Radu Grosu SUNY at Stony Brook

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.