Minimization of Weighted Automata

Transcription

1 Minimization of Weighted Automata Andreas Maletti Universitat Rovira i Virgili Tarragona, Spain Wrocław May 19, 2010 Minimization of Weighted Automata Andreas Maletti 1

2 In Collaboration with ZOLTÁN ÉSIK, University of Szeged, Hungary JOHANNA HÖGBERG, Umeå University, Sweden MARKUS HOLZER, Universität Giessen, Germany JONATHAN MAY, USC, Los Angeles, CA, USA HEIKO VOGLER, TU Dresden, Germany Minimization of Weighted Automata Andreas Maletti 2

3 Table of Contents 1 Quick Recall: Unweighted Automata 2 Weighted Automata 3 Weighted Tree Automata 4 Some Experimental Results Minimization of Weighted Automata Andreas Maletti 3

4 Quick Recall: Unweighted Automata Unweighted automata Example (Unweighted automaton) F J M B E I L Q A D H R P 0 C G 2 Minimization of Weighted Automata Andreas Maletti 4

5 Quick Recall: Unweighted Automata Unweighted automata (Cont d) Minimization methods HOPCROFT s algorithm for DFA O(m log n) [HOPCROFT 1971] TARJAN s algorithm for NFA O(m log n) (bisimulation minimization) [TARJAN 1987] NFA minimization is PSPACE-complete [JIANG, RAVIKUMAR 1993] Special cases Cover automata O(m log n) [KÖRNER 2002] Hyper-minimization O(m log n) [GAWRYCHOWSKI, JEŻ 2009; HOLZER, 2010] Minimization of Weighted Automata Andreas Maletti 5

6 Quick Recall: Unweighted Automata Unweighted automata (Cont d) Minimization methods HOPCROFT s algorithm for DFA O(m log n) [HOPCROFT 1971] TARJAN s algorithm for NFA O(m log n) (bisimulation minimization) [TARJAN 1987] NFA minimization is PSPACE-complete [JIANG, RAVIKUMAR 1993] Special cases Cover automata O(m log n) [KÖRNER 2002] Hyper-minimization O(m log n) [GAWRYCHOWSKI, JEŻ 2009; HOLZER, 2010] Minimization of Weighted Automata Andreas Maletti 5

7 Quick Recall: Unweighted Automata Minimization Procedures Hopcroft (for DFA) s 1 a t 1 s 2 a t 2 Tarjan (for NFA) Minimization of Weighted Automata Andreas Maletti 6

8 Quick Recall: Unweighted Automata Minimization Procedures Hopcroft (for DFA) s 1 a t 1 s 2 a t 2 Tarjan (for NFA) Minimization of Weighted Automata Andreas Maletti 6

9 Quick Recall: Unweighted Automata Minimization Procedures Hopcroft (for DFA) Tarjan (for NFA) s 1 a t 1 a s 2 a t 2 Minimization of Weighted Automata Andreas Maletti 6

10 Quick Recall: Unweighted Automata Minimization Procedures Hopcroft (for DFA) Tarjan (for NFA) s 1 a t 1 a s 2 a t 2 Minimization of Weighted Automata Andreas Maletti 6

11 Weighted Automata Table of Contents 1 Quick Recall: Unweighted Automata 2 Weighted Automata 3 Weighted Tree Automata 4 Some Experimental Results Minimization of Weighted Automata Andreas Maletti 7

12 Syntax Weighted Automata Example a/1 a/ a/1 Definition (Weighted automaton) (Q, Σ, µ, I, F) Q finite set of states Σ alphabet µ: Q Σ Q A I : Q A initial weights F : Q A final weights References BERSTEL, REUTENAUER: Rational series and their languages. Springer, 1988 KUICH, SALOMAA: Semirings, automata, languages. Springer, 1986 Minimization of Weighted Automata Andreas Maletti 8

13 Syntax Weighted Automata Example a/1 a/ a/1 Definition (Weighted automaton) (Q, Σ, µ, I, F) Q finite set of states Σ alphabet µ: Q Σ Q A I : Q A initial weights F : Q A final weights References BERSTEL, REUTENAUER: Rational series and their languages. Springer, 1988 KUICH, SALOMAA: Semirings, automata, languages. Springer, 1986 Minimization of Weighted Automata Andreas Maletti 8

