Research in dependent type theories [M-L71a] has, in the past, concentrated on its use in the presentation of theorems and theorem-proving. This thesis is concerned mainly with the exploitation of the computational aspects of type theory for programming, in a context where the properties of programs may readily be specified and established. In particular, it develops technology for programming with dependent inductive families of datatypes and proving those programs correct. It demonstrates the considerable advantage to be gained by indexing data structures with pertinent characteristic information whose soundness is ensured by typechecking, rather than human effort. Type theory traditionally presents safe and terminating computation on inductive datatypes by means of elimination rules which serve as induction principles and, via their associated reduction behaviour, recursion operators [Dyb91]. In the programming language arena, these appear somewhat cumbersome and give rise to unappealing code, complicated by the inevitable interaction between case analysis on dependent types and equational reasoning on their indices which must appear explicitly in the terms. Thierry Coquand’s proposal [Coq92] to equip type theory directly with the kind of

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We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed -calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO. Keywords: Hilbert and Natural-Deduction proof systems for Modal Logics, Logical Frameworks, Typed -calculus, Proof Assistants. Introduction In this paper we address the issue of designing proof development environments (i.e. "proof editors" or, even better, "proof assistants") for Modal Logics, in the style of [11, 12]. To this end, we explore the possibility of using Logical Frameworks (LF's) based on Type Theory...

First-order unification algorithms (Robinson, 1965) are traditionally implemented via general recursion, with separate proofs for partial correctness and termination. The latter tends to involve counting the number of unsolved variables and showing that this total decreases each time a substitution enlarges the terms. There are many such proofs in the literature, for example, (Manna & Waldinger, 1981; Paulson, 1985; Coen, 1992; Rouyer, 1992; Jaume, 1997; Bove, 1999). This paper

This paper makes several contributions towards a clarified view of schema-based program development. First, we propose that schemata can be understood, formalized, and used in a simple way: program development schemata are derived rules. We mean this in the standard sense of a derived rule of inference in logic. A schema like Figure i can be formulated as a rule stating that the conclusion follows from the premises defining F, G, and the applicability conditions. By deriving the rule in an axiomatic theory, we validate a semantic statement about it: the conclusion of the rule holds in every model where both the axioms of the theory and the premises of the rule are true. Hence, by selecting a language to work in we control which development schemata are formalizable, and by selecting a theory we determine which schemata are derivable

We report on a case study in using logical frameworks to support the formalization of programming calculi and their application to deduction-based program synthesis. Within a conservative extension of higher-order logic implemented in the Isabelle system, we derived rules for program development that can simulate those of the deductive tableau proposed by Manna and Waldinger. We have used the resulting theory to synthesize a library of verified programs, focusing on sorting algorithms. Our experience suggests that the methodology we propose is well suited both to implement and use programming calculi, extend them, partially automate them, and even formally reason about their correctness. 1 Introduction Over the last few decades, a variety of methodologies for deductive software synthesis, transformation, and refinement from specification have been suggested, e.g., [4, 5, 8, 9, 12]. Our research investigates general frameworks that support such program development formalisms. That is, ...

Based on a representation of primitive proof objects as - terms, which has been built into the theorem prover Isabelle recently, we propose a generic framework for program extraction. We show how this framework can be used to extract functional programs from proofs conducted in a constructive fragment of the object logic Isabelle/HOL. A characteristic feature of our implementation of program extraction is that it produces both a program and a correctness proof. Since the extracted program is available as a function within the logic, its correctness proof can be checked automatically inside Isabelle.

. Formal Synthesis is a methodology developed at Kent for combining circuit design and verification, where a circuit is constructed from a proof that it meets a given formal specification. We have reinterpreted this methodology in Isabelle's theory of higher-order logic so that circuits are incrementally built during proofs using higher-order resolution. Our interpretation simplifies and extends Formal Synthesis both conceptually and in implementation. It also supports integration of this development style with other proof-based synthesis methodologies and leads to techniques for developing new classes of circuits, e.g., recursive descriptions of parametric designs. Keywords: Hardware verification and synthesis, theorem proving, higher-order logic, higherorder unification. 1. Introduction Verification by formal proof is time intensive and this is a burden in bringing formal methods into software and hardware design. One approach to reducing the verification burden is to combine develop...

Since the early days of programming and automated reasoning, researchers have developed methods for systematically constructing programs from their specifications. Especially the last decade has seen a flurry of activities including the advent of specialized conferences, such as LOPSTR, covering the synthesis of programs in computational logic. In this paper we analyze and compare three state-of-the-art methods for synthesizing recursive programs in computational logic. The three approaches are constructive/deductive synthesis, schema-guided synthesis, and inductive synthesis. Our comparison is carried out in a systematic way where, for each approach, we describe the key ideas and synthesize a common running example. In doing so, we explore the synergies between the approaches, which we believe are necessary in order to achieve progress over the next decade in this field.