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10
From Total Equational to Partial First Order Logic
, 1998
"... The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to pa ..."
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Cited by 20 (8 self)
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The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to partiality, like (variants of) error algebras and ordersortedness are also discussed, showing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragment are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Some more powerful algebraic frameworks are sketched at the end. Equational specifications introduced in last chapter, are a powerful tool to represent the most common data types used in programming languages and their semantics. Indeed, Bergstra and Tucker have shown in a series of papers (see [BT87] for a complete exposition of results) that a data type is semicompu...
Collapsing Partial Combinatory Algebras
 HigherOrder Algebra, Logic, and Term Rewriting
, 1996
"... Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pc ..."
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Cited by 17 (2 self)
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Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We establish several facts concerning final collapses (maximal identification of elements). `En passant' we find another example of a pca that cannot be extended to a total one. 1
Proof Systems for Structured Specifications and Their Refinements
, 1999
"... Reasoning about specifications is one of the fundamental activities in the process of formal program development. This ranges from proving the consequences of a specification, during the prototyping or testing phase for a requirements speci cation, to proving the correctness of refinements (or imple ..."
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Cited by 13 (6 self)
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Reasoning about specifications is one of the fundamental activities in the process of formal program development. This ranges from proving the consequences of a specification, during the prototyping or testing phase for a requirements speci cation, to proving the correctness of refinements (or implementations) of specifications. The main proof techniques for algebraic specifications have their origin in equational Horn logic and term rewriting. These proof methods have been well studied in the case of nonstructured speci cations (see Chapters 9 and 10). For large systems of specifications built using the structuring operators of speci cation languages, relatively few proof techniques have been developed yet; for such proof systems, see [SB83, HST94, Wir91, Far92, Cen94, HWB97]. In this chapter we focus on proof systems designed particularly for modular specifications. The aim is to concentrate on the structuring concepts, while abstracting as much as possible from the par...
Constructing type systems over an operational semantics
 Journal of Symbolic Computation
, 1992
"... Type theories in the sense of MartinLof and the NuPRL system are based on taking as primitive a typefree programming language given by an operational semantics, and defining types as partial equivalence relations on the set of closed terms. The construction of a type system is based on a general f ..."
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Type theories in the sense of MartinLof and the NuPRL system are based on taking as primitive a typefree programming language given by an operational semantics, and defining types as partial equivalence relations on the set of closed terms. The construction of a type system is based on a general form of inductive definition that may either be taken as acceptable in its own right, or further explicated in terms of other patterns of induction. One such account, based on a general theory of inductivelydefined relations, was given by Allen. An alternative account, based on an essentially settheoretic argument, is presented. 1
Completing Partial Combinatory Algebras with Unique HeadNormal Forms
, 1996
"... In this note, we prove that having unique headnormal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CLterms as well as the pca of natural numbers with partial recursive function application can be exte ..."
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Cited by 3 (1 self)
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In this note, we prove that having unique headnormal forms is a sufficient condition on partial combinatory algebras to be completable. As application, we show that the pca of strongly normalizing CLterms as well as the pca of natural numbers with partial recursive function application can be extended to total combinatory algebras. 1.
Extending Partial Combinatory Algebras
, 1999
"... Introduction Consider a structure A = hA; s; k; \Deltai, where A is some set containing the distinguished elements s; k, equipped with a binary operation \Delta on A, called application, which may be partial. Notation 1.1. 1 Instead of a \Delta b we write ab; and in writing applicative expression ..."
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Introduction Consider a structure A = hA; s; k; \Deltai, where A is some set containing the distinguished elements s; k, equipped with a binary operation \Delta on A, called application, which may be partial. Notation 1.1. 1 Instead of a \Delta b we write ab; and in writing applicative expressions, the usual convention of association to the left is employed. So for elements a; b; c 2 A, the expression aba(ac) is short for ((a \Delta b) \Delta a) \Delta (a \Delta c). 2 ab # will mean that ab is defined; ab " means that ab is not defined. Obviously, an applicative expression
PROPOSITIONAL LOGIC
, 2008
"... Intuitionistic logic is an important variant of classical logic, but it is not as wellunderstood, even in the propositional case. In this thesis, we describe a faithful representation of intuitionistic propositional formulas as tree automata. This representation permits a number of consequences, inc ..."
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Intuitionistic logic is an important variant of classical logic, but it is not as wellunderstood, even in the propositional case. In this thesis, we describe a faithful representation of intuitionistic propositional formulas as tree automata. This representation permits a number of consequences, including a characterization theorem for free Heyting algebras, which are the intutionistic analogue of free Boolean algebras, and a new algorithm for solving equations over intuitionistic propositional logic. BIOGRAPHICAL SKETCH
BIOGRAPHICAL SKETCH
, 2008
"... Intuitionistic logic is an important variant of classical logic, but it is not as wellunderstood, even in the propositional case. In this thesis, we describe a faithful representation of intuitionistic propositional formulas as tree automata. This representation permits a number of consequences, in ..."
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Intuitionistic logic is an important variant of classical logic, but it is not as wellunderstood, even in the propositional case. In this thesis, we describe a faithful representation of intuitionistic propositional formulas as tree automata. This representation permits a number of consequences, including a characterization theorem for free Heyting algebras, which are the intutionistic analogue of free Boolean algebras, and a new algorithm for solving equations over intuitionistic propositional logic.