14 Semantics Weighted Automata Weight structure: Semiring A = (A, +,, 0, 1) Example a/1 a/ a/1 Definition (Semantics) h µ : Σ A Q h µ (ε) q = I(q) h µ (wa) q = h µ (w) p µ(p, a, q) p Q Example (using natural numbers) ( ) 1 h µ (a) = h 1 µ (aa) = ( ) 2 1 h µ (aaa) = ( ) 3 2 Minimization of Weighted Automata Andreas Maletti 9

15 Semantics Weighted Automata Weight structure: Semiring A = (A, +,, 0, 1) Example a/1 a/ a/1 Definition (Semantics) h µ : Σ A Q h µ (ε) q = I(q) h µ (wa) q = h µ (w) p µ(p, a, q) p Q Example (using natural numbers) ( ) 1 h µ (a) = h 1 µ (aa) = ( ) 2 1 h µ (aaa) = ( ) 3 2 Minimization of Weighted Automata Andreas Maletti 9

16 Semantics Weighted Automata Weight structure: Semiring A = (A, +,, 0, 1) Example a/1 a/ a/1 Definition (Semantics) h µ : Σ A Q h µ (ε) q = I(q) h µ (wa) q = h µ (w) p µ(p, a, q) p Q Example (using natural numbers) ( ) 1 h µ (a) = h 1 µ (aa) = ( ) 2 1 h µ (aaa) = ( ) 3 2 Minimization of Weighted Automata Andreas Maletti 9

17 Semantics Weighted Automata Weight structure: Semiring A = (A, +,, 0, 1) Example a/1 a/ a/1 Definition (Semantics) h µ : Σ A Q h µ (ε) q = I(q) h µ (wa) q = h µ (w) p µ(p, a, q) p Q Example (using natural numbers) ( ) 1 h µ (a) = h 1 µ (aa) = ( ) 2 1 h µ (aaa) = ( ) 3 2 Minimization of Weighted Automata Andreas Maletti 9

18 Overview Weighted Automata Results Method Nondet. Det. Complexity Reference Pushing & HOPCROFT x O(m log n) MOHRI Forward Bisimulation x x O(m log n) BUCHHOLZ Backward Bisimulation x O(m log n) BUCHHOLZ Backward Simulation x O(mn) RANZATO,... Full minimization x x O(mn 2 ) BERSTEL,... Notation m: number of transitions n: number of states Minimization of Weighted Automata Andreas Maletti 10

19 Weighted Automata Pushing & HOPCROFT [MOHRI] 1. Push Move weights toward the front 2. Minimize Minimize as unweighted automaton; treat weight as part of label Example (ala Eisner) a/3 a/2, b/3 b/ b/2 a/6 3 b/9 4 b/1 Minimization of Weighted Automata Andreas Maletti 11

20 Weighted Automata Pushing & HOPCROFT [MOHRI] 1. Push Move weights toward the front 2. Minimize Minimize as unweighted automaton; treat weight as part of label Example (ala Eisner) a/3 a/2, b/3 b/ b/2 a/ b/3 3 b/1 Minimization of Weighted Automata Andreas Maletti 11

21 Weighted Automata Pushing & HOPCROFT [MOHRI] 1. Push Move weights toward the front 2. Minimize Minimize as unweighted automaton; treat weight as part of label Example (ala Eisner) a/3 a/2, b/3 b/ b/2 3 a/2 3 b/3 4 b/1 Minimization of Weighted Automata Andreas Maletti 11

22 Weighted Automata Pushing & HOPCROFT [MOHRI] 1. Push Move weights toward the front 2. Minimize Minimize as unweighted automaton; treat weight as part of label Example (ala Eisner) a/1 3 a/2, b/3 b/ b/2 3 a/2 3 b/3 4 b/1 Minimization of Weighted Automata Andreas Maletti 11

23 Weighted Automata Pushing & HOPCROFT [MOHRI] 1. Push Move weights toward the front 2. Minimize Minimize as unweighted automaton; treat weight as part of label Example (ala Eisner) a/1 a/2, b/3 b/ b/2 a/2 3 b/3 4 b/1 Minimization of Weighted Automata Andreas Maletti 11

24 Weighted Automata Pushing & HOPCROFT [MOHRI] 1. Push Move weights toward the front 2. Minimize Minimize as unweighted automaton; treat weight as part of label Example (ala Eisner) a/1 a/2, b/3 b/ b/2 a/2 3 b/3 4 b/1 Minimization of Weighted Automata Andreas Maletti 11

