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Three New Algorithms for Regular Language Enumeration Erkki Makinen University of Tempere Tempere, Finland Margareta Ackerman University of Waterloo Waterloo, ON

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What kind of words does this NFA accepts? 0 1 1 1 0 1 0 A B C D E

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ε 0 00 11 000 110 0000 1100 1110 00000.... Cross-section problem: enumerate all words of length n accepted by the NFA in lexicographic order. 0 1 1 1 0 1 0 A B C D E

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ε 0 00 11 000 110 0000 1100 1110 00000.... Enumeration problem: enumerate the first m words accepted by the NFA in length-lexicographic order. 0 1 1 1 0 1 0 A B C D E

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ε 0 00 11 000 110 0000 1100 1110 00000.... Min-word problem: find the first word of length n accepted by the NFA. 0 1 1 1 0 1 0 A B C D E

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Applications Correctness testing, provides evidence that an NFA generates the expected language. An enumeration algorithm can be used to verify whether two NFAs accept the same language (Conway, 1971). A cross-section algorithm can be used to determine whether every word accepted by a given NFA is a power - a string of the from w n for n>1, |w|>0. (Anderson, Rampersad, Santean, and Shallit, 2007) A cross-section algorithm can be used to solve the “k- subset of an n-set” problem: Enumerate all k-subset of a set in alphabetical order. (Ackerman & Shallit, 2007)

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Objectives Find algorithms for the three problems that are Asymptotically efficient in – Size of the NFA (s states and d transitions) – Output size (t) – The length of the words in the cross-section (n) Efficient in practice

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Previous Work A cross-section algorithm, where finding each consecutive word is super-exponential in the size of the cross-section (Domosi, 1998). A cross-section algorithm that is exponential in n (length of words in the cross-section) is found in the Grail computation package. – “Breast-First-Search” approach – Trace all paths of length n in the NFA, storing the paths that end at a final state. – O(dσ n+1 ), where d is the number of transitions in the NFA and σ is the alphabet size.

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Previous Polynomial Algorithms: Makinen, 1997 Dynamic programming solution – Min-word O(dn 2 ) – Cross-section O(dn 2 +dt) – Enumeration O(d(e+t)) e: the number of empty cross-section encountered d: the number of transitions in the NFA n: the length of words in the cross-section t: the number of characters in the output Quadratic in n

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Previous Polynomial Algorithms: Ackerman and Shallit, 2007 Linear in the length of words in the cross-section – Min-word: O(s 2.376 n) – Cross-section: O(s 2.376 n+dt) – Enumeration: O(s 2.376 c+dt) c: the number of cross-section encountered d: the number of transitions in the NFA n: the length of words in the cross-section t: the number of characters in the output Linear in n

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Previous Polynomial Algorithms: Ackerman and Shallit, 2007 The algorithm uses “smart breadth first search,” following only those paths that lead to a final state. Main idea: compute a look-ahead matrix, used to determine whether there is a path of length i starting at state s and ending at a final state. In practice, Makinen’s algorithm (slightly modified) is usually more efficient, except on some boundary cases.

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Contributions Present 3 algorithms for each of the enumeration problems, including: O(dn) algorithm for min-word O(dn+dt) algorithm for cross-section Algorithms with improved practical performance for each of the enumeration problems

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Contributions: Detailed We present three sets of algorithms 1.AMSorted: - An efficient min-word algorithm, based on Makinen’s original algorithm. - A cross-section and enumeration algorithms based on this min-word algorithm. 2.AMBoolean: - A more efficient min-word algorithm, based on minWordAMSorted. - A cross-section and enumeration algorithms based on this min-word algorithm. 3. Intersection-based: - An elegant min-word algorithm. - A cross-section algorithm based on this min-word algorithm.

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Key ideas behind our first two algorithms -Makinen’s algorithm uses simple dynamic programming, which is efficient in practice on most NFAs. -The algorithm by Ackerman & Shallit uses “smart breadth first search,” following only those paths that lead to a final state. -We build on these ideas to yield algorithms that are more efficient both asymptotically and in practice.

