Valence automata as a generalization of automata with storage

Transcription

1 Valence automata as a generalization of automata with storage Georg Zetzsche Technische Universität Kaiserslautern D-CON 2013 Georg Zetzsche (TU KL) Valence automata D-CON / 23

2 Example (Pushdown automaton) a, λ, A a, A, λ q 0 λ, λ, λ q 1 b, λ, B b, B, λ Georg Zetzsche (TU KL) Valence automata D-CON / 23

3 Example (Pushdown automaton) a, λ, A a, A, λ q 0 λ, λ, λ q 1 L tww rev w P ta, bu u b, λ, B b, B, λ Georg Zetzsche (TU KL) Valence automata D-CON / 23

4 Example (Pushdown automaton) a, λ, A a, A, λ q 0 λ, λ, λ q 1 L tww rev w P ta, bu u b, λ, B b, B, λ Example (Blind counter automaton) a, 1, 0 b, 0, 1 c, 1, 1 λ, 0, 0 λ, 0, 0 q 0 q 1 q 2 Georg Zetzsche (TU KL) Valence automata D-CON / 23

5 Example (Pushdown automaton) a, λ, A a, A, λ q 0 λ, λ, λ q 1 L tww rev w P ta, bu u b, λ, B b, B, λ Example (Blind counter automaton) a, 1, 0 b, 0, 1 c, 1, 1 λ, 0, 0 λ, 0, 0 q 0 q 1 q 2 L ta n b n c n n ě 0u Georg Zetzsche (TU KL) Valence automata D-CON / 23

6 Example (Partially blind counter automaton) a, 1 q 0 λ, 0 λ, 1 q 1 b, 1 L tw P ta, bu p a ě p b for any prefix p of wu Georg Zetzsche (TU KL) Valence automata D-CON / 23

7 Automata models that extend finite automata by some storage mechanism: Pushdown automata Blind counter automata Partially blind counter automata Turing machines Georg Zetzsche (TU KL) Valence automata D-CON / 23

8 Automata models that extend finite automata by some storage mechanism: Pushdown automata Blind counter automata Partially blind counter automata Turing machines Each storage mechanism consists of: States: set S of states Operations: partial maps α 1,..., α n : S Ñ S Georg Zetzsche (TU KL) Valence automata D-CON / 23

9 Model States Operations Pushdown automata S Γ push a :w ÞÑ wa, a P Γ pop a :wa ÞÑ w, a P Γ Georg Zetzsche (TU KL) Valence automata D-CON / 23

10 Model States Operations Pushdown automata S Γ push a :w ÞÑ wa, a P Γ pop a :wa ÞÑ w, a P Γ Blind counter automata S Z n inc i :px 1,..., x n q ÞÑ px 1,..., x i ` 1,..., x n q dec i :px 1,..., x n q ÞÑ px 1,..., x i 1,..., x n q Georg Zetzsche (TU KL) Valence automata D-CON / 23

11 Model States Operations Pushdown automata S Γ push a :w ÞÑ wa, a P Γ pop a :wa ÞÑ w, a P Γ Blind counter automata Partially blind counter automata S Z n S N n inc i :px 1,..., x n q ÞÑ px 1,..., x i ` 1,..., x n q dec i :px 1,..., x n q ÞÑ px 1,..., x i 1,..., x n q inc i :px 1,..., x n q ÞÑ px 1,..., x i ` 1,..., x n q dec i :px 1,..., x n q ÞÑ px 1,..., x i 1,..., x n q Georg Zetzsche (TU KL) Valence automata D-CON / 23

12 Model States Operations Pushdown automata S Γ push a :w ÞÑ wa, a P Γ pop a :wa ÞÑ w, a P Γ Blind counter automata Partially blind counter automata S Z n S N n inc i :px 1,..., x n q ÞÑ px 1,..., x i ` 1,..., x n q dec i :px 1,..., x n q ÞÑ px 1,..., x i 1,..., x n q inc i :px 1,..., x n q ÞÑ px 1,..., x i ` 1,..., x n q dec i :px 1,..., x n q ÞÑ px 1,..., x i 1,..., x n q Observation Here, a sequence β 1,..., β k of operations is valid if and only if β 1 β k id Georg Zetzsche (TU KL) Valence automata D-CON / 23

