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-Automata

Games, Logic and Automata SeminarRotem Zach

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Overview

• We will present several models of -automata

• We will prove the equivalence of their nondeterministic forms

• We will prove the equivalence of their deterministic forms

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Notation

• Given an alphabet – = set of finite words over – = set of infinite words over

• = Concatenation of and • = Class of regular languages

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Recap

• -automaton is a quintuple • = acceptance component• A run is an infinite sequence of states• = States occurring infinitely often in

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Büchi Acceptance [Büchi 1962]

• Define a set • A word is accepted if there exists a run s.t.

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Example

• Which language is recognized?

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Deterministic Büchi Automata

• Theorem: Deterministic Büchi Automata are weaker than nondeterministic Büchi Automata

• Proof: • Can’t express • For , enters an state after letters• For , will visit an state after • Can create which will be accepted

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Muller Acceptance [Muller 1963]

• Define a set • A word is accepted if there exists a run s.t.

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Example

• Define for ?

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Büchi Muller

• Büchi automaton = • Define

• Preserves determinism:– Deterministic Deterministic– Nondeterministic Nondeterministic

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Muller Büchi

• The Büchi automaton guesses the set which will be

• It also guesses from when on only the states in will be seen

• It verifies this guess by holding in memory which states were visited

• When all was seen, it resets the memory• The reset states are the new accepting states

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Muller Büchi

• To implement, for each create a new copy of • Simultaneously make the two guesses and

move from to • These new states also contain a memory

component , to store which states in were visited

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Muller Büchi

• If originally n states and m accepting sets, then

• Always converts to nondeterministic

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Rabin Acceptance [Rabin 1969]

• Define , • A word α is accepted if there exists a run s.t.

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Example

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Streett Acceptance [Streett 1982]

• Define , • A word is accepted if there exists a run s.t.

• Dual to Rabin acceptance

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Example

• Previous language• Define

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Rabin, Streett Muller

• Go over every possible choice for and check the acceptance conditions

• For Rabin:

• For Streett:

• Preserves determinism

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Muller Rabin, Streett

• Convert Muller to Büchi

• For Rabin: • For Streett:

• Always converts to nondeterministic (because Muller Büchi)

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Nondeterministic Equivalence

• We have proved the theorem:• Nondeterministic Büchi, Muller, Rabin and

Streett automata recognize the same class of ω-languages

• Denote this class of languages as -REG

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Parity Condition [Mostowski 1984]

• Special case of Rabin when

• Associate colors with states– for – for

• A word α is accepted if there exists a run s.t.

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Example

• Which language is accepted if ?• Which language is accepted if ?

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D’ Muller D’ Rabin

• We’ll use latest appearance records as states– Permutations of the Muller automaton’s states

extended by a hit position (#)• Let the original states be • If the original automaton moved and in the

new automaton the state is then after the transition it is

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D’ Muller D’ Rabin

• = permutations of • ()

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D’ Muller D’ Rabin

• For the run of the Muller automaton, assume • For the run of the Rabin automaton,

eventually every state in will be before #• Thus, and k• Infinitely often – When this happens the states in are

• Finitely often

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D’ Muller D’ Rabin

• The Muller automaton accepts if • Let • The Rabin automaton will visit a state in

infinitely often• But will only visit states in finitely often• So it will accept

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D’ Muller D’ Rabin

• If then

– There are accepting pairs

• Preserves determinism

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D’ Muller D’ Parity

• In the previous transformation :

• So we have also proved D’ Muller D’ Parity

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Closure Under Complement

• For a deterministic Muller automaton the acceptance condition is equivalent to:A word is accepted if for the (only) run

• Thus:A word isn’t accepted if for the run

• To accept the complement,

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D’ Muller -> D’ Streett

• Rabin acceptance:

• Streett acceptance:

• Given a deterministic Rabin/Streett keeping the same F yields a deterministic Streett /Rabin accepting the complemented L

• Muller Complement’s Muller Complement’s Rabin Streett

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D’ Rabin D’ Streett

• Rabin Complement’s Streett Complement’s Muller Muller Streett

• Streett Complement’s Rabin Complement’s Muller Muller Rabin

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Deterministic Equivalence

• We have proved the theorem:• Deterministic Muller, Rabin, Streett and parity

automata recognize the same class of -languages

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Lower Bounds [Löding 1999]

• The following transformations:– deterministic Rabin deterministic Streett– nondeterministic Büchi deterministic Rabin

• Have a lower bound of given

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Sources• Grädel, Thomas, Wilke (Eds.): Automata, Logics, and Infinite Games: A

Guide to Current Research, 2002• J.R. Büchi, On a decision method in restricted second order arithmetic,

1962• D.E. Muller, Infinite sequences and finite machines, 1963• M.O. Rabin, Decidability of second order theories and automata on

infinite trees, 1969• R.S. Streett, Propositional dynamic logic of looping and converse is

elementary decidable, 1982• A.W. Mostowski. Regular expressions for infnite trees and a standard

form of automata, 1984• C. Löding, Optimal bounds for the transformation of omega-automata,

1999

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Questions?