Sequent Calculi for Skeptical Reasoning in Predicate Default Logic and Other Nonmonotonic Logics ∗

Sequent Calculi for Skeptical Reasoning in Predicate Default Logic and Other Nonmonotonic Logics ∗ Robert Saxon Milnikel ([email protected]) Department of Mathematics, Kenyon College Abstract. Sequent calculi for skeptical consequence in predicate default logic, predicate stable model logic programming, and infinite autoepistemic theories are presented and proved sound and complete. While skeptical consequence is decidable in the finite propositional case of all three formalisms, the move to predicate or 1 infinite theories increases the complexity of skeptical reasoning to being Π1-complete. This implies the need for sequent rules with countably many premises, and such rules are employed. Keywords: Default logic, Stable models, Autoepistemic logic, Sequent calculus AMS Subject Classifications: 03B42, 68N17, 68T27 1. Introduction Skeptical consequence is a notion common to all forms of nonmonotonic reasoning. Every nonmonotonic formalism permits different world views to be justified using the same set of facts and principles; the skeptical consequences of a framework are the notions common to all world views associated with that framework. Our purpose in this paper is to present a Gentzen-style sequent calculus (incorporating some infinitary rules) which will allow us to deduce the skeptical consequences of a given framework. Such sequent calculi (with purely finite rules) were defined for several types of nonmonotonic systems by Bonatti and Olivetti in [6], but they restricted their attention to finite propositional systems for which skeptical consequence is decidable. We will adapt and extend their systems to accommodate infinite predicate systems. We will focus on three types of nonmonotonic reasoning: stable model logic programming (due to Gelfond and Lifschitz, [8]), default logic (due to Reiter, [22]), and autoepistemic logic (due to Moore, [19]). In all three cases, when one steps from the finite and propositional to the predicate and potentially infinite, finding the set of skeptical 1 consequences of a framework goes from being decidable to being Π1- complete, at the same level of the computability hierarchy as true arithmetic. This result was proved for stable model logic program- ∗ This paper grew directly out of the author’s dissertation, written under the direction of Anil Nerode. c 2004 Kluwer Academic Publishers. Printed in the Netherlands. fulldefaultsequent.tex; 18/06/2004; 16:12; p.1 2 R.S. Milnikel ming by Marek, Nerode, and Remmel in [13], but it translates to the other systems quite easily. (The reader should be aware that there will be a few computability theoretic ideas and motivations discussed in this introductory section, but that they may be considered “deep background” and will not be a part of the exposition of the main ideas.) 1 0 Π1 sets correspond to finite-path computable (or Π1) subtrees of ω<ω in a very natural way. See [7] for an excellent exposition. This makes skeptical consequence a natural fit for sequent calculi with infinitary rules, since a sequent proof is, at its core, a finite-path tree. Bonatti and Olivetti also addressed credulous consequence (“Can this notion be a part of some world view?”) in their paper, but in the cases they were interested in, this question was also decidable. In our more 1 general context, credulous reasoning is Σ1-complete, not a natural type of question to address with trees-as-proofs. (One could write a sequent 1 calculus for credulous reasoning in Π2 logic, but this would take us too far afield.) Why do we plan to address the nonmonotonic formalisms listed earlier and not others? Default logic, autoepistemic logic, and circumscription are historically the most established frameworks, with stable model logic programming now as widely studied as those three. We will begin with stable model logic programming because it has the simplest classical monotonic base over which the nonmonotonicity is layered. This will permit us to focus on the distinctly nonmonotonic parts of the skeptical sequent calculus in our first encounter with such systems. The reason for excluding circumscription (due to McCarthy, [15]) is easily stated: Skeptical consequence for predicate circumscription was 1 1 shown by Schlipf in [23] to be Π2-complete, not Π1-complete like the 1 other systems mentioned. As I said earlier, a Π2 sequent calculus would take us far beyond the intended scope of this paper. We will also not be discussing Marek, Nerode, and Remmel’s nonmonotone rule systems (see [12]) or McDermott and Doyle’s nonmonotonic modal logics (see [16]). The reasons are twofold in each case. Nonmonotone rule systems have not entered the mainstream of the nonmonotonic reasoning literature, and are also essentially isomorphic to propositional stable model logic programming. The systems of McDermott and Doyle are also not as central to the field as some other formalisms, and would benefit from a more specific study in the context of Artemov’s