13 Definition A monoid is a set M together with an associative binary operation : M ˆ M Ñ M and a neutral element 1 P M (a1 1a a for any a P M). Georg Zetzsche (TU KL) Valence automata D-CON / 23

14 Definition A monoid is a set M together with an associative binary operation : M ˆ M Ñ M and a neutral element 1 P M (a1 1a a for any a P M). Storage mechanisms as monoids Let S be a set of states and α 1,..., α n : S Ñ S partial maps. The set of all compositions of α 1,..., α n is a monoid M. The identity map is the neutral element of M. M is a decription of the storage mechanism. Georg Zetzsche (TU KL) Valence automata D-CON / 23

15 Valence automata Common generalization: Valence Automata Valence automaton over M: Finite automaton with edges p w m ÝÝÑq, w P Σ, m P M. Georg Zetzsche (TU KL) Valence automata D-CON / 23

16 Valence automata Common generalization: Valence Automata Valence automaton over M: Finite automaton with edges p w m ÝÝÑq, w P Σ, m P M. w 1 m 1 w 2 m 2 w n m n Run q 0ÝÝÝÑq1 ÝÝÝÑ ÝÝÝÑq n is accepting for w 1 w n if q 0 is the initial state, qn is a final state, and Georg Zetzsche (TU KL) Valence automata D-CON / 23

17 Valence automata Common generalization: Valence Automata Valence automaton over M: Finite automaton with edges p w m ÝÝÑq, w P Σ, m P M. w 1 m 1 w 2 m 2 w n m n Run q 0ÝÝÝÑq1 ÝÝÝÑ ÝÝÝÑq n is accepting for w 1 w n if q 0 is the initial state, qn is a final state, and m1 m n 1. Georg Zetzsche (TU KL) Valence automata D-CON / 23

18 Classical results can now be generalized: Questions Which storage mechanisms increase the expressive power? Georg Zetzsche (TU KL) Valence automata D-CON / 23

19 Classical results can now be generalized: Questions Which storage mechanisms increase the expressive power? For which storage mechanisms can we determinize? Georg Zetzsche (TU KL) Valence automata D-CON / 23

20 Classical results can now be generalized: Questions Which storage mechanisms increase the expressive power? For which storage mechanisms can we determinize? For which can we avoid silent transitions? Georg Zetzsche (TU KL) Valence automata D-CON / 23

21 Classical results can now be generalized: Questions Which storage mechanisms increase the expressive power? For which storage mechanisms can we determinize? For which can we avoid silent transitions? For which do we have a Parikh theorem? Georg Zetzsche (TU KL) Valence automata D-CON / 23

22 Classical results can now be generalized: Questions Which storage mechanisms increase the expressive power? For which storage mechanisms can we determinize? For which can we avoid silent transitions? For which do we have a Parikh theorem? For which can we compute the downward closure? Georg Zetzsche (TU KL) Valence automata D-CON / 23

23 Classical results can now be generalized: Questions Which storage mechanisms increase the expressive power? For which storage mechanisms can we determinize? For which can we avoid silent transitions? For which do we have a Parikh theorem? For which can we compute the downward closure? For which can we decide, for example, emptiness? Georg Zetzsche (TU KL) Valence automata D-CON / 23

24 Classical results can now be generalized: Questions Which storage mechanisms increase the expressive power? For which storage mechanisms can we determinize? For which can we avoid silent transitions? For which do we have a Parikh theorem? For which can we compute the downward closure? For which can we decide, for example, emptiness? Georg Zetzsche (TU KL) Valence automata D-CON / 23

