9th International Conference Developments in Language Theory Palermo, 4–8 July 2005 Adding Monotonic Counters to Automata and Transition Graphs Wong Karianto karianto@informatik.rwth-aachen.de Lehrstuhl f¨ ur Informatik VII
Motivations: Parikh Automata (Klaedtke & Rueß, ICALP 2003) Idea: how to recognize a language with arithmetical properties , such as L abc := { a k b k c k | k ≥ 1 } ? � Use a finite automaton recognizing a + b + c + . � Assign a vector to each input symbol. � Put a Presburger constraint on the summed vector ( x, y, z ) : x = y = z . ( a, (1 , 0 , 0)) ( b, (0 , 1 , 0)) ( c, (0 , 0 , 1)) ( a, (1 , 0 , 0)) ( b, (0 , 1 , 0)) ( c, (0 , 0 , 1)) RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 2
Outline 1. Parikh automata � Semi-linear sets and Parikh’s theorem � Parikh automata and the Chomsky hierarchy 2. Monotonic-counter extensions of (infinite) graphs � Some classes of infinite graphs � Monotonic-counter extensions 3. Reachability problem RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 3
Part 1 Parikh Automata RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 4
Semi-Linear Sets and Parikh’s Theorem A ⊆ N n linear : A = { ¯ x 0 + k 1 ¯ x 1 + . . . + k m ¯ x m | k 1 , . . . , k m ∈ N } x m ∈ N n for some ¯ x 0 , ¯ x 1 , . . . , ¯ Semi-linear set : finite union of linear sets. Example: B := { ( x 1 , x 2 , x 3 ) ∈ N 3 | x 1 < x 2 < x 3 } is linear: { (0 , 1 , 2) + k 1 (0 , 0 , 1) + k 2 (0 , 1 , 1) + k 3 (1 , 1 , 1) | k 1 , k 2 , k 3 ∈ N } . Properties of semi-linear sets: � effective closure under Boolean operations [Ginsburg & Spanier] � equivalence to Presburger-definable sets [Ginsburg & Spanier] Parikh’s theorem : The Parikh image of any context-free language is effectively semi-linear. RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 5
Parikh Automata ( a, (1 , 0 , 0)) ( b, (0 , 1 , 0)) ( c, (0 , 0 , 1)) ( a, (1 , 0 , 0)) ( b, (0 , 1 , 0)) ( c, (0 , 0 , 1)) Parikh finite automaton ( Parikh-FA ) ( A , C ) of dimension n ≥ 1 over Σ : � finite automaton A over Σ × D ( D ⊆ N n finite, nonempty) � semi-linear set C ⊆ N n Word u := a 1 · · · a m is accepted iff � v := ( a 1 , ¯ d 1 ) · · · ( a m , ¯ d m ) ∈ L ( A ) exists, for some ¯ d 1 , . . . , ¯ d m ∈ D , � and Φ( v ) := ¯ d 1 + · · · + ¯ d m ∈ C . ❆ ❑ ❆ extended Parikh mapping Φ: (Σ × D ) ∗ → N n RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 6
Emptiness Problem Lemma (Klaedtke & Rueß): If the Parikh image of L ⊆ (Σ × D ) ∗ is effectively semi-linear, then also its extended Parikh image Φ( L ) . Theorem (Klaedtke & Rueß): The emptiness problem for Parikh-FA’s is decidable. Proof idea. L ( A , C ) � = ∅ Φ( L ( A )) ∩ C � = ∅ iff � �� � – Both sets are semi-linear. – Intersection of semi-linear sets is effectively semi-linear. = ⇒ (Non-)Emptiness is decidable. � RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 7
Parikh Automata and the Chomsky Hierarchy Automata of the Chomsky hierarchy as the automaton component A : Parikh-TM Turing machines 1. { a k b k c k | k ≥ 1 } Parikh-LBA 2. { ww R | w ∈ { a, b } ∗ } linear-bounded automata (3) 3. semi-linearity of Parikh-PDA Parikh-PDA (1) recognizable languages pushdown automata (2) Parikh-FA (1) finite automata RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 8
Part 2 Monotonic-Counter Extensions of (Infinite) Graphs RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 9
Monotonic-Counter Graphs Monotonic-counter extension of G : Σ -labeled graph � G := ( � V , ( � (Σ × D ) -labeled graph E a ) a ∈ Σ ) with V := V × N n and (( α, ¯ � y )) ∈ � G := ( V, ( E ( a, ¯ d ) ) ( a, ¯ d ) ∈ Σ × D ) x ) , ( β, ¯ E a iff � ( α, β ) ∈ E ( a, ¯ d ) and ( D ⊆ N n finite, nonempty) x + ¯ d , for some ¯ y = ¯ ¯ d ∈ D . a a a (0 , 0) (1 , 0) (2 , 0) · · · • b b b a a a ( a, (1 , 0)) ( b, (0 , 1)) (0 , 1) (1 , 1) (2 , 1) · · · b b b a a a (0 , 2) (1 , 2) (2 , 2) · · · b b b · · · · · · · · · RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 10
