Buchi Automaton Construction
Learn how to convert Linear Temporal Logic (LTL) formulas to Buchi automata step by step. Understand the concept of observations, consistency, and maximality in constructing Buchi automata. Visual examples provided for better understanding.
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Converting LTL to Buchi Wishnu Prasetya wishnu@cs.uu.nl www.cs.uu.nl/docs/vakken/pv
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 symbols , each is a set of propositions. Sentence = (abstract) execution. Steps: Construct GNBA Convert to NBA Optimize 2
Restricting to X/U All LTL formulas can be expressed with just X and U. <> = true U [] = ( <> ) W = [] \/ U Let s assume that your input formula is expressed in this form of LTL. 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
Closure closure( ) is the set of all subformulas of (incl itself) negations of subformulas Example: = p U q closure( ) = { p, q, p, q, p U q , (p U q) } Only the value of the formulas in the closure can affect the value of . 5
Observation Example: = p U q closure( ) = { p, q, p, q, p U q , (p U q) } An observation B is in principle a subset of the closure, but we want it to be consistent and maximal . { p, q, p U q } OK { p, p } inconsistent { p } not maximal 6
Consistency of the Bs 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 Consistent with respect to until . For any f U g closure( ) : g B f U g B f U g B and g B f B 7
Maximality Every observation B should be maximal For every f closure( ), either f B or f B. closure( ) = { p, q, p, q, Ex. = p U q p U q , (p U q) } { p , q , (p U q) } { p , q , p U q } 5 Observations (blue). Red ones may look like observations, but are inconsistent. { 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) } { p , q , p U q } 8
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 V = the set of postive (not in negation) propositions in B. If this arrow is consistent Acceptance states? 9
The arrows B V C is consistent if (1) : Xf B f C fUg B g B or ( f B and f U g C ) Example: { p , q , p U q } {p} { p , q , (p U q) } {p} { p , q , p U q } {p} { p , q , p U q } { p , q , (p U q) } 10
The arrows B V C is consistent if (1) : Xf B f C f U g B g B or ( f B and f U g C ) Furthermore (2) : Xf B f C (f U g) B ( f B and g B ) or ( f B and g B and (f U g ) C ) ) 11
{ 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 } (the labels on the arrows are removed for readability) 12
Enforcing eventuality For each f U g closure( ), add an accepting group: F (f U g) = { B | B Q /\ g B } { B | B Q /\ f Ug B } where Q is the set of states of GNBA of that we are constructing. (Q = the set of all observations ) 13
{ 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 } 14
From GNBA to NBA GNBA with 2x accepting groups. single accentance group of the new automaton dashed-red arrows are dropped 15
Can we make it deterministic? In ordinary automaton, DFA can be converted to an equivalent NDFA (equivalent = generating the same sentences). a b For Buchi? b b No deterministic Buchi can generate the sentences of this Buchi NBA is really more powerful than DBA. 16
How big are they? NDGBA generated by our procedure |M| = 2| |. Converting to NDBA multiplies the number of states with C, where C is the number of U in 17
