Learning Regular ω-languages
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1 Learning Regular ω-languages
2 The Goal Device an algorithm for learning an unknown regular ω-language L
3 3 The Goal Device an algorithm for learning an unknown regular ω-language L set of infinite words (ω-words)
4 4 The Goal Device an algorithm for learning accepted by a finite automaton an unknown regular ω-language L
5 5 The Goal Device an algorithm for learning an unknown regular ω-language L Learner
6 6 The Goal Device an algorithm for learning an unknown regular ω-language L L Learner Teacher
7 7 The Goal Device an algorithm for learning an unknown regular ω-language L membership queries equivalence queries L Learner Teacher
8 8 Coping with ω-words Is abcccccabdebbbaaaabcdaa in L? L Learner Teacher
9 9 Coping with ω-words Is abcccccabdebbbaaaabcdaa in L? L Learner Teacher
10 0 Coping with ω-words Is wingardium laviosa ω in L? L Learner Teacher
11 Coping with ω-words Is wingardium laviosa ω in L? wingardium laviosa laviosa laviosa laviosa laviosa L Learner Teacher
12 Coping with ω-words Is wingardium laviosa ω in L? wingardium laviosa laviosa laviosa laviosa laviosa THM: Two regular ω-languages are equivalent iff L they agree on the set of ultimately periodic words Learner Teacher
13 3 The Goal Is uv ω in L? L Learner Teacher
14 4 The Goal Is uv ω in L? Yes / No L Learner Teacher
15 5 The Goal Is uv ω in L? Yes / No Is H same as L? L Learner Teacher
16 6 The Goal Is uv ω in L? Yes / No Is H same as L? L Learner Yes / No, c.e: uv ω Teacher
17 The Goal Is uv ω in L? Yes / No Is H same as L? L Learner Yes / No, c.e: uv ω Teacher H s.t. H =L 7
18 8 Motivation Same problem for regular (finitary) languages is solved by the L* algorithm [Angluin] L* has found applications in many areas including AI, neural networks, geometry, data mining, verification and synthesis and many more. Reactive systems concerns ω-languages, using L* for this limits application to safety (and excludes liveness, fairness)
19 Previous Work on Learning ω-langs. DP / DM DS / DR all regular ω-languages DB DC DWP DWB DWC Safe strictly less expressive 9
20 Previous Work on Learning ω-langs. DP / DM DS / DR all regular ω-languages DB DC DWP DWB DWC From Prefixes From Lassos [de la Higuera & Janodet, 004] [Jayasrirani et al, 0] [Saoudi & Yokomori, 993] Safe strictly less expressive 0
21 Previous Work on Learning ω-langs. DP / DM DS / DR all regular ω-languages DB DC From Lassos [Maler & Pnueli,995] DWP DWB DWC From Prefixes From Lassos [de la Higuera & Janodet, 004] [Jayasrirani et al, 0] [Saoudi & Yokomori, 993] Safe strictly less expressive
22 Previous Work on Learning ω-langs. goal DP / DM DS / DR all regular ω-languages DB DC From Lassos [Maler & Pnueli,995] DWP DWB DWC From Prefixes From Lassos [de la Higuera & Janodet, 004] [Jayasrirani et al, 0] [Saoudi & Yokomori, 993] Safe strictly less expressive
23 Previous Work on Learning ω-langs. goal DP / DM DS / DR all regular ω-languages DB DC From Lassos [Maler & Pnueli,995] DWP DWB DWC From Prefixes From Lassos [de la Higuera & Janodet, 004] [Jayasrirani et al, 0] [Saoudi & Yokomori, 993] Safe strictly less expressive 3
24 Previous Work on Learning ω-langs. goal DP / DM DS / DR all regular ω-languages DB DC From Lassos [Maler & Pnueli,995] DWP DWB DWC From Prefixes From Lassos [de la Higuera & Janodet, 004] [Jayasrirani et al, 0] [Saoudi & Yokomori, 993] Safe strictly less expressive 4
25 5 However L $ [Calbrix, Nivat & Podelski 93] [Maler & Staiger 93] Syntactic FORC Recurrent FDFA
26 6 However May be exp. bigger than L $ [Calbrix, Nivat & Podelski 93] [Maler & Staiger 93] Syntactic FORC Recurrent FDFA
27 7 However May be exp. bigger than L $ [Calbrix, Nivat & Podelski 93] [Maler & Staiger 93] Syntactic FORC May be quadr. bigger than Recurrent FDFA
28 8 Main Contribution Learner L ω a learning algorithm for all 3 representations (periodic, syntactic, recurrent)
29 9 Why is it difficult? L* 96 ω- aut.
