❘❙ ❚❯ ❱ Further Applications In DNA-computing , there is a fast progressing work by Ehud Shapiro’s group on “ soup of automata ” (Programmable and autonomous computing machine made of biomolecules, Nature 414, no.1 (November 22, 2001) 430–434; DNA molecule provides a computing machine with both data and fuel, Proc. National Acad. Sci. USA 100 (2003) 2191–2196, etc). They have produced a solution containing ❖◗P identical DNA-based automata per l. WAW 2007, Turku, Finland, 29.03.07 – p.10/28
❨❩ ❪ ❬❭ Further Applications In DNA-computing , there is a fast progressing work by Ehud Shapiro’s group on “ soup of automata ” (Programmable and autonomous computing machine made of biomolecules, Nature 414, no.1 (November 22, 2001) 430–434; DNA molecule provides a computing machine with both data and fuel, Proc. National Acad. Sci. USA 100 (2003) 2191–2196, etc). They have produced a solution containing ❲◗❳ identical DNA-based automata per l. These automata can work in parallel on different inputs (DNA strands), thus ending up in different and unpredictable states. WAW 2007, Turku, Finland, 29.03.07 – p.10/28
❵❛ ❜❝ ❞ Further Applications In DNA-computing , there is a fast progressing work by Ehud Shapiro’s group on “ soup of automata ” (Programmable and autonomous computing machine made of biomolecules, Nature 414, no.1 (November 22, 2001) 430–434; DNA molecule provides a computing machine with both data and fuel, Proc. National Acad. Sci. USA 100 (2003) 2191–2196, etc). They have produced a solution containing ❫◗❴ identical DNA-based automata per l. These automata can work in parallel on different inputs (DNA strands), thus ending up in different and unpredictable states. One has to feed the automata with an reset sequence (again encoded by a DNA-strand) in order to get them ready for a new use. WAW 2007, Turku, Finland, 29.03.07 – p.10/28
The ˇ Cerný conjecture Clearly, from the viewpoint of the above applications (as well as from the mathematical point of view) the following question is of importance: WAW 2007, Turku, Finland, 29.03.07 – p.11/28
❡ The ˇ Cerný conjecture Clearly, from the viewpoint of the above applications (as well as from the mathematical point of view) the following question is of importance: Suppose a synchronizing automaton has states. What is the length of its shortest reset sequence? WAW 2007, Turku, Finland, 29.03.07 – p.11/28
❢ The ˇ Cerný conjecture Clearly, from the viewpoint of the above applications (as well as from the mathematical point of view) the following question is of importance: Suppose a synchronizing automaton has states. What is the length of its shortest reset sequence? In the example above the automaton has 4 states and there is a reset sequence of length 9. In fact, this was the shortest possible reset sequence. WAW 2007, Turku, Finland, 29.03.07 – p.11/28
❤ ♦ ✐ ❥ ❧ ❦ ❣ ❥ ❦ ❤ ❧ ❣ The ˇ Cerný conjecture Clearly, from the viewpoint of the above applications (as well as from the mathematical point of view) the following question is of importance: Suppose a synchronizing automaton has states. What is the length of its shortest reset sequence? In the example above the automaton has 4 states and there is a reset sequence of length 9. In fact, this was the shortest possible reset sequence. In 1964, ˇ Cerný conjectured that every synchronizing automaton with states has a reset sequence of length — as in our example where . ♠✬♥ ❣◗✐ WAW 2007, Turku, Finland, 29.03.07 – p.11/28
