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144
KripkeStyle Models for Typed Lambda Calculus
 Annals of Pure and Applied Logic
, 1996
"... The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripkestyle models. In categorical terms, our Kripke ..."
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Cited by 44 (3 self)
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The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripkestyle models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those familiar with Kripke models of modal or intuitionistic logics, Kripke lambda models are likely to seem adequately \semantic." However, when viewed as cartesian closed categories, they do not have the property variously referred to as concreteness, wellpointedness, or having enough points. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We also develop some properties of logical relations ...
On Behavioural Abstraction and Behavioural Satisfaction in HigherOrder Logic
, 1996
"... The behavioural semantics of specifications with higherorder logical formulae as axioms is analyzed. A characterization of behavioural abstraction via behavioural satisfaction of formulae in which the equality symbol is interpreted as indistinguishability, which is due to Reichel and was recently g ..."
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Cited by 25 (5 self)
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The behavioural semantics of specifications with higherorder logical formulae as axioms is analyzed. A characterization of behavioural abstraction via behavioural satisfaction of formulae in which the equality symbol is interpreted as indistinguishability, which is due to Reichel and was recently generalized to the case of firstorder logic by Bidoit et al, is further generalized to this case. The fact that higherorder logic is powerful enough to express the indistinguishability relation is used to characterize behavioural satisfaction in terms of ordinary satisfaction, and to develop new methods for reasoning about specifications under behavioural semantics. 1 Introduction An important ingredient in the use of algebraic specifications to describe data abstractions is the concept of behavioural equivalence between algebras, which seems to appropriately capture the "black box" character of data abstractions, see e.g. [GGM76], [GM82], [ST87] and [ST95]. Roughly speaking (since there ...
Proof Transformations in HigherOrder Logic
, 1987
"... We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, ..."
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Cited by 21 (5 self)
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We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, nonanalytic proofs are represented by natural deductions. A nondeterministic translation algorithm between expansion proofs and Hdeductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cutelimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higherorder system, but is proven for the firstorder fragment. We extend the translations to a nonanalytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a nonextensional equality is definable in our system of type theory. Next we extend analytic and nonanalytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 19 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
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 19 (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...
Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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Cited by 18 (6 self)
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
Secondorder logic and foundations of mathematics
 The Bulletin of Symbolic Logic
, 2001
"... We discuss the differences between firstorder set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if secondorder logic is understood in its full semantics capable of characterizing categorical ..."
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Cited by 17 (3 self)
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We discuss the differences between firstorder set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if secondorder logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. Firstorder set theory and secondorder logic are not radically different: the latter is a major fragment of the former. 1
HigherOrder Tableaux
, 1995
"... Even though higherorder calculi for automated theorem proving are rather old, tableau calculi have not been investigated yet. This paper presents two free variable tableau calculi for higherorder logic that use higherorder unification as the key inference procedure. These calculi differ in the ..."
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Cited by 16 (6 self)
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Even though higherorder calculi for automated theorem proving are rather old, tableau calculi have not been investigated yet. This paper presents two free variable tableau calculi for higherorder logic that use higherorder unification as the key inference procedure. These calculi differ in the treatment of the substitutional properties of equivalences. The first calculus is equivalent in deductive power to the machineoriented higherorder refutation calculi known from the literature, whereas the second is complete with respect to Henkin's general models.