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Cesare Tinelli - One of the best experts on this subject based on the ideXlab platform.

  • combining decision procedures for positive theories sharing constructors
    Lecture Notes in Computer Science, 2002
    Co-Authors: Franz Baader, Cesare Tinelli
    Abstract:

    This paper addresses the following Combination Problem: given two equational theories E 1 and E 2 whose positive theories are decidable, how can one obtain a decision procedure for the positive theory of E 1 ∪ E 2 ? For theories over disjoint signatures, this Problem was solved by Baader and Schulz in 1995. This paper is a first step towards extending this result to the case of theories sharing constructors. Since there is a close connection between positive theories and unification Problems, this also extends to the non-disjoint case the work on combining decision procedures for unification modulo equational theories.

  • deciding the word Problem in the union of equational theories sharing constructors
    Rewriting Techniques and Applications, 1999
    Co-Authors: Franz Baader, Cesare Tinelli
    Abstract:

    The main contribution of this paper is a new method for combining decision procedures for the word Problem in equational theories sharing "constructors." The notion of constructors adopted in this paper has a nice algebraic definition and is more general than a related notion introduced in previous work on the Combination Problem.

  • a new approach for combining decision procedure for the word Problem and its connection to the nelson oppen Combination method
    Conference on Automated Deduction, 1997
    Co-Authors: Franz Baader, Cesare Tinelli
    Abstract:

    The Nelson-Oppen Combination method can be used to combine decision procedures for the validity of quantifier-free formulae in first-order theories with disjoint signatures, provided that the theories to be combined are stably infinite. We show that, even though equational theories need not satisfy this property, Nelson and Oppen''s method can be applied, after some minor modifications, to combine decision procedures for the validity of quantifier-free formulae in equational theories. Unfortunately, and contrary to a common belief, the method cannot be used to combine decision procedures for the word Problem. We present a method that solves this kind of Combination Problem. Our method is based on transformation rules and also applies to equational theories that share a finite number of constant symbols.

Franz Baader - One of the best experts on this subject based on the ideXlab platform.

  • combining decision procedures for positive theories sharing constructors
    Lecture Notes in Computer Science, 2002
    Co-Authors: Franz Baader, Cesare Tinelli
    Abstract:

    This paper addresses the following Combination Problem: given two equational theories E 1 and E 2 whose positive theories are decidable, how can one obtain a decision procedure for the positive theory of E 1 ∪ E 2 ? For theories over disjoint signatures, this Problem was solved by Baader and Schulz in 1995. This paper is a first step towards extending this result to the case of theories sharing constructors. Since there is a close connection between positive theories and unification Problems, this also extends to the non-disjoint case the work on combining decision procedures for unification modulo equational theories.

  • deciding the word Problem in the union of equational theories sharing constructors
    Rewriting Techniques and Applications, 1999
    Co-Authors: Franz Baader, Cesare Tinelli
    Abstract:

    The main contribution of this paper is a new method for combining decision procedures for the word Problem in equational theories sharing "constructors." The notion of constructors adopted in this paper has a nice algebraic definition and is more general than a related notion introduced in previous work on the Combination Problem.

  • a new approach for combining decision procedure for the word Problem and its connection to the nelson oppen Combination method
    Conference on Automated Deduction, 1997
    Co-Authors: Franz Baader, Cesare Tinelli
    Abstract:

    The Nelson-Oppen Combination method can be used to combine decision procedures for the validity of quantifier-free formulae in first-order theories with disjoint signatures, provided that the theories to be combined are stably infinite. We show that, even though equational theories need not satisfy this property, Nelson and Oppen''s method can be applied, after some minor modifications, to combine decision procedures for the validity of quantifier-free formulae in equational theories. Unfortunately, and contrary to a common belief, the method cannot be used to combine decision procedures for the word Problem. We present a method that solves this kind of Combination Problem. Our method is based on transformation rules and also applies to equational theories that share a finite number of constant symbols.

Jonathan W Schooler - One of the best experts on this subject based on the ideXlab platform.

  • the easy part of the hard Problem a resonance theory of consciousness
    Frontiers in Human Neuroscience, 2019
    Co-Authors: Tam Hunt, Jonathan W Schooler
    Abstract:

    Synchronization, harmonization, vibrations, or simply resonance in its most general sense seems to have an integral relationship with consciousness itself. One of the possible “neural correlates of consciousness” in mammalian brains is a specific Combination of gamma, beta and theta electrical synchrony. More broadly, we see similar kinds of resonance patterns in living and non-living structures of many types. What clues can resonance provide about the nature of consciousness more generally? This paper provides an overview of resonating structures in the fields of neuroscience, biology and physics and offers a possible solution to what we see as the “easy part” of the “Hard Problem” of consciousness, which is generally known as the “Combination Problem.” The Combination Problem asks: how do micro-conscious entities combine into a higher-level macro-consciousness? The proposed solution in the context of mammalian consciousness suggests that a shared resonance is what allows different parts of the brain to achieve a phase transition in the speed and bandwidth of information flows between the constituent parts. This phase transition allows for richer varieties of consciousness to arise, with the character and content of that consciousness in each moment determined by the particular set of constituent neurons. We also offer more general insights into the ontology of consciousness and suggest that consciousness manifests as a continuum of increasing richness in all physical processes, distinguishing our view from emergentist materialism. We refer to this approach, a meta-synthesis, as a (general) resonance theory of consciousness. We offer some suggestions for testing the theory.

