notion to Aristotele. Thedevelopment of modern logic highlighted the fundamental duality betweenabduction 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= : Thus, as pointed out since the earliest papers on modern abduction [13, 2],any deductive system that can be used to generate consequences can alsobe used to perform abduction.A peculiar feature of abductive reasoning is that, usually, explanationsare required to respect some fundamental conditions, in order to be acceptedas \interesting". The three main restrictions imposed on explanations foran observation � in the context of a theory � are the following ones:(i) is consistent with �, i.e. � 6j= : .(ii) is a minimal explanation, i.e. for all 0 that are explanations of � inthe context of � (�; 0 j= �), if j= 0 then j= 0 � .(iii) has some restricted syntactical form; for example, it is a prenexformula whose matrix is a conjunction of literals.Condition (i) rules out trivial explanations, condition (ii) redundant onesand condition (iii) is motivated by the fact that if is an explanation for� in the context of �, that is consistent with � and minimal, then isequivalent to � � �. In this work any satisfying condition (iii) and suchthat �; j= � is considered as an explanation for � in the context of �. Whenan explanation is required to be either minimal or consistent with �, it willbe explicitly stated.Most proof theoretic methods for performing abduction are based onresolution (see for example [11, 12, 2, 8, 20]). An exhaustive approachwas introduced by [12] and recently described again in [11], that consistsin generating all the resolvents of the clause form of the theory and thenegation of the observation, selecting the minimal ones and negating them.In order to reduce the set of hypotheses that are generated, [2] constructshypotheses from the dead ends of linear resolution proofs. The minimalones are �ltered after the generation of all the solutions. This method isnot complete: it does not compute all the minimal explanations for theabductive problem, but only the explanations that the authors call basic,2

i.e. explanations that cannot be further explained in any non-trivial way inthe context of the same theory. For example, if � = fq � p; r � qg, thenthere are three minimal and consistent conjunctions of literals that explainthe observation p, namely p itself, q and r. However, p is considered trivial(and it is also non basic); q is not basic, because r is a non-trivial explanationfor it in the context of �. And in fact, :q is not a dead-end of a resolutionfrom the clause form of � and :p.This corresponds to the usual reading of clauses in a logic program asrules, where the head and tail of a rule have a di�erent epistemologicalmeaning: the tail is an explanation of the head. In this sense, if � is writtenas a logic program, containing the clauses p q and q r, then clearly qis an intermediate explanation for p, because it can be further explained byr. The basicality condition, however, seems to be meaningful only when itis stated in relation with theories represented by logic programs. In fact, ifthe theory � considered before is rewritten as f:q _ p;:r _ qg, there seemsto be no reasons for preferring the explanation r to q. Indeed, in some casesq may even be preferred to r: for example, because the set of consequencesof � [ fqg is smaller, i.e. q is a weaker assumption that, when added to �,derives p. The abduction method based on performing resolution steps as faras possible, however, does not calculate q as one of the minimal explanationsfor p in �.Finally, it is to be noted that there are abductive problems that do notadmit non-trivial and consistent explanations (a simple example is when � =fpg and � = q) and problems that have only non-basic minimal explanations.Consider for example the theory � = fq � p; q � rg and the observationp. There are exactly two minimal (and non-trivial) explanations for thisproblem that are consistent with �, i.e. q and r, but neither of them isbasic. In this case, resolution tout court from � and :p does not terminate.Other methods based on linear resolution are de�ned, for example, in[20] and [8]. In [20] an extension of SLD-resolution