25 Deterministic valence automata A valence automaton is called deterministic, if Georg Zetzsche (TU KL) Valence automata D-CON / 23

26 Deterministic valence automata A valence automaton is called deterministic, if every edge is of the form p a m ÝÝÑq with a P Σ. Georg Zetzsche (TU KL) Valence automata D-CON / 23

27 Deterministic valence automata A valence automaton is called deterministic, if every edge is of the form p a m ÝÝÑq with a P Σ. for each state p P Q and each letter a P Σ, there is at most one edge pýýñq a m for some m P M, q P Q. Georg Zetzsche (TU KL) Valence automata D-CON / 23

28 Deterministic valence automata A valence automaton is called deterministic, if every edge is of the form p a m ÝÝÑq with a P Σ. for each state p P Q and each letter a P Σ, there is at most one edge pýýñq a m for some m P M, q P Q. detvapmq languages accepted by deterministic valence automata over M. Georg Zetzsche (TU KL) Valence automata D-CON / 23

29 Questions When does VApMq contain non-regular languages? When does detvapmq VApMq? Georg Zetzsche (TU KL) Valence automata D-CON / 23

30 Questions When does VApMq contain non-regular languages? When does detvapmq VApMq? Theorem The following statements are equivalent: 1 VApMq REG. 2 detvapmq VApMq. 3 Every finitely generated submonoid of M possesses only finitely many right-invertible elements. Georg Zetzsche (TU KL) Valence automata D-CON / 23

31 Questions When does VApMq contain non-regular languages? When does detvapmq VApMq? Theorem The following statements are equivalent: 1 VApMq REG. 2 detvapmq VApMq. 3 Every finitely generated submonoid of M possesses only finitely many right-invertible elements. (1) ô (3) has been shown independently by Elaine Render in Georg Zetzsche (TU KL) Valence automata D-CON / 23

32 RpMq tx P M Dy P M : xy 1u Rpxq ty P M xy 1u Georg Zetzsche (TU KL) Valence automata D-CON / 23

33 RpMq tx P M Dy P M : xy 1u LpMq tx P M Dy P M : yx 1u Rpxq ty P M xy 1u Lpxq ty P M yx 1u Georg Zetzsche (TU KL) Valence automata D-CON / 23

34 RpMq tx P M Dy P M : xy 1u LpMq tx P M Dy P M : yx 1u Rpxq ty P M xy 1u Lpxq ty P M yx 1u JpMq tx P M Dy, z P M : yxz 1u Georg Zetzsche (TU KL) Valence automata D-CON / 23

35 RpMq tx P M Dy P M : xy 1u LpMq tx P M Dy P M : yx 1u Rpxq ty P M xy 1u Lpxq ty P M yx 1u JpMq tx P M Dy, z P M : yxz 1u Lemma (Dichotomy) For each monoid M, exactly one of the following holds: 1 RpMq LpMq JpMq is a finite group. 2 There are infinite subsets S Ď RpMq, S 1 Ď LpMq such that Rpsq X Rptq H for any s, t P S and Lpsq X Lptq H for any s, t P S 1. Georg Zetzsche (TU KL) Valence automata D-CON / 23

36 Silent transitions Definition Transitions p λ m ÝÝÑq are called silent or λ-transitions. Georg Zetzsche (TU KL) Valence automata D-CON / 23

37 Silent transitions Definition Transitions p λ m ÝÝÑq are called silent or λ-transitions. VA`pMq Languages accepted without λ-transitions. Georg Zetzsche (TU KL) Valence automata D-CON / 23

38 Silent transitions Definition Transitions p λ m ÝÝÑq are called silent or λ-transitions. VA`pMq Languages accepted without λ-transitions. Important problem When can λ-transitions be eliminated? Without λ-transitions, decide membership using exponential number of queries to the word problem. Elimination can be regarded as a precomputation. Georg Zetzsche (TU KL) Valence automata D-CON / 23

