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-Automata. Ekaterina Mineev. Today:. 1 Introduction - notation - -Automata overview. Today:. 1 Introduction - notation - -Automata overview 2 Nondeterministic models - B ü chi acceptance - Muller acceptance - Rabin acceptance - Streett acceptance - parity condition.
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-Automata Ekaterina Mineev
Today: 1 Introduction - notation - -Automata overview
Today: 1 Introduction - notation - -Automata overview 2 Nondeterministic models - Büchi acceptance - Muller acceptance - Rabin acceptance - Streett acceptance - parity condition
Today(cont.): 2.1 Equivalency of nondeterministic models
Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - Büchi condition - equivalency of deterministic* models
Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - Büchi condition - equivalency of deterministic* models 4 Some lower bound for transformations
Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - Büchi condition - equivalency of deterministic* models 4 Some lower bound for transformations 5 Weak acceptance conditions - Staiger-Wagner acceptance
Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - Büchi condition - equivalency of deterministic* models 4 Some lower bound for transformations 5 Weak acceptance conditions - Staiger-Wagner acceptance 6 Conclusion
Notation • := {0, 1, 2, 3, …}
Notation • := {0, 1, 2, 3, …} • - finite alphabet
Notation • := {0, 1, 2, 3, …} • - finite alphabet • * - set of finite words over
Notation • := {0, 1, 2, 3, …} • - finite alphabet • * - set of finite words over • - set of infinite words (-words) over
Notation • := {0, 1, 2, 3, …} • - finite alphabet • * - set of finite words over • - set of infinite words (-words) over • u, v, w – finite words
Notation • := {0, 1, 2, 3, …} • - finite alphabet • * - set of finite words over • - set of infinite words (-words) over • u, v, w – finite words • , , - infinite words
Notation • := {0, 1, 2, 3, …} • - finite alphabet • * - set of finite words over • - set of infinite words (-words) over • u, v, w – finite words • , , - infinite words • = (0)(1)(2)…with (i)
Notation • := {0, 1, 2, 3, …} • - finite alphabet • * - set of finite words over • - set of infinite words (-words) over • u, v, w – finite words • , , - infinite words • = (0)(1)(2)…with (i) • , - runs of automata
Notation • := {0, 1, 2, 3, …} • - finite alphabet • * - set of finite words over • - set of infinite words (-words) over • u, v, w – finite words • , , - infinite words • = (0)(1)(2)…with (i) • , - runs of automata • -language – set of -words
Notation(cont.) • ||a– number of occurrences of a in
Notation(cont.) • ||a– number of occurrences of a in • Occ() := {ai (i)=a}
Notation(cont.) • ||a– number of occurrences of a in • Occ() := {ai (i)=a} • Inf () := {ai j>i (j)=a}
Notation(cont.) • ||a– number of occurrences of a in • Occ() := {ai (i)=a} • Inf () := {ai j>i (j)=a} • 2M– powerset of a set M
Notation(cont.) • ||a– number of occurrences of a in • Occ() := {ai (i)=a} • Inf () := {ai j>i (j)=a} • 2M– powerset of a set M • REG – class of regular languages
Notation(cont.) • ||a– number of occurrences of a in • Occ() := {ai (i)=a} • Inf () := {ai j>i (j)=a} • 2M– powerset of a set M • REG – class of regular languages • L(A) := {*A accepts } - -language recognized by A
-Automata -Automaton is (Q, , , qI, Acc)
-Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states
-Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states • - finite alphabet
-Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states • - finite alphabet • : Q 2Q/Q – state transition function
-Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states • - finite alphabet • : Q 2Q/Q – state transition function • qIQ – initial state
-Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states • - finite alphabet • : Q 2Q/Q – state transition function • qIQ – initial state • Acc – acceptance component
-Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states • - finite alphabet • : Q 2Q/Q – state transition function • qIQ – initial state • Acc – acceptance component can be given in different way!!!
-Automata -Automaton is (Q, , , qI, Acc) • Q – finite set of states • - finite alphabet • : Q 2Q/Q – state transition function • qIQ – initial state • Acc – acceptance component can be given in different way!!! |A| = |Q| - size of automaton Acc sometimes used too
Büchi acceptance -Automaton (Q, , , qI, FQ) is Büchi if Acc is Büchi acceptance:
Büchi acceptance -Automaton (Q, , , qI, FQ) is Büchi if Acc is Büchi acceptance: A word * is accepted by A iff there exists a run of A on satisfying the condition: Inf()F
Example 1 L := {{a, b}| ends with aor with (ab)}
Büchi acceptance(cont.) • is accepted by A iff some run of A on visit some final state qF infinitely often, i.e. W(q0, q)W(q, q)
Büchi acceptance(cont.) • is accepted by A iff some run of A on visit some final state qF infinitely often, i.e. W(q0, q)W(q, q) The Büchi recognizable -languages are the -languages of the form: L=ki=1 UiVi with k and Ui , Vi REG for i=1, 2, 3, …
Büchi acceptance(cont.) The family of -languages is also called the -Kleene closure of the class of regular languages denoted -KC(REG)
Muller acceptance -Automaton (Q, , , qI, F 2Q) is Muller if Acc is Muller acceptance:
Muller acceptance -Automaton (Q, , , qI, F 2Q) is Muller if Acc is Muller acceptance: A word * is accepted by A iff there exists a run of A on satisfying the condition: Inf()F
Example 2 L := {{a, b}| ends with aor with (ab)} F = { {qa}, {qa,qb} }
Büchi and Muller automata Nondeterministic Büchi automata and nondeterministic Muller automata are equivalent in expressive power
Büchi and Muller automata Nondeterministic Büchi automata and nondeterministic Muller automata are equivalent in expressive power One direction is simple: F := { KQ | KF }
Büchi and Muller automata Nondeterministic Büchi automata and nondeterministic Muller automata are equivalent in expressive power One direction is simple: F := { KQ | KF } Second is complex and multiples states number exponentially
Rabin acceptance -Automaton (Q, , , qI, ), = {(E1, F1),…,(Ek, Fk)}with Ei, Fi Q is Rabin if Acc is Rabin acceptance:
Rabin acceptance -Automaton (Q, , , qI, ), = {(E1, F1),…,(Ek, Fk)}with Ei, Fi Q is Rabin if Acc is Rabin acceptance: A word * is accepted by A iff there exists a run of A on satisfying the condition: (E,F) . (Inf()E = ) (Inf()F )