is used, coping also withnegation as failure. [8] de�nes a modi�cation of linear resolution that is com-plete w.r.t. the generation of interesting consequences. Most of the workson abduction in the context of logic programming, however, are stronglyin uenced by the linear resolution view, even in the de�nition of the basicconcepts involved.The above sketchy observations suggest that investigating the nature ofabduction outside the context of resolution based logic programming stilldeserves attention, in order to characterize abductive explanations withouttailoring them to any �xed method of computation.In this work, we propose a semantical characterization of abductive ex-planations based on Kleene's three-valued logic. Minimal three-valued mod-els are in fact shown to have interesting properties in relation to abduc-tive reasoning. This highlights a new connection between abductive andnonmonotonic reasoning: some formalisms for nonmonotonic reasoning are3

based on the selection of the best - according to some conditions - amongminimal models of the theory (generally two valued models are considered,but see also [5]). In this work we show that abductive explanations are tobe chosen among the minimal three-valued models of the abductive problem(the de�nition of general criteria for selecting the best explanation is still anopen question). This establishes a relation between the minimization prob-lem in abductive reasoning and three-valued semantics, in the same sense asnon-monotonic reasoning deals with minimization in two-valued semantics.The fundamental results are stated for propositional logic �rst, and thenextended to a �rst order restricted language.2 Kleene's three-valued logicWe consider a �rst order logical language L�, without function symbolswith arity > 0. Atomic formulae, literals, ground expressions and sentencesare de�ned as usual. A clause is a purely universal prenex sentence, whosematrix is a disjunction of literals: 8y1 � � � 8yr(L1_� � �_Ln). A clause will beidenti�ed with the set of its literals and set operations on clauses are de�nedas usual. Analogously, a conjunction of literals is said to be included inanother conjunction of literals 0 if all the literals in occur also in 0.Two disjunctions (conjunctions) are considered equal - and the equalitysign '=' will be used - if they disjoin (conjoin) sets of equal elements. Afundamental clause is a non tautological clause where no literal is repeated.A fundamental conjunction is a non contradictory conjunction of literalswhere no literal is repeated.A ground sentence � of L� is said to be in conjunctive normal form(CNF) (resp. disjunctive normal form (DNF)) if either � is a fundamentaldisjunction (resp. conjunction), or � is a conjunction (resp. disjunction) oftwo or more fundamental disjunctions (resp. conjunctions), none of which isa logical consequence of another one. Consequently, the transformation of� into CNF (DNF) requires the elimination of tautologies (contradictions)and elements that include other elements. We convene that the empty dis-junction is equivalent to the atom false and the empty conjunction to theatom true.In the following we assume that a theory � is a �nite set of purelyuniversal sentences of L�. � is clearly equivalent to a set of clauses in L�.A Herbrand interpretation for the language L� is an interpretation of L�whose domain is the Herbrand universe of L�. The ground instantiation ��of a theory � is the set of all ground instances of formulae in �, obtained bysubstituting elements of the Herbrand universe for variables in all possibleways. Clearly, �� is �nite and a Herbrand interpretation of � is a modelof � i� it is a model of ��. Hence, over its Herbrand universe, each purelyuniversal theory is equivalent to a propositional theory.4