39 Monoids defined by graphs By graphs, we mean undirected graphs with loops allowed. Georg Zetzsche (TU KL) Valence automata D-CON / 23

40 Monoids defined by graphs By graphs, we mean undirected graphs with loops allowed. Notation B: monoid for partially blind counter Z: monoid for blind counter, i.e. the group of integers Georg Zetzsche (TU KL) Valence automata D-CON / 23

41 Monoids defined by graphs By graphs, we mean undirected graphs with loops allowed. Notation B: monoid for partially blind counter Z: monoid for blind counter, i.e. the group of integers Intuition To each graph Γ, we associate a monoid MΓ: For each unlooped vertex, we have a copy of B For each looped vertex, we have a copy of Z MΓ consists of sequences of such elements An edge between vertices means that elements can commute Georg Zetzsche (TU KL) Valence automata D-CON / 23

42 Examples Georg Zetzsche (TU KL) Valence automata D-CON / 23

43 Examples Z 3 Georg Zetzsche (TU KL) Valence automata D-CON / 23

44 Examples Z 3 Blind multicounter Georg Zetzsche (TU KL) Valence automata D-CON / 23

45 Examples Z 3 Blind multicounter Georg Zetzsche (TU KL) Valence automata D-CON / 23

46 Examples Z 3 B B B Blind multicounter Georg Zetzsche (TU KL) Valence automata D-CON / 23

47 Examples Z 3 B B B Blind multicounter Pushdown Georg Zetzsche (TU KL) Valence automata D-CON / 23

48 Examples Z 3 B B B Blind multicounter Pushdown Georg Zetzsche (TU KL) Valence automata D-CON / 23

49 Examples Z 3 B B B Blind multicounter Pushdown B 3 Georg Zetzsche (TU KL) Valence automata D-CON / 23

50 Examples Z 3 B B B Blind multicounter Pushdown B 3 Partially blind multicounter Georg Zetzsche (TU KL) Valence automata D-CON / 23

51 Examples Z 3 B B B Blind multicounter Pushdown B 3 Partially blind multicounter Georg Zetzsche (TU KL) Valence automata D-CON / 23

52 Examples Z 3 B B B Blind multicounter Pushdown B 3 Partially blind multicounter Georg Zetzsche (TU KL) Valence automata D-CON / 23

53 Examples Z 3 B B B Blind multicounter Pushdown B 3 pb Bq ˆ pb Bq Partially blind multicounter Georg Zetzsche (TU KL) Valence automata D-CON / 23

54 Examples Z 3 B B B Blind multicounter Pushdown B 3 pb Bq ˆ pb Bq Partially blind multicounter Infinite tape (TM) Georg Zetzsche (TU KL) Valence automata D-CON / 23

55 Examples Z 3 B B B Blind multicounter Pushdown B 3 pb Bq ˆ pb Bq Partially blind multicounter Infinite tape (TM) Georg Zetzsche (TU KL) Valence automata D-CON / 23

56 Examples Z 3 B B B Blind multicounter Pushdown B 3 pb Bq ˆ pb Bq Partially blind multicounter Infinite tape (TM) Georg Zetzsche (TU KL) Valence automata D-CON / 23

57 Examples Z 3 B B B Blind multicounter Pushdown B 3 pb Bq ˆ pb Bq Partially blind multicounter Infinite tape (TM) Georg Zetzsche (TU KL) Valence automata D-CON / 23

58 Examples Z 3 B B B Blind multicounter Pushdown B 3 pb Bq ˆ pb Bq Partially blind multicounter Infinite tape (TM) Georg Zetzsche (TU KL) Valence automata D-CON / 23

59 Theorem Let Γ be a graph such that between any two looped vertices, there is an edge and between any two unlooped vertices, there is no edge. Georg Zetzsche (TU KL) Valence automata D-CON / 23

