not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a meta-logical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, and experimental results related to the areas of programming languages, type theory, and logics. Design: The meta-logical framework extends the logical framework LF [HHP93] by a meta-logic M + 2. This design is novel and unique since it allows higher-order encodings of deductive systems and induction principles to coexist. On the one hand, higher-order representation techniques lead to concise and direct encodings of programming languages and logic calculi. Inductive de nitions on the other hand allow the formalization of properties about deductive systems, such as the proof that an operational semantics preserves types or the proof that a logic is is a proof calculus whose proof terms are recursive functions that may be consistent.M +

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.

Higher-order representation techniques allow elegant encodings of logics and programming languages in the logical framework LF, but unfortunately they are fundamentally incompatible with induction principles needed to reason about them. In this paper we develop a meta-logic M_2 which allows inductive reasoning over LF encodings, and describe its implementation in Twelf, a special-purpose automated theorem prover for properties of logics and programming languages. We have used Twelf to automatically prove a number of non-trivial theorems, including type preservation for Mini-ML and the deduction theorem for intuitionistic propositional logic.

Abstract. Combining Higher Order Abstract Syntax (HOAS) and induction is well known to be problematic. We have implemented a tool called Hybrid, within Isabelle HOL, which does allow object logics to be represented using HOAS, and reasoned about using tactical theorem proving in general and principles of (co)induction in particular. In this paper we describe Hybrid, and illustrate its use with case studies. We also provide some theoretical adequacy results which underpin our practical work. 1 Introduction Many people are concerned with the development of computing systems which can be used to reason about and prove properties of programming languages. However, developing such systems is not easy. Difficulties abound in both practical implementation and underpinning theory. Our paper makes both a theoretical and practical contribution to this research area. More precisely, this paper concerns how to reason about object level logics with syntax involving variable binding--note that a programming language can be presented as an example of such an object logic. Our contribution is the provision of a mechanized tool, Hybrid, which has been coded within Isabelle HOL, and- provides a form of logical framework within which the syntax of an object

Intuitionistic and linear logics can be used to specify the operational semantics of transition systems in various ways. We consider here two encodings: one uses linear logic and maps states of the transition system into formulas, and the other uses intuitionistic logic and maps states into terms. In both cases, it is possible to relate transition paths to proofs in sequent calculus. In neither encoding, however, does it seem possible to capture properties, such as simulation and bisimulation, that need to consider all possible transitions or all possible computation paths. We consider augmenting both intuitionistic and linear logics with a proof theoretical treatment of definitions. In both cases, this addition allows proving various judgments concerning simulation and bisimulation (especially for noetherian transition systems). We also explore the use of infinite proofs to reason about infinite sequences of transitions. Finally, combining definitions and induction into sequent calculus proofs makes it possible to reason more richly about properties of transition systems completely within the formal setting of sequent calculus.

Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K.

We present a meta-logic that contains a new quantifier (for encoding "generic judgment") and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite π-calculus within this meta-logic. Since we

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

david.baelde at ens-lyon.org dale.miller at inria.fr Abstract. The first-order theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and?), we add least and greatest fixed point operators. The resulting logic, which we call µMALL = , satisfies two fundamental proof theoretic properties. In particular, µMALL = satisfies cut-elimination, which implies consistency, and has a complete focused proof system. This second result about focused proofs provides a strong normal form for cut-free proof structures that can be used, for example, to help automate proof search. We then consider applying these two results about µMALL = to derive a focused proof system for an intuitionistic logic extended with induction and co-induction. The traditional approach to encoding intuitionistic logic into linear logic relies heavily on using the exponentials, which unfortunately weaken the focusing discipline. We get a better focused proof system by observing that certain fixed points satisfy the structural rules of weakening and contraction (without using exponentials). The resulting focused proof system for intuitionistic logic is closely related to the one implemented in Bedwyr, a recent model checker based on logic programming. We discuss how our proof theory might be used to build a computational system that can partially automate induction and co-induction. 1