① r q s t ✉ ♣ t s ✉ q ♣ The ˇ Cerný conjecture Clearly, from the viewpoint of the above applications (as well as from the mathematical point of view) the following question is of importance: Suppose a synchronizing automaton has states. What is the length of its shortest reset sequence? In the example above the automaton has 4 states and there is a reset sequence of length 9. In fact, this was the shortest possible reset sequence. In 1964, ˇ Cerný conjectured that every synchronizing automaton with states has a reset sequence of length — as in our example where . ✈✬✇ ♣◗r The simply looking conjecture is still open in general!! WAW 2007, Turku, Finland, 29.03.07 – p.11/28
⑧⑨ ③ ⑩ The ˇ Cerný conjecture ⑤⑦⑥ The best upper bound known so far is ②④③ (J.-E. Pin, 1983). WAW 2007, Turku, Finland, 29.03.07 – p.12/28
❺❻ ❷ ❼ The ˇ Cerný conjecture ❸⑦❹ The best upper bound known so far is ❶④❷ (J.-E. Pin, 1983). It is also known that the problem is hard from the computational complexity point of view. WAW 2007, Turku, Finland, 29.03.07 – p.12/28
➅ ➄ ➅ ➆ ➄ ❾ ➁➂ ➃ The ˇ Cerný conjecture ❿⑦➀ The best upper bound known so far is ❽④❾ (J.-E. Pin, 1983). It is also known that the problem is hard from the computational complexity point of view. Given a DFA and a positive integer , the problem whether or not has a reset word of length is NP-complete (D. Eppstein, 1990; P . Goralˇ cik and V. Koubek, 1995; A. Salomaa, 2003). WAW 2007, Turku, Finland, 29.03.07 – p.12/28
➐ ➏ ➏ ➎ ➏ ➈ ➋➌ ➍ ➎ ➏ ➎ ➎ The ˇ Cerný conjecture ➉⑦➊ The best upper bound known so far is ➇④➈ (J.-E. Pin, 1983). It is also known that the problem is hard from the computational complexity point of view. Given a DFA and a positive integer , the problem whether or not has a reset word of length is NP-complete (D. Eppstein, 1990; P . Goralˇ cik and V. Koubek, 1995; A. Salomaa, 2003). Given a DFA and a positive integer , the problem whether or not the shortest reset word for has length is co-NP-hard (W. Samotij, 2007). WAW 2007, Turku, Finland, 29.03.07 – p.12/28
Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. WAW 2007, Turku, Finland, 29.03.07 – p.13/28
Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In this of aperiodic talk we concentrate on the class ➑➓➒ automata . WAW 2007, Turku, Finland, 29.03.07 – p.13/28
➨ ➛ ➙ ➤ ➤ ➛ ➩ ➝ ➞ ➝ ➜ ➙ ➜ ↕ ➫ Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In this of aperiodic talk we concentrate on the class ➔➓→ automata . Recall that the transition monoid of a DFA consists of all transformations ➣✂↔ ➙✝➛ induced by words . ➟➡➠➢ ➥➧➦ WAW 2007, Turku, Finland, 29.03.07 – p.13/28
➼ ➺ ➬ ➻ ➷ ➶ ➸ ➴ ➸ ➵ ➶ ➻ ➺ ➼ ➽ Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In this of aperiodic talk we concentrate on the class ➭➓➯ automata . Recall that the transition monoid of a DFA consists of all transformations ➲✂➳ ➸✝➺ induced by words . A ➾➡➚➪ ➹➧➘ monoid is said to be aperiodic if all its subgroups are singletons. WAW 2007, Turku, Finland, 29.03.07 – p.13/28
Ï Ð Ù Õ ❮ Ø ❮ Õ ❰ Ð Ú Ñ ❰ Ï ❒ Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In this of aperiodic talk we concentrate on the class ➮➓➱ automata . Recall that the transition monoid of a DFA consists of all transformations ✃✂❐ ❮✝❰ induced by words . A Ò➡ÓÔ Ö➧× monoid is said to be aperiodic if all its subgroups are singletons. A DFA is called aperiodic (or counter-free ) if its transition monoid is aperiodic. WAW 2007, Turku, Finland, 29.03.07 – p.13/28
è ã í â ì è à ë à ß á â á ã ä Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In this of aperiodic talk we concentrate on the class Û➓Ü automata . Recall that the transition monoid of a DFA consists of all transformations Ý✂Þ à✝á induced by words . A å➡æç é➧ê monoid is said to be aperiodic if all its subgroups are singletons. A DFA is called aperiodic (or counter-free ) if its transition monoid is aperiodic. Synchronization issues remain difficult when restricted to . Û➓Ü WAW 2007, Turku, Finland, 29.03.07 – p.13/28
