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Efficient Algorithms for Counting Automata
Mikšaník, David ; Holík, Lukáš (oponent) ; Lengál, Ondřej (vedoucí práce)
Counting automata (CAs) are classical finite automata extended with bounded counters. They still denote the class of regular languages but in a more compact way than finite automata. Since CAs are a recent model, there is a gap in the knowledge of efficient algorithms implementing various operations on the CAs. In this thesis, we mainly focus on an existing subclass of CAs called monadic counting automata (MCAs), i.e., CAs with counting loops on character classes, which are common in practice (e.g., detection of packets in network traffic, log analysis). For this subclass, we efficiently solve the emptiness and inclusion problems. Moreover, we provide two extensions of the class of MCAs (but not beyond the class of CAs) and efficiently solve the emptiness problem for them. MCAs naturally arise from regular expressions that are extended by the counting operator limited only to character classes. Thus our algorithm solving the inclusion problem of MCAs can be used in a new method for solving the inclusion problem of such regular expressions. We experimentally evaluated this method on regular expressions from a wide range of applications and compared it with the naive method. The experiments show that the method using our algorithm is less prone the explode. It also outperforms the naive method if the regular expressions contain counting operators with large bounds. As expected, for the easy cases, the naive method is still faster than the method based on our algorithm.
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The combinatorics of pattern-avoiding matrices
Mikšaník, David ; Jelínek, Vít (vedoucí práce) ; Klazar, Martin (oponent)
Permutační matice P částečné vynechává kvazi-permutační matici A (jinými slovy, 01- matici takovou, že každý sloupec a řádek matice A obsahuje nejvýše jednu nenulovou hod- notu), pokud neexistuje podmatice P′ matice P stejné velikosti jako A splňující Ai,j ≤ P′ i,j pro každé indexy i a j. Kvazi-permutační matice A a B jsou částečně Wilf-ekvivalentní, pokud pro každé n ∈ N počet permutačních matic řádu n částečné vynechávajících A je stejný jako počet permutačních matic řádu n částečně vynechávajících B. Tyto pojmy zobecňují známý koncept vynechávání permutací a Wilfovy ekvivalence permutací. Stě- žejní oblast výzkumu je klasifikace permutací řádu k do tříd Wilfovy ekvivalence. Tato klasifikace je známa pro k = 1, 2, . . . , 7. V naší práci studujeme stejný problém pro kvazi- permutační matice. Konkrétně, klasifikujeme všech 371 kvazi-permutačních matic veli- kosti nejvýše 4×4 do říd částečné Wilfovy ekvivalence (dvě kvazi-permutační matice patří do stejné třídy právě tehdy, když jsou částečně Wilf-ekvivalentní). V průběhu odvodíme několik obecných výsledků o tom, jak zkonstruovat z jedné či dvou kvazi-permutačních matic více kvazi-permutačních matic, které jsou po dvou částečně Wilf-ekvivalentní. 1
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Efficient Algorithms for Counting Automata
Mikšaník, David ; Holík, Lukáš (oponent) ; Lengál, Ondřej (vedoucí práce)
Counting automata (CAs) are classical finite automata extended with bounded counters. They still denote the class of regular languages but in a more compact way than finite automata. Since CAs are a recent model, there is a gap in the knowledge of efficient algorithms implementing various operations on the CAs. In this thesis, we mainly focus on an existing subclass of CAs called monadic counting automata (MCAs), i.e., CAs with counting loops on character classes, which are common in practice (e.g., detection of packets in network traffic, log analysis). For this subclass, we efficiently solve the emptiness and inclusion problems. Moreover, we provide two extensions of the class of MCAs (but not beyond the class of CAs) and efficiently solve the emptiness problem for them. MCAs naturally arise from regular expressions that are extended by the counting operator limited only to character classes. Thus our algorithm solving the inclusion problem of MCAs can be used in a new method for solving the inclusion problem of such regular expressions. We experimentally evaluated this method on regular expressions from a wide range of applications and compared it with the naive method. The experiments show that the method using our algorithm is less prone the explode. It also outperforms the naive method if the regular expressions contain counting operators with large bounds. As expected, for the easy cases, the naive method is still faster than the method based on our algorithm.
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