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Radu Grosu SUNY at Stony Brook

Finite Automata as Linear Systems Observability, Reachability and More. Radu Grosu SUNY at Stony Brook. a. x 2. b. x 1. x 3. c. a. Minimal DFA are Not Minimal NFA (Arnold, Dicky and Nivat’s Example). b. x 3. b. a. x 1. x 2. x 4. c. c. L = a (b* + c*). b. a. x 2. c. a. b.

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Radu Grosu SUNY at Stony Brook

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

  2. a x2 b x1 x3 c a Minimal DFA are Not Minimal NFA (Arnold, Dicky and Nivat’s Example) b x3 b a x1 x2 x4 c c L = a (b* + c*)

  3. b a x2 c a b x1 x3 x5 c a c x4 b Minimal NFA: How are they Related? (Arnold, Dicky and Nivat’s Example) c a x2 b b a x1 x3 x5 c a b x4 c L = ab+ac + ba+bc + ca+cb No homomorphism of either automaton onto the other.

  4. c a b a x2 x5 b c a b a b x1 x3 x8 x1 x6 x8 c c a a b c x4 x7 c b Minimal NFA: How are they Related? (Arnold, Dicky and Nivat’s Example) Carrez’s solution:Take both in aterminal NFA. Is this the best one can do? No! One can useuse linear (similarity) transformations.

  5. Convergence of Theories • Hybrid Systems Computation and Control: • 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.

  6. 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!

  7. 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.

  8. Finite Automata as Linear Systems

  9. Finite Automata as Linear Systems

  10. Finite Automata as Linear Systems

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

  12. Polynomials and their Operations

  13. Polynomials and their Operations

  14. Boolean Semimodules

  15. Boolean Semimodules

  16. Boolean Semimodules

  17. Observability

  18. b a b a x3 x1 x2 Observability L1

  19. Linear Dependence

  20. Linear Dependence

  21. Linear Dependence

  22. Linear Dependence

  23. Basis in Boolean Semimodule

  24. Basis in Boolean Semimodule

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

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

  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 Rows L2

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

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

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

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

  35. 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

  36. 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

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

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

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

  41. 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).

  42. Reachability: Dual of Observability

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

  44. 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.

  45. c a b a x2 x23 b c a b a b x1 x3 x5 x1 x24 x5 c c a a b c x4 x34 c b Arnold, Dicky & Nivat’s Example Revisited (Obsrvability Reduction)

  46. Arnold, Dicky & Nivat’s Example Revisited (Reachability Reduction) b a c a x2 x23 b c a b b a x1 x3 x5 x1 x24 x5 c c a c a b x4 x34 b c

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