ï î ð ó ô õ Aperiodic Automata Good news: Recently A. Trakhtman has proved that every synchronizing aperiodic automaton with states admits a reset word of length at most ïòñ . WAW 2007, Turku, Finland, 29.03.07 – p.14/28
� ö ö ÿ ý þ ý ü û ú ø ÷ ö Aperiodic Automata Good news: Recently A. Trakhtman has proved that every synchronizing aperiodic automaton with states admits a reset word of length at most ÷òù . Bad news: No precise bound for , the minimum length of reset words for synchronizing aperiodic automata with states, has been found so far. WAW 2007, Turku, Finland, 29.03.07 – p.14/28
✠ ✁ ✏ ✎ ☞ ✍ ✌ ✁ ☛ ✁ ✠ ☛ ✁ ✁ ☞ ☞ ☛ ✡ ✞ ✁ ✂ ✄ ✠ ✝ ✟ ✠ ✡ Aperiodic Automata Good news: Recently A. Trakhtman has proved that every synchronizing aperiodic automaton with states admits a reset word of length at most ✂✆☎ . Bad news: No precise bound for , the minimum length of reset words for synchronizing aperiodic automata with states, has been found so far. (Trakhtman) WAW 2007, Turku, Finland, 29.03.07 – p.14/28
✜ ✚ ✛ ✑ ✑ ✣ ✚ ✑ ✑ ✛ ✜ ✢ ✛ ✣ ✤ ✘ ✤ ✑ ✘ ✦ ✧ ✑ ✣ ✑ ✒ ✓ ✕ ✚ ✖ ✗ ✥ ✑ ✘ ✙ ✘ ✙ Aperiodic Automata Good news: Recently A. Trakhtman has proved that every synchronizing aperiodic automaton with states admits a reset word of length at most ✒✆✔ . Bad news: No precise bound for , the minimum length of reset words for synchronizing aperiodic automata with states, has been found so far. (Trakhtman) (Ananichev) WAW 2007, Turku, Finland, 29.03.07 – p.14/28
★ ✱ ★ ✶ ★ ✲ ★ ✱ ★ ✲ ✳ ✴ ✲ ✵ ✶ ✯ ✷ ★ ✯ ✵ ✵ ★ ✳ ✹ ✩ ✪ ✬ ✱ ✭ ✮ ★ ✸ ✯ ✰ ✯ ✰ Aperiodic Automata Good news: Recently A. Trakhtman has proved that every synchronizing aperiodic automaton with states admits a reset word of length at most ✩✆✫ . Bad news: No precise bound for , the minimum length of reset words for synchronizing aperiodic automata with states, has been found so far. (Trakhtman) (Ananichev) The gap between the upper and the lower bounds is rather drastic. WAW 2007, Turku, Finland, 29.03.07 – p.14/28
✺ ✻ ✺ ✿ Aperiodic Automata Producing lower bounds for is difficult because ✼✾✽ it is quite difficult to produce aperiodic automata. WAW 2007, Turku, Finland, 29.03.07 – p.15/28
❍ ❀ ❁ ❀ ❄ ❈ ■ ● ❊ Aperiodic Automata Producing lower bounds for is difficult because ❂✾❃ it is quite difficult to produce aperiodic automata. The question of whether or not a given DFA ❅❇❆ ❉❋❊ is aperiodic is PSPACE-complete (Cho and Huynh, 1991). WAW 2007, Turku, Finland, 29.03.07 – p.15/28
❱ ❖ ❏ ❑ ❏ ❳ ◆ ❨ ❳ ❲ ❘ ◗ ❲ ❚ ❙ ❯ Aperiodic Automata Producing lower bounds for is difficult because ▲✾▼ it is quite difficult to produce aperiodic automata. The question of whether or not a given DFA ❖❇P ❘❋❙ is aperiodic is PSPACE-complete (Cho and Huynh, 1991). Practically, there is no way to check the aperiodicity of avoiding the calculation of its transition monoid, and the cardinality of the monoid can reach . WAW 2007, Turku, Finland, 29.03.07 – p.15/28
❢ ❴ ❩ ❬ ❩ ❤ ❫ ✐ ❤ ❣ ❜ ❛ ❣ ❞ ❝ ❡ Aperiodic Automata Producing lower bounds for is difficult because ❭✾❪ it is quite difficult to produce aperiodic automata. The question of whether or not a given DFA ❴❇❵ ❜❋❝ is aperiodic is PSPACE-complete (Cho and Huynh, 1991). Practically, there is no way to check the aperiodicity of avoiding the calculation of its transition monoid, and the cardinality of the monoid can reach . Hence, no hope that experiments can help. WAW 2007, Turku, Finland, 29.03.07 – p.15/28
