No abstract

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 termlevel abstractions (λ-abstraction) and proof-level abstractions (eigenvariables). When one wishes to use such logical theories to support reasoning about properties of computation, the usual quantifiers and proof-level abstractions do not seem adequate: proof-level abstraction of variables with scope over sequents (global scope) as well as over only formulas (local scope) seem required for many examples. We will present a sequent calculus that provides this local notion of proof-level abstraction via generic judgment and a new quantifier, ∇, which explicitly manipulates such local scope. 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 abstractions, and we illustrate the uses of ∇ with the π -calculus and an encoding of provability of an object-logic.In specifying and reasoning about computations involving abstractions, one needs to encode both the static structure of such abstractions and their dynamic structure during computation. One successful approach to such an encoding, generally called λ-tree syntax [Miller 2000] (a proof search approach to higherorder abstract syntax [Pfenning and Elliott 1988]), uses λ-terms to encode the static structure of abstractions and universally quantified judgments to encode their dynamic structure. Consider in more detail the role of the universal quantifier and eigenvariables in proof search and the specification of computations.There are, of course, at least a few ways to prove the universally quantified formula ∀ τ x.B. The extensional approach attempts to prove B[t/x] for all (closed) terms t of type τ . This rule might involve an infinite number of premises if the domain of the type τ is infinite. If the type τ is defined inductively, a proof by induction can replace the need for infinite premises with finite premises (the base cases and inductive cases) but with the need to discover invariants. Another more intensional approach, however, involves introducing a new variable, say, c : τ , that has not been introduced before in the proof, and attempting to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are called eigenvariables, and they are used to prove universally quantified formulas generically.In Gentzen's original presentation of the sequent calculus [Gentzen 1969], eigenvariables are immutable during proof search: once an eigenvariable is introduced (reading proofs bottom-up), it is not used as a site for substitution. In other words, eigenvariables did not vary during proof search: rather they acted more as new, scoped constants. ...

Abstract. The calculus of structures is a framework for specifying logical systems, which is similar to the one-sided sequent calculus but more general. We present a system of inference rules for propositional classical logic in this new framework and prove cut elimination for it. The system enjoys a decomposition theorem for derivations that is not available in the sequent calculus. The main novelty of our system is that all the rules are local : contraction, in particular, is reduced to atomic form. This should be interesting for distributed proof-search and also for complexity theory, since the computational cost of applying each rule is bounded.

Abstract. We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search.