A framework for defining logics

Abstract

The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed λ-calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Löf's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools such as proof editors and proof checkers can be constructed.

An Authoring Tool for Informal and Formal Requirements Specifications

“…GF, like other logical frameworks in the LF [6] tradition, uses a higher-order type theory with dependent types. In this type theory, it is possible to define logical calculi, as well as mathematical theories, simply by type signatures.…”

On the Specification of Sequent Systems

“…Such meta-logics or logical frameworks have been mostly based on intuitionistic logic (see, for example, [FM88,NM88,Har93]) or dependent types (see [Pfn89]) in which quantification at (non-predicate) higher-order types is available. These computer systems have been used as meta-languages to automate various aspects of different logics.…”

Small Proof Witnesses for LF

“…To illustrate the notation and explain the problem of small proof witnesses, we will first give an example of encoding the natural deduction calculus in the logical framework LF using higher-order logic programming following the methodology in Harper et al [12]. For more information on how to encode formal systems in LF, see for example Pfenning [21].…”

“…The Twelf system [22], an implementation of the logical framework LF [12], provides a general safety infrastructure to represent and execute safety policies via a higher-order logic program interpretation and has been employed in several proof-carrying code projects [4,8,3,9]. Higher-order logic programming extends first order logic programming along two orthogonal dimensions: First, dynamic assumptions may be generated and used during proof search.…”