We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science --- LICS'96 (E. Clarke editor), pp. 264--275, New Brunswick, NJ, July 27--30 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cut-elimination. 1 Introduction A logical framework is a formal system desig...

Rewriting logic [72] is proposed as a logical framework in which other logics can be represented, and as a semantic framework for the specification of languages and systems. Using concepts from the theory of general logics [70], representations of an object logic L in a framework logic F are understood as mappings L ! F that translate one logic into the other in a conservative way. The ease with which such maps can be defined for a number of quite different logics of interest, including equational logic, Horn logic with equality, linear logic, logics with quantifiers, and any sequent calculus presentation of a logic for a very general notion of "sequent," is discussed in detail. Using the fact that rewriting logic is reflective, it is often possible to reify inside rewriting logic itself a representation map L ! RWLogic for the finitely presentable theories of L. Such a reification takes the form of a map between the abstract data types representing the finitary theories of...

In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...

One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fully-expansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...

Given the different perspectives from which a complex software system has to be analyzed, the multiplicity of formalisms is unavoidable. This poses two important technical challenges: how to rigorously meet the need to interrelate formalisms, and how to reduce the duplication of effort in tool and specification building across formalisms. These challenges could be answered by adequate formal meta-tools that, when given the specification of a formal inference system, generate an efficient inference engine, and when given a specification of two formalisms and a translation, generate an actual translator between them. Similarly, module composition operations that are logic-independent, but that at present require costly implementation efforts for each formalism, could be provided for logics in general by module algebra generator meta-tools. The foundations of meta-tools of this kind can be based on a metatheory of general logics. Their actual design and implementation can be based on appropriate logical frameworks having efficient implementations. This paper explains how the reflective logical framework of rewriting logic can be used, in conjunction with an efficient reflective implementation such as the Maude language, to design formal meta-tools such as those described above. The feasibility of these ideas and techniques has been demonstrated by a number of substantial experiments in which new formal tools and new translations between formalisms, efficient enough to be used in practice, have been generated. 1

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. 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...

. 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 ...

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.

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