A Set of Postulates for the Foundation of Logic

Church, Alonzo

doi:10.2307/1968337

Cited by 372 publications

(126 citation statements)

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“…Church [21] proposed a version of higher-order logic, called simple type theory (in the following referred to as HOL), which he built on top of the simply typed λ-calculus [19,20]. The simply typed λ-calculus augments the untyped λ-calculus, as studied by Alonzo Church in the 1930s, with simple types.…”

Section: Classical Higher-order Logicmentioning

confidence: 99%

Sweet SIXTEEN: Automation via Embedding into Classical Higher-Order Logic

Steen

Benzmüller

2016

LLP

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Abstract. An embedding of many-valued logics based on SIXTEEN in classical higher-order logic is presented. SIXTEEN generalizes the fourvalued set of truth degrees of Dunn/Belnap's system to a lattice of sixteen truth degrees with multiple distinct ordering relations between them. The theoretical motivation is to demonstrate that many-valued logics, like other non-classical logics, can be elegantly modeled (and even combined) as fragments of classical higher-order logic. Equally relevant are the pragmatic aspects of the presented approach: interactive and automated reasoning in many-valued logics, which have broad applications in computer science, artificial intelligence, linguistics, philosophy and mathematics, become readily enabled with state of the art reasoning tools for classical higher-order logic.

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Section: Classical Higher-order Logicmentioning

confidence: 99%

Sweet SIXTEEN: Automation via Embedding into Classical Higher-Order Logic

Steen

Benzmüller

2016

LLP

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“…Its aim is to provide a universal model that is terse enough for mathematical proofs of general properties about aggregate programming and the aggregate/local relationship, just as λ-calculus [36] provides for functional programming, π -calculus for parallel programming [37] or Featherweight Java [38] for object-oriented programming.…”

Section: Foundation: Field Calculusmentioning

confidence: 99%

Space–time programming

Beal

Viroli

2015

Phil. Trans. R. Soc. A.

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Computation increasingly takes place not on an individual device, but distributed throughout a material or environment, whether it be a silicon surface, a network of wireless devices, a collection of biological cells or a programmable material. Emerging programming models embrace this reality and provide abstractions inspired by physics, such as computational fields, that allow such systems to be programmed holistically, rather than in terms of individual devices. This paper aims to provide a unified approach for the investigation and engineering of computations programmed with the aid of space-time abstractions, by bringing together a number of recent results, as well as to identify critical open problems.

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“…This dominance was also fuelled by the discovery of the so-called CurryHoward-de Bruijn (CHdB) correspondence (see [3,13,4], also [42,9,41]) between Gentzen's intuitionistic Natural Deduction (ND, [8]) and Church's simply-typed λ-calculus ( [1]) and the cordial endorsement from computer science that followed (see e.g. [35]).…”

Section: Introductionmentioning

confidence: 99%

Algorithmic Theories of Problems. A Constructive and a Non-Constructive Approach

Pezlar

2017

LLP

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Abstract. In this paper we examine two approaches to the formal treatment of the notion of problem in the paradigm of algorithmic semantics. Namely, we will explore an approach based on Martin-Löf's Constructive Type Theory (CTT), which can be seen as a direct continuation of Kolmogorov's original calculus of problems, and an approach utilizing Tichý's Transparent Intensional Logic (TIL), which can be viewed as a non-constructive attempt of interpreting Kolmogorov's logic of problems. In the last section we propose Kolmogorov and CTT-inspired modifications to TILbased approach. The focus will be on non-empirical (i.e., mathematical and logical) problems only.

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A Set of Postulates for the Foundation of Logic

Cited by 372 publications

References 0 publications

Sweet SIXTEEN: Automation via Embedding into Classical Higher-Order Logic

Sweet SIXTEEN: Automation via Embedding into Classical Higher-Order Logic

Space–time programming

Algorithmic Theories of Problems. A Constructive and a Non-Constructive Approach

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