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Makinen’s original min-word algorithm 0 3 1 2 A B C 123 A -(3,C) B 0(2,B)(0,A) C 1(1,B) S[i] stores a representation of the minimal word w of length i that appears on a path from S to a final state. 1

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Makinen’s original min-word algorithm 0 3 1 2 A B C 123 A -(3,C) B 0(2,B)(0,A) C 1(1,B) The minimal word of length n can be found by tracing back from the last column of the start state. 1

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Makinen’s original min-word algorithm Initialize the first column For columns i = 2...n – For each state S Find S[i] by comparing all words of length i appearing on paths from S to a final state. 123 A-(3,C) B0(2,B)(0,A) C1(1,B) 0 3 1 2 A B C 1

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Makinen’s original min-word algorithm Initialize the first column For columns i = 2...n – For each state S Find S[i] by comparing all words of length i appearing on paths from S to a final state. i operations 123 A-(3,C) B0(2,B)(0,A) C1(1,B) 0 3 1 2 A B C 1

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Makinen’s original min-word algorithm Initialize the first column For columns i = 2...n – For each state S Find S[i] by comparing all words of length i appearing on paths from S to a final state. i operations Theorem: Makinen’s original min-word algorithm is O(dn 2 ).

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New min-word algorithm: MinWordAMSorted Idea: Sort every columns by the words that the entries represent. 0 3 1 2 A B C 1 123 A-(3,C) B0(2,B)(0,A) C1(1,B) 321 031 120

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New min-word algorithm: MinWordAMSorted We define an order on {S[i] : S a state in N}. If A[1]=a and B[1]=b, where a** 1, A[i] = (a, A’) and B[i] = (b, B’) – If a B[i].
**

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New min-word algorithm: MinWordAMSorted Initialize the first column For columns i = 2...n – For each state S Find S[i] using only column i-1 and the edges leaving S. – Sort column i 123 A-(3,C) B0(2,B)(0,A) C1(1,B) 0 3 1 2 A B C 1

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New min-word algorithm: MinWordAMSorted Initialize the first column For columns i = 2...n – For each state S Find S[i] using only column i-1 and the edges leaving S. – Sort column i d operations s log s operations Theorem: The algorithm minWordAMSorted is O((s log s +d) n).

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New cross-section algorithm: crossSectionAMSorted A state S is i-complete if there exists a path of length i from state S to a final state. To enumerate all words of length n: 1. Call minWordAMSorted (create a table) 2. Perform a “smart BFS”: - Begin at the start state. - Follow only those paths of length n that end at a final state, by using the table to identify i-complete states. O((s log s +d) n). O(dt) Theorem: The algorithm crossSectionAMSorted is O(n (s log s + d) + dn).

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New enumeration algorithm: enumAMSorted Run the cross-section algorithm until the required number of words are listed, while reusing the table. Theorem: The algorithm enumAMSorted is O(c (s log s + d)+ dt). c: the number of cross-section encountered d: the number of transitions in the NFA t: the number of characters in the output

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What have we got so far? c: the number of cross-section encountered e: the number of empty cross-section encountered d: the number of transitions in the NFA n: the length of words in the cross-section t: the number of characters in the output

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New min-word algorithm: minWordAMBoolean Idea: instead of using a table to find the minimal word, construct a table whose only purpose is to determine i-complete states. Can be done using a similar algorithm to minWordAMSorted, but more efficiently, since there is no need to sort.

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New min-word algorithm: minWordAMBoolean 0 3 1 A B C 123 A FTT B TTT C TTF

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Fill in the first column For i=2... n – For every state S Determine whether S is i-complete using only the transitions leaving S and column i-1 Starting at the start state, follow minimal transitions to paths that can complete a word of length n (using the table). 123 AFTT BTTT CTTF 0 3 1 A B C

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New min-word algorithm: minWordAMBoolean Fill in the first column For i=2... n – For every state S Determine whether S is i-complete using only the transitions leaving S and column i-1 Starting at the start state, follow minimal transitions to paths that can complete a word of length n (using the table). 123 AFTT BTTT CTTF 0 3 1 A B C d operations

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New min-word algorithm: minWordAMBoolean Fill in the first column For i=2... n – For every state S Determine whether S is i-complete using only the transitions leaving S and column i-1 Starting at the start state, follow minimal transitions to paths that can complete a word of length n (using the table). d operations Theorem: The algorithm minWordAMBoolean is O(dn).

