Vector logics: The matrix-vector representation of logical calculus

Mizraji, Eduardo

doi:10.1016/0165-0114(92)90216-q

Cited by 48 publications

(37 citation statements)

References 2 publications

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“…Both dynamical system processing models were essentially grounded in Smolensky's and Mizraji's tensor product representations of symbolic content in neural activation spaces (Dolan and Smolensky 1989;Mizraji 1989Mizraji , 1992Smolensky 1990). The first model represented the syntactic categories of the disambiguated grammars as linearly independent filler vectors and positions in a labeled binary tree as a basis of three-dimensional space.…”

Section: Discussionmentioning

confidence: 99%

“…In response to these criticisms, Dolan and Smolensky (1989); Smolensky (1990) and independently Mizraji (1989Mizraji ( , 1992 suggested a unifying framework for distributed representations of structured symbolic information such as lists or phrase structure trees. According to Smolensky (1990), this approach comprises three steps:…”

Section: Dynamical System Modelsmentioning

confidence: 99%

“…In this representation, the tensor products are then given by Kronecker products (Mizraji 1989(Mizraji , 1992) of filler and role vectors, f i r k ; where…”

Section: Tensor Product Top-down Recognizermentioning

confidence: 99%

“…Using a particular ERP experiment on the processing of German subjectobject ambiguities Bader and Meng 1999;beim Graben et al 2000;Vos et al 2001;Frisch et al 2002Frisch et al , 2004Schlesewsky and Bornkessel 2006), a locally ambiguous context-free grammar and two appropriate deterministic pushdown recognizers are constructed from a Government and Binding (Chomsky 1981;Haegeman 1994) representation of the stimulus material. Subsequently, we present two different implementations of these automata by nonlinear dynamical systems, resulting from universal tensor product representations (Dolan and Smolensky 1989;Mizraji 1989;Smolensky 1990;Mizraji 1992;Smolensky and Legendre 2006;Smolensky 2006). The first model is a parallel distributed processing (PDP) model (Rumelhart et al 1986) in a high-dimensional activation vector space.…”

Section: Introductionmentioning

confidence: 99%

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Towards dynamical system models of language-related brain potentials

2008

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Event-related brain potentials (ERP) are important neural correlates of cognitive processes. In the domain of language processing, the N400 and P600 reflect lexicalsemantic integration and syntactic processing problems, respectively. We suggest an interpretation of these markers in terms of dynamical system theory and present two nonlinear dynamical models for syntactic computations where different processing strategies correspond to functionally different regions in the system's phase space.

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Section: Discussionmentioning

confidence: 99%

Section: Dynamical System Modelsmentioning

confidence: 99%

“…In this representation, the tensor products are then given by Kronecker products (Mizraji 1989(Mizraji , 1992) of filler and role vectors, f i r k ; where…”

Section: Tensor Product Top-down Recognizermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Towards dynamical system models of language-related brain potentials

2008

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“…It allows the explicit construction of neural realizations for highly structured mental representations by means of filler/role decompositions and tensor product representations (cf. Mizraji (1989Mizraji ( , 1992 for a related approach). Moreover ICS suggests a dual aspect interpretation: at the macroscopic, symbolic level, cognitive computations are performed by the complex dynamics of distributed activation patterns; at the microscopic, connectionist level, these patterns are generated by deterministic evolution laws governing neural network dynamics (Smolensky and Legendre 2006a, b;Smolensky 2006;.…”

Section: Introductionmentioning

confidence: 99%

Unifying syntactic theory and sentence processing difficulty through a connectionist minimalist parser

Gerth

Graben

2009

Cogn Neurodyn

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Syntactic theory provides a rich array of representational assumptions about linguistic knowledge and processes. Such detailed and independently motivated constraints on grammatical knowledge ought to play a role in sentence comprehension. However most grammar-based explanations of processing difficulty in the literature have attempted to use grammatical representations and processes per se to explain processing difficulty. They did not take into account that the description of higher cognition in mind and brain encompasses two levels: on the one hand, at the macrolevel, symbolic computation is performed, and on the other hand, at the microlevel, computation is achieved through processes within a dynamical system. One critical question is therefore how linguistic theory and dynamical systems can be unified to provide an explanation for processing effects. Here, we present such a unification for a particular account to syntactic theory: namely a parser for Stabler's Minimalist Grammars, in the framework of Smolensky's Integrated Connectionist/Symbolic architectures. In simulations we demonstrate that the connectionist minimalist parser produces predictions which mirror global empirical findings from psycholinguistic research.

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A dynamical approach to logical decisions

Mizraji

Lin

1997

Complexity

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We propose a first approximation to the modeling of cognitive decisions based on the theory of associative memories. The basic structure are the matrix memories of Anderson‐Kohonen adapted to include the modulation of associations via tensorial preprocessing of inputs. This view admits an easy implementation of logical operations as matrix operators. From this “vectorial logic” springs a variety of models for fuzzy decision processes. Degrees of fuzziness are introduced at two different levels: (a) by the operators of modal logic, and (b) by using logical variables as vectors with projections inside the interval [0, 1]. The outcomes of this vectorial logic can be projected onto unit vectors yielding scalar difference equations. As examples we study the dynamics of contradictory self‐referential systems and processes leading to competition between options. These models exhibit a variety of dynamical patterns that include stable steady states, oscillations, and deterministic chaos. © 1997 John Wiley & Sons, Inc.

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Vector logics: The matrix-vector representation of logical calculus

Cited by 48 publications

References 2 publications

Towards dynamical system models of language-related brain potentials

Towards dynamical system models of language-related brain potentials

Unifying syntactic theory and sentence processing difficulty through a connectionist minimalist parser

A dynamical approach to logical decisions

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