Simon J. Ambler

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.

Multi-level meta-reasoning with higher-order abstract syntax

Combining Higher Order Abstract Syntax (HOAS) and (co)- induction is well known to be problematic. In previous work [1] we have described the implementation of a tool called Hybrid, within Isabelle HOL, which allows object logics to be represented using HOAS, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. In this paper we describe how to use it in a multi-level reasoning fashion, similar in spirit to other metalogics such FOλΔN and Twelf. By explicitly referencing provability, we solve the problem of reasoning by (co)induction in presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications. We demonstrate the method by formally verifying the correctness of a compiler for (a fragment) of Mini-ML, following [10]. To further exhibit the flexibility of our system, we modify the target language with a notion of non-well-founded closure, inspired by Milner & Tofte [16] and formally verify via co-induction a subject reduction theorem for this modified language.

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.

Duality and the completeness of the modal m-calculus

We consider the modal μ-calculus due to Kozen, which is a finitary modal logic with least and greatest fixed points of monotone operators. We extend the existing duality between the category of modal algebras with homomorphisms and the category of descriptive modal frames with contractions to the case of having fixed point operators. As a corollary, we obtain completeness results for two proof systems due to Kozen (finitary and infinitary) with respect to certain classes of modal frames. The rules are sound in every model, not only for validity.

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].

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.

Generalized logic and the representation of rings

Following the progression towards weaker logics, a number of authors have considered the notion of a ‘sheaf over a quantale’ or, equivalently, a ‘quantale valued set’. In this paper, we use ideas from enriched category theory to motivate the definition of a ‘quantic sheaf’. Given a localic subquantale of Q, a quantic sheaf over Q gives a sheaf in the usual sense. As an application, we derive a series of sheaf representations for commutative rings including the familiar Pierce representation.

On Duality for the Modal µ-Calculus

We consider the modal μ-calculus due to Kozen, which is a finitary modal logic with least and greatest fixed points of monotone operators. We extend the existing duality between the category of modal algebras with homomorphisms and the category of descriptive modal frames with contractions to the case of having fixed point, operators. As a corollary, we obtain a completeness result for Kozens original system with respect to a certain class of modal frames.