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Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
The tractable cognition thesis
 Cognitive Science: A Multidisciplinary Journal
, 2008
"... The recognition that human minds/brains are finite systems with limited resources for computation has led some researchers to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. This thesis, if true, serves cognitive psychology by constra ..."
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The recognition that human minds/brains are finite systems with limited resources for computation has led some researchers to advance the Tractable Cognition thesis: Human cognitive capacities are constrained by computational tractability. This thesis, if true, serves cognitive psychology by constraining the space of computationallevel theories of cognition. To utilize this constraint, a precise and workable definition of “computational tractability ” is needed. Following computer science tradition, many cognitive scientists and psychologists define computational tractability as polynomialtime computability, leading to the PCognition thesis. This article explains how and why the PCognition thesis may be overly restrictive, risking the exclusion of veridical computationallevel theories from scientific investigation. An argument is made to replace the PCognition thesis by the FPTCognition thesis as an alternative formalization of the Tractable Cognition thesis (here, FPT stands for fixedparameter tractable). Possible objections to the Tractable Cognition thesis, and its proposed formalization, are discussed, and existing misconceptions are clarified.
Computability and Incomputability
"... The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal cr ..."
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The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal credit. This is incorrect. It was Turing alone who achieved the characterization, in the opinion of Gödel. We also explore Turing’s oracle machine and its analogous properties in analysis. Keywords: Turing amachine, computability, ChurchTuring Thesis, Kurt Gödel, Alan Turing, Turing omachine, computable approximations,
A Random Walk in Statistical Physics
, 2001
"... This thesis deals with some aspects of the physics of disordered systems. It consists of four papers and an introductory part. An introduction ..."
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This thesis deals with some aspects of the physics of disordered systems. It consists of four papers and an introductory part. An introduction
Language Learning and Nonlinear Dynamical Systems
, 2003
"... stems can be mapped directly to any member of a large class of lowdimensional chaotic dynamical systems. The importance of this is that for a given chaotic dynamical system to be a model of a given language, we may set up a target probability distribution over its statespace, such that if it visit ..."
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stems can be mapped directly to any member of a large class of lowdimensional chaotic dynamical systems. The importance of this is that for a given chaotic dynamical system to be a model of a given language, we may set up a target probability distribution over its statespace, such that if it visits its statespace according to this distribution it will generate the language that is desired. A specific dynamical system is chosen as a model for learning. This is known as the 2d tentmap. We derive a learning algorithm for this particular dynamical system. This algorithm is based on a matrix approximation of the FrobeniusPerron operator. Examples of learning regular, contextfree and contextsensitive languages are provided. BIOGRAPHICAL SKETCH Mark Andrews was born in Ireland in 1973. In 1995, he graduated with a BA from the National University of Ireland. The same year, he began graduate study in Cornell Univeristy in Ithaca, New York. In 1998, he received a M.Sc from Cornell. In
The origins of combinatorics on words
, 2007
"... We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early ..."
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We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a byproduct of investigations on various combinatorial objects. For example, paths in graphs are encoded by words in a natural way, and conversely, the Cayley graph of a group or a semigroup encodes words by paths. We give in this text an account of this twosided interaction.
Automata and Formal Languages
, 2003
"... This article provides an introduction to the theory of automata and formal languages. The elements are presented in a historical perspective and the links with other areas are underlined. In particular, applications of the field to linguistics, software design, text processing, computational alg ..."
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This article provides an introduction to the theory of automata and formal languages. The elements are presented in a historical perspective and the links with other areas are underlined. In particular, applications of the field to linguistics, software design, text processing, computational algebra or computational biology are given.
Stochastic ContextFree Grammars
, 2004
"... Introduction: Strings, Grammars & Formal Languages Stochastic contextfree grammars, or SCFGs, are generative systems of stochastic languages. In other words, a SCFG specifies a probability distribution over the set of all possible strings that are concatenated from a finite alphabet . Definitio ..."
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Introduction: Strings, Grammars & Formal Languages Stochastic contextfree grammars, or SCFGs, are generative systems of stochastic languages. In other words, a SCFG specifies a probability distribution over the set of all possible strings that are concatenated from a finite alphabet . Definition 1 An alphabet is a finite set of symbols. It denoted here by }. Definition 2 The kth power of an alphabet , denoted , is the set of strings of length k made from the elements of . Example 1 For the alphabet 1}, where V = 2. will be set of strings, made from the elements of , that are of length 1, or 1}. Likewise, 01, 10, 11}, {000, 001, 010, 011, 100, 101, 110, 111}, etc. Definition 3 # , or Kleene closure of , is the union of all powers of the alphabet, or # = . . . . Definition 4 A formal language L is defined as L # . In other words, L is a (possibly infinite) set of strings, each formed by concatenation from a finite set of
PhraseStructure . . . Stochastic Grammars Grammar Induction . . . Conclusion Title Page
"... Stochastic contextfree grammars, or SCFGs, are generative systems of stochastic languages. In other words, a SCFG specifies a probability distribution over the set of (all possible) strings that are concatenated from a finite alphabet V. ..."
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Stochastic contextfree grammars, or SCFGs, are generative systems of stochastic languages. In other words, a SCFG specifies a probability distribution over the set of (all possible) strings that are concatenated from a finite alphabet V.
Internal Report, PLUS ESPRIT project P5254. Communicative Activity Analysis of a Wizard of Oz Experiment
"... In this chapter we will describe and analyze a simulation of a factual information seeking system for a subsection of the "yellow pages". The subsection contains information about (i) car hire (ii) restaurants, and (iii) personal insurance. The simulation was ..."
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In this chapter we will describe and analyze a simulation of a factual information seeking system for a subsection of the "yellow pages". The subsection contains information about (i) car hire (ii) restaurants, and (iii) personal insurance. The simulation was