15. Theorietag Automaten und Formale Sprachen Lauterbad, 28–29 September 2005 On the Parikh Images of Level-Two Pushdown Automata Wong Karianto karianto@informatik.rwth-aachen.de Lehrstuhl f¨ ur Informatik VII
Motivations Correspondence between automata (and formal language) theory and number theory: RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 2
Motivations Correspondence between automata (and formal language) theory and number theory: Automata theory Number theory (Parikh mapping) finite automata, ✲ semi-linear sets pushdown automata RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 2
Motivations Correspondence between automata (and formal language) theory and number theory: Automata theory Number theory (Parikh mapping) finite automata, ✲ semi-linear sets pushdown automata higher-order pushdown ✲ ??? automata ( HOPDA ) HOPDA: finite-state automata with a stack of stacks of . . . of stacks RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 2
Motivations Correspondence between automata (and formal language) theory and number theory: Automata theory Number theory (Parikh mapping) finite automata, ✲ semi-linear sets pushdown automata higher-order pushdown ✲ ??? automata ( HOPDA ) HOPDA: finite-state automata with a stack of stacks of . . . of stacks In this talk: HOPDA of level 2 ( 2-PDA ) Two questions for a class characterizing the Parikh images of 2-PDA’s: � Can all sets from this class be generated (via the Parikh mapping)? � Does the Parikh image of each 2-PDA belong to this class? RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 2
Outline � Semi-linear sets and Parikh’s theorem � Level 2 pushdown automata � Semi-polynomial sets � From semi-polynomial sets to 2-PDA’s � From 2-PDA’s to semi-polynomial sets? � Conclusions RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 3
Semi-Linear Sets A ⊆ N n linear : A = { ¯ x 0 + k 1 ¯ x 1 + . . . + k m ¯ x m | k 1 , . . . , k m ∈ N } � ∈ N n for some ¯ x 0 , ¯ x 1 , . . . , ¯ x m � �� ✕ ✁ ✁ ❆ ❑ ✁ ❆ constant vector periods 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 } . RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 4
Semi-Linear Sets A ⊆ N n linear : A = { ¯ x 0 + k 1 ¯ x 1 + . . . + k m ¯ x m | k 1 , . . . , k m ∈ N } � ∈ N n for some ¯ x 0 , ¯ x 1 , . . . , ¯ x m � �� ✕ ✁ ✁ ❆ ❑ ✁ ❆ constant vector periods 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] RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 4
Parikh Mapping and Parikh’s Theorem Σ = { a 1 , . . . , a n } � Parikh mapping Φ: Σ ∗ → N n Φ( w ) := ( | w | a 1 , . . . , | w | a n ) . � Φ( w ) : the Parikh image of w � Φ( L ) := { Φ( w ) | w ∈ L } ⊆ N n : the Parikh image of L RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 5
Parikh Mapping and Parikh’s Theorem Σ = { a 1 , . . . , a n } � Parikh mapping Φ: Σ ∗ → N n Φ( w ) := ( | w | a 1 , . . . , | w | a n ) . � Φ( w ) : the Parikh image of w � Φ( L ) := { Φ( w ) | w ∈ L } ⊆ N n : the Parikh image of L Theorem (Parikh (1961)): The Parikh image of any context-free language is effectively semi-linear. RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 5
Higher-Order Pushdown Automata � Finite-state automata augmented with a nested pushdown stack, i.e., a stack of stacks of . . . stacks � Level n HOPDA: n -fold nested stacks RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 6