A three-valued Herbrand interpretation M for the language L� is a setof ground literals, where no atom A occurs both positive and negative. Theinterpretation of an atom A inM , M(A), is a mapping onto the set of truthvalues fT; �; Fg, where T and F correspond to classical truth and falsity,and � is meant to capture the notion of absence of information. M (A) isde�ned by stipulating that:M(A) = 8><>: T if A 2M� if A 62M and :A 62MF otherwise, i.e. if :A 2MIn Kleene's three-valued logic, it is convened that F < � < T . The inter-pretation M is extended recursively to a truth valuation on the set of allground sentences of the language by establishing that [9]:M (:�) = 8><>: T if M(�) = F� if M(�) = �F otherwise, i.e. if M(�) = TM (� ^ ) = min(M(�);M ( ))M (� _ ) = max(M(�);M ( ))M (� � ) = max(M(:�);M ( ))A three-valued Herbrand interpretation M is a model of a purely uni-versal set of sentences � (M j=3 �) if M(�) = T for each � 2 ��. When� = f�g, we shall write M j=3 � instead of M j=3 f�g. A formula � is athree-valued consequence of � (� j=3 �) i� for any three-valued Herbrandinterpretation M , if M j=3 � then M j=3 �.If I and J are three-valued Herbrand interpretations, we say that Jinformationally extends I i� I � J . A three-valued model I of � is minimalif for no model J of �, J � I.Kleene's logic has the following properties:a. Kleene's three-valued logic has no logical theorems: for any �, 6j=3 �.In fact, no formula is true in the empty model.1b. Contradictions are never true, i.e. A^:A has no three-valued models.c. If I � J and � is a sentence, then I j=3 � implies J j=3 �. (Persistence[10]).As two-valued models are also three-valued models, � j=2 � implies� j=3 �, but the converse is not always true. In particular, CNF (�) j=3 �1In order to have both tautologies and a deduction theorem the logic is often extendedwith a weak negation (see [4, 5]). Another way for closing Kleene's three-valued logicunder classical consequence, is by means of the supervaluation truth de�nition, originallyintroduced by Bas van Fraassen [7] 5

may be false. For example, if � = p � (p^q), so that CNF (�) = :p_q, andM = fqg, then M j=3 CNF (�) but M (�) = �. However, � j=3 CNF (�),DNF (�) j=3 � and � j=3 DNF (�).If M is any �nite Herbrand interpretation, then Th(M) denotes theconjunction of the literals in M . Note that any Herbrand interpretation ofa purely universal formula in L� is �nite. The following properties specifythe relation between j=3 and the classical consequence relation. If M is any�nite Herbrand interpretation and �, are formulae, then:1. If M j=3 � then Th(M) j= �.2. If � is not valid and Th(M) j= �, then M j=3 CNF (�).3. If is not valid and � j= , then � j=3 CNF ( ).4. If � j=3 , then � j= The above properties show that Kleene's semantics is sensitive to normalform transformations and tautology elimination, that are the basic opera-tions for checking propositional derivability. It is this feature that makes itvaluable for the computation of abductive hypotheses.3 Abduction in the ground case: prime implicants,prime implicates and minimal explanationsLet � be a sentence in L�. A prime implicant of � is a conjunction of literals such that j= � and if 0 is any conjunction of literals such that 0 j= �and j= 0, then 0 j= . The notion of propositional prime implicant isdue to Quine [17, 18] and it has been extended to �rst order only recently[11]. It is of central interest in abduction. In fact the set of the primeimplicants of � � � coincides with the set of the minimal explanations for �in the context of �.In the propositional case the computation of the set of all prime im-plicants of a sentence is a matter of syntactical transformations of such asentence into normal forms, as the following theorem states.Theorem 1 Let be any ground formula and 1; : : : ; n all the prime im-plicants of . Then 1 _ : : : _ n = DNF (CNF ( )).(Note that the above symbol = is equality, not logical equivalence.) Thisis not surprising: deduction itself, in the propositional case, is very mucha matter of pure syntactical transformations of formulae (checking whether is valid amounts to checking whether CNF ( ) = true, or, in terms ofrefutation, whether DNF (:�) = false. See also [1]).The following theorems state the fundamental connection between three-valued semantics and abduction. 6