25 Weighted Automata Pushing & HOPCROFT [MOHRI] 1. Push Move weights toward the front 2. Minimize Minimize as unweighted automaton; treat weight as part of label Example (ala Eisner) a/1 a/2, b/3 b/ b/2 3 a/2, b/3 Minimization of Weighted Automata Andreas Maletti 11

26 Weighted Automata Pushing & HOPCROFT [MOHRI] 1. Push Move weights toward the front 2. Minimize Minimize as unweighted automaton; treat weight as part of label Example (ala Eisner) 3 a/1, b/2 a/2, b/3 b/ Minimization of Weighted Automata Andreas Maletti 11

27 Weighted Automata Pushing & HOPCROFT [MOHRI] Prerequisites (ala EISNER) 1 Automaton deterministic 2 Semiring multiplicatively cancellative 3 Semiring allows greedy factorization Definition (Greedy factorization) There exists a mapping f : A 2 A such that for all a, b, c A with c 0: If a c and b c, then c a f (a, b) = c b f (b, a) a 1 f(a, b) 1 b 1 f(b, a) 1 Minimization of Weighted Automata Andreas Maletti 12

28 Weighted Automata Forward Bisimulation [BUCHHOLZ] Definition (Forward bisimulation) Equivalence relation on states such that F(p) = F(p ) and µ(p, a, r) = µ(p, a, r) r [q] r [q] for every p p, state q, and symbol a. Example 3 a/1 a/2, b/3 b/ b/2 a/1 3 a/1, b/3 4 b/1 Minimization of Weighted Automata Andreas Maletti 13

29 Weighted Automata Forward Bisimulation [BUCHHOLZ] Definition (Forward bisimulation) Equivalence relation on states such that F(p) = F(p ) and µ(p, a, r) = µ(p, a, r) r [q] r [q] for every p p, state q, and symbol a. Example 3 a/1 a/2, b/3 b/ b/2 a/1 3 a/1, b/3 4 b/1 Minimization of Weighted Automata Andreas Maletti 13

30 Weighted Automata Forward Bisimulation [BUCHHOLZ] Definition (Forward bisimulation) Equivalence relation on states such that F(p) = F(p ) and µ(p, a, r) = µ(p, a, r) r [q] r [q] for every p p, state q, and symbol a. Example 3 a/1 a/2, b/3 b/ b/2 3 a/2, b/3 Minimization of Weighted Automata Andreas Maletti 13

31 Weighted Automata Forward Bisimulation [BUCHHOLZ] Definition (Forward bisimulation) Equivalence relation on states such that F(p) = F(p ) and µ(p, a, r) = µ(p, a, r) r [q] r [q] for every p p, state q, and symbol a. Example 3 a/1 a/2, b/3 b/ b/2 3 a/2, b/3 Minimization of Weighted Automata Andreas Maletti 13

32 Weighted Automata Forward Bisimulation [BUCHHOLZ] Definition (Forward bisimulation) Equivalence relation on states such that F(p) = F(p ) and µ(p, a, r) = µ(p, a, r) r [q] r [q] for every p p, state q, and symbol a. Example 3 a/1, b/2 a/2, b/3 b/ Minimization of Weighted Automata Andreas Maletti 13

33 Weighted Automata Unweighted Forward Simulation [ABDULLA et. al.] Definition (Forward bisimulation unweighted) Reflexive, symmetric, and transitive relation on states such that F(p) = F(p ) and q q : µ(p, a, q) µ(p, a, q ) for every p p, state q, and symbol a. Minimization of Weighted Automata Andreas Maletti 14

34 Weighted Automata Unweighted Forward Simulation [ABDULLA et. al.] Definition (Forward bisimulation unweighted) Reflexive, symmetric, and transitive relation on states such that F(p) = F(p ) and q q : µ(p, a, q) µ(p, a, q ) for every p p, state q, and symbol a. Minimization of Weighted Automata Andreas Maletti 14

35 Weighted Automata Unweighted Forward Simulation [ABDULLA et. al.] Definition (Forward simulation) Pre-order on states such that F(p) F(p ) and q q : µ(p, a, q) µ(p, a, q ) for every p p, state q, and symbol a. Example 1 0 a/1 1 a/1, b/1 2 b/1 5 b/1 b/1 3 b/ Minimization of Weighted Automata Andreas Maletti 14