Bernd Fiedler - One of the best experts on this subject based on the ideXlab platform.

  • Ideal decompositions and computation of tensor normal forms
    2012
    Co-Authors: Bernd Fiedler
    Abstract:

    Abstract. Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[Sr] of a symmetric group Sr. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W ⊥ of W within the dual space K[Sr] ∗ of K[Sr] yield linear identities needed for a treatment of the term Combination Problem for the coordinates of the T. We give the structure of these W for every situation which appears in symbolic tensor calculations by computer. Characterizing idempotents of such W can be determined by means of an ideal decomposition algorithm which works in every semisimple ring up to an isomorphism. Furthermore, we use tools such as the Littlewood-Richardson rule, plethysms and discrete Fourier transforms for Sr to increase the efficience of calculations. All described methods were implemented in a Mathematica package called PERMS. 1. The Term Combination Problem for Tensors The use of computer algebra systems for symbolic calculations with tensor expressions is very important in differential geometry, tensor analysis and general relativity theory. The investigations of this paper1 are motivated by the following term Combination Problem or normal form Problem which occurs within such calculations. Let us consider real or complex linear Combinations n∑ (1.1) τ = , αi ∈ R, C i=1 αiT(i) of expressions T(i) which are formed from the coordinates of certain tensors A, B, C,... by multiplication and, possibly, contractions of some pairs of indices. An example of such an expression is (1.2) A iabc A a jkd Bbd e Cec. In (1.2) we use Einstein’s summation convention. Further we assume that each of the numbers of A, B, C,... is constant if we run through the set of the T(i). Now we aim to carry out symbolic calculations with expressions of the type (1.1), (1.2

  • Séminaire Lotharingien de Combinatoire 45 (2001), Article B45g IDEAL DECOMPOSITIONS AND COMPUTATION OF TENSOR NORMAL FORMS
    2008
    Co-Authors: Bernd Fiedler
    Abstract:

    Abstract. Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[Sr] of a symmetric group Sr. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W ⊥ of W within the dual space K[Sr] ∗ of K[Sr] yield linear identities needed for a treatment of the term Combination Problem for the coordinates of the T. We give the structure of these W for every situation which appears in symbolic tensor calculations by computer. Characterizing idempotents of such W can be determined by means of an ideal decomposition algorithm which works in every semisimple ring up to an isomorphism. Furthermore, we use tools such as the Littlewood-Richardson rule, plethysms and discrete Fourier transforms for Sr to increase the efficience of calculations. All described methods were implemented in a Mathematica package called PERMS. 1. The Term Combination Problem for Tensors The use of computer algebra systems for symbolic calculations with tensor expressions is very important in differential geometry, tensor analysis and general relativity theory. The investigations of this paper1 are motivated by the following term Combination Problem or normal form Problem which occurs within such calculations. Let us consider real or complex linear Combinations n� (1.1) τ = αiT(i) , αi ∈ R, C i=1 of expressions T(i) which are formed from the coordinates of certain tensors A, B, C,... by multiplication and, possibly, contractions of some pairs of indices. An example of such an expression is (1.2) A iabc A a jkd Bbd e C ec. In (1.2) we use Einstein’s summation convention. Further we assume that each of the numbers of A, B, C,... is constant if we run through the set of the T(i). Now we aim to carry out symbolic calculations with expressions of the type (1.1), (1.2

Tam Hunt - One of the best experts on this subject based on the ideXlab platform.

  • the easy part of the hard Problem a resonance theory of consciousness
    Frontiers in Human Neuroscience, 2019
    Co-Authors: Tam Hunt, Jonathan W Schooler
    Abstract:

    Synchronization, harmonization, vibrations, or simply resonance in its most general sense seems to have an integral relationship with consciousness itself. One of the possible “neural correlates of consciousness” in mammalian brains is a specific Combination of gamma, beta and theta electrical synchrony. More broadly, we see similar kinds of resonance patterns in living and non-living structures of many types. What clues can resonance provide about the nature of consciousness more generally? This paper provides an overview of resonating structures in the fields of neuroscience, biology and physics and offers a possible solution to what we see as the “easy part” of the “Hard Problem” of consciousness, which is generally known as the “Combination Problem.” The Combination Problem asks: how do micro-conscious entities combine into a higher-level macro-consciousness? The proposed solution in the context of mammalian consciousness suggests that a shared resonance is what allows different parts of the brain to achieve a phase transition in the speed and bandwidth of information flows between the constituent parts. This phase transition allows for richer varieties of consciousness to arise, with the character and content of that consciousness in each moment determined by the particular set of constituent neurons. We also offer more general insights into the ontology of consciousness and suggest that consciousness manifests as a continuum of increasing richness in all physical processes, distinguishing our view from emergentist materialism. We refer to this approach, a meta-synthesis, as a (general) resonance theory of consciousness. We offer some suggestions for testing the theory.