Synchronizing Automata Preserving a Chain of Partial Orders

Synchronizing Automata Preserving a Chain of Partial Orders M. V. Volkov Ural State University, Ekaterinburg, Russia CIAA 2007, Prague, Finland, 16.07.07 p.1/25

Synchronizing automata We consider DFA:. CIAA 2007, Prague, Finland, 16.07.07 p.2/25

Synchronizing automata We consider DFA: The DFA is called synchronizing if there exists a word whose action resets, that is, leaves the automaton in one particular state no matter which state in it started at.. CIAA 2007, Prague, Finland, 16.07.07 p.2/25

) ( $ " & # #!! Synchronizing automata We consider DFA: The DFA is called synchronizing if there exists a word whose action resets, that is, leaves the automaton in one particular state no matter which state in it started at.. Here stands for.. %'& CIAA 2007, Prague, Finland, 16.07.07 p.2/25

10 /. * * 4 / 3 2 - = -3 < 0 9 5- - 7 8 8 2 6 6 2 Synchronizing automata We consider DFA:,-. * + The DFA is called synchronizing if there exists a word whose action resets, that is, leaves the automaton in one particular state no matter which state in it started at.. Here stands for. 5 +. :';. 5 ; Any word with this property is said to be a reset word for the automaton. CIAA 2007, Prague, Finland, 16.07.07 p.2/25

?? >? Synchronizing automata 1 2 4 >? 3 > > CIAA 2007, Prague, Finland, 16.07.07 p.3/25

A A @ A A A A @ A A A @ @ Synchronizing automata 1 2 4 @ A 3 @ @ A reset sequence of actions is. Applying it at any state brings the automaton to the state 2. CIAA 2007, Prague, Finland, 16.07.07 p.3/25

Synchronizing Automata The notion was formalized in 1964 in a paper by Jan Černý (Poznámka k homogénnym eksperimentom s konecnými automatami, Mat.-Fyz. Cas. Slovensk. Akad. Vied. 14 (1964) 208 216) though implicitly it had been studied since 1956. The idea of synchronization is pretty natural and of obvious importance: we aim to restore control over a device whose current state is not known. CIAA 2007, Prague, Finland, 16.07.07 p.4/25

Engineering Applications In the 80s, the notion was reinvented by engineers working in robotics or, more precisely, robotic manipulation which deals with part handling problems in industrial automation such as part feeding, fixturing, loading, assembly and packing (and which is therefore of utmost and direct practical importance). Suppose that one of the parts of a certain device has the following shape: CIAA 2007, Prague, Finland, 16.07.07 p.5/25

Engineering Applications Assume that only four initial orientations of the part shown above are possible, namely, the following ones: CIAA 2007, Prague, Finland, 16.07.07 p.6/25

Engineering Applications Assume that only four initial orientations of the part shown above are possible, namely, the following ones: Suppose that prior the assembly the part should take the bump-left orientation (the second one on the picture). Thus, one has to construct an orienter which action will put the part in the prescribed position independently of its initial orientation. CIAA 2007, Prague, Finland, 16.07.07 p.6/25

Engineering Applications We put parts to be oriented on a conveyer belt which takes them to the assembly point and let the stream of the details encounter a series of passive obstacles of two types (high and low) placed along the belt. CIAA 2007, Prague, Finland, 16.07.07 p.7/25

B B Engineering Applications We put parts to be oriented on a conveyer belt which takes them to the assembly point and let the stream of the details encounter a series of passive obstacles of two types (high and low) placed along the belt. A high obstacle is high enough so that any part on the belt encounters this obstacle by its rightmost low angle. CIAA 2007, Prague, Finland, 16.07.07 p.7/25

C C C C F ED Engineering Applications We put parts to be oriented on a conveyer belt which takes them to the assembly point and let the stream of the details encounter a series of passive obstacles of two types (high and low) placed along the belt. A high obstacle is high enough so that any part on the belt encounters this obstacle by its rightmost low angle. Being curried by the belt, the part then is forced to turn clockwise. CIAA 2007, Prague, Finland, 16.07.07 p.7/25

G G G G G G J IH Engineering Applications We put parts to be oriented on a conveyer belt which takes them to the assembly point and let the stream of the details encounter a series of passive obstacles of two types (high and low) placed along the belt. A high obstacle is high enough so that any part on the belt encounters this obstacle by its rightmost low angle. Being curried by the belt, the part then is forced to turn clockwise. CIAA 2007, Prague, Finland, 16.07.07 p.7/25

