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Converting LTL to Büchi Automata: A Comprehensive Guide

This guide explains the process of converting Linear Temporal Logic (LTL) formulas into Büchi automata, a critical step in formal verification. It covers the construction of a Generalized Nondeterministic Büchi Automaton (GNBA), conversion to a Nondeterministic Büchi Automaton (NBA), and optimization techniques. The document delves into concepts such as consistent observations, maximality, and eventuality, providing examples for clarity. Understanding these concepts is essential for applying LTL in the context of model checking and automated analysis of systems.

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Converting LTL to Büchi Automata: A Comprehensive Guide

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  1. Converting LTL to Buchi Wishnu Prasetya wishnu@cs.uu.nl www.cs.uu.nl/docs/vakken/pv

  2. Converting LTL to Buchi • Given an LTL formula , construct a Buchi automaton M that accepts the same sentences as . • Recall: “sentence” is a sequence of ‘something’, each is a set of propositions. Sentence = (abstract) execution. • Steps: • Construct GNBA • Convert to NBA • Optimize

  3. Idea.. {q} {p} {q} B = {  p , q , Xq } B = {  p , q , Xq } B = { p , q , Xq } To help us, each state s will be labeled with an “observation” B. It is a consistent set of formulas. Any infinite sequence starting from s must satisfy all formulas in B. The set of candidate “observations” for a given  is finite; and we can figure out how to connect them with arrows.

  4. Restricting to X/U • All LTL formulas can be expressed with just X and U. • Let’s assume that your input formula is expressed in this form of LTL. <> = true U  []  =  ( <>  ) W = []  \/ U

  5. Closure • closure() is the set of all • subformulas of  (incl  itself) • negations of subformulas • Example:  = pUq • Only the value of the formulas in the closure can affect the value of . closure() = { p, q, p, q, pUq , (pUq) }

  6. Observation • Example:  = pUq • An ‘observation’ B is in principle a subset of the closure, but we want it to be ‘consistent’ and ‘maximal’. • { p, q, pUq }  OK • { p, p }  inconsistent • { p }  not maximal • We’ll only take consistent observations. closure() = { p, q, p, q, pUq , (pUq) }

  7. Consistency of the B’s • An observation B must be consistent with respect to propositional logic: • f and f cannot be both in B • f /\ g  B  f,g  B • Locally consistent with respect to “until”. For any fUg  closure( ) : • gB  fUg  B • fUg  B and gB  f  B

  8. Maximality • Every observation B should be maximal • Ex. For everyf  closure(), eitherfBor fB. closure() = { p, q, p, q, pUq , (pUq) }  = pUq { p , q , (p U q) } { p , q , p U q } { p , q , (p U q) } { p , q , p U q } 8 maximal subsets, 3 are inconsistent. {p , q , (p U q)} {p , q , p U q } {p , q , (p U q) } {p , q , p U q }

  9. Constructing the automaton A • States: observations from closures() • Initial states: all states that contain  • Arrows: for any pairs observations B,C add this arrow:B  V  C • If this arrow is ‘consistent’ • V = the set of propositions in B. • Acceptance states?

  10. The arrows • B  V  C is consistent if : • Example: (note the bi-implication!) • XfB  fC • fU g  B  gB or fB and fUg  C { p , q , p U q } { p , q , (p U q) } { p , q , p U q } {p , q , p U q } {p , q , (p U q) }

  11. The arrows • B  V  C is consistent if : • The bi-implications thus also imply: • XfB  fC • fU g  B  gB or fB and fUg  C •  XfB  fC •  (fU g )  B  gB and ( fB or  (fUg )  C )

  12. { p , q , p U q } {p , q , (p U q) } { p , q , p U q } { p , q , (p U q) } {p , q , p U q }

  13. Enforcing eventuality • For each fUg closure(), add an accepting group:where Q is the set of states of GNBA of  that we are constructing.(btw = the set of all ‘observations’) F(fUg) = { B | B Q /\ g B }  { B | B Q /\ fUgB }

  14. { p , q , p U q } {p , q , (p U q) } { p , q , p U q } { p , q , (p U q) } {p , q , p U q }

  15. From GNBA to NBA GNBA with 2x accepting groups. single accentance group of the new automaton dashed-red arrows are dropped

  16. Can we make it deterministic? • In ordinary automaton, DFA can be converted to an equivalent NDFA (equivalent = generating the same sentences). • For Buchi? • NBA is really more powerful than DBA. a b b b No deterministic Buchi can generate the sentences of this Buchi

  17. How big are they? • NGBA generated by our procedure  |M| = 2||. • Converting to GBA multiplies the number of states with C+1, where C is the number of U in  • There are LTL formulas of polynomial size, whose NBA will have at least exponential number of states.

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