Some Classes of Infinite Graphs with Finite Representations Vertices: regular sets over alphabet Γ Edges: automaton-definable relations over words, e.g.: � pushdown graphs [Muller & Schupp]: transitions of ε -free pushdown automata � prefix-recognizable graphs [Caucal]: generalized prefix rewriting rules � synchronized rational graphs or automatic graphs [Frougny & Sakarovitch, Blumensath & Gr¨ adel]: synchronized rational relations � rational graphs [Morvan]: rational relations RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 11
Hierarchy of Graph Classes R-MC rational graphs SR-MC 1. infinite two-dimensional grid synchronized rational graphs 2. decidability of the reachability (2) problem PR-MC (1) 3. graphs with vertices of prefix-recognizable graphs (3) unbounded degree 4. graphs with repetition-free PD-MC (1) cycles of unbounded length pushdown graphs (4) F-MC (1) finite graphs RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 12
Synchronized Rational Graphs Vertices: regular set V over alphabet Γ Edges: synchronized rational relations, i.e. edge relation E a is recognized by a finite-state automaton working on pairs ( X 1 · · · X m , Y 1 · · · Y n ) ∈ Γ ∗ × Γ ∗ with two one-way input tapes and simultaneously moving input heads. X 1 X 2 X 3 X m · · · ⋄ · · · ⋄ finite memory new symbol Y 1 Y 2 Y 3 Y m Y m +1 Y n · · · · · · RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 13
Synchronized Rational Graphs: Example a XXX · · · a XX b a X b XXY · · · ε b XY b c Y b XY Y · · · c Y Y b c Y Y Y · · · RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 14
Monotonic-Counter Extensions of Synchronized Rational Graphs (Σ × D ) -labeled Σ -labeled SRMC graph G := ( � � V , ( � E a ) a ∈ Σ ) with � V := V × N n synchronized rational graph y )) ∈ � G := ( V, ( E ( a, ¯ d ) ) ( a, ¯ d ) ∈ Σ × D ) and (( α, ¯ x ) , ( β, ¯ E a iff � ( α, β ) ∈ E ( a, ¯ d ) and ( D ⊆ N n finite, nonempty) x + ¯ d , for some ¯ d ∈ D . y = ¯ ¯ Proposition: � G is synchronized rational. Proof sketch. Encode vertex ( α, ( x 1 , . . . , x n )) of � G by means of word # 1 · · · # 1 · · · # n · · · # n α � �� � � �� � x 1 x n Define automaton for � n α, # y 1 1 · · · # y n E a working on pairs (# x 1 1 · · · # x n n β ) : 1. Guess a vector ¯ x + ¯ d ∈ D and check whether ¯ d = ¯ y . 2. Simulate the automaton for E ( a, ¯ d ) on ( α, β ) . Bounded delay sufficient since D and Γ are finite. � RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 15
Part 3 Reachability Problem RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 16
Logical Decision Problems over Transition Graphs rational graphs undecidable FO theory decidable synchronized rational graphs undecidable reachability decidable undecidable PR-MC MSO theory decidable prefix-recognizable graphs PD-MC pushdown graphs F-MC finite graphs RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 17
Prefix-Recognizable Graphs → V with U, V ⊆ Γ ∗ regular a − rule U Σ := { a, b } Γ := { Z } ✁ ✁ ☛ → Z + } Prefix-rewriting system R := { ε a → Z , ε b − − Prefix-recognizable graph G = ( V, E a , E b ) defined by R : � V = Γ ∗ , � E a = { ( Z i , Z i +1 ) | i ∈ N } , and � E b = { ( Z i , Z j ) | i, j ∈ N and i < j } . b b · · · a a a a Z 2 Z 3 ε Z · · · · · · b b RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 18
Monotonic-Counter Extensions of Prefix-Recognizable Graphs (Σ × D ) -labeled Σ -labeled PRMC graph G := ( � � V , ( � E a ) a ∈ Σ ) with � V := V × N n prefix-recognizable graph y )) ∈ � G := ( V, ( E ( a, ¯ d ) ) ( a, ¯ d ) ∈ Σ × D ) and (( α, ¯ x ) , ( β, ¯ E a iff � ( α, β ) ∈ E ( a, ¯ d ) and ( D ⊆ N n finite, nonempty) x + ¯ d , for some ¯ y = ¯ ¯ d ∈ D . Reachability problem for � G : Given: regular sets U, U ′ ⊆ V of vertices in G and semi-linear sets C, C ′ ⊆ N n x ) ∈ U × C and Question: are there vertices ( α, ¯ y ) ∈ U ′ × C ′ in � ( β, ¯ G such that ( β, ¯ y ) is reachable from ( α, ¯ x ) ? RWTH Aachen – Wong Karianto Adding Monotonic Counters to Automata and Transition Graphs – p. 19
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