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New cross-section algorithm: crossSectionAMBoolean Extend to a cross-section algorithm using the same approach as the Sorted algorithm. To enumerate all words of length n: – Call minWordAMBoolean (create a table) – Perform a “smart BFS”: - Begin at the start state. - Follow only those paths of length n that end at a final state, by using the table to identify i-complete states. O(dn). O(dt) Theorem: The algorithm crossSectionAMBoolean is O(dn+dt).

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New enumeration algorithm: enumAMBoolean Run the cross-section algorithm until the required number of words are listed, while reusing the table. Theorem: The algorithm enumAMBoolean is O(de+ dn). e: the number of empty cross-section encountered d: the number of transitions in the NFA n: the length of words in the cross-section t: the number of characters in the output

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What have we got so far? c: the number of cross-section encountered e: the number of empty cross-section encountered d: the number of transitions in the NFA n: the length of words in the cross-section t: the number of characters in the output

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We present surprisingly elegant min-word and cross-section algorithms that have the asymptotic efficiency of the Boolean-based algorithms. However, these algorithms are not as efficient in practice as the Boolean-based and Sorted- based algorithms. Intersection-Based Algorithms

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Let N be the input NFA, and A be the NFA that accepts the language of all words of length n. 1.Let C = N x A 2.Remove all states of C that cannot be reached from the final states of C using reversed transitions. 3. Starting at the start state, follow the minimal n consecutive transitions to a final state. New min-word algorithm: minWordIntersection

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Let N be the input NFA, and A be the NFA that accepts the language of all words of length n. 1.Let C = N x A 2.Remove all states of C that cannot be reached from the final states of C using reversed transitions. 3. Starting at the start state, follow the minimal n consecutive transitions to a final state. New min-word algorithm: minWordIntersection 1 0 1 A B C Automaton N Let n = 2 0 00 1 11 Automaton A 0

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Let N be the input NFA, and A be the NFA that accepts the language of all words of length n. 1.Let C = N x A 2.Remove all states of C that cannot be reached from the final states of C using reversed transitions. 3. Starting at the start state, follow the minimal n consecutive transitions to a final state. New min-word algorithm: minWordIntersection Automaton N 0 0 1 1 Automaton C 1 0 1 A B C 0

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Let N be the input NFA, and A be the NFA that accepts the language of all words of length n. 1.Let C = N x A 2.Remove all states of C that cannot be reached from the final states of C using reversed transitions. 3. Starting at the start state, follow the minimal n consecutive transitions to a final state. New min-word algorithm: minWordIntersection 1 1

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Let N be the input NFA, and A be the NFA that accepts the language of all words of length n. 1.Let C = N x A 2.Remove all states of C that cannot be reached from the final states of C using reverse transitions. 3. Starting at the start state, follow the minimal n consecutive transitions to a final state. New min-word algorithm: minWordIntersection 1 1 Thus the minimal word of length 2 accepted by N is “11”

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1.Let C = N x A 2.Remove all states of C that cannot be reached from the final states of C using reverse transitions. 3. Starting at the start state, Follow the minimal n consecutive transitions to final. Asymptotic running time of minWordIntersection Each step is proportional to size of C, which is O(nd). Theorem: The algorithm minWordIntersection is O(dn). Theorem: The algorithm minWordIntersection is O(dn). Concatenate n copies of N.

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To enumerate all words of length n, perform BFS on C = N x A, and remove all states not reachable from final state removed (using reverse transitions). Since all paths of length n starting at the start state lead to a final state, there is no need to check for i-completness. New cross-section algorithm: crossSectionIntersection Theorem: The algorithm crossSectionIntersection is O(dn+dt).

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We compared Makinen’s, Ackerman-Shallit, AMSorted, and AMBoolean, and Intersection-based algorithms. Tested the algorithms on a variety of NFAs: dense, sparse, few and many final states, different alphabet size, worst case for Makinen’s algorithm, ect… Here are the best performing algorithms: – Min-word: AMSorted – Cross-section: AMBoolean – Enumeration: AMBoolean Practical Performance

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Summary c: the number of cross-section encountered e: the number of empty cross-section encountered d: the number of transitions in the NFA n: the length of words in the cross-section t: the number of characters in the output : most efficient in practice

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Extending the intersection-based cross-section algorithm to an enumeration algorithm. Lower bounds. Can better results be obtained using a different order? Restricting attention to a smaller family of NFAs. Open problems

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