Engineering Applications A low obstacle has the same effect whenever the part is in the bump-down orientation; otherwise it does not touch the part which therefore passes by without changing the orientation. The following schema summarizes how the obstacles effect the orientation of the part in question: low HIGH low HIGH HIGH low HIGH low CIAA 2007, Prague, Finland, 16.07.07 p.8/25

Engineering Applications We met this picture a few slides ago: CIAA 2007, Prague, Finland, 16.07.07 p.9/25

L L K L L L L K L L L K K Engineering Applications We met this picture a few slides ago: 1 2 4 K L 3 K K this was our example of a synchronizing automaton, and we saw that is a reset sequence of actions. CIAA 2007, Prague, Finland, 16.07.07 p.9/25

N N M N N N N M N N N M M Engineering Applications We met this picture a few slides ago: 1 2 4 M N 3 M M this was our example of a synchronizing automaton, and we saw that is a reset sequence of actions. Hence the series of obstacles low-high-high-high-low-high-high-high-low yields the desired sensorless orienter. CIAA 2007, Prague, Finland, 16.07.07 p.9/25

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). CIAA 2007, Prague, Finland, 16.07.07 p.10/25

UT SR V 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. OQP CIAA 2007, Prague, Finland, 16.07.07 p.10/25

\[ ZY ] 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. WQX CIAA 2007, Prague, Finland, 16.07.07 p.10/25

cb a` d 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. CIAA 2007, Prague, Finland, 16.07.07 p.10/25 ^Q_

The Černý conjecture Clearly, from the viewpoint of the above applications (as well as from the mathematical point of view) the following question is of importance: CIAA 2007, Prague, Finland, 16.07.07 p.11/25

e The Černý 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? CIAA 2007, Prague, Finland, 16.07.07 p.11/25

f The Černý 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. CIAA 2007, Prague, Finland, 16.07.07 p.11/25

g g l kj ho l kj h i The Černý 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, Černý conjectured that every synchronizing automaton with states has a reset sequence of length as in our example where. gqi m'n CIAA 2007, Prague, Finland, 16.07.07 p.11/25

p p u ts qx u ts q r The Černý 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, Černý conjectured that every synchronizing automaton with states has a reset sequence of length as in our example where. pqr v'w The simply looking conjecture is still open in general!! CIAA 2007, Prague, Finland, 16.07.07 p.11/25

The Černý conjecture The best upper bound known so far is (J.-E. Pin, 1983). z ~} y{z CIAA 2007, Prague, Finland, 16.07.07 p.12/25

The Černý conjecture The best upper bound known so far is (J.-E. Pin, 1983). It is also known that the problem is hard from the computational complexity point of view. ˆ ƒ ~ {ƒ CIAA 2007, Prague, Finland, 16.07.07 p.12/25

The Černý conjecture The best upper bound known so far is (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. Goralčik and V. Koubek, 1995; A. Salomaa, 2003). Ž Š ~Œ {Š CIAA 2007, Prague, Finland, 16.07.07 p.12/25

š œ š š š The Černý conjecture The best upper bound known so far is (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. Goralčik 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). ~ { CIAA 2007, Prague, Finland, 16.07.07 p.12/25

Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. CIAA 2007, Prague, Finland, 16.07.07 p.13/25

Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In particular, consider the class of aperiodic automata. Ÿž CIAA 2007, Prague, Finland, 16.07.07 p.13/25

³ ² Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In particular, consider the class of aperiodic automata. Recall that the transition monoid of a DFA consists of all transformations induced by words. ª «± Ÿ CIAA 2007, Prague, Finland, 16.07.07 p.13/25

¾½ ¼» Ç ¼ Æ Â ½ º Å Â Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In particular, consider the class of aperiodic 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. ÀÁ» ¹º» º Ã±Ä µÿ CIAA 2007, Prague, Finland, 16.07.07 p.13/25

ÑÐ Ï Î Ú Ï Ù Õ Ð Í Ø Õ Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In particular, consider the class of aperiodic 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. A DFA is called aperiodic (or counter-free) if its transition monoid is aperiodic. Ê Ë Ò ÓÔ Î ÌÍ Î Í Ö± ÈŸÉ CIAA 2007, Prague, Finland, 16.07.07 p.13/25

äã â á í â ì è ã à ë è Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. In particular, consider the class of aperiodic 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. A DFA is called aperiodic (or counter-free) if its transition monoid is aperiodic. Synchronization issues remain difficult when restricted to Ý Þ å æç á ßà á. ÛŸÜ à é±ê ÛŸÜ CIAA 2007, Prague, Finland, 16.07.07 p.13/25

