Ranking based Techniques for Disambiguating B¨ uchi Automata Hrishikesh Karmarkar Supratik Chakraborty January 29, 2011 Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Non-deterministic B¨ uchi automata over words (NBW) A 5 − tuple (Σ , Q , Q 0 , δ, F ), a q1 where Σ : Input alphabet b q2 Q : Finite set of states Q 0 ⊆ Q : Initial states q3 δ ⊆ Q × Σ × Q : State qs transition relation F : Set of final/accepting q4 states q5 Final state a, b Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Runs and acceptance A run of A on α ∈ Σ ω is a a q1 sequence ρ : N → Q such that b q2 ρ (0) ∈ Q 0 ρ ( i + 1) ∈ δ ( ρ ( i ) , α ( i )) q3 qs q4 q5 Final state a, b α = abbbbb · · · , ρ 1 = q 1 q 2 q 2 q 2 q 2 q 2 q 2 · · · Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Runs and acceptance A run of A on α ∈ Σ ω is a a q1 sequence ρ : N → Q such that b q2 ρ (0) ∈ Q 0 ρ ( i + 1) ∈ δ ( ρ ( i ) , α ( i )) q3 An automaton may have qs several runs on α . q4 q5 Final state a, b α = abbbbb · · · , ρ 1 = q 1 q 2 q 2 q 2 q 2 q 2 q 2 · · · ρ 2 = q 1 q 1 q 2 q 2 q 2 q 2 q 2 · · · Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Runs and acceptance A run of A on α ∈ Σ ω is a a q1 sequence ρ : N → Q such that b q2 ρ (0) ∈ Q 0 ρ ( i + 1) ∈ δ ( ρ ( i ) , α ( i )) q3 An automaton may have qs several runs on α . q4 ρ is accepting iff inf( ρ ) ∩ F � = ∅ q5 α is accepted by A Final state a, b ( α ∈ L ( A )) iff there is an accepting run of A on α . α = abbbbb · · · , ρ 1 = q 1 q 2 q 2 q 2 q 2 q 2 q 2 · · · ρ 2 = q 1 q 1 q 2 q 2 q 2 q 2 q 2 · · · Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Ambiguous automata A is ambiguous if there exists α ∈ L ( A ) such that there are ≥ 2 accepting runs of A on α Otherwise, A is unambiguous . Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Ambiguous automata A is ambiguous if there exists α ∈ L ( A ) such that there are ≥ 2 accepting runs of A on α Otherwise, A is unambiguous . An ambiguous NBW a q1 b α = ab ω q2 ρ 1 = q 1 q ω 2 q3 qs ρ 2 = q 1 q 1 q ω 2 q4 q5 Final state a, b Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Strongly Unambiguous automata Final run of A on α : A run ρ starting from any state in Q such that inf( ρ ) ∩ F � = ∅ . A word �∈ L ( A ) may have 0 or more final runs A word ∈ L ( A ) has ≥ 1 final runs Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Strongly Unambiguous automata Final run of A on α : A run ρ starting from any state in Q such that inf( ρ ) ∩ F � = ∅ . A word �∈ L ( A ) may have 0 or more final runs A word ∈ L ( A ) has ≥ 1 final runs NBW A is strongly unambiguous if for every α ∈ Σ ω , there is exactly one final run. Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Strongly Unambiguous automata Final run of A on α : A run ρ starting from any state in Q such that inf( ρ ) ∩ F � = ∅ . A word �∈ L ( A ) may have 0 or more final runs A word ∈ L ( A ) has ≥ 1 final runs NBW A is strongly unambiguous if for every α ∈ Σ ω , there is exactly one final run. Not all unambiguous automata are strongly unambiguous. Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Strongly Unambiguous automata Final run of A on α : A run ρ b a starting from any state in Q such that inf( ρ ) ∩ F � = ∅ . A word �∈ L ( A ) may have 0 or a b a b more final runs A word ∈ L ( A ) has ≥ 1 final runs NBW A is strongly unambiguous a b Deterministic (hence Strongly if for every α ∈ Σ ω , there is unambiguous) but unambiguous exactly one final run. not strongly unambiguous Not all unambiguous automata are strongly unambiguous. Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Containment relations UNBW: Unambiguous NBW, SUNBW: Strongly unambiguous NBW, DBW: Deterministic B¨ uchi automata over words Expressive power-wise DBW � NBW ≡ UNBW ≡ SUNBW Automata structure-wise NBW DBW UNBW SUNBW Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
