The Experts below are selected from a list of 108 Experts worldwide ranked by ideXlab platform
Wilfried Sieg - One of the best experts on this subject based on the ideXlab platform.
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CiE - What Is the Concept of Computation
Sailing Routes in the World of Computation, 2018Co-Authors: Wilfried SiegAbstract:This essay examines the two central and equivalent ways of introducing rigorous notions of computation as starting-points. The first way explicates Effective Calculability of number theoretic functions as the (uniform) Calculability of their values in formal calculi; Godel, Church and Kleene initially pursued this way. The other way views mechanical procedures as transforming finite configurations via production rules, a path Turing and Post took. For both explications, one can prove important mathematical theorems, namely, absoluteness and reducibility results. These results are of great interest, but problematic when viewed as justifying the adequacy of the rigorous notions. However, the theorems and their proofs reveal features that motivate the abstract concept of a computable dynamical system. That is, so I argue, the appropriate structural axiomatization of computation.
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Church Without Dogma: Axioms for Computability
2018Co-Authors: Wilfried SiegAbstract:Church's and Turing's theses dogmatically assert that an informal notion of Effective Calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of Calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses. To investigate Effective Calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. The analysis leads to axioms for discrete dynamical systems (representing human and machine computations) and allows the reduction of models of these axioms to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations
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Church Without Dogma: Axioms for computability
2013Co-Authors: Wilfried SiegAbstract:Abstract. Church’s and Turing’s theses assert dogmatically that an informal notion of Effective Calculability is adequately captured by a particular mathematical concept of computabilty. I present analyses of Calculability that are embedded in a rich historical and philosophical context, lead to precise concepts, and dispense with theses. To investigate Effective Calculability is to analyze processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (or discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. Representing human and machine computations by discrete dynamical systems, the boundedness and locality conditions can be captured through axioms for Turing computors and Gandy machines; models o
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Computability Theory *
2011Co-Authors: Wilfried SiegAbstract:Computability is perhaps the most significant and distinctive notion modern logic has introduced; in the guise of decidability and Effective Calculability it has a venerable history within philosophy and mathematics. Now it is also the basic theoretical concept for computer science, artificial intelligence and cognitiv
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ON COMPUTABILITY
2011Co-Authors: Wilfried SiegAbstract:Computability is perhaps the most significant and distinctive notion modern logic has introduced; in the guise of decidability and Effective Calculability it has a venerable history within philosophy and mathematics. Now it is also the basic theoretical concept for computer science, artificial intelligence and cognitive science
John V. Tucker - One of the best experts on this subject based on the ideXlab platform.
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Logical approaches to computational barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, Wales, July 2006, Proceedings
Lecture Notes in Computer Science, 2006Co-Authors: Arnold Beckmann, Ulrich Berger, Benedikt Löwe, John V. TuckerAbstract:Heap-Abstraction for an Object-Oriented Calculus with Thread Classes.- From Constructibility and Absoluteness to Computability and Domain Independence.- Datatype-Generic Reasoning.- The Logical Strength of the Uniform Continuity Theorem.- Elementary Algebraic Specifications of the Rational Function Field.- Random Closed Sets.- Deep Inference and Its Normal Form of Derivations.- Logspace Complexity of Functions and Structures.- Prefix-Like Complexities and Computability in the Limit.- Partial Continuous Functions and Admissible Domain Representations.- An Invariant Cost Model for the Lambda Calculus.- On the Complexity of the Sperner Lemma.- The Church-Turing Thesis: Consensus and Opposition.- Godel and the Origins of Computer Science.- The Role of Algebraic Models and Type-2 Theory of Effectivity in Special Purpose Processor Design.- Turing Universality in Dynamical Systems.- Every Sequence Is Decompressible from a Random One.- Reversible Conservative Rational Abstract Geometrical Computation Is Turing-Universal.- LJQ: A Strongly Focused Calculus for Intuitionistic Logic.- Bohm Trees, Krivine's Machine and the Taylor Expansion of Lambda-Terms.- What Does the Incompleteness Theorem Add to the Unsolvability of the Halting Problem?.- An Analysis of the Lemmas of Urysohn and Urysohn-Tietze According to Effective Borel Measurability.- Enumeration Reducibility with Polynomial Time Bounds.- Coinductive Proofs for Basic Real Computation.- A Measure of Space for Computing over the Reals.- On Graph Isomorphism for Restricted Graph Classes.- Infinite Time Register Machines.- Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems.- Forcing with Random Variables and Proof Complexity.- Complexity-Theoretic Hierarchies.- Undecidability in the Homomorphic Quasiorder of Finite Labeled Forests.- Lower Bounds Using Kolmogorov Complexity.- The Jump Classes of Minimal Covers.- Space Bounds for Infinitary Computation.- From a Zoo to a Zoology: Descriptive Complexity for Graph Polynomials.- Towards a Trichotomy for Quantified H-Coloring.- Two Open Problems on Effective Dimension.- Optimization and Approximation Problems Related to Polynomial System Solving.- Uncomputability Below the Real Halting Problem.- Constraints on Hypercomputation.- Martingale Families and Dimension in P.- Can General Relativistic Computers Break the Turing Barrier?.- Degrees of Weakly Computable Reals.- Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis.- A Subrecursive Refinement of the Fundamental Theorem of Algebra.- An Introduction to Program and Thread Algebra.- Fast Quantifier Elimination Means P = NP.- Admissible Representations in Computable Analysis.- Do Noetherian Modules Have Noetherian Basis Functions?.- Inverting Monotone Continuous Functions in Constructive Analysis.- Partial Recursive Functions in Martin-Lof Type Theory.- Partially Ordered Connectives and ?1 1 on Finite Models.- Upper and Lower Bounds for the Computational Power of P Systems with Mobile Membranes.- Godel's Conflicting Approaches to Effective Calculability.- Co-total Enumeration Degrees.- Relativized Degree Spectra.- Phase Transition Thresholds for Some Natural Subclasses of the Computable Functions.- Non-deterministic Halting Times for Hamkins-Kidder Turing Machines.- Kurt Godel and Computability Theory.- A Computability Theory of Real Numbers.- Primitive Recursive Selection Functions over Abstract Algebras.
