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Understanding Finite Automata as Linear Systems: Observability and Reachability Insights

This paper explores the convergence of finite automata and linear systems, highlighting their relationship in control and automata theory. It emphasizes that finite automata can be framed as time-invariant linear systems over semimodules, allowing the application of linear techniques to automata. The discussion includes observability and reachability reductions through linear transformations, exemplifying applications in modeling embedded and biological systems. Notably, techniques such as automata minimization and the "Z"-transform of automata are explored, showcasing the rich interplay between these disciplines.

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Understanding Finite Automata as Linear Systems: Observability and Reachability Insights

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  1. Finite Automata as Linear Systems Observability, Reachability and More Radu GrosuSUNY at Stony Brook

  2. Convergence of Theories • HSCC Conference: a witness of the fascinating • convergence between control and automata theory. • Hybrid Automata: an outcomeof this convergence • modeling formalism for systems exhibiting both discrete and continuous behavior, • successfully used to modeland analyze embedded and biological systems.

  3. voltage(mv) Stimulated time(ms) Lack of Common Foundation for HA • Mode dynamics: • Linear system (LS) • Mode switching: • Finite automaton(FA) • Different techniques: • LS reduction • FA minimization • LS & FA taught separately: No common foundation!

  4. Main Conjecture of this Talk • Finite automata can be conveniently regarded as time invariant linear systems over semimodules: • linear systems techniques generalize to automata • Examples of such techniques include: • linear transformations of automata, • minimization and determinization of automata as observability and reachability reductions • “Z”-transform of automata to compute associated regular expression through Gaussian elimination.

  5. Finite Automata as Linear Systems

  6. Finite Automata as Linear Systems

  7. Finite Automata as Linear Systems

  8. b a b a x3 x1 x2 Finite Automata as Linear Systems L1

  9. Polynomials and their Operations

  10. Polynomials and their Operations

  11. Boolean Semimodules

  12. Boolean Semimodules

  13. Boolean Semimodules

  14. Observability

  15. b a b a x3 x1 x2 Observability L1

  16. Linear Dependence

  17. Linear Dependence

  18. Linear Dependence

  19. Linear Dependence

  20. Basis in Boolean Semimodule

  21. Basis in Boolean Semimodule

  22. b a b a x3 x1 x2 Basis in Boolean Semimodule L1

  23. b a b a x3 x1 x2 Basis in Boolean Semimodule L1

  24. a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2

  25. a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2

  26. a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2

  27. a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2

  28. a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2

  29. a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2

  30. a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Columns L2

  31. a b a x2 x4 a x1 b b a x3 x5 b b a Mixed Observability Reduction L2

  32. a b a x2 x4 a x1 b b a L2 x3 x5 b b a DFA L21 by rows NFA L22 by columns NFA L23 mixed a a,b b a,b a,b a,b a,b b b b x1 x3 x3 x2 x3 x2 x2 x1 x1 a Original and Reduced Automata L2

  33. a b a x2 x4 a b x1 b a x3 x5 b b L2 a DFA L21 by rows NFA L22 by columns NFA L23 mixed a a,b b a,b a,b a,b a,b b b b x1 x3 x3 x2 x3 x2 x2 x1 x1 Original and Reduced Automata L2 a

  34. a a a a x1 x2 x3 x4 b b b b b x5 x6 x7 L3 a Row Basis but No Column Basis

  35. a a a a x1 x2 x3 x4 b b b b b x5 x6 x7 L3 a Row Basis but No Column Basis

  36. a a a a x1 x2 x3 x4 b b b b b x5 x6 x7 L3 a Row Basis but No Column Basis

  37. a a a a x1 x2 x3 x4 b b b b b x5 x6 x7 L3 a Row Basis but No Column Basis

  38. Observabilty Reduction • Theorem (Cover):Finding a (possibly mixed) basis T • for OL is equivalent to finding a minimal cover for OL. • either as itsset basis coveror asitsKarnaugh cover. • Theorem (Complexity):Determining a cover T for OL • is NP-complete (set basis problem complexity). • Theorem (Rank): The row (= column) rank of OL is the • size of the set coverT (size of Karnaugh cover).

  39. Reachability: Dual of Observability

  40. b a b a x3 x1 x2 Reachability: Dual of Observability L1

  41. Observabilty, Reachability and More • DFA Minimization: Is aparticular caseof observability • reduction (single initial state requires distinctness only) • NFA Determinization: Is a particular case of reachability • transformation(take all distinct columns as “basis”) • Minimal automata: Are related by linear maps (but not • by graph isomorphisms!). Better definition of minimality • Other techniques: Are easily formalized in this setting: • Pumping lemma, NFA to RE, Z-transforms, etc.

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