30 30 Why is it difficult? L* works due to the Myhill-Nerode Theorem, stating a -to- relationship between states of the minimal DFA (deterministic finite automaton) and the syntactic right congruence ~ of a language. a b c 4 4 b a 3 c a L* 96 ω- aut.
31 3 Why is it difficult? L* works due to the Myhill-Nerode Theorem, stating a -to- relationship between states of the minimal DFA (deterministic finite automaton) and the syntactic right congruence ~ of a language. a b c 4 4 b a 3 c a L* But for ω-langs, the syntactic right congruence is not informative enough! 96 ω- aut.
32 3 The Syntactic Right Congruence For finite words x» L y iff v *. xvl, yvl
33 33 The Syntactic Right Congruence For finite words x» L y iff v *. xvl, yvl Example: L = { w w has an even number of a s}
34 34 The Syntactic Right Congruence For finite words x» L y iff v *. xvl, yvl Example: L = { w w has an even number of a s} Then» L has two equivalence classes: words with an odd number of a s and b 0 a b words with an even number of a s a
35 35 The Syntactic Right Congruence For finite words: x» L y iff v *. xvl, yvl For ω-words: x» L y iff w!. xwl, ywl
36 36 The Syntactic Right Congruence For finite words: x» L y iff v *. xvl, yvl For ω-words: x» x» L y iff w! L y iff u,v *.. xuv xwl!, ywl yuv! L
37 37 The Syntactic Right Congruence For ω-words: x» L y iff u,v *. xuv! L, yuv! L Example: L = { w w has finitely many a s}
38 38 The Syntactic Right Congruence For ω-words: x» L y iff u,v *. xuv! L, yuv! L Example: L = { w w has finitely many a s} Then xuv! L iff uv! has finitely may a s.
39 39 The Syntactic Right Congruence For ω-words: x» L y iff u,v *. xuv! L, yuv! L Example: L = { w w has finitely many a s} Then xuv! L iff uv! has finitely may a s. Regardless of x. Thus» L has just one equivalence class.
40 40 The Syntactic Right Congruence For ω-words: x» L y iff u,v *. xuv! L, yuv! L Example: L = { w w has finitely many a s} Then xuv! L iff uv! has finitely may a s. Regardless of x. Thus» L has just one equivalence class.
41 The Syntactic Right Congruence For ω-words: x» L y iff u,v *. xuv! L, yuv! L Example: L = { w w has finitely many a s} Then xuv! L iff uv! has finitely may a s. Regardless of x. Thus» L has just one equivalence class. But clearly an ω-automaton for L requires at least two states. 4
42 4 Solution Foundation Families of DFAs (FDFA) A non-traditional representation of ω-automata inspired by Maler & Staiger s 97 families of right congruences (FORC) and their canonical representation of a Syntactic FORC for which they have shown a Myhill-Nerode Theorem
43 Family of Right Congruences [MS97] (»,¼,¼,¼ 3,¼ 4,¼ 5 ) Leading Progress Leading Right Congruence 4 5 Plus some restriction (details ommited) 43
44 44 Family of DFAs (FDFA) (M, P, P, P 3, P 4, P 5 ) Leading Progress Leading DFA M
45 45 Family of DFAs (FDFA) (M, P, P, P 3, P 4, P 5 ) Leading Progress P Leading DFA M
46 46 Family of DFAs (FDFA) (M, P, P, P 3, P 4, P 5 ) Leading Progress P P Leading DFA M
47 47 Family of DFAs (FDFA) (M, P, P, P 3, P 4, P 5 ) Leading Progress P P 3 3 P 3 Leading DFA M
48 48 Family of DFAs (FDFA) (M, P, P, P 3, P 4, P 5 ) Leading Progress P P P 3 Leading DFA M 4 P 4
49 49 Family of DFAs (FDFA) (M, P, P, P 3, P 4, P 5 ) Leading Progress P P P 3 Leading DFA M 4 P 4 P 5
50 Family of DFAs (FDFA) (M, P, P, P 3, P 4, P 5 ) Leading Progress P P P 3 Leading DFA M 4 P 4 P 5 That restriction removed. 50
51 FDFA Exact Acceptance? (u,v) M, P, P, P 3, P 4, P 5 uv! P P P3 M P4 P5
52 FDFA Exact Acceptance? (u,v) M, P, P, P 3, P 4, P 5 uv! P P u P3 M P4 P5
53 FDFA Exact Acceptance? (u,v) M, P, P, P 3, P 4, P 5 uv! P P u P3 M P4 v P5
54 FDFA Exact Acceptance? (u,v) M, P, P, P 3, P 4, P 5 uv! P P u P3 M P4 v P5
55 FORC and FDFA Example L = { w w has finitely many a s} FORC FDFA Progress No a s Some a s Leading a,b All prefixes are equally good
56 FORC and FDFA Example L = { w w has finitely many a s} FORC FDFA Progress No a s Some a s b a a,b P a,b Leading All prefixes are equally good
57 FORC and FDFA Example L = { w w has finitely many a s} FORC FDFA Progress No a s Some a s b a a,b P a,b Leading (abba,bbb) All prefixes are equally good (abba,bab)
58 58 4-state automaton 0,, 0 0
59 59 4-state automaton Some periods of are easy to find: ,, 0 0
60 60 4-state automaton Some periods of are easy to find: ,, 0 0 Some are hard: 0
61 6 4-state automaton Some periods of are easy to find: ,, 0 0 Some are hard:
62 6 4-state automaton Some periods of are easy to find: ,, 0 0 Some are hard: The DFA for periods of has 3 states => L $ > 3
63 63 4-state automaton All periods that loop back are easy to find: ,, 0 0
64 4-state automaton All periods that loop back are easy to find: ,, 0 0 The DFA that accepts only periods of that loop back has only 5 states! 64
65 65 Saturation (Consistency) How can we use this without losing saturation?
