Learning by Answer Sets

This paper presents a novel application of answer set programming to concept learning in nonmonotonic logic programs. Given an extended logic program as a background theory, we introduce techniques for inducing new rules using answer sets of the program. The produced new rules explain positive/negative examples in the context of inductive logic programming. The result of this paper combines techniques of two important fields of logic programming in the context of nonmonotonic inductive logic programming. Introduction Nonmonotonic logic programming (NMLP) introduces mechanisms of representing incomplete knowledge and reasoning with commonsense. An example of such extensions is extended logic programs with the answer set semantics (Gelfond and Lifschitz 1991). On the other hand, inductive logic programming (ILP) (Muggleton 1992) realizes inductive learning in logic programming. It is concerned with learning a general theory from examples and background knowledge. NMLP realizes commonsense reasoning in logic programming, but a program never derives information that is not specified in the program. By contrast, ILP constructs new rules from given examples, but the present ILP mostly considers Horn logic programs and has limited applications to nonmonotonic theories. Thus, both NMLP and ILP have limitations in their present frameworks and complement each other. Since both commonsense reasoning and machine learning are indispensable for constructing powerful knowledge systems, combining techniques of the two fields in the context of nonmonotonic inductive logic programming (NMILP) is important. Such combination will extend the representation language on the ILP side, while it will introduce a learning mechanism to programs on the NMLP side. This position paper presents techniques of realizing inductive learning in nonmonotonic logic programs. We consider an extended logic program as a background theory, and introduce a method of learning new rules using answer Copyright c © 2001, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. sets of the program. The produced new rules explain positive/negative examples in the context of inductive logic programming. From the viewpoint of answer set programming (ASP), it provides a novel application of ASP to concept learning in nonmonotonic logic programs. Preliminaries A program considered in this paper is an extended logic program (ELP) (Gelfond and Lifschitz 1991), which is a set of rules of the form: L0 ← L1, . . . , Lm, not Lm+1, . . . , not Ln (1) where each Li is a literal and not is negation as failure (NAF). The literal L0 is the head and the conjunction L1, . . . , Lm, not Lm+1, . . . , not Ln is the body. The conjunction in the body is identified with the set of conjuncts. For a rule R, head(R) and body(R) denote the head and the body of R, respectively. The head is possibly empty and a rule with the empty head is called an integrity constraint. A rule with the empty body L ← is identified with the literal L and is called a fact. A program (rule, literal) is ground if it contains no variable. Any rule with variables is considered as a shorthand of its ground instances. A program or a rule is NAF-free if it contains no not (i.e., m = n for the rule (1)). Let Lit be the set of all ground literals in the language of a program. Any element in Lit = Lit ∪ {notL | L ∈ Lit } is called an LP-literal and an LP-literal notL is called an NAF-literal. When K is an LP-literal, it is defined |K| = K if K is a literal; and |K| = L if K = notL. For an LP-literal L, pred(L) denotes the predicate of L and const(L) denotes the set of constants appearing in L. A set S(⊆ Lit) satisfies the conjunction C = (L1, . . . , Lm, not Lm+1, . . . , not Ln) (written as S |= C) if {L1, . . . , Lm} ⊆ S and {Lm+1, . . . , Ln }∩S = ∅. S satisfies a ground rule R (written as S |= R) if S |= body(R) implies S |= head(R). In particular, S satisfies the ground integrity constraint ← L1, . . . , Lm, not Lm+1, . . . , not Ln if {L1, . . . , Lm} 6⊆ S or {Lm+1, . . . , Ln } ∩ S 6= ∅. The semantics of ELPs is given by the answer set semantics (Gelfond and Lifschitz 1991). The answer sets of a program are defined by the following two steps. First, let P be an NAF-free program and S ⊂ Lit. Then, S is a consistent answer set of P if S is a minimal set which satisfies every ground rule from P and does not contain both L and ¬L for any L ∈ Lit. Next, let P be any program and S ⊂ Lit. Then, define the NAF-free program P as follows: a rule L0 ← L1, . . . , Lm is in P iff there is a ground rule of the form (1) from P such that {Lm+1, . . . , Ln } ∩ S = ∅. Then, S is a consistent answer set of P if S is a consistent answer set of P . A consistent answer set is simply called an answer set hereafter. For an answer set S, we define S = S ∪ {notL | L ∈ Lit \ S }. A program P is consistent if it has a (consistent) answer set; otherwise P is inconsistent. Throughout the paper, a program is assumed to be consistent unless stated otherwise. A program P is called categorical if it has a unique consistent answer set (Baral and Gelfond 1994). If a rule R (resp. a conjunction C) is satisfied in every answer set of P , it is written as P |= R (resp. P |= C); otherwise P 6|= R (resp. P 6|= C). In particular, P |= L if a literal L is included in every answer set of P ; otherwise P 6|= L. Learning from Positive Examples A typical induction problem is as follows. Given a background program P and a ground literal L which represents a (positive) example, construct a rule R satisfying