r ✈ ❥ ❦ ❥ ✇ ♥ ① ♦ ② ① q ✇ t s ✉ Aperiodic Automata Producing lower bounds for is difficult because ❧✾♠ it is quite difficult to produce aperiodic automata. The question of whether or not a given DFA ♦❇♣ r❋s is aperiodic is PSPACE-complete (Cho and Huynh, 1991). Practically, there is no way to check the aperiodicity of avoiding the calculation of its transition monoid, and the cardinality of the monoid can reach . Hence, no hope that experiments can help. On the other hand, all attempts to reduce the upper bound have failed so far. WAW 2007, Turku, Finland, 29.03.07 – p.15/28
❻ ❺ ③ ④ ③ ❻ ⑦ ❼ ❽ ⑧ ❼ ⑩ ❶ ❸ ❷ ❹ Aperiodic Automata Producing lower bounds for is difficult because ⑤✾⑥ it is quite difficult to produce aperiodic automata. The question of whether or not a given DFA ⑧❇⑨ ❶❋❷ is aperiodic is PSPACE-complete (Cho and Huynh, 1991). Practically, there is no way to check the aperiodicity of avoiding the calculation of its transition monoid, and the cardinality of the monoid can reach . Hence, no hope that experiments can help. On the other hand, all attempts to reduce the upper bound have failed so far. The idea: consider certain properties that guarantee aperiodicity and are easier to check. WAW 2007, Turku, Finland, 29.03.07 – p.15/28
➍ ➈ ➄ ➆ ➉ ➁ ➎ ➂ ➇ ➂ ➇ ➄ ➍ ➃ ➇ ➑ ➆ ➆ ➄ ➉ ➁ ➏ ➅ ➄ ➂ ➃ ➂ ➀ ➇ ➎ Monotonicity is monotonic if A DFA admits a linear ❾❇❿ ➁❋➂ order such that, for , the transformation of preserves : ➉➋➊➌ ➍➓➒ ➆➐➏ WAW 2007, Turku, Finland, 29.03.07 – p.16/28
➤ ➟ ➛ ➝ ➠ ↔ ➥ ↕ ➞ ↕ ➞ ➛ ➤ ➙ ➞ ➧ ➝ ➝ ➛ ➠ ↔ ➦ ➜ ➛ ↕ ➙ ↕ ➣ ➞ ➥ Monotonicity is monotonic if A DFA admits a linear ➔❇→ ↔❋↕ order such that, for , the transformation of preserves : ➠➋➡➢ ➤➓➨ ➝➐➦ Monotonic automata are aperiodic (known and easy). WAW 2007, Turku, Finland, 29.03.07 – p.16/28
➷ ➶ ➯ ➺ ➷ ➶ ➷ ➘ ➵ ➽ ➲ ➷ ➻ ➪ ➺ ➵ ➽ ➹ ➲ ➻ ➪ ➻ ➲ ➯ ➭ ➵ ➳ ➲ ➳ ➼ ➻ ➵ ➺ ➷ ➸ Monotonicity is monotonic if A DFA admits a linear ➩❇➫ ➯❋➲ order such that, for , the transformation of preserves : ➽➋➾➚ ➪➓➴ ➺➐➹ Monotonic automata are aperiodic (known and easy). WAW 2007, Turku, Finland, 29.03.07 – p.16/28
✃ ❒ Ø Ö Ö ❐ Ï Ð Õ Ï Õ Ð ❐ ❮ ❮ Ò Ñ Ò Ð × Ï ❐ Ð ✃ Ý ❰ ❮ ❐ ❒ Ú ➱ Þ ❮ Monotonicity is monotonic if A DFA admits a linear ➬❇➮ ✃❋❐ order such that, for , the transformation of preserves : Ò➋ÓÔ Õ➓Ù Ï➐× Monotonic automata are aperiodic (known and easy). ÚÜÛ WAW 2007, Turku, Finland, 29.03.07 – p.16/28
ç å ï ã è í ê â ç ã í î è ð å ê å ä ã æ ö á ò ä ã å ô é â õ ô ç è è î Monotonicity is monotonic if A DFA admits a linear ß❇à â❋ã order such that, for , the transformation of preserves : ê➋ëì í➓ñ ç➐ï Monotonic automata are aperiodic (known and easy). òÜó òÜó WAW 2007, Turku, Finland, 29.03.07 – p.16/28
✞ ÿ ✝ ✎ ú � ÿ û ✟ ✟ û ✠ ý ✂ ✞ û � ✝ ÿ � ✆ ✂ ú ✒ ù ☞ ü û ý þ ✎ ✑ ✏ ✎ ÿ ✌ � ✁ ü ☞ ý ý Monotonicity is monotonic if A DFA admits a linear ÷❇ø ú❋û order such that, for , the transformation of preserves : ✂☎✄ ✝☛✡ Monotonic automata are aperiodic (known and easy). ☞✍✌ ☞✍✌ WAW 2007, Turku, Finland, 29.03.07 – p.16/28
✬ ✛ ✰ ✗ ✯ ✢ ✮ ✩ ✱ ✰ ✥ ✣ ✩ ✘ ✣ ★ ✢ ✛ ✥ ✪ ★ ✘ ✣ ✗ ✴ ✖ ✮ ✚ ✘ ✛ ✜ ✰ ✳ ✲ ✰ ✢ ✯ ✣ ✤ ✚ ✮ ✛ ✘ Monotonicity is monotonic if A DFA admits a linear ✓✕✔ ✗✙✘ order such that, for , the transformation of preserves : ✥☎✦✧ ★☛✭ ✢✫✪ Monotonic automata are aperiodic (known and easy). ✮✍✯ ✮✍✯ WAW 2007, Turku, Finland, 29.03.07 – p.16/28
✻ ❀ ✸ ❊ ✽ ❉ ❄ ❈ ❆ ❋ ❄ ❃ ✹ ✾ ❃ ✽ ✻ ❀ ❅ ✹ ● ✾ ❊ ✸ ❏ ✷ ■ ✺ ✹ ✻ ✼ ❈ ❊ ✹ ❉ ✽ ❈ ✾ ✿ ✺ ❍ ✻ ❊ ✾ Monotonicity is monotonic if A DFA admits a linear ✵✕✶ ✸✙✹ order such that, for , the transformation of preserves : ❀☎❁❂ ❃☛❇ ✽✫❅ Monotonic automata are aperiodic (known and easy). ❈✍❉ ❈✍❉ – contradiction! WAW 2007, Turku, Finland, 29.03.07 – p.16/28