What this talk is about Given an NBW, construct UNBW accepting the same language and using as few states as possible. Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
What this talk is about Given an NBW, construct UNBW accepting the same language and using as few states as possible. Relevant earlier work: Arnold 1983: UNBW expressively equivalent to NBW Carton & Michel 2003: Effective construction of SUNBW, size bound O ((12 n ) n ) K¨ ahler and Wilke 2008: Effective construction of UNBW, size bound O ((3 n ) n ). Bousquet and L¨ oding 2010: Equivalence and inclusion problems for SUNBW are poly-time Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
What this talk is about Given an NBW, construct UNBW accepting the same language and using as few states as possible. Relevant earlier work: Arnold 1983: UNBW expressively equivalent to NBW Carton & Michel 2003: Effective construction of SUNBW, size bound O ((12 n ) n ) K¨ ahler and Wilke 2008: Effective construction of UNBW, size bound O ((3 n ) n ). Bousquet and L¨ oding 2010: Equivalence and inclusion problems for SUNBW are poly-time Our contribution Effective construction of UNBW, size bound O ( n 2 . (0 . 76 n ) n ) Same as best known bound for NBW complementation! Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Why care about disambiguation? Of course, a theoretically interesting problem Can it lead to a better understanding of what kinds of NBW admit easy determinization? Practical application? Seek inputs from the audience. Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Run DAGs Run DAG for a ω q1 a q1 q1 q2 b q2 q2 q3 q1 q3 q4 q3 q1 q2 qs q4 q5 q1 q2 q3 q4 q5 q2 q3 q4 q5 q3 q4 q5 Vertices with qs not Final state shown for clarity a, b Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Ranking run DAGs Intuitively, assign a metric to each vertex in run DAG such that the metric changes in a desirable way only along “good” runs. Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Ranking run DAGs Intuitively, assign a metric to each vertex in run DAG such that the metric changes in a desirable way only along “good” runs. Early work by Michel (1984?), Klarlund (1991): Ranking functions/progress measures for B¨ uchi complementation Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Ranking run DAGs Intuitively, assign a metric to each vertex in run DAG such that the metric changes in a desirable way only along “good” runs. Early work by Michel (1984?), Klarlund (1991): Ranking functions/progress measures for B¨ uchi complementation Recent spurt of work triggered by similar metrics defined by Kupferman & Vardi (2001 onwards) Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Ranking run DAGs Intuitively, assign a metric to each vertex in run DAG such that the metric changes in a desirable way only along “good” runs. Early work by Michel (1984?), Klarlund (1991): Ranking functions/progress measures for B¨ uchi complementation Recent spurt of work triggered by similar metrics defined by Kupferman & Vardi (2001 onwards) Schewe (2009) used this approach to match upper bound of NBW complementation within O ( n 2 ) of lower bound Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Ranking run DAGs Intuitively, assign a metric to each vertex in run DAG such that the metric changes in a desirable way only along “good” runs. Early work by Michel (1984?), Klarlund (1991): Ranking functions/progress measures for B¨ uchi complementation Recent spurt of work triggered by similar metrics defined by Kupferman & Vardi (2001 onwards) Schewe (2009) used this approach to match upper bound of NBW complementation within O ( n 2 ) of lower bound We use Kupferman-Vardi style rankings Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Kupferman-Vardi style ranking n : Number of states in NBW V : Set of run DAG vertices r : V → { 1 , 2 , . . . 2 n + 1 } : Ranking function Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B¨ uchi Automata
Recommend
More recommend
