On the Boundary of Regular Languages . Galina Jir askov a, Jozef - PowerPoint PPT Presentation

. On the Boundary of Regular Languages . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) Slovak Academy of Sciences and ˇ Saf´ arik University, Koˇ sice 1377 Tartu 1033 Kaunas 711 Hamburg 951 Rotterdam 1241 Krakow 177 1093 Brno 343 ��� ��� ��� ��� ��� ��� Debrecen 135 Graz 471 Milan 982 Montreal 6700 Marseilles 1365 Waterloo 7260 Vatican 1024 7136 Los Angeles 9985 Θεσσαλουικη 909 2170 8862 8288 Durban 8801 . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . Outline . Motivation and history Two problems by JS 2010 . . L ∗ c ∗ 1 bd( L ) = L ∗ ∩ ( L c ) ∗ . . 2 Known results Our results . . tight bounds for bd( L ) 1 . . 5-letter alphabet 2 . . optimal size (?) 3 Applications . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . Motivation I . . J. Brzozowski, E. Grant, J. Shallit: Outline . Closures in formal languages Motivation and history and Kuratowski’s theorem Two problems by JS 2010 [DLT 09, IJFCS 11] . . L ∗ c ∗ 1 concepts of ”open” and ”closed” bd( L ) = L ∗ ∩ ( L c ) ∗ . . 2 L ⊆ Σ ∗ is closed if L = L ∗ L is open if L c closed Known results Our results natural analogues of classical THMs . . . tight bounds for bd( L ) 1 . . 5-letter alphabet 2 . . . In point-set topology: optimal size (?) 3 . bd ( S ) = closure ( S ) ∩ closure ( S c ) Applications . S = { ( x , y ): x 2 + y 2 ≤ 1 } ⇒ bd ( S ) = { ( x , y ): x 2 + y 2 = 1 } . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . Motivation II . . A. Salomaa, K. Salomaa, S. Yu: Outline . State complexity Motivation and history of combined operations [TCS 07] Two problems by JS 2010 . . L ∗ c ∗ operation composition complexity 1 bd( L ) = L ∗ ∩ ( L c ) ∗ . . 3 / 4 · 2 mn 3 / 4 · 2 mn 2 ( K ∩ L ) ∗ 3 / 4 · 2 mn ≤ 3 / 4 · 2 m + n ( K ∪ L ) ∗ . Known results Our results . Combined operations [SSY 07, ...] . . tight bounds for bd( L ) 1 . . . 5-letter alphabet 2 comb. operations complexity . . optimal size (?) 3 2 O ( m + n ) without c and ∩ Applications 2 poly ( mn ) without c . L ∗ c ∗ 2 Θ( n log n ) [JS 12] . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . . Outline Motivation III (for me:) . . Motivation and history An article by Hor´ ak about Paul Erd¨ os: Two problems by JS 2010 A pop-singer needs crowds; . . L ∗ c ∗ 1 the larger, the better... bd( L ) = L ∗ ∩ ( L c ) ∗ . . 2 A researcher needs Known results to be acknowledged by 5 people; Our results he knows them by name. . . tight bounds for bd( L ) 1 Horak’s fives: . . 5-letter alphabet 2 Erd¨ os, Erd¨ os, Erd¨ os, Erd¨ os, Erd¨ os. . . optimal size (?) 3 My fives? Applications ... . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . . Outline Known results . . Motivation and history operation complexity Two problems by JS 2010 L c n [folklore] . . L ∗ c ∗ K ∩ L [RS 59, Ma 70] mn 1 bd( L ) = L ∗ ∩ ( L c ) ∗ . . 3 / 4 · 2 n 2 L ∗ [YZS 94] . Known results Our results . . tight bounds for bd( L ) 1 . . 5-letter alphabet 2 . . optimal size (?) 3 Applications . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . . Outline Known results . . Motivation and history operation complexity Two problems by JS 2010 L c n [folklore] . . L ∗ c ∗ K ∩ L [RS 59, Ma 70] mn 1 bd( L ) = L ∗ ∩ ( L c ) ∗ . . 3 / 4 · 2 n 2 L ∗ [YZS 94] . Known results . Trivial upper bound on sc of bd( L ): Our results . . . tight bounds for bd( L ) operation complexity 1 . . 5-letter alphabet 2 3 / 4 · 2 n L ∗ . . optimal size (?) 3 L c ∗ 3 / 4 · 2 n bd( L ) = L ∗ ∩ L c ∗ 9 / 16 · 4 n Applications . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . Triv. upper bound for bd( L ) is 9 / 16 · 4 n . . Outline Question: Is it attainable??? . Answer: Almost!!! . Motivation and history Two problems by JS 2010 . . L ∗ c ∗ 1 bd( L ) = L ∗ ∩ ( L c ) ∗ . . 2 Known results Our results . . tight bounds for bd( L ) 1 . . 5-letter alphabet 2 . . optimal size (?) 3 Applications . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . Triv. upper bound for bd( L ) is 9 / 16 · 4 n . . Outline Question: Is it attainable??? . Answer: Almost!!! . Motivation and history . Two problems by JS 2010 If L is accepted by an n -state DFA . . L ∗ c ∗ with k final states, then 1 . bd( L ) = L ∗ ∩ ( L c ) ∗ . . 2 sc ( L ∗ ∩ L c ∗ ) ≤ 2 + 2 n − k 2 n − 1 − 3 n − k 2 k − 1 Known results + 2 k − 1 2 n − 1 − 3 k − 1 2 n − k Our results ( n − 1 ) . . + 4 n − 1 − tight bounds for bd( L ) 1 , . . k − 1 5-letter alphabet 2 . . optimal size (?) 3 which is maximal if k = 2, and it equals Applications 3 / 8 · 4 n + 2 n − 2 − 2 · 3 n − 2 − n + 2. . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . Triv. upper bound for bd( L ) is 9 / 16 · 4 n . . Outline Question: Is it attainable??? . Answer: Almost!!! . Motivation and history . Two problems by JS 2010 If L is accepted by an n -state DFA . . L ∗ c ∗ with k final states, then 1 . bd( L ) = L ∗ ∩ ( L c ) ∗ . . 2 sc ( L ∗ ∩ L c ∗ ) ≤ 2 + 2 n − k 2 n − 1 − 3 n − k 2 k − 1 Known results + 2 k − 1 2 n − 1 − 3 k − 1 2 n − k Our results ( n − 1 ) . . + 4 n − 1 − tight bounds for bd( L ) 1 , . . k − 1 5-letter alphabet 2 . . optimal size (?) 3 which is maximal if k = 2, and it equals Applications 3 / 8 · 4 n + 2 n − 2 − 2 · 3 n − 2 − n + 2. . . . This upper bound is tight!!! ( | Σ | ≥ 5) . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages

. On the Boundary of Regular Languages . . . Outline . Motivation and history Two problems by JS 2010 . . L ∗ c ∗ 1 bd( L ) = L ∗ ∩ ( L c ) ∗ . . 2 Known results Our results . . tight bounds for bd( L ) 1 . . 5-letter alphabet 2 . . optimal size (?) 3 Applications . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . Galina Jir´ askov´ a, Jozef Jir´ asek (and his PC:-) On the Boundary of Regular Languages