The Complexity of Circumscriptive Inference in ... - Semantic Scholar
The Complexity of Circumscriptive Inference in Post’s Lattice∗ Michael Thomas Institut f¨ ur Theoretische Informatik, Gottfried Wilhelm Leibniz Universit¨ at Appelstr. 4, 30167 Hannover, Germany [email protected]
Abstract. Circumscription is one of the most important formalisms for reasoning with incomplete information. It is equivalent to reasoning under the extended closed world assumption, which allows to conclude that the facts derivable from a given knowledge base are all facts that satisfy a given property. In this paper, we study the computational complexity of several formalizations of inference in propositional circumscription for the case that the knowledge base is described by a propositional theory using only a restricted set of Boolean functions. To systematically cover all possible sets of Boolean functions, we use Post’s lattice. With its help, we determine the complexity of circumscriptive inference for all but two possible classes of Boolean functions. Each of these problems is shown to be either Πp2 -complete, coNP-complete, or contained in L. In particular, we show that in the general case, unless P = NP, only literal theories admit polynomial-time algorithms, while for some restricted variants the tractability border is the same as for classical propositional inference.
1
Introduction
Circumscription is a non-monotonic logic introduced by McCarthy for first-order theories to overcome the ‘qualification problem’ which is concerned with the impossibility of representing all conditions for the successful performance of an action [17]. Circumscription allows to conclude that the objects that can be shown to have a certain property P by reasoning from a given knowledge base Γ are all objects that satisfy P . Moreover, circumscription has been shown to coincide with reasoning under the extended closed world assumption, in which all formulae involving only propositions from P that cannot be derived from Γ are assumed to be false [12]. To date, circumscription has become one of the most well developed and extensively studied formalisms for non-monotonic reasoning. Given a theory Γ containing a predicate P , circumscribing P amounts to selecting only the models of Γ in which P is assigned the value true on a minimal set of tuples. The key intuition behind this rationale is that minimal models have as few ‘exceptions’ as possible and, thus, embody common sense. In propositional ∗
Supported by DFG grant VO 630/6-1.
logic, P is simply a set of propositions; whence propositional circumscription asks for the minimal models of Γ w. r. t. the coordinatewise partial order induced on P by 0 < 1. The remaining propositions are partitioned into sets Q and Z where propositions in Q are fixed and propositions in Z are allowed to vary in minimizing the extent of P . We write
Abstract. Circumscription is one of the most important formalisms for reasoning with incomplete information. It is equivalent to reasoning under the extended closed world assumption, which allows to conclude that the facts derivable from a given knowledge base are all facts that satisfy a given property. In this paper, we study the computational complexity of several formalizations of inference in propositional circumscription for the case that the knowledge base is described by a propositional theory using only a restricted set of Boolean functions. To systematically cover all possible sets of Boolean functions, we use Post’s lattice. With its help, we determine the complexity of circumscriptive inference for all but two possible classes of Boolean functions. Each of these problems is shown to be either Πp2 -complete, coNP-complete, or contained in L. In particular, we show that in the general case, unless P = NP, only literal theories admit polynomial-time algorithms, while for some restricted variants the tractability border is the same as for classical propositional inference.
1
Introduction
Circumscription is a non-monotonic logic introduced by McCarthy for first-order theories to overcome the ‘qualification problem’ which is concerned with the impossibility of representing all conditions for the successful performance of an action [17]. Circumscription allows to conclude that the objects that can be shown to have a certain property P by reasoning from a given knowledge base Γ are all objects that satisfy P . Moreover, circumscription has been shown to coincide with reasoning under the extended closed world assumption, in which all formulae involving only propositions from P that cannot be derived from Γ are assumed to be false [12]. To date, circumscription has become one of the most well developed and extensively studied formalisms for non-monotonic reasoning. Given a theory Γ containing a predicate P , circumscribing P amounts to selecting only the models of Γ in which P is assigned the value true on a minimal set of tuples. The key intuition behind this rationale is that minimal models have as few ‘exceptions’ as possible and, thus, embody common sense. In propositional ∗
Supported by DFG grant VO 630/6-1.
logic, P is simply a set of propositions; whence propositional circumscription asks for the minimal models of Γ w. r. t. the coordinatewise partial order induced on P by 0 < 1. The remaining propositions are partitioned into sets Q and Z where propositions in Q are fixed and propositions in Z are allowed to vary in minimizing the extent of P . We write
