Roy L. Crole

Algebraic and coalgebraic methods in the mathematics of program construction

Ordered Sets and Complete Lattices.- Algebras and Coalgebras.- Galois Connections and Fixed Point Calculus.- Calculating Functional Programs.- Algebra of Program Termination.- Exercises in Coalgebraic Specification.- Algebraic Methods for Optimization Problems.- Temporal Algebra.

New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic

Abstract This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi (in “Proceedings, 4th Annual Symposium on Logic in Computer Science,” pp. 14–23, IEEE Comput. Soc. Press, Washington, 1989) and contains a version of Martin-Lofs “iteration type” (in “Proceedings, Workshop on Semantics in Programming Laguages,” Chalmers University, 1983) . The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a category-theoretic version of the “logical relations” method ( Plotkin, unpublished lecture notes from CSLI Summer School, 1985 ).

Combining Higher Order Abstract Syntax with Tactical Theorem Proving and (Co)Induction

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.

A Hybrid Encoding of Howe's Method for Establishing Congruence of Bisimilarity

Abstract We give a short description of Hybrid, a new tool for interactive theorem proving. It provides a form of Higher Order Abstract Syntax (HOAS) combined consistently with induction and coinduction. We present a case study illustrating the use of Hybrid for reasoning about the lazy lambda-calculus. In particular, we prove that the standard notion of simulation is a precongruence. Although such a proof is not new, the development is non-trivial, and we attempt to illustrate the advantages of using Hybrid, as well as some issues which are being addressed as further work.

A definitional approach to primitivexs recursion over higher order abstract syntax

It is well known that there are problems associated with formal systems which attempt to combine higher order abstract syntax (HOAS) with principles of induction and recursion. We describe a formal system, called Bsyntax, which we have implemented in Isabelle HOL. Our contribution is to prove the existence of a combinator for primitive recursion with parameters over HOAS. The definition of the combinator is facilitated by the use of terms with infinite contexts. In particular, our work is purely definitional, and is thus consistent with classical logic and choice. An immediate payoff is that we obtain a primitive recursive definition of higher order substitution. We give a presheaf model of Bsyntax, providing additional semantic validation of Bsyntaxs principles of recursion. We outline an application of our work to mechanized reasoning about the compiler intermediate language MIL-lite [2].

New foundations for fixpoint computations

A novel higher-order typed constructive predicate logic for fixpoint computations which exploits the categorical semantics of computations introduced by E. Moggi (1989) and contains a strong version of P. Martin-Lofs (1983) iteration type is introduced. The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a category-theoretic version of the logical relations method.<<ETX>>

A Sound Metalogical Semantics for Input/Output Effects

We study the longstanding problem of semantics for input/output (I/O) expressed using side-effects. Our vehicle is a small higher-order imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity and prove it is a congruence using Howes method.

Mechanized Operational Semantics via (Co)Induction

We give a fully automated description of a small programming language PL in the theorem prover Isabelle98. The language syntax and semantics are encoded, and we formally verify a range of semantic properties. This is achieved via uniform (co)inductive methods. We encode notions of bisimulation and contextual equivalence. The main original contribution of this paper is a fully automated proof that PL bisimulation coincides with PL contextual equivalence.

Completeness of bisimilarity for contextual equivalence in linear theories

In this paper, we develop new variations of methods from operational semantics, and show how to apply these to a linear type theory which has a lazy operational semantics. In particular, we consider how one can establish contextual equivalences in a linear theory with function types and tensor types by instead establishing bisimulations. Thus bisimilarity is sound for contextual equivalence. Further, we show that bisimilarity is complete for contextual equivalence. We shall show that the notion of a program context in the linear setting is non-trivial. In particular, we give a definition of linear context which is amenable to mechanization in a theorem prover, and explain why a more naive approach to dealing with linear contexts would not be so tractable. The central contributions of the paper are: the formulation, in a linear setting, of a good notion of program context and the associated contextual equivalence; an adaptation of Howe’s method; the notion of a linear precongruence; and a proof that bisimilarity is sound and complete for contextual equivalence.

Nominal Lambda Calculus: An Internal Language for FM-Cartesian Closed Categories

Reasoning about atoms (names) is difficult. The last decade has seen the development of numerous novel techniques. For equational reasoning, Clouston and Pitts introduced Nominal Equational Logic (NEL), which provides judgements of equality and freshness of atoms. Just as Equational Logic (EL) can be enriched with function types to yield the lambda-calculus (LC), we introduce NLC by enriching NEL with (atom-dependent) function types and abstraction types. We establish meta-theoretic properties of NLC; define -cartesian closed categories, hence a categorical semantics for NLC; and prove soundness & completeness by way of NLC-classifying categories. A corollary of these results is that NLC is an internal language for -cccs. A key feature of NLC is that it provides a novel way of encoding freshness via dependent types, and a new vehicle for studying the interaction of freshness and higher order types.