▲ ❑ ▼ P Monotonicity For monotonic automata the synchronization problem is easy: Ananichev and (2004) observed that every monotonic synchronizing automaton with states has a reset word of length and the bound is tight. ▲❖◆ WAW 2007, Turku, Finland, 29.03.07 – p.17/28
❘ ◗ ❙ ❯ Monotonicity For monotonic automata the synchronization problem is easy: Ananichev and (2004) observed that every monotonic synchronizing automaton with states has a reset word of length and the bound is tight. ❘❖❚ A much more difficult case of so-called 0-monotonic automata was analyzed by Ananichev in 2005. WAW 2007, Turku, Finland, 29.03.07 – p.17/28
❞ ❛ ❡❢ ❝ ❝ ❜ ❛ ❴ ❵ ❣ ❪ ❤ ❴ ❫ ❧ ❩ ♠ ❳ ❫ ❡ ❲ ❢ ❝ ❱ ❣ ❫ Monotonicity For monotonic automata the synchronization problem is easy: Ananichev and (2004) observed that every monotonic synchronizing automaton with states has a reset word of length and the bound is tight. ❲❖❨ A much more difficult case of so-called 0-monotonic automata was analyzed by Ananichev in 2005. A DFA is 0- monotonic if it has a unique sink ❬✕❭ ❫✙❴ and admits a linear order preserved by the restrictions of the transformations to ✐☎❥❦ . WAW 2007, Turku, Finland, 29.03.07 – p.17/28
⑤ ② ⑥⑦ ④ ④ ③ ② ✇ ① ⑧ ✉ ⑨ ✇ ✈ ❸ r ❹ ♣ ✈ ⑥ ♦ ⑦ ④ ♥ ⑧ ✈ Monotonicity For monotonic automata the synchronization problem is easy: Ananichev and (2004) observed that every monotonic synchronizing automaton with states has a reset word of length and the bound is tight. ♦❖q A much more difficult case of so-called 0-monotonic automata was analyzed by Ananichev in 2005. A DFA is 0- monotonic if it has a unique sink s✕t ✈✙✇ and admits a linear order preserved by the restrictions of the transformations to ⑩☎❶❷ . Clearly, 0-monotonic automata are in a 1-1 correspondence with incomplete monotonic automata. WAW 2007, Turku, Finland, 29.03.07 – p.17/28
➂ ➅ ➈ ➇ ➇ ➆ ➅ ➃ ➄ ➋ ➁ ➌ ➃ ➂ ➐ ❾ ➑ ❼ ➂ ➉ ❻ ➊ ➇ ❺ ➋ ➉➊ Monotonicity For monotonic automata the synchronization problem is easy: Ananichev and (2004) observed that every monotonic synchronizing automaton with states has a reset word of length and the bound is tight. ❻❖❽ A much more difficult case of so-called 0-monotonic automata was analyzed by Ananichev in 2005. A DFA is 0- monotonic if it has a unique sink ❿✕➀ ➂✙➃ and admits a linear order preserved by the restrictions of the transformations to ➍☎➎➏ . Clearly, 0-monotonic automata are in a 1-1 correspondence with incomplete monotonic automata. Again, it is known and easy to check that 0-monotonic automata are aperiodic. WAW 2007, Turku, Finland, 29.03.07 – p.17/28
➒ ➓ ➓ ➒ ➓ ➓ ➓ ➓ ➒ ➒ ➒ ➓ ➒ Monotonicity 5 1 2 3 4 6 0 ➒→➔ WAW 2007, Turku, Finland, 29.03.07 – p.18/28
↔ ➣ ➙ ↔ ↔ ↔ ↔ ↔ ➣ ➣ ↔ ➣ ➣ ➛ ➙ ➝ ➢ ➧ ➭ ➣ Monotonicity 5 1 2 3 4 6 0 ➣→↕ This 0-monotonic automaton is the first in Ananichev’s ➤➦➥ ➨➫➩ series that yields the lower bound . ➜➞➝ ➟➡➠ WAW 2007, Turku, Finland, 29.03.07 – p.18/28
➚ ➲ ➘ ➾ ➻ ❒ ➯ ➵ ➸ ➵ ➱ ➷ ➲ ➲ ➯ ➲ ➲ ➴ ➲ ➲ ➲ ➯ ➯ ➯ ➯ ➯ ➬ ➯ ➹ ➶ ➮ ➾ ➶ Monotonicity 5 1 2 3 4 6 0 ➯→➳ This 0-monotonic automaton is the first in Ananichev’s ➚➦➪ ➹➫➘ series that yields the lower bound . ➺➞➻ ➼➡➽ It has 7 states and its shortest reset word is of length . ➴❐✃ WAW 2007, Turku, Finland, 29.03.07 – p.18/28
ß Ó ß Ó Ù × Õ Ó Ú Û Ó Ý Õ Þ Ø Ý Ù à Ó Ù Û Ö Õ Ó Ô Ò Ñ ß Û ❰ Ô ❮ Ú Generalized Monotonicity An equivalence relation on the state set of a DFA is a congruence if implies ×❐Ø ÚÜÛ Ï✕Ð Ò✙Ó for all and all . ×❐Ø WAW 2007, Turku, Finland, 29.03.07 – p.19/28
ò ã ð ø æ å ï ð ø é è í î ð ö ë í ô ô ç ç ï í é ò û æ ï í ú ï ô é ù ä ð ø æ è ê ö è ò æ é ò ð î í ç ä ô ê ç é è é å ë í â ç á ñ ç í ó æ ï í ç ì í ð ï î ç ò ç í ë é ç î ò ö Generalized Monotonicity An equivalence relation on the state set of a DFA is a congruence if implies ë❐ì îÜï ã✕ä æ✙ç for all and all . ë❐ì . The quotient is the -class containing the state õ÷ö is the DFA where and the function is defined by õ÷ö the rule for all and . õ÷ö õ÷ö WAW 2007, Turku, Finland, 29.03.07 – p.19/28