Higher-Order Pushdown Automata � Finite-state automata augmented with a nested pushdown stack, i.e., a stack of stacks of . . . stacks � Level n HOPDA: n -fold nested stacks � Background: ◮ Maslov (1976): Formal definition; correspondence with generalized indexed languages ◮ Damm and Goerdt (1982): automaton characterization of the OI hierarchy ◮ Engelfriet (1983): correspondence to complexity classes ◮ Carayol and W¨ ohrle (2003): correspondence to a hierarchy of infinite graphs introduced by Caucal RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 6
Level 2 PDA Stack alphabet Γ with initial symbol ⊥ : � Level 1 stack (1-stack): [ Z m · · · Z 1 ] ; Z m is the topmost symbol. � Level 2 stack (2-stack): [ s r , . . . , s 1 ] , where s 1 , . . . , s r are 1-stacks, and s r is the topmost 1-stack. � Empty level 2 stack [[ ε ]] ; initial level 2 stack [[ ⊥ ]] . RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 7
Level 2 PDA Stack alphabet Γ with initial symbol ⊥ : � Level 1 stack (1-stack): [ Z m · · · Z 1 ] ; Z m is the topmost symbol. � Level 2 stack (2-stack): [ s r , . . . , s 1 ] , where s 1 , . . . , s r are 1-stacks, and s r is the topmost 1-stack. � Empty level 2 stack [[ ε ]] ; initial level 2 stack [[ ⊥ ]] . � Instructions on 1-stacks: push and pop � Instructions on level 2 stacks: ◮ push and pop on the topmost 1-stack ◮ copy the topmost 1-stack ◮ remove the topmost 1-stack � Access: only to the topmost symbol of the topmost 1 -stack ! RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 7
Example: A Non-Semi-Linear Language L quad := { a k b k 2 | k ∈ N } ⊆ { a, b } ∗ Take Γ := {⊥ , Z, Z 2 } and process input a k b k 2 as follows: [[ Z 2 k ⊥ ]] RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8
Example: A Non-Semi-Linear Language L quad := { a k b k 2 | k ∈ N } ⊆ { a, b } ∗ Take Γ := {⊥ , Z, Z 2 } and process input a k b k 2 as follows: [[ Z 2 Z 2 k ⊥ ]] RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8
Example: A Non-Semi-Linear Language L quad := { a k b k 2 | k ∈ N } ⊆ { a, b } ∗ Take Γ := {⊥ , Z, Z 2 } and process input a k b k 2 as follows: [[ Z 2 Z 2 k ⊥ ] , [ Z 2 Z 2 k ⊥ ]] RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8
Example: A Non-Semi-Linear Language L quad := { a k b k 2 | k ∈ N } ⊆ { a, b } ∗ Take Γ := {⊥ , Z, Z 2 } and process input a k b k 2 as follows: [[ ZZ 2( k − 1) ⊥ ] , Z 2 k [ Z 2 ⊥ ]] RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8
Example: A Non-Semi-Linear Language L quad := { a k b k 2 | k ∈ N } ⊆ { a, b } ∗ Take Γ := {⊥ , Z, Z 2 } and process input a k b k 2 as follows: [[ Z ⊥ ] , [ ZZZ ⊥ ] , [ ZZZZZ ⊥ ] , . . . Z 2( k − 1) [ Z ⊥ ] , Z 2 k [ Z 2 ⊥ ]] Number of Z ’s above Z 2 : � k − 1 i =0 (2 i + 1) = k 2 RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8
Example: A Non-Semi-Linear Language L quad := { a k b k 2 | k ∈ N } ⊆ { a, b } ∗ Take Γ := {⊥ , Z, Z 2 } and process input a k b k 2 as follows: [[ Z ⊥ ] , [ ZZZ ⊥ ] , [ ZZZZZ ⊥ ] , . . . Z 2( k − 1) [ Z ⊥ ] , Z 2 k [ Z 2 ⊥ ]] Number of Z ’s above Z 2 : � k − 1 i =0 (2 i + 1) = k 2 Φ( L quad ) = { ( x 1 , x 2 ) ∈ N 2 | x 2 1 = x 2 } = ⇒ not semi-linear (proof by growth rate arguments) RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8
Semi-Polynomial Sets � A ⊆ N n linear: ¯ x ∈ A iff k 1 , . . . , k m ∈ N exist such that x = ¯ ¯ x 0 + k 1 ¯ x 1 + . . . + k m ¯ x m . RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 9
Semi-Polynomial Sets � A ⊆ N n linear: ¯ x ∈ A iff k 1 , . . . , k m ∈ N exist such that x = ¯ ¯ x 0 + k 1 ¯ x 1 + . . . + k m ¯ x m . � Replace the vectors with their components: x = ( x 01 , . . . , x 0 n ) + ( k 1 x 11 , . . . , k 1 x 1 n ) + . . . + ( k m x m 1 , . . . , k m x mn ) ¯ and replace linear terms k i x ij with polynomial ones . RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 9
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