Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions Generalized Automata over the Reals Klaus Meer Brandenburgische Technische Universit¨ at, Cottbus-Senftenberg, Germany Metafinite 2017, Reykjavik, June 2017 (joint work with Ameen Naif) Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions 1. Introduction One of the most basic algorithm models in Computer Science Finite Automata over finite alphabets Related theory well developed, for example, with respect to structural characterizations of languages accepted, decidability and complexity of important computational problems, relation to logic etc. Decidability questions are treated in Turing machine framework. Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions In recent years lot of interest in Theoretical Computer Science in alternative computation models: Neural Networks Quantum computing membrane computing analogue computing algebraic models .. Underlying data structure often more general than finite alphabets, for example real and complex numbers Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions In recent years lot of interest in Theoretical Computer Science in alternative computation models: Neural Networks Quantum computing membrane computing analogue computing algebraic models .. Underlying data structure often more general than finite alphabets, for example real and complex numbers Thus natural idea: extend concept of finite automata to more such general structures Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions At least two lines of motivation: several generalized automata models have been introduced in areas like program verification, database theory: automata for words and trees on infinite structures define and study finite automata model in algebraic computability theory following approach by Blum & Shub & Smale: BSS-model for real and complex numbers (compare Quantum Finite Automata) Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions At least two lines of motivation: several generalized automata models have been introduced in areas like program verification, database theory: automata for words and trees on infinite structures define and study finite automata model in algebraic computability theory following approach by Blum & Shub & Smale: BSS-model for real and complex numbers (compare Quantum Finite Automata) Generalized automata model over arbitrary structures introduced by Gandhi & Khoussainov & Liu, TAMC 2012: - homogenizes (some of the) previous automata concepts; - authors ask in particular for studying their extended automata model in BSS framework Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions At least two lines of motivation: several generalized automata models have been introduced in areas like program verification, database theory: automata for words and trees on infinite structures define and study finite automata model in algebraic computability theory following approach by Blum & Shub & Smale: BSS-model for real and complex numbers (compare Quantum Finite Automata) Generalized automata model over arbitrary structures introduced by Gandhi & Khoussainov & Liu, TAMC 2012: - homogenizes (some of the) previous automata concepts; - authors ask in particular for studying their extended automata model in BSS framework starting point of this work! Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions 2. Generalized Finite Automata over R Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D , finite sets of (binary) functions and relations; universe can be infinite, uncountable Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions 2. Generalized Finite Automata over R Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D , finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D ∗ , i.e., finite strings with components from universe; Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions 2. Generalized Finite Automata over R Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D , finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D ∗ , i.e., finite strings with components from universe; automata have accepting/rejecting states Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions 2. Generalized Finite Automata over R Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D , finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D ∗ , i.e., finite strings with components from universe; automata have accepting/rejecting states ! each single step allows limited way of computation and testing using functions and relations from S as well as a constant number of registers Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions 2. Generalized Finite Automata over R Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D , finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D ∗ , i.e., finite strings with components from universe; automata have accepting/rejecting states ! each single step allows limited way of computation and testing using functions and relations from S as well as a constant number of registers Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions 2. Generalized Finite Automata over R Basic ideas of how automata introduced by Gandhi et al. work: structure S consists of universe D , finite sets of (binary) functions and relations; universe can be infinite, uncountable automata process words in D ∗ , i.e., finite strings with components from universe; automata have accepting/rejecting states ! each single step allows limited way of computation and testing using functions and relations from S as well as a constant number of registers Results by Gandhi et al. deal with countable universes Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions Most interesting structures for us are the reals and complex numbers with basic operations: S R := ( R , + , − , • , pr , ≥ , =) reals as ring with order; S C := ( C , + , − , • , pr , =) complex numbers as ring with equality; Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions Most interesting structures for us are the reals and complex numbers with basic operations: S R := ( R , + , − , • , pr , ≥ , =) reals as ring with order; S C := ( C , + , − , • , pr , =) complex numbers as ring with equality; gives possibility to compare generalized finite automata with BSS theory Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions Most interesting structures for us are the reals and complex numbers with basic operations: S R := ( R , + , − , • , pr , ≥ , =) reals as ring with order; S C := ( C , + , − , • , pr , =) complex numbers as ring with equality; gives possibility to compare generalized finite automata with BSS theory Important: All statements below about decidability, computability, complexity etc. to be understood in BSS model over R and C Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions Most interesting structures for us are the reals and complex numbers with basic operations: S R := ( R , + , − , • , pr , ≥ , =) reals as ring with order; S C := ( C , + , − , • , pr , =) complex numbers as ring with equality; gives possibility to compare generalized finite automata with BSS theory Important: All statements below about decidability, computability, complexity etc. to be understood in BSS model over R and C Gandhi & Khoussainov & Liu approach applied to S R : Real GKL automata Klaus Meer Generalized Automata over the Reals
Introduction Generalized Finite Automata Results I Periodic GKL automata Conclusions How real GKL automata work: Klaus Meer Generalized Automata over the Reals
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