In order to reason about specifications of computations that are given via the proof search or logic programming paradigm one needs to have at least some forms of induction and some principle for reasoning about the ways in which terms are built and the ways in which computations can progress. The literature contains many approaches to formally adding these reasoning principles with logic specifications. We choose an approach based on the sequent calculus and design an intuitionistic logic F Oλ ∆IN that includes natural number induction and a notion of definition. We have detailed elsewhere that this logic has a number of applications. In this paper we prove the cut-elimination theorem for F Oλ ∆IN, adapting a technique due to Tait and Martin-Löf. This cut-elimination proof is technically interesting and significantly extends previous results of this kind. 1

this paper, we do this by adding the #-quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type o, and for all types # not containing o, # is a constant of type (# o) o. The expression # #x.B is ACM Transactions on Computational Logic, Vol. V, No. N, October 2003. 4 usually abbreviated as simply # x.B or as if the type information is either simple to infer or not important

A powerful and declarative means of specifying computations containing abstractions involves meta-level, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and to each formula in the sequent (used to account for generic variables locally scoped over the formula). A new quantifier, ∇, is introduced to explicitly manipulate the local signature. Intuitionistic logic extended with ∇ satisfies cut-elimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving name abstractions, and we illustrate using the π-calculus and the encoding of objectlevel provability.

Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles are based on a proof theoretic (rather than set-theoretic) notion of definition [13, 20, 25, 51]. Definitions are akin to (stratified) logic programs, where the left and right rules for defined atoms allow one to view theories as “closed ” or defining fixed points. The use of definitions makes it possible to reason intensionally about syntax, in particular enforcing free equality via unification. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and co-inductively about properties of computational system making full use of higher-order abstract syntax. Consistency is guaranteed via cut-elimination, where we give the first, to our knowledge, cut-elimination procedure in the presence of general inductive and co-inductive definitions. 1

Abstract. Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the meta-theory of linear logic can be used to draw conclusions about the specified sequent calculus. For example, derivability of one proof system from another can be decided by a simple procedure that is implemented via bounded logic programming-style search. Also, simple and decidable conditions on the linear logic presentation of inference rules, called homogeneous and coherence, can be used to infer that the initial rules can be restricted to atoms and that cuts can be eliminated. In the present paper we introduce Llinda, a logical framework based on linear logic augmented with inference rules for definition (fixed points) and induction. In this way, the above properties can be proved entirely inside the framework. To further illustrate the power of Llinda, we extend the definition of coherence and provide a new, semi-automated proof of cut-elimination for Girard’s Logic of Unicity (LU). 1

What is the semantics of Negation-as-Failure in logic programming? We try to answer this question by proof-theoretic methods. A rule based sequent calculus is used in which a sequent is provable if, and only if, it is true in all three-valued models of the completion of a logic program. The main theorem is that proofs in the sequent calculus can be transformed into SLDNF-computations if, and only if, a program has the cut-property. A fragment of the sequent calculus leads to a sound and complete semantics for SLDNFresolution with substitutions. It turns out that this version of SLDNF-resolution is sound and complete with respect to three-valued possible world models of the completion for arbitrary logic programs and arbitrary goals. Since we are dealing with possibly nonterminating computations and constructive proofs, three-valued possible world models seem to be an appropriate semantics.

In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic and show that LZF is a conservative extension of ZF 0 i.e., the Zermelo-Fraenkel set theory without the axiom of regularity. We formulate LZF as a sequent calculus with abstraction terms and prove the partial cut-elimination theorem for it. The cut-elimination result ensures the subterm property for those formulas which contain only terms corresponding to sets in ZF 0 . This implies that LZF is a conservative extension of ZF 0 and therefore the former is consistent relative to the latter. 1 Introduction In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic. The fine distinction provided by linear logic between contractible and non-contractible formulas makes it possible to have more sets than in classical set theory. For the sake of simplicity, LZF is built on top of ZF 0 i.e., ZF without the axiom of regulari...

In recent years, intuitionistic logic and type systems have been used in numerous computational logical systems as frameworks for the specification of natural deduction proof systems. As we shall illustrate here, linear logic can be similarly used to specify the more general setting of sequent calculus proof systems and provides rich forms of analysis and deduction of properties of the specified systems. We shall present several example encodings of sequent calculus proof systems using the Forum presentation of linear logic: linear logic is a resource conscious logic developed by Girard, and Forum is an abstract logic programming language associated to it, due to Miller. We start by proposing an encoding of sequents, rules and systems. Then a correctness result is proved for these encodings and it is observed that meta-level proofs match closely the object-level ones. The encoding of an object-level proof system as Forum clauses has certain advantages over encoding them as inference figures. For example, Forum specifications do not deal with context explicitly and instead it only focuses on the formulas that are directly involved in the inference rule. The distinction between making the inference rule additive or multiplicative is achieved in inference rule figures by explicitly presenting contexts and either splitting or copying them. The Forum clause representation achieves the same distinction using meta-level additive or multiplicative connectives. Object-level quantifiers can be handled directly using the meta-level quantification. Similarly, the structural rules of contraction and weakening can be captured together using the ? modal. Finally, since the encoding of proof systems is natural and direct, we are able to use the rich meta-theory of linear logic to help ...

The operational semantics of a computation system is often presented as inference rules or, equivalently, as logical theories. Specifications can be made more declarative and high-level if syntactic details concerning bound variables and substitutions are encoded directly into the logic using term-level abstractions (#-abstraction) and proof-level abstractions (eigenvariables). When one wishes to reason about relations defined using term-level abstractions, generic judgment are generally required.

Abstract. This paper presents a cut-elimination proof for the logic LG ω, which is an extension of a proof system for encoding generic judgments, the logic FOλ ∆ ∇ of Miller and Tiu, with an induction principle. The logic LG ω, just as FOλ ∆ ∇ , features extensions of first-order intuitionistic logic with fixed points and a “generic quantifier”, ∇, which is used to reason about the dynamics of bindings in object systems encoded in the logic. A previous attempt to extend FOλ ∆ ∇ with an induction principle has been unsuccessful in modeling some behaviours of bindings in inductive specifications. It turns out that this problem can be solved by relaxing some restrictions on ∇, in particular by adding the axiom B ≡ ∇x.B, where x is not free in B. We show that by adopting the equivariance principle, the presentation of the extended logic can be much simplified. This paper contains the technical proofs for the results stated in [14]; readers are encouraged to consult [14] for motivations and examples for LG ω. 1