36 Weighted Automata Unweighted Forward Simulation [ABDULLA et. al.] Definition (Forward simulation) Pre-order on states such that F(p) F(p ) and q q : µ(p, a, q) µ(p, a, q ) for every p p, state q, and symbol a. Example 1 0 a/1 1 a/1, b/1 2 b/1 5 b/1 b/1 3 b/ Minimization of Weighted Automata Andreas Maletti 14

37 Weighted Automata Unweighted Forward Simulation [ABDULLA et. al.] Definition (Forward simulation) Pre-order on states such that F(p) F(p ) and q q : µ(p, a, q) µ(p, a, q ) for every p p, state q, and symbol a. Note Does in general not preserve the language! Minimization of Weighted Automata Andreas Maletti 14

38 Weighted Automata Backward Simulation [ABDULLA et. al.] Definition (Backward (bi)simulation) A forward (bi)simulation on the reversed automaton. Example a/1 a/ Theorem Reducing automaton by with a backward simulation preserves the language. 1 a/ Note Slightly more general than backward bisimulation. Minimization of Weighted Automata Andreas Maletti 15

39 Weighted Automata Backward Simulation [ABDULLA et. al.] Definition (Backward (bi)simulation) A forward (bi)simulation on the reversed automaton. Example a/1 a/ Theorem Reducing automaton by with a backward simulation preserves the language. 1 a/ Note Slightly more general than backward bisimulation. Minimization of Weighted Automata Andreas Maletti 15

40 Weighted Automata Full Minimization [BERSTEL et. al.] Requirements weight structure is a field Procedure prefix compression: select prefix-closed set W such that all h µ (w) with w W are linearly independent suffix compression References BERSTEL, REUTENAUER: Rational series and their languages. Springer, 1988 BERSTEL, REUTENAUER: Noncommutative Rational Series With Applications, Cambridge Univ. Press, 2010 Minimization of Weighted Automata Andreas Maletti 16

41 Weighted Automata Simulation vs. Equivalence Theorem Two equivalent automata can be joined by forward and backward bisimulation (and weight inversion). Extensions natural numbers integers rings,... References BERSTEL, REUTENAUER: Rational series and their languages. Springer, 1988 BEAL, LOMBARDY, SAKAROVITCH: On the Equivalence of Z-Automata, ICALP 2005 Minimization of Weighted Automata Andreas Maletti 17

42 Weighted Automata Simulation vs. Equivalence Theorem Two equivalent automata can be joined by forward and backward bisimulation (and weight inversion). Extensions natural numbers integers rings,... References BERSTEL, REUTENAUER: Rational series and their languages. Springer, 1988 BEAL, LOMBARDY, SAKAROVITCH: On the Equivalence of Z-Automata, ICALP 2005 Minimization of Weighted Automata Andreas Maletti 17

43 Weighted Tree Automata Table of Contents 1 Quick Recall: Unweighted Automata 2 Weighted Automata 3 Weighted Tree Automata 4 Some Experimental Results Minimization of Weighted Automata Andreas Maletti 18

44 Syntax Weighted Tree Automata Example (Transition) r σ q q p Definition (Weighted tree automaton) (Q, Σ, (µ k ) k N, F) Q finite set of states Σ ranked alphabet µ k : Q k Σ k Q A F : Q A final weights References BERSTEL, REUTENAUER: Recognizable formal power series on trees. Theor. Comput. Sci. 18, 1982 BORCHARDT: The theory of recognizable tree series. PhD thesis, 2004 Minimization of Weighted Automata Andreas Maletti 19

45 Weighted Tree Automata Syntax Illustration Example 3 6 f/0.3 f/ a/1 b/1 a/1 a/1 Minimization of Weighted Automata Andreas Maletti 20

46 Semantics Weighted Tree Automata Definition Let t T Σ (Q) and W = pos(t). Run on t: map r : W Q with r(w) = t(w) if t(w) Q Weight of r wt(r) = w W t(w) Σ Recognized tree series ( M, t) = µ k (r(w1),..., r(wk), t(w), r(w)) r run on t F(r(ε)) wt(r) Minimization of Weighted Automata Andreas Maletti 21

47 Semantics Weighted Tree Automata Definition Let t T Σ (Q) and W = pos(t). Run on t: map r : W Q with r(w) = t(w) if t(w) Q Weight of r wt(r) = w W t(w) Σ Recognized tree series ( M, t) = µ k (r(w1),..., r(wk), t(w), r(w)) r run on t F(r(ε)) wt(r) Minimization of Weighted Automata Andreas Maletti 21

