Blondeel, Marjon

De Cock, Martine

Schockaert, Steven

Vermeir, Dirk

English

Online Research @ Cardiff

Towards a deeper understanding of nonmonotonic reasoning with degrees

Ghent University Academic Bibliography

Towards a Deeper Understanding of Nonmonotonic Reasoning with DegreesMarjon Blondeel ∗Advisors:Steven Schockaert †, Dirk Vermeir ‡, Martine De Cock §1 IntroductionAnswer set programming (ASP) [Baral, 2003] is a form ofdeclarative programming that can be used to model combina-torial search problems in a concise and declarative manner.One of the advantages of ASP is the fact that it works non-monotonically. On the contrary, classical logic works mono-tonically: when new knowledge is added, the set of conclu-sions that can be inferred grows. Hence it is not really suitableto imitate human reasoning since humans constantly revisetheir knowledge when new information becomes available.In ASP, nonmonotonicity is obtained by using a special op-erator “not”, the negation-as-failure operator. An expressionof the form not a is “true” when we fail to derive a, whereas¬a, the negation of a, is “true” if we can derive ¬a.For example, consider the following ASP program.r1 : suitable ← certificate ∧ not criminal-recordr2 : certificate ← trueRule r1 informally means that a person is suitable for a job ifthere is no reason to think that he has a criminal record andif we can establish that he has a certificate. A rule such as r2is called a fact; the head “certificate” is unconditionally true.Given such a program, the idea is to find a minimal set ofliterals that can be derived from the program. These “answersets” then correspond to the solutions of the original searchproblem.Unfortunately, ASP is not directly suitable for expressingcontinuous optimization problems since it is limited to ex-pressing problems in Boolean logic. For example, supposeone wants to travel by car from one city to another in win-ter. The driving time that is needed to do this depends onseveral factors; for instance the amount of snow, the distanceand the traffic. These concepts are a matter of degree ratherthan Boolean properties, thus we cannot directly use ASP to∗Vrije Universiteit Brussel, Department of Computer Science,Pleinlaan 2, 1050 Brussel, Belgium, mblondee@vub.ac.be, Fundedby a joint Research Foundation-Flanders (FWO) project†Cardiff University, School of Computer Scienceand Informatics, 5 The Parade, Cardiff, CF24 3AA,s.schockaert@cs.cardiff.ac.uk‡Vrije Universiteit Brussel, Department of Computer Science,Pleinlaan 2, 1050 Brussel, Belgium, dvermeir@vub.ac.be§Ghent University, Department of Applied Mathematics, Com-puter Science and Statistics (S9), Krijgslaan 281, 9000 Gent, Bel-gium, martine.decock@ugent.bemodel this problem. One solution to this problem is to allowpropositions to be true to a certain degree in [0, 1] and to gen-eralize the syntax and semantics of ASP using fuzzy logics[Ha´jek, 1998]. We can then write a ruledriving time ← f(snow, distance, traffic)where “driving time”, “snow” , “distance” and “traffic” nowhave to be seen as atoms that can be assigned a degree in[0, 1]. The function f defines how these degrees have to becombined to establish the driving time.Fuzzy answer set programming (FASP) (see [Blondeel etal., 2013c] for an introduction to the topic) is a generalizationof answer set programming (ASP) based on fuzzy logics. InFASP, a continuous search problem is translated into a set ofrules α ← β where the body β and the head α are built fromliterals, expressions of the form not a with a a literal, con-stants and connectives that can in principal be interpreted byarbitrary [0, 1]n → [0, 1]-mappings. Such a rule now intu-itively means that the truth degree of α must be greater thanor equal to the truth degree of β. Answer sets of FASP pro-grams are then mappings from the set of literals into [0, 1].Since it is a relatively new concept, little is known about thecomputational complexity of FASP and almost no techniquesare available to compute answer sets of FASP programs. Fur-thermore, the connections of FASP to other paradigms ofnonmonotonic reasoning with continuous values are largelyunexplored. In our dissertation, we contribute to the ongoingresearch on FASP on two different levels:• Complexity issues (Section 2) We have pinned downthe complexity of the direct syntactical generalizationof classical ASP to FASP, and we have developed animplementation into bilevel linear programming for thistype of programs. For another class of FASP programswith more syntactic freedom, we showed a connectionto an existing open problem about the complexity of in-teger equations, indicating that settling the complexityof FASP programs in this case will not be an easy task.• Connection to fuzzy modal logics (Section 3) We com-bined the paradigms of fuzzy logic and autoepistemiclogic into fuzzy autoepistemic logic, and showed thatthe latter generalizes FASP. Furthermore, we have intro-duced generalizations of classical propositional modallogics of belief, and we have obtained a generalizationof a known result on the relationship between stable ex-pansions, belief sets and “only believing” operators. Ourongoing work deals with fully developing this “only be-lieving” logic to obtain a better understanding of the na-ture of fuzzy autoepistemic logic and hence of FASP.2 Complexity issuesWe have analyzed the computational complexity of FASPunder Łukasiewicz semantics in [Blondeel et al., 2013b].