KMS states on Quantum Grammars

We consider quantum (unitary) continuous time evolution of spins on a lattice together with quantum evolution of the lattice itself. In physics such evolution was discussed in connection with quantum gravity. It is also related to what is called quantum circuits, one of the incarnations of a quantum computer. We consider simpler models for which one can obtain exact mathematical results. We prove existence of the dynamics in both Schroedinger and Heisenberg pictures, construct KMS states on appropriate C*-algebras. We show (for high temperatures) that for each system where the lattice undergoes quantum evolution, there is a natural scaling leading to a quantum spin system on a fixed lattice, defined by a renormalized Hamiltonian.Comment: 22 page

Algebraic properties of operator precedence languages

This paper presents new results on the algebraic ordering properties of operator precedence grammars and languages. This work was motivated by, and applied to, the mechanical acquisition or inference of operator precedence grammars. A new normal form of operator precedence grammars called homogeneous is defined. An algorithm is given to construct a grammar, called max-grammar, generating the largest language which is compatible with a given precedence matrix. Then the class of free grammars is introduced as a special subclass of operator precedence grammars. It is shown that operator precedence languages corresponding to a given precedence matrix form a Boolean algebra

From indexed grammars to generating functions

We extend the Chomsky/Sch\"utzenberger method of computing the growth series of an unambiguous context-free language to the larger class of indexed languages. We illustrate the technique with numerous examples.Comment: 23 pages, 3 figure

Assumptions behind grammatical approaches to code-switching: when the blueprint is a red herring

Many of the so-called ‘grammars’ of code-switching are based on various underlying assumptions, e.g. that informal speech can be adequately or appropriately described in terms of ‘‘grammar’’; that deep, rather than surface, structures are involved in code-switching; that one ‘language’ is the ‘base’ or ‘matrix’; and that constraints derived from existing data are universal and predictive. We question these assumptions on several grounds. First, ‘grammar’ is arguably distinct from the processes driving speech production. Second, the role of grammar is mediated by the variable, poly-idiolectal repertoires of bilingual speakers. Third, in many instances of CS the notion of a ‘base’ system is either irrelevant, or fails to explain the facts. Fourth, sociolinguistic factors frequently override ‘grammatical’ factors, as evidence from the same language pairs in different settings has shown. No principles proposed to date account for all the facts, and it seems unlikely that ‘grammar’, as conventionally conceived, can provide definitive answers. We conclude that rather than seeking universal, predictive grammatical rules, research on CS should focus on the variability of bilingual grammars

Noncommutative algebras, context-free grammars and algebraic Hilbert series

In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.Comment: 26 pages, to appear in Journal of Symbolic Computatio

Methods for Structural Pattern Recognition: Complexity and Applications

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