48 Semantics Weighted Tree Automata Definition Let t T Σ (Q) and W = pos(t). Run on t: map r : W Q with r(w) = t(w) if t(w) Q Weight of r wt(r) = w W t(w) Σ Recognized tree series ( M, t) = µ k (r(w1),..., r(wk), t(w), r(w)) r run on t F(r(ε)) wt(r) Minimization of Weighted Automata Andreas Maletti 21

49 Weighted Tree Automata Semantics Illustration Example (Automaton) 3 6 f/0.3 f/ a/1 b/1 a/1 a/1 Example (Runs) Input tree: f Runs: 6 with weight 0 a b 1 2 Minimization of Weighted Automata Andreas Maletti 22

50 Weighted Tree Automata Semantics Illustration Example (Automaton) 3 6 f/0.3 f/ a/1 b/1 a/1 a/1 Example (Runs) Input tree: f Runs: f with weight a b a b Minimization of Weighted Automata Andreas Maletti 22

51 Weighted Tree Automata Semantics Illustration Example (Automaton) 3 6 f/0.3 f/ a/1 b/1 a/1 a/1 Example (Runs) Input tree: f Runs: f with weight 1 a b 1 b Minimization of Weighted Automata Andreas Maletti 22

52 Weighted Tree Automata Semantics Illustration Example (Automaton) 3 6 f/0.3 f/ a/1 b/1 a/1 a/1 Example (Runs) Input tree: f Runs: f with weight 1 a b 1 2 Minimization of Weighted Automata Andreas Maletti 22

53 Weighted Tree Automata Semantics Illustration Example (Automaton) 3 6 f/0.3 f/ a/1 b/1 a/1 a/1 Example (Runs) Input tree: f Runs: 3 with weight 0.3 a b 1 2 Minimization of Weighted Automata Andreas Maletti 22

54 Overview Weighted Tree Automata Results Method Nondet. Det. Complexity Reference Det. Minimization x O(rmn) Forw. Bisimulation x x O(rm log n) HÖGBERG,... Backw. Bisimulation x O(r 2 m log n) HÖGBERG,... Backw. Simulation x O(r 2 mn) ABDULLA,... Full minimization x x P BOZAPALIDIS Notation m: number of transitions n: number of states r: maximal rank of the input symbols Minimization of Weighted Automata Andreas Maletti 23

55 Weighted Tree Automata Forward Bisimulation [HÖGBERG, et. al.] Definition (Forward bisimulation) Equivalence relation on states such that F(p) = F(p ) and µ(..., p,..., a, r) = µ(..., p,..., a, r) r [q] r [q] for every p p, symbol a, and states q and.... Minimization of Weighted Automata Andreas Maletti 24

56 Weighted Tree Automata Backward Bisimulation [HÖGBERG, et. al.] Definition (Backward bisimulation) Equivalence relation on states such that µ(q 1,..., q k, a, p) = µ(q 1,..., q k, a, p ) q 1 q k B 1 B k q 1 q k B 1 B k for every p p, symbol a, and blocks B 1,..., B k. Minimization of Weighted Automata Andreas Maletti 25

57 Weighted Tree Automata Det. Minimization Overview Applicability Deterministic wta Commutative semifield (i.e. multiplicative inverses) Roadmap MYHILL-NERODE congruence relation [BORCHARDT 2003] Determine signs of life Refinement Minimization of Weighted Automata Andreas Maletti 26

58 Weighted Tree Automata MYHILL-NERODE congruence Definition p q: there exists nonzero a such that for every context C ( M, C[p]) = a ( M, C[q]) Notes Semifields are zero-divisor free Element a is unique if p is not dead Minimization of Weighted Automata Andreas Maletti 27

59 Weighted Tree Automata MYHILL-NERODE congruence Definition p q: there exists nonzero a such that for every context C ( M, C[p]) = a ( M, C[q]) Notes Semifields are zero-divisor free Element a is unique if p is not dead Minimization of Weighted Automata Andreas Maletti 27

60 Signs of Life Weighted Tree Automata Definition Sign of life of q Q: context C such that ( M, C[q]) 0 Example 6 3 f/0.3 f/ a/1 b/1 State Sign of life State Sign of life 1 f (, b) 2 f (1, ) 3 6 Minimization of Weighted Automata Andreas Maletti 28