Arnold Beckmann - One of the best experts on this subject based on the ideXlab platform.
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Logical approaches to computational barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, Wales, July 2006, Proceedings
Lecture Notes in Computer Science, 2006Co-Authors: Arnold Beckmann, Ulrich Berger, Benedikt Löwe, John V. TuckerAbstract:Heap-Abstraction for an Object-Oriented Calculus with Thread Classes.- From Constructibility and Absoluteness to Computability and Domain Independence.- Datatype-Generic Reasoning.- The Logical Strength of the Uniform Continuity Theorem.- Elementary Algebraic Specifications of the Rational Function Field.- Random Closed Sets.- Deep Inference and Its Normal Form of Derivations.- Logspace Complexity of Functions and Structures.- Prefix-Like Complexities and Computability in the Limit.- Partial Continuous Functions and Admissible Domain Representations.- An Invariant Cost Model for the Lambda Calculus.- On the Complexity of the Sperner Lemma.- The Church-Turing Thesis: Consensus and Opposition.- Godel and the Origins of Computer Science.- The Role of Algebraic Models and Type-2 Theory of Effectivity in Special Purpose Processor Design.- Turing Universality in Dynamical Systems.- Every Sequence Is Decompressible from a Random One.- Reversible Conservative Rational Abstract Geometrical Computation Is Turing-Universal.- LJQ: A Strongly Focused Calculus for Intuitionistic Logic.- Bohm Trees, Krivine's Machine and the Taylor Expansion of Lambda-Terms.- What Does the Incompleteness Theorem Add to the Unsolvability of the Halting Problem?.- An Analysis of the Lemmas of Urysohn and Urysohn-Tietze According to Effective Borel Measurability.- Enumeration Reducibility with Polynomial Time Bounds.- Coinductive Proofs for Basic Real Computation.- A Measure of Space for Computing over the Reals.- On Graph Isomorphism for Restricted Graph Classes.- Infinite Time Register Machines.- Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems.- Forcing with Random Variables and Proof Complexity.- Complexity-Theoretic Hierarchies.- Undecidability in the Homomorphic Quasiorder of Finite Labeled Forests.- Lower Bounds Using Kolmogorov Complexity.- The Jump Classes of Minimal Covers.- Space Bounds for Infinitary Computation.- From a Zoo to a Zoology: Descriptive Complexity for Graph Polynomials.- Towards a Trichotomy for Quantified H-Coloring.- Two Open Problems on Effective Dimension.- Optimization and Approximation Problems Related to Polynomial System Solving.- Uncomputability Below the Real Halting Problem.- Constraints on Hypercomputation.- Martingale Families and Dimension in P.- Can General Relativistic Computers Break the Turing Barrier?.- Degrees of Weakly Computable Reals.- Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis.- A Subrecursive Refinement of the Fundamental Theorem of Algebra.- An Introduction to Program and Thread Algebra.- Fast Quantifier Elimination Means P = NP.- Admissible Representations in Computable Analysis.- Do Noetherian Modules Have Noetherian Basis Functions?.- Inverting Monotone Continuous Functions in Constructive Analysis.- Partial Recursive Functions in Martin-Lof Type Theory.- Partially Ordered Connectives and ?1 1 on Finite Models.- Upper and Lower Bounds for the Computational Power of P Systems with Mobile Membranes.- Godel's Conflicting Approaches to Effective Calculability.- Co-total Enumeration Degrees.- Relativized Degree Spectra.- Phase Transition Thresholds for Some Natural Subclasses of the Computable Functions.- Non-deterministic Halting Times for Hamkins-Kidder Turing Machines.- Kurt Godel and Computability Theory.- A Computability Theory of Real Numbers.- Primitive Recursive Selection Functions over Abstract Algebras.
