Rigorous Approximated Determinization of Weighted Automata - PowerPoint PPT Presentation

Rigorous Approximated Determinization of Weighted Automata Benjamin Aminof (Hebrew University) Orna Kupferman (Hebrew University) Robby Lampert (Weizmann Institute) Israel

Outline  Weighted automata  Determinizability of weighted automata  Mohri’s determinization algorithm  Approximated-determinization algorithm  Correctness and termination  Summary  Future work

Weighted Automata (WFA) b,2 weight functions A : a,1 c,1 q 1 /0 q 0 q 3 /0 c: transitions ! R ¸ 0 q 2 /0 d,1 a,1 f: accepting states ! R ¸ 0 b,1  w=abc cost(w)=(1+2+1)+0=4  w=abbd cost(w)=(1+1+1+1)+0=4  w=abb cost(w)=min{5,3}=3

Weighted Automata – language  A run of A on a word w=w 1 …w n is a sequence r=r 0 r 1 r 2 … r n over Q such that r 0 2 Q 0 and for all 1 · i · n, w i r i-1 r i we have . r n  A run r is accepting $ r n is accepting. (standard finite-word accepting condition)  L( A )={w: A has an accepting run on w}

Weighted Automata – costs  A cost of a run r=r 0 r 1 r 2 … r n is w i cost(r) = ∑ i=1 c( ) + f( ) r n r i-1 r i n  defined only for accepting runs  A cost of a word w=w 1 …w n is cost(w)=min accepting runs r of A on w cost(r)  If w 62 L( A ) then cost(w)= 1 .

Weighted Automata – more  A WFA A is trim if each of its states is reachable from some initial state, and has a reachable accepting state.  A WFA A is unambiguous (single-run) if it has at most one accepting run on every word.

Applications of WFA  formal verification of quantitative properties  automatic speech recognition  image compression  pattern matching (widely used in computational biology)  …

A 1 is non-determinizable b,2 A 1 : a,1 c,1 q 1 q 0 q 3 /0 q 2 d,1 a,1  cost(ab k c)=2k+2, cost(ab k d)=k+2 b,1  After reading the word ab k , the difference between the costs of reading c and d is k.  For i ≠ j, a deterministic WFA must be in different states after reading ab i and ab j .  A deterministic WFA must have 1 states.

Determinizability  Weighted automata are not necessarily determinizable.  To decide whether a given weighted automaton is determinizable is an open question.  A sufficient condition for determinizability + algorithm [Mohri ’97].

A sufficient condition [Mohri ’97]  The twins property: For every two states q,q’ 2 Q, and two words u,v 2 Σ * , if q,q’ 2 δ (Q 0 ,u), q 2 δ (q,v), and then cost(q,v,q)=cost(q’,v,q’). q’ 2 δ (q’,v),  In case the automaton is trim (no empty u v q states) and unambiguous (single-run), q 0 the twins property is a characterization. u v q’

Determinization algorithm [Mohri ’97] - example word / cost A 2 : ac 8 a,3 c,5 q 1 b,2 bc 7 q 0 q 3 /0 a,4 ad 8 q 2 d,4 b,3 bd 7 4-3 0+5 min A 2 ’: {(q 1 ,0), {3,4} {(q 3 ,0)} 0 (q 2 ,1)} /0 0 a,? 3 c,? 5 0+0 {(q 0 ,0)} d,? 5 b,? 2 {(q 1 ,0), {(q 3 ,0)} 0 (q 2 ,1)} /0 0 min 1+4 {2,3} 3-2

Determinization algorithm - another example word / cost c,2 A 3 : ac i 3+2i+1 a,1 d,2 q 1 bc i 2+2i b,4 q 0 q 3 /0 a,3 ac i d 3+2i q 2 /1 d,1 b,1 bc i d 2+2i c,2 c,2 {(q 1 ,0), {(q 1 ,0), {(q 3 ,0)} A 3 ’: c,2 a,1 (q 2 ,2)} (q 2 ,2)} d,2 /0 /3 /3 {(q 3 ,0)} {(q 0 ,0)} /0 b,1 {(q 1 ,3), {(q 1 ,3), {(q 3 ,0)} c,2 d,1 (q 2 ,0)} (q 2 ,0)} /0 /1 /1 c,2

Determinization algorithm - non-determinizable example b,2 word / cost A 1 : a,1 c,1 q 1 ab i c 2+2i q 0 q 3 /0 ab i d 2+i q 2 d,1 a,1 b,1 A 1 ’: … {(q 1 ,0), {(q 1 ,1), b,1 {(q 1 ,2), {(q 1 ,3), a,1 b,1 b,1 {(q 0 ,0)} (q 2 ,0)} (q 2 ,0)} (q 2 ,0)} (q 2 ,0)}