61 Stages Weighted Tree Automata Definition Stage (Π, sol, f, r): (i) refinement of Π (ii) sol(f ) = { } (iii) for live q with p = r([q]) (iv) Π congruence ( M, sol(p)[q]) = f (q) ( M, sol(p)[p]) (v) for symbol σ and context C with live δ σ (C[q]) f (q) 1 c σ (C[q]) f (δ σ (C[q])) = c σ (C[p]) f (δ σ (C[p])) where δ σ : Q k Q and c σ : Q k A Minimization of Weighted Automata Andreas Maletti 29

62 Stages Weighted Tree Automata Definition Stable stage (Π, sol, f, r): (i) refinement of Π (ii) sol(f) = { } (iii) for live q with p = r([q]) (iv) Π congruence ( M, sol(p)[q]) = f (q) ( M, sol(p)[p]) (v) for symbol σ and context C with live δ σ (C[q]) f (q) 1 c σ (C[q]) f (δ σ (C[q])) = c σ (C[p]) f (δ σ (C[p])) where δ σ : Q k Q and c σ : Q k A Minimization of Weighted Automata Andreas Maletti 29

63 Weighted Tree Automata Refining a Stage Definition Refinement of (Π, sol, f, r): Partition Π with p Π q if (i) p Π q (ii) δ σ (C[p]) Π δ σ (C[q]) (iii) if δ σ (C[p]) is live, then f (p) 1 c σ (C[p]) f (δ σ (C[p])) = f (q) 1 c σ (C[q]) f (δ σ (C[q])) for states p and q, symbol σ, and context C Minimization of Weighted Automata Andreas Maletti 30

64 Weighted Tree Automata Complete Algorithm Algorithm (Π, sol, D) COMPUTESOL(M) 2: repeat (Π, sol, f, r) COMPLETE(M, Π, sol, D) 4: Π REFINE(M, Π, sol, f, r, D) until Π = Π 6: return minimized wta Notes Algorithm runs in O(rmn 4 ) Can be optimized to run in O(rmn) [ 2008] Returns equivalent minimal deterministic wta Minimization of Weighted Automata Andreas Maletti 31

65 Weighted Tree Automata Complete Algorithm Algorithm (Π, sol, D) COMPUTESOL(M) 2: repeat (Π, sol, f, r) COMPLETE(M, Π, sol, D) 4: Π REFINE(M, Π, sol, f, r, D) until Π = Π 6: return minimized wta Notes Algorithm runs in O(rmn 4 ) Can be optimized to run in O(rmn) [ 2008] Returns equivalent minimal deterministic wta Minimization of Weighted Automata Andreas Maletti 31

66 Weighted Tree Automata Simulation vs. Equivalence Theorem Two equivalent tree automata over fields can be joined by forward and backward bisimulation (and weight inversion). Extensions natural numbers integers rings,... References BOZAPALIDIS: Effective Construction of the Syntactic Algebra of a Recognizable Series on Trees. Acta Inf ÉSIK, : Simulations of weighted tree automata, 2010 Minimization of Weighted Automata Andreas Maletti 32

67 Weighted Tree Automata Simulation vs. Equivalence Theorem Two equivalent tree automata over fields can be joined by forward and backward bisimulation (and weight inversion). Extensions natural numbers integers rings,... References BOZAPALIDIS: Effective Construction of the Syntactic Algebra of a Recognizable Series on Trees. Acta Inf ÉSIK, : Simulations of weighted tree automata, 2010 Minimization of Weighted Automata Andreas Maletti 32

68 Some Experimental Results Table of Contents 1 Quick Recall: Unweighted Automata 2 Weighted Automata 3 Weighted Tree Automata 4 Some Experimental Results Minimization of Weighted Automata Andreas Maletti 33

69 Experiments Some Experimental Results State Count Original Minimal Reduction to % % % % % % State & Transition Count Error Original Minimal Reduction to 10 4 (727, 6485) (629, 6131) (87%, 95%) 10 2 (727, 6485) (525, 3425) (72%, 53%) Minimization of Weighted Automata Andreas Maletti 34

70 Experiments Some Experimental Results State Count Original Minimal Reduction to % % % % % % State & Transition Count Error Original Minimal Reduction to 10 4 (727, 6485) (629, 6131) (87%, 95%) 10 2 (727, 6485) (525, 3425) (72%, 53%) Minimization of Weighted Automata Andreas Maletti 34

71 The End Some Experimental Results Thank You! Minimization of Weighted Automata Andreas Maletti 35