Matteo De Benedetto - One of the best experts on this subject based on the ideXlab platform.
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Explication as a Three-Step Procedure: the case of the Church-Turing Thesis
European Journal for Philosophy of Science, 2021Co-Authors: Matteo De BenedettoAbstract:In recent years two different axiomatic characterizations of the intuitive concept of Effective Calculability have been proposed, one by Sieg and the other by Dershowitz and Gurevich. Analyzing them from the perspective of Carnapian explication, I argue that these two characterizations explicate the intuitive notion of Effective Calculability in two different ways. I will trace back these two ways to Turing’s and Kolmogorov’s informal analyses of the intuitive notion of Calculability and to their respective outputs: the notion of computorability and the notion of algorithmability. I will then argue that, in order to adequately capture the conceptual differences between these two notions, the classical two-step picture of explication is not enough. I will present a more fine-grained three-step version of Carnapian explication, showing how with its help the difference between these two notions can be better understood and explained.
Benedikt Löwe - One of the best experts on this subject based on the ideXlab platform.
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Logical approaches to computational barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, Wales, July 2006, Proceedings
Lecture Notes in Computer Science, 2006Co-Authors: Arnold Beckmann, Ulrich Berger, Benedikt Löwe, John V. TuckerAbstract:Heap-Abstraction for an Object-Oriented Calculus with Thread Classes.- From Constructibility and Absoluteness to Computability and Domain Independence.- Datatype-Generic Reasoning.- The Logical Strength of the Uniform Continuity Theorem.- Elementary Algebraic Specifications of the Rational Function Field.- Random Closed Sets.- Deep Inference and Its Normal Form of Derivations.- Logspace Complexity of Functions and Structures.- Prefix-Like Complexities and Computability in the Limit.- Partial Continuous Functions and Admissible Domain Representations.- An Invariant Cost Model for the Lambda Calculus.- On the Complexity of the Sperner Lemma.- The Church-Turing Thesis: Consensus and Opposition.- Godel and the Origins of Computer Science.- The Role of Algebraic Models and Type-2 Theory of Effectivity in Special Purpose Processor Design.- Turing Universality in Dynamical Systems.- Every Sequence Is Decompressible from a Random One.- Reversible Conservative Rational Abstract Geometrical Computation Is Turing-Universal.- LJQ: A Strongly Focused Calculus for Intuitionistic Logic.- Bohm Trees, Krivine's Machine and the Taylor Expansion of Lambda-Terms.- What Does the Incompleteness Theorem Add to the Unsolvability of the Halting Problem?.- An Analysis of the Lemmas of Urysohn and Urysohn-Tietze According to Effective Borel Measurability.- Enumeration Reducibility with Polynomial Time Bounds.- Coinductive Proofs for Basic Real Computation.- A Measure of Space for Computing over the Reals.- On Graph Isomorphism for Restricted Graph Classes.- Infinite Time Register Machines.- Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems.- Forcing with Random Variables and Proof Complexity.- Complexity-Theoretic Hierarchies.- Undecidability in the Homomorphic Quasiorder of Finite Labeled Forests.- Lower Bounds Using Kolmogorov Complexity.- The Jump Classes of Minimal Covers.- Space Bounds for Infinitary Computation.- From a Zoo to a Zoology: Descriptive Complexity for Graph Polynomials.- Towards a Trichotomy for Quantified H-Coloring.- Two Open Problems on Effective Dimension.- Optimization and Approximation Problems Related to Polynomial System Solving.- Uncomputability Below the Real Halting Problem.- Constraints on Hypercomputation.- Martingale Families and Dimension in P.- Can General Relativistic Computers Break the Turing Barrier?.- Degrees of Weakly Computable Reals.- Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis.- A Subrecursive Refinement of the Fundamental Theorem of Algebra.- An Introduction to Program and Thread Algebra.- Fast Quantifier Elimination Means P = NP.- Admissible Representations in Computable Analysis.- Do Noetherian Modules Have Noetherian Basis Functions?.- Inverting Monotone Continuous Functions in Constructive Analysis.- Partial Recursive Functions in Martin-Lof Type Theory.- Partially Ordered Connectives and ?1 1 on Finite Models.- Upper and Lower Bounds for the Computational Power of P Systems with Mobile Membranes.- Godel's Conflicting Approaches to Effective Calculability.- Co-total Enumeration Degrees.- Relativized Degree Spectra.- Phase Transition Thresholds for Some Natural Subclasses of the Computable Functions.- Non-deterministic Halting Times for Hamkins-Kidder Turing Machines.- Kurt Godel and Computability Theory.- A Computability Theory of Real Numbers.- Primitive Recursive Selection Functions over Abstract Algebras.