Łukasiewicz logic is a particular kind of fuzzy logic that isoften used in applications because it preserves many desir-able properties from classical logic. Given a FASP programP , a literal l and a value λl ∈ [0, 1] ∩Q, we are interested inthe following decision problems:1. Existence: Does there exist an answer set I of P ?2. Set-membership: Does there exist an answer set I of Psuch that I(l) ≥ λl?3. Set-entailment: Is I(l) ≥ λl for each answer set I ofP ?For programs with rules of the forma1 ⊕ . . .⊕ an ← b1 ⊗ . . .⊗ bm ⊗ not c1 ⊗ . . .⊗ not ckwhere ⊕ and ⊗ are resp. the Łukasiewicz disjunction andconjunction, we showed that the complexity of the main rea-soning tasks is located at the first level of the polynomial hi-erarchy although for classical ASP it is located at the secondlevel. This is due to the fact that we can use linear program-ming to verify yes instances of the decision problems in poly-nomial time. Moreover, we provided a reduction from reason-ing with such FASP programs to bilevel linear programming,thus opening the door to practical applications.For the subclass of programs in which negation-as-failureis not allowed and where there is at most one literal in thehead of each rule, we showed that the answer set can be foundin polynomial time. Surprisingly, when allowing disjunc-tions to occur in the body of rules – a syntactic generalizationwhich does not affect the expressivity of ASP in the classi-cal case – the picture changes drastically. Reasoning tasksare then located at the second level of the polynomial hierar-chy and for simple FASP programs, i.e. programs in whichnegation-as-failure and negation do not occur and in whicheach rule has exactly one atom in the head, we have not beenable to pin down the complexity yet. More specifically, wehave an algorithm to find the unique answer set of these pro-grams in pseudo polynomial time but whether this problem isNP-complete remains an open problem. Moreover, the con-nection to an existing open problem about the complexity ofinteger equations [Bjorklund et al., 2003] suggests that theproblem of fully characterizing the complexity of FASP inthis more general setting is not likely to have an easy solu-tion. Another possibly even more intriguing and fundamentalopen research question is whether there exists a FASP pro-gram in which each head contains at most one literal withoutanswer sets. Other future work is to study the complexityunder other semantics.3 Connection to fuzzy modal logicsAutoepistemic logic is an important formalism for nonmono-tonic reasoning. It extends propositional logic by offering theability to reason about an agent’s (lack of) beliefs. More pre-cisely, these beliefs are a set of sentences in a propositionallanguage augmented by a modal operator B. Moreover, au-toepistemic logic is well known to generalize the stable modelsemantics of ASP [Gelfond and Lifschitz, 1988]. In [Blon-deel et al., 2013a] we combined the ideas of autoepistemiclogic and fuzzy logic to a fuzzy autoepistemic logic whichcan be used to reason about one’s beliefs in the degrees towhich properties are satisfied. We showed that, like in theclassical case, fuzzy autoepistemic logic generalizes FASP.On the other hand, there are well known links betweenautoepistemic logic and several nonmonotonic modal logicsystems. In ongoing work we are investigating how fuzzyautoepistemic logic can be studied in many-valued modalsettings. In particular, we have introduced generalizationsof classical propositional modal logics of belief based onfinitely-valued Łukasiewicz logic for which we obtainedcompleteness w.r.t appropriate Kripke style semantics andfor which we could prove NP-completeness for the satisfia-bility problem. We obtained a generalization of Levesque’s[Levesque, 1990] result on the relationship between stableexpansions, belief sets and “only believing” operators. In fu-ture work, by fully developing this ”only believing” logic wewant to obtain a better understanding of the nature of fuzzyautoepistemic logic and hence of FASP.References[Baral, 2003] C. Baral. Knowledge Representation, Reason-ing and Declarative Problem Solving. Cambridge Univer-sity Press, 2003.[Bjorklund et al., 2003] H. Bjorklund, S. Sandberg, andS. Vorobyov. Complexity of model checking by iterativeimprovement: the pseudo-boolean framework. In Proc. ofthe 5th Andrei Ershov Memorial Conference “Perspectivesof System Informatics”, pages 381–394, 2003.[Blondeel et al., 2013a] M. Blondeel, S. Schockaert, M. DeCock, and D. Vermeir. Fuzzy autoepistemic logic and itsrelation to fuzzy answer set programming. To appear inFuzzy Sets and Systems, 2013.[Blondeel et al., 2013b] M. Blondeel, S. Schockaert, D. Ver-meir, and M. De Cock. Complexity of fuzzy answer setprogramming under Łukasiewicz semantics. Submitted,2013.[Blondeel et al., 2013c] M. Blondeel, S. Schockaert, D. Ver-meir, and M. De Cock. Fuzzy answer set programming:An introduction. In Soft Computing: State of the ArtTheory and Novel Applications, volume 291 of Studies inFuzziness and Soft Computing. Springer, 2013.[Gelfond and Lifschitz, 1988] M. Gelfond and V. Lifschitz.The stable model semantics for logic programming. InProc. of the 5th International Conference and Symposiumon Logic Programming, pages 1070–1080, 1988.[Ha´jek, 1998] P. Ha´jek. Metamathematics of Fuzzy Logic.Trends in Logic, 1998.[Levesque, 1990] H.J. Levesque. All I know: A study in au-toepistemic logic. Artificial Intelligence, pages 263–309,1990.

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Towards a deeper understanding of nonmonotonic reasoning with degrees

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