60 Theorem Let Γ be a graph such that between any two looped vertices, there is an edge and between any two unlooped vertices, there is no edge. Georg Zetzsche (TU KL) Valence automata D-CON / 23

61 Theorem Let Γ be a graph such that between any two looped vertices, there is an edge and between any two unlooped vertices, there is no edge. Then VApMΓq VA`pMΓq if and only if Γ does not contain as an induced subgraph. Georg Zetzsche (TU KL) Valence automata D-CON / 23

62 Theorem Let Γ be a graph such that between any two looped vertices, there is an edge and between any two unlooped vertices, there is no edge. Then VApMΓq VA`pMΓq if and only if Γ does not contain as an induced subgraph. Georg Zetzsche (TU KL) Valence automata D-CON / 23

63 By reduction to an undecidable problem from group theory, we obtain: Lemma Let Γ be a graph with as an induced subgraph. Then VApMΓq contains an undecidable language. Hence, VA`pMΓq Ĺ VApMΓq. Georg Zetzsche (TU KL) Valence automata D-CON / 23

64 Definition Let C be the smallest class of monoids such that 1 P C if M P C, then M ˆ Z P C if M P C, then M B P C Georg Zetzsche (TU KL) Valence automata D-CON / 23

65 Definition Let C be the smallest class of monoids such that 1 P C if M P C, then M ˆ Z P C if M P C, then M B P C Lemma Let Γ be a graph such that between any two looped vertices, there is an edge between any two unlooped vertices, there is no edge Then, MΓ P C. does not appear as an induced subgraph Georg Zetzsche (TU KL) Valence automata D-CON / 23

66 Definition Let C be the smallest class of monoids such that 1 P C if M P C, then M ˆ Z P C if M P C, then M B P C Interpretation of C C corresponds to the class of storage mechanisms obtained by adding a blind counter (M ˆ Z) and building stacks (M B). Georg Zetzsche (TU KL) Valence automata D-CON / 23

67 Proof. Let Γ pv, Eq, V L Y U, looped and unlooped vertices. For each x P L, let νpxq Npxq X U Georg Zetzsche (TU KL) Valence automata D-CON / 23

68 Proof. Let Γ pv, Eq, V L Y U, looped and unlooped vertices. For each x P L, let νpxq Npxq X U and for x, y P L : x ď y iff νpxq Ď νpyq. Georg Zetzsche (TU KL) Valence automata D-CON / 23

69 Proof. Let Γ pv, Eq, V L Y U, looped and unlooped vertices. For each x P L, let νpxq Npxq X U and for x, y P L : x ď y iff νpxq Ď νpyq. Since is not an induced subgraph, ď is a total preorder. Georg Zetzsche (TU KL) Valence automata D-CON / 23

70 Proof. Let Γ pv, Eq, V L Y U, looped and unlooped vertices. For each x P L, let νpxq Npxq X U and for x, y P L : x ď y iff νpxq Ď νpyq. Since is not an induced subgraph, ď is a total preorder. Let m P L be maximal. Georg Zetzsche (TU KL) Valence automata D-CON / 23

71 Proof. Let Γ pv, Eq, V L Y U, looped and unlooped vertices. For each x P L, let νpxq Npxq X U and for x, y P L : x ď y iff νpxq Ď νpyq. Since is not an induced subgraph, ď is a total preorder. Let m P L be maximal. If νpmq U, then Γ pγztmuq ˆ m. Georg Zetzsche (TU KL) Valence automata D-CON / 23

72 Proof. Let Γ pv, Eq, V L Y U, looped and unlooped vertices. For each x P L, let νpxq Npxq X U and for x, y P L : x ď y iff νpxq Ď νpyq. Since is not an induced subgraph, ď is a total preorder. Let m P L be maximal. If νpmq U, then Γ pγztmuq ˆ m. MΓ MpΓztmuq ˆ Z. Georg Zetzsche (TU KL) Valence automata D-CON / 23