logic of proofs (see [3]). Why does the move from the propositional to the predicate entail such an enormous increase in complexity? Because in general, it also implies a move from a finite set of rules to an infinite one. In predicate logic programming, an open variable is viewed as an abbreviation for all values which that variable might take on. The standard approach fulldefaultsequent.tex; 18/06/2004; 16:12; p.2 Predicate Nonmonotone Sequent Calculi 3 is to make each of those possible instantiations explicit, turning a finite predicate logic program over an infinite domain into an infinite program. In default logic, there has been debate since Reiter first defined the framework in 1980 about how to treat unbounded variables in the rules. Reiter ([22]) advocated treating open variables (at least in the negative premises of a default rule) in the same way that they are treated in logic programs: as abbreviations for the same rule with each ground term of the language substituted. Again, this can turn a finite default theory into an infinite grounded one. This is the definition we will use in this paper. However, there is some quite justified criticism of this ap- : MP (x) proach. Under these definitions, the default theory ( , ¬P (a)) P (x) does not imply (∀x)[P (x) ↔ x 6= a]. Lifschitz, in [10], defines exten- sions (the possible world views associated with default logic) relative to fixed domains. For finite theories and finite domains, everything is decidable, but over infinite domains this is no longer the case. In [18], it is shown that skeptical reasoning over countable domains using 1 Lifschitz’ definition of extension is Π2-complete, the same level as for circumscription. Because nonmonotonic logics deal not only with proof but with lack of proof, we will need not only standard monotone sequent calculi, but also rule systems for showing a lack of proof. (We will call these antisequent calculi, using the terminology of Bonatti from [5].) While propositional provability and lack of provability are decidable, predicate provability is only recursively enumerable, and hence predicate non- 1 provability is co-r.e. Just as Σ1 sets do not lend themselves naturally 1 to tree-based proofs, neither do co-r.e. sets. However, while Σ1 sets 1 required a jump to Π2 logic, co-r.e. sets are easily accommodated in 1 the Π1 framework within which we will already be working. Bringing such enormously powerful logical machinery to bear on such a relatively simple problem may seem like overkill, but it works out quite naturally. The reader is thus warned that infinitary proofs will appear throughout the paper, even when talking about something as simple as lack of a standard predicate logic proof. When discussing autoepistemic logic, we limit ourselves to the propositional case, but with a potentially infinite theory. This puts us at the 1 same Π1 level of complexity as for stable model logic programming and default logic. Predicate autoepistemic logic is discussed in the literature (see, for example, [9]), and the interested reader should not find it difficult to combine the rules specific to predicate logic (from the discussion of default logic) with the results about propositional autoepistemic logic. fulldefaultsequent.tex; 18/06/2004; 16:12; p.3 4 R.S. Milnikel We will begin in Section 2 with some preliminary definitions and results about the types of nonmonotonic systems we will consider in this paper. Section 3 presents rudimentary sequent and antisequent calculi for Horn programs, continues with a sequent calculus for skeptical reasoning in stable model logic programming, and concludes with a proof of the soundness and completeness of this calculus. Section 4 concentrates on finding an antisequent calculus for classical predicate logic, and concludes with a very brief treatment of a sequent calculus for skeptical reasoning in default logic. The nonmonotonic aspects of stable model logic programming and default logic are so similar that the sequent calculi are nearly identical and no proof is necessary for the soundness and completeness theorems in this case. Section 5 presents a sequent calculus for skeptical reasoning in infinite propositional autoepistemic theories, and proves soundness and completeness theorems for this calculus. Section 6 looks toward further directions for research. 2. Preliminaries We will be working with systems based on a variety of languages, both predicate and propositional. We will assume that the reader has some familiarity with classical propositional and predicate logic, including the notion of a Herbrand base for a given predicate language, the standard sequent calculus LK for propositional logic, and the modal operator L. 2.1. Stable Model Logic Programming Logic programming is a broad and widely studied subfield of computer science. We will concentrate here on only one aspect – finding models of a program given a particular interpretation of negation. For a more general introduction, see [2], [11], or [21].

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