✂ ✁ ✡ ✝ ✓ ✆ ✂ ☎ ✂ ✌ ✕ � ✕ ✎ ✌ ✕ þ ☛ ✡ ✡ ✌ ☞ ✁ ✌ ✡ ✆ ☛ ✡ ✑ ✞ ✓ ✆ ÿ ✑ ✓ ✆ ÿ ✘ ✁ ☞ ✡ ✗ ✞ ✡ ✑ ✖ ✁ ✑ ✎ ✝ ✍ ✎ ✌ ☞ ☛ ✡ ✂ ✎ ✎ ✆ ✎ ✂ ☎ ✙ � ✙ ✙ ý ✙ ü ✆ ✂ ✎ ☎ ☎ ☞ ✎ ☞ ✁ ☞ ✡ ✂ ✟ ✌ ✎ ☞ ✏ ☛ ✎ ✂ ✡ ✞ ✆ ✂ ☛ ✂ Generalized Monotonicity An equivalence relation on the state set of a DFA is a congruence if implies ✞✠✟ þ✕ÿ ✁✄✂ for all and all . ✞✠✟ . The quotient is the -class containing the state ✒✔✓ is the DFA where and the function is defined by ✒✔✓ the rule for all and . ✒✔✓ ✒✔✓ 3 4 1 2 WAW 2007, Turku, Finland, 29.03.07 – p.19/28
✧ ✥ ✩ ✶ ★ ✦ ✬ ✦ ✰ ✷ ✤ ✥ ✦ ✰ ✷ ✜ ✲ ✬ ✭ ✰ ✬ ✪ ✷ ✬ ✴ ✬ ✴ ✪ ✶ ★ ✭ ✢ ✶ ★ ✺ ✰ ✥ ✮ ✬ ✹ ★ ✬ ✴ ✸ ✢ ✮ ✴ ✦ ✩ ★ ✱ ✲ ✰ ✲ ✬ ✦ ✲ ✲ ★ ✦ ✦ ✧ ✻ ✤ ✻ ✻ ✛ ✻ ✚ ✧ ✲ ✥ ✮ ✧ ✮ ✲ ✲ ✥ ✮ ✬ ✦ ✫ ✰ ✭ ✮ ✳ ✭ ✲ ✦ ✬ ✪ ★ ✦ ✲ Generalized Monotonicity An equivalence relation on the state set of a DFA is a congruence if implies ✪✠✫ ✭✯✮ ✜✣✢ ✥✄✦ for all and all . ✪✠✫ . The quotient is the -class containing the state ✵✔✶ is the DFA where and the function is defined by ✵✔✶ the rule for all and . ✵✔✶ ✵✔✶ 3 4 1 2 WAW 2007, Turku, Finland, 29.03.07 – p.19/28
▲ ❋ ▼ ❁ ❉ ❅ ❈ ❄ ❂ ❃ ❂ ▼ ✿ ❁ ❀ ❊ ❋ ▼ ✾ ❁ ❈ ❍ ❋ ◆ ❊ ❆ ❉ ❍ ❂ ❄ ❈ ❏ ❆ ▲ ❄ ❈ ❏ ▲ ❄ ❂ P ❁ ❊ ❈ ❖ ❍ ❈ ❋ ❃ ❏ ◗ ❍ ❄ ● ◗ ❋ ◗ ❈ ❂ ◗ ❅ ❍ ❄ ❂ ❃ ◗ ❀ ❍ ❍ ✽ ◗ ✼ ❂ ❉ ❈ ❂ ❏ ❍ ❃ ❊ ❍ ❍ ❁ ❊ ❈ ❇ ❂ ❍ ❋ ❊ ■ ❉ ❍ ❂ ❈ ❆ ❄ ✿ Generalized Monotonicity An equivalence relation on the state set of a DFA is a congruence if implies ❆✠❇ ❉✯❊ ✾✣✿ ❁✄❂ for all and all . ❆✠❇ . The quotient is the -class containing the state ❑✔▲ is the DFA where and the function is defined by ❑✔▲ the rule for all and . ❑✔▲ ❑✔▲ 3,4 3 4 1,2 1 2 WAW 2007, Turku, Finland, 29.03.07 – p.19/28
❘ ❯ ❳ ❲ ❨ ❩ Generalized Monotonicity Let be a congruence on a DFA . ❙✣❚ ❱✄❲ WAW 2007, Turku, Finland, 29.03.07 – p.20/28
❝ ❜ ❬ ❴ ❡ ❫ ❞ ❛ ❵ Generalized Monotonicity Let be a congruence on a DFA . The ❭✣❪ ❴✄❵ - monotonic if there exists a (partial) automaton is order on the set such that: WAW 2007, Turku, Finland, 29.03.07 – p.20/28
♣ ❥ ❢ ♦ ❥ ✐ q ❧ ❦ ♠ ♥ ❥ ♦ ♦ ♣ Generalized Monotonicity Let be a congruence on a DFA . The ❣✣❤ ❥✄❦ - monotonic if there exists a (partial) automaton is order on the set such that: 1) the order is contained in (as a subset of ) and its restriction to any -class is a linear order; WAW 2007, Turku, Finland, 29.03.07 – p.20/28
② ④ ④ ✈ ⑤ ⑤ ✇ ✈ ✈ ⑤ ④ ❻ ✈ ✈ ⑦ ③ ② ✇ ① ⑧ ✉ ① ⑦ r ⑥ Generalized Monotonicity Let be a congruence on a DFA . The s✣t ✈✄✇ - monotonic if there exists a (partial) automaton is order on the set such that: 1) the order is contained in (as a subset of ) and its restriction to any -class is a linear order; 2) for each , the transformation ⑨❶⑩❷ ❸❺❹ preserves . WAW 2007, Turku, Finland, 29.03.07 – p.20/28
➀ ➀ ➀ ➈ ➅ ➉ ➆ ➂ ➃ ➀ ➁ ➆ ➈ ➅ ➅ ➏ ➀ ➄ ➃ ➁ ➂ ➆ ❿ ➅ ➅ ❼ ➇ Generalized Monotonicity Let be a congruence on a DFA . The ❽✣❾ ➀✄➁ - monotonic if there exists a (partial) automaton is order on the set such that: 1) the order is contained in (as a subset of ) and its restriction to any -class is a linear order; 2) for each , the transformation ➊❶➋➌ ➍❺➎ preserves . Clearly, for being universal, -monotonic automata are precisely monotonic automata. WAW 2007, Turku, Finland, 29.03.07 – p.20/28
➙ ➥ ➔ ➜ ➔ ➣ ➙ ↔ ➛ → ➝ ➔ ➔ ➛ ➔ ➝ ➙ ➛ ➙ ↕ ↔ → ➣ ➙ ➓ ➙ ➙ ➐ ➞ Generalized Monotonicity Let be a congruence on a DFA . The ➑✣➒ ➔✄→ - monotonic if there exists a (partial) automaton is order on the set such that: 1) the order is contained in (as a subset of ) and its restriction to any -class is a linear order; 2) for each , the transformation ➟❶➠➡ ➢❺➤ preserves . Clearly, for being universal, -monotonic automata are precisely monotonic automata. On the other hand, for being the equality, every DFA is -monotonic. WAW 2007, Turku, Finland, 29.03.07 – p.20/28
