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

Bedwyr is a generalization of logic programming that allows model checking directly on syntactic expressions possibly containing bindings. This system, written in OCaml, is a direct implementation of two recent advances in the theory of proof search. The first is centered on the fact that both finite success and finite failure can be captured in the sequent calculus by incorporating inference rules for definitions that allow fixed points to be explored. As a result, proof search in such a sequent calculus can capture simple model checking problems as well as may and must behavior in operational semantics. The second is that higherorder abstract syntax is directly supported using term-level λ-binders and the quantifier known as ∇. These features allow reasoning directly on expressions containing bound variables. 2

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nigam at lix.inria.fr dale.miller at inria.fr Abstract. We consider the problem of automating and checking the use of previously proved lemmas in the proof of some main theorem. In particular, we call the collection of such previously proved results a table and use a partial order on the table’s entries to denote the (provability) dependency relationship between tabled items. Tables can be used in automated deduction to store previously proved subgoals and in interactive theorem proving to store a sequence of lemmas introduced by a user to direct the proof system towards some final theorem. Tables of literals can be incorporated into sequent calculus proofs using two ideas. First, cuts are used to incorporate tabled items into a proof: one premise of the cut requires a proof of the lemma and the other branch of the cut inserts the lemma into the set of assumptions. Second, to ensure that lemma is not reproved, we exploit the fact that in focused proofs, atoms can have different polarity. Using these ideas, simple logic engines that do focused proof search (such as logic programming interpreters) are able to check proofs for correctness with guarantees that previous work is not redone. We also discuss how a table can be seen as a proof object and discuss some possible uses of tables-as-proofs. 1

Logical frameworks based on intuitionistic or linear logics with higher-type quantification have been successfully used to give high-level, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that the subject-reduction theorem holds. One challenge in developing a framework for such reasoning is that higherorder abstract syntax (HOAS), an elegant and declarative treatment of object-level abstraction and substitution,is difficult to treat in proofs involving induction. In this paper, we present a meta-logic that can be used to reason about judgments coded using HOAS; this meta-logic is an extension of a simple intuitionistic logic that admits higher-order quantification over simply typed-terms (key ingredients for HOAS) as well as induction and a notion of definition. The latter concept of a definition is a proof-theoretic device that allows certain theories to be treated as “closed ” or as defining fixed points. The resulting meta-logic can specify various logical frameworks and a large range of judgments regarding programming languages and inference systems. We illustrate this point through examples, including the admissibility of cut for a simple logic and subject reduction, determinacy of evaluation, and the equivalence of SOS and natural semantics presentations of evaluation for a simple functional programming language. 1.

Logic programming can be given a foundation in sequent calculus by viewing computation as the process of building a cut-free sequent proof bottom-up. The first accounts of logic programming as proof search were given in classical and intuitionistic logic. Given that linear logic allows richer sequents and richer dynamics in the rewriting of sequents during proof search, it was inevitable that linear logic would be used to design new and more expressive logic programming languages. We overview how linear logic has been used to design such new languages and describe briefly some applications and implementation issues for them.

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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.

Some recent theoretical work in proof search has illustrated that it is possible to combine the following two computational principles into one computational logic. 1. A symmetric treatment of finite success and finite failure. This allows capturing both aspects of may and must behavior in operational semantics and mixing model checking and logic programming. 2. Direct support for λ-tree syntax, as in λProlog, via term-level λ-binders, higher-order pattern unification, and the ∇-quantifier. All these features have a clean proof theory. The combination of these features allow, for example, specifying rather declarative approaches to model checking syntactic expressions containing bindings. The Bedwyr system is intended as an implementation of these computational logic principles. Why the name Bedwyr? In the legend of King Arthur and the round table, several knights shared in the search for the holy grail. The name of one of them, Parsifal, is used for an INRIA team associated with the “Slimmer ” effort. Bedwyr was another one of those knights. Wikipedia (using the spelling “Bedivere”) mentions that Bedwyr appears in Monty Python and the Holy Grail where he is “portrayed as a master of the extremely odd logic in the ancient times, whom occasionally blunders. ” Bedwyr is a re-implementation and rethinking ∗ Support has been obtained for this work from the following sources: from INRIA through

. Forum [36], a powerful logic formalism based on Higher Order Linear Logic, is particularly suited to specify and reason about complex programs and systems. Ehhf [12], a subset of Forum, models many interesting logic programming extensions towards O.O. and concurrent systems and can be viewed as a very high level logic programming specification language. The paper presents some results in this direction, namely the specification in Ehhf of Chimera, an Active, Object-Oriented and Deductive Database System. Keywords: Linear Logic, Object-Oriented and Deductive Databases. 1 Introduction Proof theory and automated deduction have provided relevant contributions to computer science, in particular in the fields of high-level programming languages and formal verification of software. Many different logics have been proposed and used for these purposes. We will work with Linear logic [22] with the aim to use it as a theoretical foundation for modern and powerful specification language...