73 Proof. Let Γ pv, Eq, V L Y U, looped and unlooped vertices. For each x P L, let νpxq Npxq X U and for x, y P L : x ď y iff νpxq Ď νpyq. Since is not an induced subgraph, ď is a total preorder. Let m P L be maximal. If νpmq U, then Γ pγztmuq ˆ m. MΓ MpΓztmuq ˆ Z. If νpmq Ĺ U, then Γ pγztuuq u for some u P U. Georg Zetzsche (TU KL) Valence automata D-CON / 23

74 Proof. Let Γ pv, Eq, V L Y U, looped and unlooped vertices. For each x P L, let νpxq Npxq X U and Since Let m P L be maximal. for x, y P L : x ď y iff νpxq Ď νpyq. is not an induced subgraph, ď is a total preorder. If νpmq U, then Γ pγztmuq ˆ m. MΓ MpΓztmuq ˆ Z. If νpmq Ĺ U, then Γ pγztuuq u for some u P U. MΓ MpΓztuuq B. Georg Zetzsche (TU KL) Valence automata D-CON / 23

75 Elimination of λ-transitions Definition A subset S Ď M is called rational if it is the homomorphic image of a regular language. Elimination of λ-transitions Approach: Between a given pair of non-λ-transitions, the set of x P M applied in between is a rational set. Transform the automaton so as to simulate the application of a rational set in one step. Georg Zetzsche (TU KL) Valence automata D-CON / 23

76 Proof ingredients Ingredients Semilinearity of languages in VApMΓq Normal form for rational subsets of MΓ Construction for VA`pMΓq VApMΓq Actual construction quite involved. Stronger claim to make induction work. Separate constructions for B, M ˆ Z, and M B. Representations of rational sets are encoded into the state or the monoid elements. Georg Zetzsche (TU KL) Valence automata D-CON / 23

77 Theorem Let Γ be a graph such that between any two distinct vertices, there is an edge. Georg Zetzsche (TU KL) Valence automata D-CON / 23

78 Theorem Let Γ be a graph such that between any two distinct vertices, there is an edge. Georg Zetzsche (TU KL) Valence automata D-CON / 23

79 Theorem Let Γ be a graph such that between any two distinct vertices, there is an edge. Then VApMΓq VA`pMΓq if and only if the number of unlooped nodes is ď 1. Georg Zetzsche (TU KL) Valence automata D-CON / 23

80 Theorem Let Γ be a graph such that between any two distinct vertices, there is an edge. Then VApMΓq VA`pMΓq if and only if the number of unlooped nodes is ď 1. Georg Zetzsche (TU KL) Valence automata D-CON / 23

81 Theorem Let Γ be a graph such that between any two distinct vertices, there is an edge. Then VApMΓq VA`pMΓq if and only if the number of unlooped nodes is ď 1. In other words: VApB r ˆ Z s q VA`pB r ˆ Z s q iff r ď 1. Georg Zetzsche (TU KL) Valence automata D-CON / 23

82 Theorem Let Γ be a graph such that between any two distinct vertices, there is an edge. Then VApMΓq VA`pMΓq if and only if the number of unlooped nodes is ď 1. In other words: VApB r ˆ Z s q VA`pB r ˆ Z s q iff r ď 1. Observation VApB ˆ Z s q VA`pB ˆ Z s q already follows from the first theorem. Georg Zetzsche (TU KL) Valence automata D-CON / 23

83 More classical results can be generalized: Work in Progress For which storage mechanisms do we have a Parikh theorem? Georg Zetzsche (TU KL) Valence automata D-CON / 23

84 More classical results can be generalized: Work in Progress For which storage mechanisms do we have a Parikh theorem? For which can we perform model checking? Georg Zetzsche (TU KL) Valence automata D-CON / 23

85 More classical results can be generalized: Work in Progress For which storage mechanisms do we have a Parikh theorem? For which can we perform model checking? For which can we compute the downward closure? Georg Zetzsche (TU KL) Valence automata D-CON / 23