➳ ➨ ➻ ➨ ➺ ➦ ➺ ➨ ➨ ➭ ➩ ➨ ➪ ➵➶ ➫ ➨ ➯ ➯ ➶ ➭ ➨ ➫ ➩ ➨ ➶ ➵ ➧ ➧ ➦ ➭ Generalized Monotonicity generalized monotonic of level We call a DFA if it has a strictly increasing chain of congruences ➫➲➯ ➳➸➵ in which is the equality, is universal, and ➻➽➼ is -monotonic for each . ➾❺➚ ➻➽➼ WAW 2007, Turku, Finland, 29.03.07 – p.21/28
❐Ò ➮ ➴ ✃ ➹ ❒ ➴ ➘ ❐ Ñ ➴ ❮ ❒ ➴ Ò ➮ Ò ➷ ➴ ➴ ➬ ➹ ➴ ➬ ➘ ➱ ➱ ➴ ➮ ➷ Generalized Monotonicity generalized monotonic of level We call a DFA if it has a strictly increasing chain of congruences ➬➲➱ ✃➸❐ in which is the equality, is universal, and ❮➽❰ is -monotonic for each . Ï❺Ð ❮➽❰ Monotonic automata are precisely generalized monotonic automata of level 1. WAW 2007, Turku, Finland, 29.03.07 – p.21/28
Ó Õ Ý Õ Ü Ü Õ Ú Õ Ø Ö Õ á Ûâ × Õ Ù Ù â Ø Õ × Ö Õ â Û Ô Ô Ó Ø Generalized Monotonicity generalized monotonic of level We call a DFA if it has a strictly increasing chain of congruences ×➲Ù Ú➸Û in which is the equality, is universal, and Ý➽Þ is -monotonic for each . ß❺à Ý➽Þ Monotonic automata are precisely generalized monotonic automata of level 1. The automaton in the example two slides ago is a generalized monotonic automaton of level 2. WAW 2007, Turku, Finland, 29.03.07 – p.21/28
ä æ ä ã ã ã ã ã ã ä ä ä ä å Generalized Monotonicity 3,4 3 4 1,2 1 2 WAW 2007, Turku, Finland, 29.03.07 – p.22/28
è é ê ë ð ñ ç ç è î ê ê è ê è è è ç ç ç ç ò é ê ì Generalized Monotonicity 3,4 3 4 1,2 1 2 Endowing with the order such that and íïî , we see that the automaton is -monotonic. WAW 2007, Turku, Finland, 29.03.07 – p.22/28
û ÿ ÷ ú ø ö ✂ û ö ö ô ü ú ö ý õ õ ö ✄ ó ÷ ô ✂ ý � ó ó ó ó ô ó ô ô õ ö ü ÿ ô þ Generalized Monotonicity 3,4 3 4 1,2 1 2 Endowing with the order such that and ùïú , we see that the automaton is -monotonic. If we order by letting , the quotient ù ✁� automaton becomes monotonic. WAW 2007, Turku, Finland, 29.03.07 – p.22/28
☞ ✞ ✟ ☞ ✠ ✞ ✏ ✡ ☛ ✞ ✆ ✝ ✌ ☛ ✞ ✍ ✎ ✝ ✒ ☎ ✟ ✆ ✒ ✍ ✑ ☎ ☎ ☎ ☎ ✆ ☎ ✆ ✆ ✝ ✞ ✌ ✏ ✆ ✞ Generalized Monotonicity 3,4 3 4 1,2 1 2 Endowing with the order such that and , we see that the automaton is -monotonic. If we order by letting , the quotient ✡✁✑ ☛✔✓ automaton becomes monotonic. It can be shown that the automaton is not monotonic. WAW 2007, Turku, Finland, 29.03.07 – p.22/28
✕ ✖ ✙ ✘✚ ✚ ✚ Generalized Monotonicity 1. The hierarchy of generalized monotonic automata is strict: there are automata of each level . ✗✁✘ WAW 2007, Turku, Finland, 29.03.07 – p.23/28
✛ ✜ ✤ ✣✥ ✥ ✥ Generalized Monotonicity 1. The hierarchy of generalized monotonic automata is strict: there are automata of each level . ✢✁✣ 2. Every generalized monotonic automaton is aperiodic. WAW 2007, Turku, Finland, 29.03.07 – p.23/28
✦ ✧ ✪ ✩✫ ✫ ✫ Generalized Monotonicity 1. The hierarchy of generalized monotonic automata is strict: there are automata of each level . ★✁✩ 2. Every generalized monotonic automaton is aperiodic. 3. Every star-free language can be recognized by a generalized monotonic automaton. WAW 2007, Turku, Finland, 29.03.07 – p.23/28
✶ ✵ ✷ ✬ ✭ ✮ ✰ ✯✱ ✱ ✱ Generalized Monotonicity 1. The hierarchy of generalized monotonic automata is strict: there are automata of each level . ✮✁✯ 2. Every generalized monotonic automaton is aperiodic. 3. Every star-free language can be recognized by a generalized monotonic automaton. However, generalized monotonic automata are not representative for the class from the ✲✴✳ synchronization point of view: Ananichev and (2005) proved that every generalized monotonic synchronizing automaton with states has a reset word of length . ✶✹✸ WAW 2007, Turku, Finland, 29.03.07 – p.23/28
