Algebraic Automata Theory
This is a self-contained, modern treatment of the algebraic theory of machines. Dr Holcombe examines various applications of the idea of a machine in biology, biochemistry and computer science and gives also a rigorous treatment of the way in which these machines can be decomposed and simulated by simpler ones. This treatment is based on fundamental ideas from modern algebra. Motivation for many of the newer results is provided by way of applications so this account should be accessible and valuable for those studying applied algebra or theoretical computer science at advanced undergraduate or beginning postgraduate level, as well as for those undertaking research in those areas.
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Machines and semigroups
The holonomy decomposition
Sequential machines and functions
admissible subset system algebraic aperiodic applicable to q apply behaviour called cascade product Clearly compatible subset system congruence consider construct define a relation denote derived sequence element epimorphism equivalence classes equivalence relation example 4.5 exists finite set free semigroup g e G given group G height function Hence holonomy decomposition theorem holonomy group holonomy transformation group identity injective function integer irreducible isomorphism Let Q mapping mation semigroups maximal image Mealy machine normal subgroup notation operation orthogonal admissible partition output partial function partial sequential function permutation group Proof Let Prove q e Q Q x Q rank recognizable set recognizable subset relational covering reset machine restricted direct product result satisfying semigroup homomorphism set of words set Q singleton skeleton subgroup subsemigroup surjective function theory transformation monoid wreath product
Page 221 - References Arbib, MA  Brains, machines and mathematics. McGraw-Hill, New York Arbib, MA  'Memory limitations of SR models'.