Determinization algorithm - a bad determinizable example b,2 word / cost A 1 : a,1 d q 1 c,1 ab i c 2+2i q 0 q 3 /0 ab i d 2+i q 2 d,1 a,1 b,1 A 1 ’: … {(q 1 ,0), {(q 1 ,1), b,1 {(q 1 ,2), {(q 1 ,3), a,1 b,1 b,1 {(q 0 ,0)} (q 2 ,0)} (q 2 ,0)} (q 2 ,0)} (q 2 ,0)}

Mohri’s algorithm - remarks  Mohri’s algorithm terminates iff the original automaton has the twins property.  For trim and unambiguous WFAs, there is a polynomial algorithm for testing the twins property.  There are determinizable WFAs that do T not satisfy the twins property.

Approximated determinization Given a WFA A and an approximation factor t≥1, construct a deterministic WFA A ’, such that for every word w we have cost( A ,w) ≤ cost( A ’,w) ≤ t ∙ cost( A ,w).  When exact determinization is impossible.  When the result of exact determinization is too large.

Succinctness Σ,0 … Ln={Σ* . a . Σn- Σ,0 a,1 Σ,0 1} A 4 : n-1 Σ,t A deterministic equivalent Σ,0 requires 2 n states + L( A 4 )=Σ A t-approximate 1 w = ε deterministic? / t cost(w)= 1 w 2 L n 2 states + \L n t w 2 Σ

Approx. determinization algorithm [Buchsbaum- Giancarlo-Westbrook ’01]  Based on Mohri’s algorithm.  Relaxes the condition for unification of states – rather than requiring residuals of corresponding states to be identical, requires them to be close (within 1+ε of the smaller one).  No guarantees about the new costs.  No sufficient condition for termination.

Our algorithm: t-determinization  Determinization up to a factor t  The new cost of any accepted word w is between cost(w) and t ¢ cost(w).  differs from Mohri’s algorithm  Weights are multiplied by t.  For each state in a subset we maintain a range of residues rather than one.  The criterion for unification of states is relaxed (they may be non-identical).

2-determinization of A 1 b,2 A 1 : a,1 c,1 q 1 q 0 q 3 /0 q 2 d,1 a,1 b,1 b,2 {(q 3 ,-2,0)} A 1 ’: /0 c,2 a,? a,2 b,? b,2 {(q 1 ,-1,0), {(q 1 ,-1,1), -1 0 -1 2 {(q 0 ,0,0)} (q 2 ,-1,0)} -1 0 (q 2 ,-2,0)} -2 0 d,2 {(q 3 ,-2,0)} residual ranges lower upper /0 contain those of bound bound t cost(w) ¢ cost(w)

2-determinization of A 2 c,2 a,2 a,1 A 5 : q 0 /0 q 1 /0 q 2 /0 b,1 b,2 c,4 {(q 0 ,-1,0), {(q 0 ,-2,0), a,2 b,2 {(q 1 ,-4,0)} {(q 1 ,-2,0)} {(q 0 ,-2,0)} a,2 (q 2 ,0,2)} (q 1 ,0,4)} /0 /0 /0 /0 /0 c,4 b,2 A 5 ’: {(q 0 ,0,0), a,4 a,2 b,4 (q 1 ,0,0)} b,2 /0 b,4 a,2 {(q 1 ,-6,0)} {(q 2 ,-4,0)} {(q 0 ,-1,0)} {(q 0 ,-2,0)} b,2 /0 /0 /0 /0

Correctness of the algorithm  Thm: If the algorithm terminates on a given WFA A , with the result A ’, then for every word w we have cost( A ,w) ≤ cost( A ’,w) ≤ t ∙ cost( A ,w).

Termination of the algorithm  Thm: If a WFA has the t-twins property, then the algorithm terminates on it.  The weights and the factor t are rational.  Thm: For trim unambiguous WFAs, a WFA is t-determinizable iff it has the t-twins property.  Thm: Deciding the t-twins property for trim unambiguous WFAs can be done in polynomial time.

Summary  Why approximate determinization?  Non-determinizable WFA  Equivalent deterministic is large  t-determinization algorithm  Weights multiplied by t  Use ranges rather than single residues  Collapse to a state whose ranges are contained in mine  A sufficient condition  The t-twins property  For unambiguous WFAs – characterizes determinizability  Decidable in polynomial time

Future work  Generalize the termination proof to the case where the weights and the factor t are real numbers ( R ¸ 0 ).  An algorithm to decide whether a WFA is determinizable. Alternatively – prove that it is undecidable.