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Programming with Higher-Order Logic

, 2012
Formal systems that describe computations over syntactic structures occur frequently in computer science. Logic programming provides a natural framework for encoding and animating such systems. However, these systems often embody variable binding, a notion that must be treated carefully at a computational level.
Miller, Dale, Gopalan, Nadathur
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Boolean-Valued Second-Order Logic

Notre Dame Journal of Formal Logic, 2015
In so-called full second-order logic, the second-order variables range over all subsets and relations of the domain in question. In so-called Henkin second-order logic, every model is endowed with a set of subsets and relations which will serve as the ...
Jouko Väänänen
exaly   +2 more sources

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On connections and higher-order logic

Journal of Automated Reasoning, 1989
Mathematics Technical ...
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Hauptsatz for higher order logic

Journal of Symbolic Logic, 1968
I shall prove in this paper that Gentzen's Hauptsatz is extendible to simple type theory, i.e., to the predicate logic obtained by admitting quantification over predicates of arbitrary finite type and generalizing the second order quantification rules to cover quantifiers of other types.
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On Higher-Order Description Logics. [PDF]

, 2009
We investigate an extension of Description Logics with higher-order capabilities, based on Henkin-style semantics. Our study starts from the observation that the various possibilities of adding higher-order constructs to a DL form a spectrum of increasing expressive power, including domain metamodeling, i.e., using concepts and roles as predicate ...
DE GIACOMO, Giuseppe, LENZERINI, Maurizio, ROSATI, Riccardo   +2 more
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Higher-Order Transformation of Logic Programs

2001
It has earlier been assumed that a compositional approach to algorithm design and program transformation is somehow unique to functional programming. Elegant theoretical results codify the basic laws of algorithmics within the functional paradigm and with this paper we hope to demonstrate that some of the same techniques and results are applicable to ...
Silvija Seres, J. Michael Spivey
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Higher-Order Logic

1998
We now introduce the logical basis for program refinement, higher-order logic. This is an extension of the simply typed lambda calculus with logical connectives and quantifiers, permitting logical reasoning about functions in a very general way. In particular, it allows quantification over higher-order entities such as predicates and relations, a ...
Ralph-Johan Back, Joakim Wright
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Connections and higher-order logic

1986
Theorem proving is difficult and deals with complex phenomena. The difficulties seem to be compounded when one works with higher-order logic, but the rich expressive power of Church's formulation [10] [3] of this language makes research on theorem proving in this realm very worthwhile.
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On nonstandard models in higher order logic

Journal of Symbolic Logic, 1984
There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn ...
Christian Hort, Horst Osswald
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Higher-Order Logic

1983
What is nowadays the central part of any introduction to logic, and indeed to some the logical theory par excellence, used to be a modest fragment of the more ambitious language employed in the logicist program of Frege and Russell. ‘Elementary’ or ‘first-order’, or ‘predicate logic’ only became a recognized stable base for logical theory by 1930, when
Johan Van Benthem, Kees Doets
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