Short Accepting Lassos & Witnesses in ω -automata R¨ udiger Ehlers Saarland University, Reactive Systems Group LATA 2010 – May 27, 2010 R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 1 / 20
ω -automata Basic properties Similar to ordinary finite automata Accept/reject infinite words w ∈ Σ ω Typical acceptance condition types: Safety, B¨ uchi, Rabin, Streett, Muller, . . . R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 2 / 20
Automata theory & model checking Overview Example acc. System to Safety word/lasso be checked automaton Product B¨ uchi automaton Neg. of the B¨ uchi Emptiness property to automaton result be checked R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 3 / 20
An example system Setting A conveyor belt merger controller g 1 r 1 w 1 t 1 � · 1 � � � � r 2 · 0 � 1 0 · 0 g 2 · 0 0 0 � · 0 � � 1 0 � init 1 0 · 0 Alphabet semantics � 0 0 � � 0 0 0 0 � r 1 g 1 � First belt � 0 0 � 1 0 � · 0 Second belt r 2 g 2 w 2 t 2 � · 0 � Grants · 1 Requests R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 4 / 20
An example system Setting An example property g 1 The system is starvation-free. r 1 r 2 The corresponding neg. automaton g 2 � � � � 1 0 · 0 � · · � , · 0 1 0 · · Alphabet semantics e d f � r 1 g 1 � � · 0 � First belt · 0 � · · � � · 0 � Second belt r 2 g 2 · · · 0 Grants Requests R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 4 / 20
The product ( w 1 , f ) ( w 2 , d ) ( t 2 , e ) ( init , e ) ( t 1 , e ) ( t 1 , f ) ( t 2 , f ) ( init , d ) ( w 1 , d ) ( w 1 , e ) ( t 1 , d ) ( w 2 , e ) ( w 2 , f ) ( t 2 , d ) R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 5 / 20
Short lassos: an example ( w 1 , f ) ( w 2 , d ) ( t 2 , e ) ( init , e ) ( t 1 , e ) ( t 1 , f ) ( t 2 , f ) ( init , d ) ( w 1 , d ) ( w 1 , e ) ( t 1 , d ) ( w 2 , e ) ( w 2 , f ) ( t 2 , d ) R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 6 / 20
Short lassos: an example ( w 1 , f ) ( w 2 , d ) ( t 2 , e ) ( init , e ) ( t 1 , e ) ( init , d ) ( t 1 , f ) ( t 2 , f ) ( w 1 , d ) ( w 1 , e ) ( t 1 , d ) ( w 2 , e ) ( w 2 , f ) ( t 2 , d ) R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 6 / 20
An alternative point of view – short witnesses A different kind of counter-examples Often, it is enough for the designer to know one erroneous example trace of the system. Such a trace can often be represented in a much shorter way. An example The conveyor belt merger behaves incorrectly with the following input/output: � 1 � ω 0 1 0 Conclusion A “witness” is often much simpler to understand by the system designer. R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 7 / 20
Defining the size of a counter-example Lassos ( w 1 , f ) ( w 2 , d ) ( t 2 , e ) ( init , e ) ( t 1 , e ) ( t 1 , f ) ( t 2 , f ) ( init , d ) ( w 1 , d ) ( w 1 , e ) ( t 1 , d ) ( w 2 , e ) ( w 2 , f ) ( t 2 , d ) This lasso is of size 3 . Witnesses For uw ω being the witness for u , w ∈ Σ ∗ , we define the size to be | u | + | w | . R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 8 / 20
Applications of short lassos & witnesses Some examples: Model checking Certificates for the satisfiability of a formula in logics such as S1S Sanity checks of specification automata . . . Consequences It makes sense to consider this problem for all commonly used types of acceptance conditions. The main question we ask here is: what is the complexity of this problem? R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 9 / 20
Previously known results Direct results on the complexity of these problems previously known Acc. cond. type Short lassos Short witnesses Safety O ( | Q | 2 ) [SE05] B¨ uchi NP-complete [KSF06] co-B¨ uchi Parity Rabin Gen. B¨ uchi NP-complete [CGMZ95] Streett Muller R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 10 / 20
Previously known results Implicit results on the complexity of these problems previously known Acc. cond. type Short lassos Short witnesses O ( | Q | 2 ) Safety O ( | Q | 2 ) B¨ uchi NP-complete co-B¨ uchi in PTIME Parity in PTIME NP-complete Rabin in PTIME NP-complete Gen. B¨ uchi NP-complete NP-complete Streett NP-complete NP-complete Muller R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 11 / 20
