Operational State Complexity under Parikh Equivalence Giovanna Lavado 1 Giovanni Pighizzini 1 Shinnosuke Seki 2 , 3 1 Dipartimento di Informatica, Universit` a degli Studi di Milano Helsinki Institute for Information Technology (HIIT) 2 3 Department of Information and Computer Science, Aalto University ICTCS 2014 Universit` a degli Studi di Perugia, Italy September 17-19, 2014 G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Standard equivalence: nfa s vs dfa s Subset construction [Rabin&Scott ’59] nfa dfa 2 n states ⇒ n states = L L Moreover, this state bound cannot be reduced [Meyer&Fischer ’71, Moore ’71] What happens if we do not care of the order of symbols in the strings? This problem is related to the concept of Parikh equivalence [Parikh ’66] G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Standard equivalence: nfa s vs dfa s Subset construction [Rabin&Scott ’59] nfa dfa 2 n states ⇒ n states = L L Moreover, this state bound cannot be reduced [Meyer&Fischer ’71, Moore ’71] What happens if we do not care of the order of symbols in the strings? This problem is related to the concept of Parikh equivalence [Parikh ’66] G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Parikh equivalence: preliminaries Σ = { a 1 , . . . , a m } alphabet of m symbols | w | a be the number of occurrences of a in w ∈ Σ ∗ Parikh map The Parikh map ψ : Σ ∗ → N m associates with a word w ∈ Σ ∗ the m -dimensional nonnegative vector ( | w | a 1 , | w | a 2 , . . . , | w | a m ) . Parikh image The Parikh image of a language L is ψ ( L ) = { ψ ( w ) | w ∈ L } . w 1 = π w 2 iff ψ ( w 1 ) = ψ ( w 2 ) L 1 = π L 2 iff ψ ( L 1 ) = ψ ( L 2 ) G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Parikh equivalence: Parikh’s theorem Theorem ([Parikh ’66]) For each context-free language L ⊆ Σ ∗ , there exists a Parikh equivalent regular language R ⊆ Σ ∗ . Example ( L = π R ) L = { a n b n | n ≥ 0 } R = ( ab ) ∗ and have the same Parikh image, namely the set { ( n , n ) | n ≥ 0 } G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
From nfa s to Parikh equivalent dfa s We have the following Parikh equivalent conversion: Theorem ( nfa to dfa ) nfa dfa √ ⇒ π n · ln n ) states e O ( n states = L 1 L 2 Moreover, this cost is tight. Quite surprisingly: Polynomial conversion If the given nfa accepts only nonunary strings then the cost reduces to a polynomial in n . G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
From nfa s to Parikh equivalent dfa s We have the following Parikh equivalent conversion: Theorem ( nfa to dfa ) nfa dfa √ ⇒ π n · ln n ) states e O ( n states = L 1 L 2 Moreover, this cost is tight. Quite surprisingly: Polynomial conversion If the given nfa accepts only nonunary strings then the cost reduces to a polynomial in n . G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Our Goal We investigate, under Parikh equivalence, the state complexity of some language operations which preserve regularity ( ∪ , ∩ , c , · , ∗ , ✁ , R , P Σ 0 ). Problem ( dfa s to dfa ) A, B dfa s C dfa ⇒ π n 1 , n 2 states L ( C ) = π L = L ( A ) , L ( B ) how many states? where: L = L ( A ) ∪ L ( B ) L = L ( A ) ∩ L ( B ) L = L ( A ) L ( B ) ... G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Standard equivalence: concatenation A , B dfa s C dfa ⇒ 2 n 1 + n 2 states n 1 , n 2 states = L ( A ) L ( B ) L ( C ) = L ( A ) L ( B ) In the worst case: ( 2 n 1 − 1 ) 2 n 2 − 1 states [Yu ’00] Under Parikh equivalence we reduce this bound. G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Standard equivalence: concatenation A , B dfa s C dfa ⇒ 2 n 1 + n 2 states n 1 , n 2 states = L ( A ) L ( B ) L ( C ) = L ( A ) L ( B ) In the worst case: ( 2 n 1 − 1 ) 2 n 2 − 1 states [Yu ’00] Under Parikh equivalence we reduce this bound. G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence One of our contribution