Nicola Olivetti

Sequent and hypersequent calculi for abelian and łukasiewicz logics

We present two embeddings of Łukasiewicz logic <b>Ł</b> into Meyer and Slaneys Abelian logic <b>A</b>, the logic of lattice-ordered Abelian groups. We give new analytic proof systems for <b>A</b> and use the embeddings to derive corresponding systems for <b>Ł</b>. These include hypersequent calculi, terminating hypersequent calculi, co-NP labeled sequent calculi, and unlabeled sequent calculi.

A sequent calculus and a theorem prover for standard conditional logics

In this paper we present a cut-free sequent calculus, called SeqS, for some standard conditional logics. The calculus uses labels and transition formulas and can be used to prove decidability and space complexity bounds for the respective logics. We also show that these calculi can be the base for uniform proof systems. Moreover, we present CondLean, a theorem prover in Prolog for these calculi.

Goal-directed proof theory

1. Introduction. 2. Intuitionistic and Classical Logics. 3. Intermediate Logics. 4. Modal Logics of Strict Implication. 5. Substructural Logics. 6. Conclusions and Further Work. Bibliography. Index.

Sequent calculi for propositional nonmonotonic logics

A uniform proof-theoretic reconstruction of the major nonmonotonic logics is introduced. It consists of analytic sequent calculi where the details of nonmonotonic assumption making are modelled by an axiomatic rejection method. Another distinctive feature of the calculi is the use of provability constraints that make reasoning largely independent of any specific derivation strategy. The resulting account of nonmonotonic inference is simple and flexible enough to be a promising playground for investigating and comparing proof strategies, and for describing the behavior of automated reasoning systems. We provide some preliminary evidence for this claim by introducing optimized calculi, and by simulating an existing tableaux-based method for circumscription. The calculi for skeptical reasoning support concise proofs that may depend on a strict subset of the given theory. This is a difficult task, given the nonmonotonic behavior of the logics.

Tableaux and sequent calculus for minimal entailment

Minimal entailment is the semantical counterpart of Circumscription and Closed World Assumption. In this paper we show that it is possible to formalize minimal entailment at the propositional level, using standard deduction methods. Firstly we present a tableau procedure which is a natural reformulation of Smullyans Analytic Tableaux. Then we introduce the sequent calculus MLK which is an extension of Gentzens LK calculus. We prove that both are sound and complete formalizations of minimal entailment. Various extensions of the proposed methods are also discussed.

Resolution and model building in the infinite-valued calculus of Lukasiewicz

Abstract We discuss resolution and its complexity in the infinite-valued sentential calculus of L ukasiewicz, with special emphasis on model building algorithms for satisfiable sets of clauses. We prove that resolution and model building are polynomially tractable in the fragments given by Horn clauses and by Krom clauses, i.e., clauses with at most two literals. Generalizing the pre-existing literature on resolution in infinite-valued logic, by a positive literal we mean a negationless formula in one variable, built only from the connectives ⊕, ⊙, ν, Λ. We prove that the expressive power of our literals encompasses all monotone McNaughton functions of one variable.

Semantic characterization of rational closure: From propositional logic to description logics

Abstract In this paper we provide a semantic reconstruction of rational closure. We first consider rational closure as defined by Lehman and Magidor [33] for propositional logic, and we provide a semantic characterization based on a minimal models mechanism on rational models. Then we extend the whole formalism and semantics to Description Logics, by focusing our attention to the standard ALC : we first naturally adapt to Description Logics Lehman and Magidors propositional rational closure, starting from an extension of ALC with a typicality operator T that selects the most typical instances of a concept C (hence T ( C ) stands for typical C). Then, for the Description Logics, we define a minimal model semantics for the logic ALC and we show that it provides a semantic characterization for the rational closure of a Knowledge base. We consider both the rational closure of the TBox and the rational closure of the ABox.

A Sequent Calculus for Skeptical Default Logic

In this paper, we contribute to the proof-theory of Reiters Default Logic by introducing a sequent calculus for skeptical reasoning. The main features of this calculus are simplicity and regularity, and the fact that proofs can be surprisingly concise and, in many cases, involve only a small part of the default theory.

Analytic tableaux calculi for KLM logics of nonmonotonic reasoning

We present tableau calculi for the logics of nonmonotonic reasoning defined by Kraus, Lehmann and Magidor (KLM). We give a tableau proof procedure for all KLM logics, namely preferential, loop-cumulative, cumulative, and rational logics. Our calculi are obtained by introducing suitable modalities to interpret conditional assertions. We provide a decision procedure for the logics considered and we study their complexity.

Weak AGM postulates and strong Ramsey Test: A logical formalization

We reformulate AGM postulates for belief revision systems that may contain conditional formulas. We show that we can establish a mapping between belief revision systems and conditionals by means of the so called Ramsey Test, without incurring Gardenfors triviality result. We then derive the conditional logic BCR from our revision postulates by means of a strong version of the Ramsey Test. We give a sound and complete axiomatization of this logic with respect to its standard selection-function models semantics, and we prove its decidability. We finally show that there is an isomorphism between belief revision systems and selection function models of BCR via a representation theorem. The logic BCR provides a logical formalization of belief revision in the language of conditional logic.