Theorem 2 Let � be a non valid ground sentence of L�, M1; : : : Mn allthe minimal three-valued models of CNF (�), and 1; : : : ; n all the primeimplicants of �. Then 1 _ : : : _ n = Th(M1) _ : : : _ Th(Mn).A method to construct the three-valued minimal models of a set of purelyuniversal clauses, based on [6], can be given and it will be described in alonger paper.A prime implicate of a formula is a clause � such that j= � and if �0is a clause such that j= �0 and �0 j= � then � j= �0. An obvious relationbetween prime implicants and prime implicates is that is a prime implicantof � i� : is a prime implicate of :�. So, Theorem 2 already gives a methodfor determining the set of prime implicates of a propositional formula �, i.e.it is f:Th(M)jM is a minimal three-valued model of :�g. The followingresult establishes a direct relation between prime implicates and minimalthree valued models.Theorem 3 Let � be a non valid ground sentence of L�, M1; : : :Mk all theminimal three-valued models of �, and �1; : : : ; �n all the prime implicates of�. Then �1 ^ : : : ^ �n = CNF (Th(M1) _ : : : _ Th(Mk)).Theorems 2 and 3 give two dual methods for computing the set of min-imal explanations for a ground event � in the context of a ground theory�:Corollary 1 Let � = f�1; : : : ; �kg be a set of ground sentences and � aground observation, such that � 6j= �. The set of minimal explanationsfor � in the context of � is fTh(M)jM is a minimal three-valued model ofCNF ((�1 ^ : : : ^ �k) � �)g.Corollary 2 Let � be a set of ground sentences and � a ground observation,such that 6j= � [ f:�g (i.e. it is not the case that � is valid and � is con-tradictory - a very trivial case is ruled out). Let M1; : : : ;Mr be the minimalthree-valued models of � [ f:�g and let CNF (Th(M1) _ : : : _ Th(Mr)) =�1 ^ : : :^ �m. The set of all the minimal explanations for � in the context of� is f:�ij1 � i � mg.Although the second way of computing minimal explanations seems tointroduce an unnecessary detour, it may be useful when the computation ofCNF ((�1 ^ : : :^�k) � �)g is quite expensive, for example when � is a logicprogram. In fact, computing CNF (Th(M1) _ : : : _ Th(Mr)) can be moredirect, as one deals only with minimal models.A method to get rid of all explanations inconsistent with � can easily beobtained. We describe it as a re�nement of Corollary 1: the set of minimalexplanations for � in the context of � = f�1; : : : ; �kg is fTh(M)jM is a7

minimal three-valued model of CNF ((�1 ^ : : : ^ �k) � �) and M is not aminimal three-valued model of CNF (:(�1 ^ : : : ^ �k))g.An immediate corollary of the above results is the following, that relatesprime implicants and prime implicates of the same formula.Corollary 3 If �1; : : : ; �n are all the prime implicants of a ground formula and �1 : : : �m are all the prime implicates of , then �1 ^ : : : ^ �m =CNF (�1 _ : : : _ �n).Here follows an example showing that two-valued minimal models cannotbe used instead of three-valued ones. Let � = fq � p;:r^:p � q;:q^p � sgand q the observation to be explained. The minimal three-valued modelsfor � [ f:qg are M1 = fr;:q;:pg, M2 = fr;:q; sg and M3 = fp;:q; sg.CNF (Th(M1)_Th(M2)_Th(M3)) = (r_ p)^ (r_ s)^:q^ (:p_ s). Thus(Corollary 2) the explanations for q are, besides the trivial one, �1 = :r^:p,�2 = :r^:s and �3 = p^:s. The minimal two-valued models of �[ f:qgare M 01 = fr;:q;:p;:sg and M 02 = fr;:q; s;:rg (in two-valued modelsnegative literals are normally omitted). CNF (Th(M 01) _ Th(M 02)) = (r _p)^ (r_s)^:q^ (:p_s)^(:p_:r)^(:s_p)^ (:s_:r). Thus, besides theexplanations obtained by means of three-valued models, also the followingconjunctions are generated: (p^r), (:p^s), (s^r), that are not explanationsfor q in �.4 The simplest �rst order caseThe results obtained so far for ground abduction can be immediately liftedto the case where the theory � contains only purely universal sentences and� is a purely existential sentence. We state here the method in terms ofprime implicates, because this does not require the introduction of any newconcept. Needless to say, the dual method in terms of prime implicants isreadily obtainable.Let � be a set of purely universal formulae. For any k = 0; 1; 2; : : :,we de�ne the extension L�k of the language L�, obtained by adding k newconstants m1; : : : ;mk. ��k is the set of all the ground instances of � in theHerbrand universe of L�k . For any sentence �, let deH(�) be obtained from� by replacing each constant mi (not in L�) by a new universally quanti�edvariable. Thus deH(�) has the form 8y1 : : : 8yr(L1 _ : : : _ Lm). The primeimplicates of � are among the elements in the set1[k=0fdeH(�) j � is a prime implicate of ��kg.In fact, if � is purely universal and sk(�) is the skolemization of the formulaein �, and sk(�) = �[m1; :::;mk] is the skolemization of �, obtained by use ofthe herbrand constants m1; :::;mk: 8

