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Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing b ..."
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Cited by 114 (22 self)
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
Algorithms: A quest for absolute definitions
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTurin ..."
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Cited by 19 (9 self)
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTuring thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.
Computing and Information Compression: A Reply
 AI Communications
, 1994
"... An earlier article [25] discusses the proposition that the storage and processing of information in computers and in brains may often be understood as information compression. A subsequent article [15] criticises the computing aspects of [25] and research on the more specific conjecture that all for ..."
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Cited by 7 (7 self)
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An earlier article [25] discusses the proposition that the storage and processing of information in computers and in brains may often be understood as information compression. A subsequent article [15] criticises the computing aspects of [25] and research on the more specific conjecture that all forms of computing and formal reasoning may usefully be understood as information compression. The present article, which is intended to be intelligible without recourse to earlier articles, answers the main points in [15], tries to correct the many inaccuracies and misconceptions in that article, and discusses related issues. Topics which are discussed include: the way theories are or should be developed; the role of evidence in motivating research; apparent shortcomings in the Turing machine concept as a reason for seeking new principles of computing; the apparent conflict between the idea of `computing as compression' and the fact that computers may create redundancy  and how the contradict...
Operations on Proofs That Can Be Specified By Means of Modal Logic
"... Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynomials which ..."
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Cited by 2 (2 self)
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Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynomials which constitute the system of proof terms in LP. Proof polynomials are built from variables and constants by three operations on proofs: "\Delta" (application), "!" (proof checker), and "+" (choice). Here constants stand for canonical proofs of "simple facts", namely instances of propositional axioms and axioms of LP in a given proof system. We show that every operation on proofs that (i) can be specified in a propositional modal language and (ii) is invariant with respect to the choice of a proof system is realized by a proof polynomial.
Kolmogorov Complexity and NonDeterminism
"... We are concerned with Kolmogorov complexity of strings produced by nondeterministic algorithms. For this, we consider binary recursively enumerable relations, named description modes. We give conditions on the class of description modes to provide a Kolmogorov entropy. Within this framework, we ..."
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Cited by 1 (1 self)
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We are concerned with Kolmogorov complexity of strings produced by nondeterministic algorithms. For this, we consider binary recursively enumerable relations, named description modes. We give conditions on the class of description modes to provide a Kolmogorov entropy. Within this framework, we prove that there is a proper hierarchy of such nondeterministic modes. Then, we give a sharp estimation on the amount of information to turn a deterministic mode into a nondeterministic one, and inversely. Lastly, we show that deterministic modes are less ecient than nondeterministic modes from some rank. 1 Introduction Shen and Uspensky in [5] compared various standard denitions of Kolmogorov complexities based on very general description modes. In essence, a description mode is a binary recursively enumerable, (r.e.), relation. So when a description mode turns out to be the graph of a function, it denotes a deterministic computation. Our starting point is to consider description...
Kolmogorov Complexity and NonDeterminism
"... We are concerned with Kolmogorov complexity of strings produced by nondeterministic algorithms. For this, we consider binary recursively enumerable relations, named description modes. We give conditions on the class of description modes to provide a Kolmogorov entropy. Within this framework, we ..."
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Cited by 1 (1 self)
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We are concerned with Kolmogorov complexity of strings produced by nondeterministic algorithms. For this, we consider binary recursively enumerable relations, named description modes. We give conditions on the class of description modes to provide a Kolmogorov entropy. Within this framework, we prove that there is a proper hierarchy of such nondeterministic modes. Then, we give a sharp estimation on the amount of information to turn a deterministic mode into a nondeterministic one, and inversely. Lastly, we show that deterministic modes are less efficient than nondeterministic modes from some rank.
Programmable reconfiguration of Physarum
, 901
"... Plasmodium of Physarum polycephalum is a large cell capable of solving graphtheoretic, optimization and computational geometry problems due to its unique foraging behavior. Also the plasmodium is unique biological substrate that mimics universal storage modification machines, namely the KolmogorovU ..."
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Plasmodium of Physarum polycephalum is a large cell capable of solving graphtheoretic, optimization and computational geometry problems due to its unique foraging behavior. Also the plasmodium is unique biological substrate that mimics universal storage modification machines, namely the KolmogorovUspensky machine. In the plasmodium implementation of the storage modification machine data are represented by sources of nutrients and memory structure by protoplasmic tubes connecting the sources. In laboratory experiments and simulation we demonstrate how the plasmodiumbased storage modification machine can be programmed. We show execution of the following operations with active zone (where computation occurs): merge two active zones, multiple active zone, translate active zone from one data site to another, direct active zone. Results of the paper bear twofold value: they provide a basis for programming unconventional devices based on biological substrates and also shed light on behavioral patterns of the plasmodium. Keywords: Physarum polycephalum, KolmogorovUspensky machine, pattern formation, morphogenesis, graph theory
Towards an ObjectOriented Programming Language for Physarum Polycephalum Computing
"... Abstract. In the paper, we present foundations of a new objectoriented programming language for Physarum polycephalum computing. Both, theoretical foundations and assumptions for a language specification are considered. Physarum polycephalum is a onecell organism. In the phase of plasmodium, its b ..."
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Abstract. In the paper, we present foundations of a new objectoriented programming language for Physarum polycephalum computing. Both, theoretical foundations and assumptions for a language specification are considered. Physarum polycephalum is a onecell organism. In the phase of plasmodium, its behavior can be regarded as a biological substrate that implements the KolmogorovUspensky machine which is the most generalized and natureoriented version of a mathematical machine. The proposed language will be used for developing programs for Physarum polycephalum by the spatial configuration of stationary nodes (inputs).