Problem ( dfa s to dfa ) A, B dfa s C dfa ⇒ π n 1 , n 2 states L ( C ) = π L = L = L ( A ) L ( B ) how many states? √ n · ln n , where n = n 1 + n 2 Upper bound: e by Parikh equivalent conversion Lower bound: n 1 n 2 states by unary case [Yu ’00] G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Unary and nonunary parts of a language a 1 a 1 a 2 q p a 2 Nonunary part: Unary parts: a 1 a 1 [ p , 1 ] [ q , 1 ] a 2 q a 1 a 1 a 1 a 2 q 0 a 2 a 2 [ q , 2 ] q p a 2 a 1 q p a 2 [ p , 2 ] a 2 a 1 a 2 L ( A ) = � m i = 0 L ( A i ) G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence: proof idea dfa s A , B n 1 , n 2 states L = L ( A ) L ( B ) Σ = { a 1 , . . . , a m } G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence: proof idea ∀ i = 1 . . . m A i , O ( n 1 ) states B i , O ( n 2 ) states unary � dfa s A , B � n 1 , n 2 states � L = L ( A ) L ( B ) Σ = { a 1 , . . . , a m } G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence: proof idea ⇒ ∀ i = 1 . . . m ∀ i = 1 . . . m = A i , O ( n 1 ) states dfa M i B i , O ( n 2 ) states L ( M i ) = L ( A i ) L ( B i ) unary O ( n 1 n 2 ) states [Yu ’00] � dfa s A , B � n 1 , n 2 states � L = L ( A ) L ( B ) Σ = { a 1 , . . . , a m } G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence: proof idea ⇒ ⇒ ∀ i = 1 . . . m ∀ i = 1 . . . m dfa M ′ = = A i , O ( n 1 ) states dfa M i L ( M ′ ) = � m i = 1 L ( M i ) B i , O ( n 2 ) states L ( M i ) = L ( A i ) L ( B i ) poly ( n 1 , n 2 ) states unary O ( n 1 n 2 ) states [Yu ’00] � dfa s A , B � n 1 , n 2 states � L = L ( A ) L ( B ) Σ = { a 1 , . . . , a m } G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence: proof idea ⇒ ⇒ ∀ i = 1 . . . m ∀ i = 1 . . . m dfa M ′ = = A i , O ( n 1 ) states dfa M i L ( M ′ ) = � m i = 1 L ( M i ) B i , O ( n 2 ) states L ( M i ) = L ( A i ) L ( B i ) poly ( n 1 , n 2 ) states unary O ( n 1 n 2 ) states [Yu ’00] � dfa s A , B � n 1 , n 2 states � L = L ( A ) L ( B ) ❅ Σ = { a 1 , . . . , a m } ❅ ❅ nonunary nfa M L ( M ) = L n 1 + n 2 states G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence: proof idea ⇒ ⇒ ∀ i = 1 . . . m ∀ i = 1 . . . m dfa M ′ = = A i , O ( n 1 ) states dfa M i L ( M ′ ) = � m i = 1 L ( M i ) B i , O ( n 2 ) states L ( M i ) = L ( A i ) L ( B i ) poly ( n 1 , n 2 ) states unary O ( n 1 n 2 ) states [Yu ’00] � dfa s A , B � n 1 , n 2 states � L = L ( A ) L ( B ) ❅ Σ = { a 1 , . . . , a m } ❅ ❅ nonunary nfa M 0 ⇒ nfa M L \ L ( M ′ ) = L ( M ) = L ( n 1 + n 2 )( m + 1 )+ 1 n 1 + n 2 states states G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence: proof idea ⇒ ⇒ ∀ i = 1 . . . m ∀ i = 1 . . . m dfa M ′ = = A i , O ( n 1 ) states dfa M i L ( M ′ ) = � m i = 1 L ( M i ) B i , O ( n 2 ) states L ( M i ) = L ( A i ) L ( B i ) poly ( n 1 , n 2 ) states unary O ( n 1 n 2 ) states [Yu ’00] � dfa s A , B � n 1 , n 2 states � L = L ( A ) L ( B ) ❅ Σ = { a 1 , . . . , a m } ❅ ❅ Parikh equivalent conversion nonunary nfa M 0 ⇒ π ⇒ nfa M dfa M ′ L \ L ( M ′ ) = = 0 L ( M ) = L poly ( n 1 , n 2 ) ( n 1 + n 2 )( m + 1 )+ 1 states n 1 + n 2 states states G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
Concatenation under Parikh equivalence: proof idea ⇒ ⇒ ∀ i = 1 . . . m ∀ i = 1 . . . m dfa M ′ = = A i , O ( n 1 ) states dfa M i L ( M ′ ) = � m i = 1 L ( M i ) B i , O ( n 2 ) states L ( M i ) = L ( A i ) L ( B i ) poly ( n 1 , n 2 ) states unary O ( n 1 n 2 ) states [Yu ’00] � ❅ dfa s A , B � dfa C ❅ n 1 , n 2 states � ❅ poly ( n 1 , n 2 ) L = L ( A ) L ( B ) ❅ � states Σ = { a 1 , . . . , a m } � ❅ ❅ Parikh equivalent conversion � nonunary nfa M 0 ⇒ π ⇒ nfa M dfa M ′ L \ L ( M ′ ) = = 0 L ( M ) = L poly ( n 1 , n 2 ) ( n 1 + n 2 )( m + 1 )+ 1 states n 1 + n 2 states states G.J. Lavado, G. Pighizzini, S. Seki Operational State Complexity under Parikh Equivalence
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