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Finite Automata as Linear Systems Observability, Reachability and More. Radu Grosu SUNY at Stony Brook. Convergence of Theories. HSCC Conference: a witness of the fascinating convergence between control and automata theory. Hybrid Automata: an outcome of this convergence
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Finite Automata as Linear Systems Observability, Reachability and More Radu GrosuSUNY at Stony Brook
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.
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!
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.
b a b a x3 x1 x2 Finite Automata as Linear Systems L1
b a b a x3 x1 x2 Observability L1
b a b a x3 x1 x2 Basis in Boolean Semimodule L1
b a b a x3 x1 x2 Basis in Boolean Semimodule L1
a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2
a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2
a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2
a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2
a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2
a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Rows L2
a b a x2 x4 a x1 b b a x3 x5 b b a Observability Reduction by Columns L2
a b a x2 x4 a x1 b b a x3 x5 b b a Mixed Observability Reduction L2
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
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
a a a a x1 x2 x3 x4 b b b b b x5 x6 x7 L3 a Row Basis but No Column Basis
a a a a x1 x2 x3 x4 b b b b b x5 x6 x7 L3 a Row Basis but No Column Basis
a a a a x1 x2 x3 x4 b b b b b x5 x6 x7 L3 a Row Basis but No Column Basis
a a a a x1 x2 x3 x4 b b b b b x5 x6 x7 L3 a Row Basis but No Column Basis
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).
b a b a x3 x1 x2 Reachability: Dual of Observability L1
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.