Our completion of the landscape All results now known Acc. cond. type Short lassos Short witnesses Safety B¨ uchi co-B¨ uchi in PTIME Parity NP-complete Rabin Gen. B¨ uchi Streett NP-complete Muller R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 12 / 20
On approximating shortest witnesses & lassos In practice For practical application, approximate shortest witnesses and lassos would usually suffice! Important question: For those problems that are not in PTIME (assuming NP � =PTIME), can they be approximated well in polynomial time? R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 13 / 20
On finding approximate short lassos Overview Generalised B¨ uchi & Streett Not approximable within Acc. cond. type Short lassos any constant in polynomial Safety time (unless P=NP). B¨ uchi co-B¨ uchi in PTIME Proof idea Parity Reduction to the Rabin E k -Vertex-Cover problem Gen. B¨ uchi This case Streett Muller R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 14 / 20
On finding approximate short lassos Overview The Muller case Not approximable within Acc. cond. type Short lassos 321 320 − ǫ (unless P=NP), Safety approximable within B¨ uchi � log 2 | Q | � in polynomial time. co-B¨ uchi in PTIME Parity Proof idea Rabin Using the connection to the Gen. B¨ uchi asymmetric metric Streett travelling salesman problem . Muller This case R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 14 / 20
On finding approximate short witnesses Overview The safety case Not approximable within Acc. cond. type Short witnesses any polynomial function Safety in polynomial time (unless B¨ uchi P=NP). co-B¨ uchi Parity NP-complete Proof idea Rabin Gen. B¨ uchi Reduction from the Streett satisfiability problem using Muller the gap technique. R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 15 / 20
Proof idea for the shortest witness case Reduction from the SAT-problem Idea: Encode potential solutions to a SAT problem as words over { 0 , 1 , # } For every clause in the SAT problem, build a block requiring that a part of the word “satisfies” the clause. For every clause, put k of these blocks in a line (for some k ∈ N ) and plug together the lines for all clauses. Example block for the clause ¬ v 1 ∨ v 2 0 , 1 1 0 . . . # 0 1 # . . . 0 , 1 0 , 1 R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 16 / 20
Shortest witness case - An example SAT instance ( v 1 ∨ v 2 ∨ ¬ v 3 ) ∧ ( ¬ v 1 ∨ v 2 ) ∧ ( ¬ v 2 ∨ v 3 ) Safety automaton 0 0 1 0 1 0 0 1 . . . . . . # # 1 1 0 1 1 1 0 0 . . . . . . 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 # 0 , 1 0 , 1 0 , 1 1 0 1 1 0 . . . . . . # # 0 1 0 0 1 . . . . . . 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 # 0 , 1 0 , 1 0 , 1 1 0 0 1 0 . . . . . . # # 0 , 1 , # 0 1 1 0 1 . . . . . . 0 , 1 0 , 1 0 , 1 0 , 1 0 , 1 # R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 17 / 20
Implications for practice Counter-example generation for model checking We can either: stick to the shortest lasso case (when applicable) try to use potentially slow techniques develop & use suitable heuristics R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 18 / 20
Outlook Implications for synthesis of open systems Finding a small implementation satisfying a specification is a hard problem, even for safety games! R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 19 / 20
References [CGMZ95] Edmund M. Clarke, Orna Grumberg, Kenneth L. McMillan, and Xudong Zhao. Efficient generation of counterexamples and witnesses in symbolic model checking. In DAC , pages 427–432, 1995. [KSF06] Orna Kupferman and Sarai Sheinvald-Faragy. Finding shortest witnesses to the nonemptiness of automata on infinite words. In Christel Baier and Holger Hermanns, editors, CONCUR , volume 4137 of LNCS , pages 492–508. Springer, 2006. [SE05] Stefan Schwoon and Javier Esparza. A note on on-the-fly verification algorithms. In Nicolas Halbwachs and Lenore D. Zuck, editors, TACAS , volume 3440 of LNCS , pages 174–190, 2005. R¨ udiger Ehlers (SB) Short Lassos & Witnesses LATA 2010 – May 27, 2010 20 / 20
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