Yet Another Generalization Surprisingly, a slight relaxation of the definition of a generalized monotonic automaton gives a much larger class of automata that strictly includes . ✺✴✻ WAW 2007, Turku, Finland, 29.03.07 – p.24/28
❊ ❆ ❀ ❃ ❉ ❅ ✿ ❁ ❃ ❇ ❃ ❃ ❈ ❃ ❉ ❊ ❄ ❍ ❀ ❋ ❊ ● ❀❈ ❈ ❀❈ ❈ ❂ ❇ ❃ ❁ ■ ❃ ✾ ✿ ❀ ❁ ■ ❇ ❃ ❀ ❄ ❅ ✿ ❃ ❃ ❆ ❀ ❃ ❈ Yet Another Generalization Surprisingly, a slight relaxation of the definition of a generalized monotonic automaton gives a much larger class of automata that strictly includes . ✼✴✽ is connected if for every A finite poset there exist such that , , and for each either or . ■❑❏ ■❑❏ WAW 2007, Turku, Finland, 29.03.07 – p.24/28
❨ ❩ ❯ ❖ ◗ ❙ ❩ ❙ ❱ ❫ ❙ ❙ ❚ ❙ ❫ ❙ ❬ ❩ ❭ P❳ ❳ ❳ P ❪ ◗ ❨ P ❲ ❙ ❘ ◗ P ❖ ◆ ◆ ❖ P ◗ ❘ ❲ P ❳ ❚ ❯ ❖ ❙ ❙ ❱ P ❙ ❲ P❳ ❳ ❙ Yet Another Generalization Surprisingly, a slight relaxation of the definition of a generalized monotonic automaton gives a much larger class of automata that strictly includes . ▲✴▼ is connected if for every A finite poset there exist such that , , and for each either or . ❫❑❴ ❫❑❴ This simply means that the Hasse diagram of is connected as a graph – one can walk from each point to each other via alternating uphill and downhill segments WAW 2007, Turku, Finland, 29.03.07 – p.24/28
♥ ❣ ✐ ❝ ❡ ❣ ♥ ❣ ❥ r ❣ ♠ ❤ ❣ r ❣ ♦ ♥ ♣ ❞❧ ❧ ❧ ❞ q ❡ ♠ ❞ ❦ ❣ ❢ ❡ ❞ ❝ ❜ ❜ ❝ ❞ ❡ ❢ ❦ ❞ ❧ ❤ ✐ ❝ ❣ ❣ ❥ ❞ ❣ ❦ ❞❧ ❧ ❣ Yet Another Generalization Surprisingly, a slight relaxation of the definition of a generalized monotonic automaton gives a much larger class of automata that strictly includes . ❵✴❛ is connected if for every A finite poset there exist such that , , and for each either or . r❑s r❑s This simply means that the Hasse diagram of is connected as a graph – one can walk from each point to each other via alternating uphill and downhill segments (like in Turku). Yliopistonmäki Karhumäki WAW 2007, Turku, Finland, 29.03.07 – p.24/28
✉ t ② ① ⑨ ⑤ ③ ⑥ ⑦ ⑧ Yet Another Generalization Let be a congruence on a DFA . We ✉✇✈ ②④③ is partially - monotonic if there exists a say that partial order on such that: WAW 2007, Turku, Finland, 29.03.07 – p.25/28
❹ ❿ ⑩ ❿ ➀ ❸ ❹ ❻ ❺ ❼ ❽ ❹ ❾ ❶ ❾ ❾ Yet Another Generalization Let be a congruence on a DFA . We ❶✇❷ ❹④❺ is partially - monotonic if there exists a say that partial order on such that: 1) the order is contained in (as a subset of ) and its restriction to any -class is connected; WAW 2007, Turku, Finland, 29.03.07 – p.25/28
➊ ➆ ➅ ➋ ➊ ➍ ➅ ➎ ➋ ➇ ➈ ➊ ➂ ➅ ➍ ➅ ➉ ➈ ➆ ➇ ➣ ➄ ➅ ➋ ➁ ➌ Yet Another Generalization Let be a congruence on a DFA . We ➂✇➃ ➅④➆ is partially - monotonic if there exists a say that partial order on such that: 1) the order is contained in (as a subset of ) and its restriction to any -class is connected; 2) for each , the transformation ➏➑➐➒ ➓→➔ preserves . WAW 2007, Turku, Finland, 29.03.07 – p.25/28
➽ ➲ ➘ ➸ ➡ ➵ ➳ ➡ ➘ ➘ ➺ ➝ ↕ ➲ ➢ ➜ ➯ ➜ ➺ ➵ ➝ ↕ ➾ ➡ ➸ ➸ ➶ ➡ ➽ ➙ ➡ ➼ ➡ ➹ ➳ ➡ ➝ ➝ ➼ ➡ ➥ ➜ ➡ ➡ ➟ ➟ ➤ ➜ ➝ ➞ ➜ ➢ ➛ ➢ ➥ ➡ ➦ ➞ ↕ ↔ ➠ Yet Another Generalization Let be a congruence on a DFA . We ↕✇➙ ➜④➝ is partially - monotonic if there exists a say that partial order on such that: 1) the order is contained in (as a subset of ) and its restriction to any -class is connected; 2) for each , the transformation ➧➑➨➩ ➫→➭ preserves . generalized partially monotonic of We call a DFA level if it has a chain of congruences ➵➻➺ in which is the equality, is universal, and ➾➪➚ is partially -monotonic for each . ➾➪➚ WAW 2007, Turku, Finland, 29.03.07 – p.25/28
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