66 66 Saturation (Consistency) How can we use this without losing saturation? We say that a language is saturated if uv! =xy! implies (u,v)l, (x,y)l
67 67 Saturation (Consistency) How can we use this without losing saturation? We say that a language is saturated if uv! =xy! implies (u,v)l, (x,y)l To achieve saturation, we change the definition of acceptance.
68 FDFA Normalized Acceptance? (u,v) M, P, P, P 3, P 4, P 5 P P P4 4 M 3 5 P3 P5
69 FDFA Normalized Acceptance? (u,v) M, P, P, P 3, P 4, P 5 Normalization seeks for the smallest repetition of the period that loops back P P P4 4 M 3 5 P3 P5
70 FDFA Normalized Acceptance? (u,v) M, P, P, P 3, P 4, P 5 Normalization seeks for the smallest repetition of the period that loops back P P P4 4 M 3 5 P3 P5
71 FDFA Normalized Acceptance? (u,v) M, P, P, P 3, P 4, P 5 Normalization seeks for the smallest repetition of the period that loops back P P P4 4 M 3 5 P3 P5
72 FDFA Normalized Acceptance? (u,v) M, P, P, P 3, P 4, P 5 Normalization seeks for the smallest repetition of the period that loops back P P P4 4 M 3 5 P3 P5
73 FDFA Normalized Acceptance? (u,v) M, P, P, P 3, P 4, P 5 Normalization seeks for the smallest repetition of the period that loops back P P P4 4 M 3 5 P3 P5 We term Recurrent FDFA the FDFA where progress DFA recognize only periods that loop back. It is saturated under normalized acceptance!
74 74 More generally Generalizing to arbitrary n we get the following lower bound May be exp. bigger than Periodic FDFA (L $ ) Syntactic FDFA Recurrent FDFA
75 75 The Learner L ω Same scheme for all 3 canonical FDFAs
76 76 Minimality? The Recurrent FDFA for a given L is not necessarily the minimal FDFA for L. According to the normalized acceptance criterion a progress DFA P u should give correct results only to extensions that close a cycle to u. On other extensions it has freedom. This has the flavor of minimization with unknowns, which is NP complete. The Recurrent FDFA chooses to treat all don t cares as rejecting.
77 77 Time Complexity Interestingly, [Klarlund, 994] has shown that choosing a leading automaton which is more refined (bigger) than the syntactic right congruence, may yield an overall smaller FDFA. If we are given such a leading automaton, we can feed it to the learning algorithm, in which case it will yield the smaller FDFA. However, the same phenomenon can cause the learning algorithm for the syntactic/recurrent FDFAs to work as hard as the one for the periodic FDFA
78 78 Time Complexity The worst case time complexity for all 3 families is thus polynomial in the size of the periodic FDFA. Some positive results on sizes obtained via recurrent FDFA learner vs. periodic FDFA learner on randomly generated Muller automata.
79 80 Future Direction Find smaller canonical representations? Find smallest FDFA? Learning the leading first? L*-learning for (traditional) ω-automata? Other types of learning for ω-languages?
80 Learning THE END Thank you for your attention! Comments or questions? Regular Languages via Alternating Automata IJCAI 5 8
81 83 Time Complexity The worst case time complexity for all 3 families is thus polynomial in the size of the periodic FDFA. Some positive results on sizes obtained via recurrent FDFA learner vs. periodic FDFA learner on randomly generated Muller automata.
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