� j= � i� � [ f:�g is unsatis�ablei� sk(� [ f:�g) is unsatis�ablei� � [ fsk(:�)g is unsatis�ablei� � [ f:�[m1; :::;mk]g is unsatis�ablei� � [ f:�[m1; :::;mk]g has no Herbrand modelsi� ��k [ f:�[m1; :::;mk]g has no Herbrand modelsi� ��k j= �[m1; :::;mk].Finally, � = deH(�[m1; :::;mk]) (it is clear that any prime implicate of apurely universal set of formulae is purely universal, too).If � is purely universal and � purely existential, then the minimal expla-nations for � in the theory � are among the negations of the prime implicatesof � [ f:�g, thus they are existential conjunctions of literals.The method described here is a simpli�ed form of reverse skolemization[3]. A much less explosive method for lifting the propositional results to �rstorder is, of course, by use of a uni�cation mechanism. The general approachto �rst order abduction will be developed in a forthcoming paper, whereneither the language nor the form of the theory/observation are submittedto restrictions.References[1] A. Avron, \Gentzen-Type Systems, Resolution and Tableaux" Jour-nal of Automated Reasoning, vol. 10, pp.265-281, 1993.[2] P. T. Cox and T. Pietrzykowski, \Causes for Events : their Com-putation and Applications", Proc. Eighth International Conferenceon Automated Deduction, Lecture notes in Computer Science, 230,Springer, Berlin, pp. 608-621, 1986.[3] P. T. Cox and T. Pietrzykowski, \A Complete Nonredundant algo-rithm for reversed skolemization", Theoretical Computer Science, 28,pp. 239-261, 1984.[4] P. Doherty, \A Constraint-Based Approach to Proof Procedures forMulti-Valued Logics", Proceedings of WOCFAI '91, (DeGlas andGabbay Eds), Angkor, Paris, 1991.[5] P. Doherty and W.Lukaszewicz, \NML3. A Non-Monotonic Logicwith Explicit Default", Journal of Applied Non-Classical Logics, vol.2, pp.9-48, 1992.[6] M. Fitting \A Kripke-Kleene semantics for logic programs" Journalof Logic Programming 2(4), pp295-312, North-Holland, 1985.9

[7] B. vanFraassen \Presupposition, Implication and Self-Reference".Journal of Philosophy, 65, pp. 135-152, 1968.[8] K. Inoue, \Linear Resolution for Consequence Finding", Arti�cialIntelligence 56, pp.301-353, 1992.[9] S.C. Kleene Introduction To Metamathematics, North-Holland Pub-lishing Company, Amsterdam, 1971.[10] T. Langholm Partiality, Truth and Persistence, Lecture Notes CSLI,Stanford, 1988.[11] P., Marquis, \Extending Abduction from Propositional to �rst or-der logic", Proc. First International Workshop on Fundamentals ofArti�cial Intelligence Research, LNAI 535, pp. 141-155, 1991.[12] C.G., Morgan, \Hypothesis Generation by Machine", Arti�cial In-telligence, 2, (1971) pp. 179-187.[13] H.E. Jr, Pople \ On the Mechanization of Abductive Logic", Proc.International Joint Conference on Arti�cial Intelligence, pp. 147-152,1973.[14] D. Poole,R.Goebel and R. Aleliunas, \Theorist: a logical reasoningsystem for default and diagnosis" In N. Cercone, G. Mc Calla, (eds)The Knowledge Frontier, Springer Verlag, pp.331-352, 1987.[15] D. Poole, \Explanation and Prediction: an architecture for defaultand abductive reasoning" Computational Intelligence, 5(1), pp. 97-110, 1989.[16] D. Poole, \Compiling a Default Reasoning System into Prolog" NewGeneration Computing, 9, pp. 3-38, OHMSHA, LTD and SpringerVerlag, 1991.[17] Quine, W. V. O., "The problem of simplifying truth functions",American Math. Monthly, 59, 521-531, 1952.[18] Quine, W. V. O., "On cores and prime implicants of truth functions",American Math. Monthly, 66, 755-760, 1959.[19] Reiter, R. and de Kleer, J., \Foundations of Assumption-Based TruthMaintenance Systems: Preliminary Report", Proc. AAAI, pp.183-188, Seattle, WA,1987.[20] M. Shanahan, \Prediction is Deduction but Explanation is Abduc-tion", Proc. of the 8th National Conference on Arti�cial Intelligence,Boston, Ma, pp. 1055-1060, 1990.10