Feferman has proposed a system, FS 0 , as an alternative framework for encoding logics and also for reasoning about those encodings. We have implemented a version of this framework and performed experiments that show that it is practical. Specifically, we describe a formalisation of predicate calculus and the development of an admissible rule that manipulates formulae with bound variables. This application will be of interest to researchers working with frameworks that use mechanisms based on substitution in the lambda calculus to implement variable binding and substitution in the declared logic directly. We suggest that meta-theoretic reasoning, even for a theory using bound variables, is not as difficult as is often supposed, and leads to more powerful ways of reasoning about the encoded theory. x 1 Introduction: why metamathematics? A logical framework is a formal theory that is designed for the purpose of describing other formal theories in a uniform way, and for making the work ...

This paper describes a reasoning system, called GETFOL, able to introspect (the code implementing) its own deductive machinery, to reason deductively about it in a declarative metatheory and to produce new executable code which can then be pushed back into the underlying implementation. In this paper we discuss the general architecture of GETFOL and the problems related to its implementation.

A metalogical framework is a logic with an associated methodology that is used to represent other logics and to reason about their metalogical properties. We propose that logical frameworks can be good metalogical frameworks when their logics support reective reasoning and their theories always have initial models. We present a concrete realization of this idea in rewriting logic. Theories in rewriting logic always have initial models and this logic supports reective reasoning. This implies that inductive reasoning is valid when proving properties about the initial models of theories in rewriting logic, and that we can use reection to reason at the metalevel about these properties. In fact, we can uniformly reect induction principles for proving metatheorems about rewriting logic theories and their parameterized extensions. We show that this reective methodology provides an eective framework for dierent, non-trivial, kinds of formal metatheoretic reasoning; one can...

. We show how logical frameworks can provide a basis for logic program synthesis. With them, we may use first-order logic as a foundation to formalize and derive rules that constitute program development calculi. Derived rules may be in turn applied to synthesize logic programs using higher-order resolution during proof that programs meet their specifications. We illustrate this using Paulson's Isabelle system to derive and use a simple synthesis calculus based on equivalence preserving transformations. 1 Introduction Background In 1969 Dana Scott developed his Logic for Computable Functions and with it a model of functional program computation. Motivated by this model, Robin Milner developed the theorem prover LCF whose logic PP used Scott's theory to reason about program correctness. The LCF project [13] established a paradigm of formalizing a programming logic on a machine and using it to formalize different theories of functional programs (e.g., strict and lazy evaluation) and the...

We describe an implementation and a proof of correctness of binary decision diagrams (BDDs), completely formalized in Coq. This allows us to run BDD-based algorithms inside Coq and paves the way for a smooth integration of symbolic model checking in the Coq proof assistant by using reflection. It also gives us, by Coq's extraction mechanism, certified BDD algorithms implemented in Caml. We also implement and prove correct a garbage collector for our implementation of BDDs inside Coq. Our experiments show that this approach works in practice, and is able to solve both relatively hard propositional problems and actual industrial hardware verification tasks.

. Feferman has proposed FS 0 , a theory of finitary inductive systems, as a framework theory that allows a user to reason both in and about an encoded theory. I look here at how practical FS 0 really is. To this end I formalise a sequent calculus presentation of classical propositional logic, and show this can be used for work in both the theory and the metatheory. the latter is illustrated with a discussion of a proof of Gentzen's Hauptsatz. Contents x 1 Introduction 2 x 1.1 Background : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 x 1.2 Outline of paper : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 x 2 The theory FS 0 and notational conventions 4 x 2.1 What is FS 0 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 x 3 An informal description of Gentzen's calculus 5 x 3.1 The language : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 x 3.2 The calculus for classical propositional logic : : : : : : : : : : : : 6 x 4 Formalising the ...

This paper presents the concepts and the semantics of a transformation-calculus TC that is generic wrt. concrete object languages. Built upon an object language description given by theory in higher-order logics (see [Andr 86]), TC provides context-sensitive rules in which requirements on the context of a redex can be imposed, and integrates a restricted form of extended rewriting. Furthermore, rules may be higher-order in order to represent tactical combinators and to model "parametric transformations". This work can be seen as a specification of transformation systems and a foundation for correctness-proofs of transformations.

We show that the generalized variant of rewriting logic where the underlying equational specifications are membership equational theories, and where the rules are conditional and can have equations, memberships and rewrites in the conditions is reflective. We also show that membership equational logic, many-sorted equational logic, and Horn logic with equality are likewise reflective. These results provide logical foundations for reflective languages and tools based on these logics, and in particular for the Maude language itself. 1

The reflective capabilities of rewriting logic and their efficient implementation in the Maude language can be exploited to endow a reflective language like Maude with a module algebra in which structured theories can be combined and transformed by means of a rich collection of module operations. We have followed this approach and we have used the specification of such a module algebra as its implementation, including a user interface and an execution environment for it. The high level at which the specification of the module algebra has been given makes this approach particularly attractive when compared to conventional implementations, because of its shorter development time and the greater flexibility, maintainability, and extensibility that it affords. We explain the general principles of the reflective design of the module algebra and explain the categorical semantics of parameterized theories, modules and views and their instantiation, and the reflective algebraic specification of the different module and view operations.

We examine a problem for machine supported metatheory. There are statements true about a theory that are true of some (but only some) extensions; however standard theory-structuring facilities do not support selective inheritance. We use the example of the deduction theorem for modal logic and show how a statement about a theory can explicitly formalize the closure conditions extensions should satisfy for it to remain true. We show how metatheories based on inductive denitions allow theories and general metatheorems to be organized this way, and report on a case study using the theory FS0 . 1 Introduction Hierarchical theory structuring plays an important role in the application of theorem provers to nontrivial problems, and many systems provide support for it. For example HOL [6], Isabelle [13] and their predecessor LCF [7] support simple theory hierarchies. In these systems a theory is a specication of a language, using types and typed constants, and a collection of rules...