î ó ð ï Aperiodic Automata Good news: Recently A. Trakhtman has proved that every synchronizing aperiodic automaton with states admits a reset word of length at most. ô õ ïòñ CIAA 2007, Prague, Finland, 16.07.07 p.14/25

ö ú ø ö ÿý þ ý 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. û ü òù Bad news: No precise bound for, the minimum length of reset words for synchronizing aperiodic automata with states, has been found so far. CIAA 2007, Prague, Finland, 16.07.07 p.14/25

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) CIAA 2007, Prague, Finland, 16.07.07 p.14/25

! 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) CIAA 2007, Prague, Finland, 16.07.07 p.14/25

# (' % $ ) - #,* + * # 3 2 # 1 - #,* + * 1 -/. #, # 0 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) 0. 4 # 0 (Ananichev) The gap between the upper and the lower bounds is rather drastic. CIAA 2007, Prague, Finland, 16.07.07 p.14/25

5 6 5 Aperiodic Automata Producing lower bounds for is difficult because it is quite difficult to produce aperiodic automata. : 798 CIAA 2007, Prague, Finland, 16.07.07 p.15/25

; < ; IH G E 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).? =9> @BA CDFE CIAA 2007, Prague, Finland, 16.07.07 p.15/25

J K J VU T S O 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. X XY W WR N L9M OBP QRFS CIAA 2007, Prague, Finland, 16.07.07 p.15/25

Z [ Z fe d c _ 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. h hi g gb ^ \9] _B` abfc CIAA 2007, Prague, Finland, 16.07.07 p.15/25

j k j vu t s o 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. On the other hand, all attempts to reduce the upper bound have failed so far. x xy w wr n l9m obp qrfs CIAA 2007, Prague, Finland, 16.07.07 p.15/25

z { z ƒ 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. On the other hand, all attempts to reduce the upper bound have failed so far. ˆ ˆ ~ 9} B Fƒ An idea: consider certain properties that guarantee aperiodicity and are easier to check. CIAA 2007, Prague, Finland, 16.07.07 p.15/25

Ž š š Ž Monotonicity A DFA order Ž is monotonic if admits a linear such that, for, the transformation of preserves : Œ FŽ ŠB Ÿž Ž œ CIAA 2007, Prague, Finland, 16.07.07 p.16/25

ª «± Monotonicity A DFA order «is monotonic if admits a linear such that, for, the transformation of preserves : F B Ÿ² «œ Monotonic automata are aperiodic (known and easy). CIAA 2007, Prague, Finland, 16.07.07 p.16/25

º ¹ ½ ¼» Á ¹» ¼ ¹» Á ¼ Â ¾ ¹ Ä ¼ Â Æ Æ Æ Æ Æ Monotonicity A DFA order ¾ À is monotonic if admits a linear such that, for, the transformation of preserves : µ F ³B ÁŸÅ ¾ Ã»œÃ Monotonic automata are aperiodic (known and easy). CIAA 2007, Prague, Finland, 16.07.07 p.16/25

ÎÍ Ì Ê Ë Ì Ñ Ð Ï Õ Í Ê Ï Ð Í Ï Õ Ð Ö Ò Í Ø Ð Ö Ë Ú Þ Monotonicity A DFA order Ò ÓÔ Ë is monotonic if admits a linear such that, for, the transformation of preserves : ÉÊFË ÇBÈ ÕŸÙ Ò Ë Ïœ Monotonic automata are aperiodic (known and easy). Ý ÚÜÛ CIAA 2007, Prague, Finland, 16.07.07 p.16/25

æå ä â ã ä é è ç í å â ç è å ç í è î ê å ð è î ã õ ò ö Monotonicity A DFA order ê ëì ã is monotonic if admits a linear such that, for, the transformation of preserves : áâfã ßBà íÿñ ê ï ã çœï Monotonic automata are aperiodic (known and easy). ô òüó ô òüó CIAA 2007, Prague, Finland, 16.07.07 p.16/25

þý ü ú û ü ÿ ý ú ÿ ý ÿ ý ÿ û û Monotonicity A DFA order û is monotonic if admits a linear such that, for, the transformation of preserves : ùúfû Bø Monotonic automata are aperiodic (known and easy). CIAA 2007, Prague, Finland, 16.07.07 p.16/25

# # $ ' $,. - ) / Monotonicity A DFA order!" is monotonic if admits a linear such that, for, the transformation of preserves : #( % &% Monotonic automata are aperiodic (known and easy). + )* + ) * + ) * + )* CIAA 2007, Prague, Finland, 16.07.07 p.16/25

76 5 3 4 5 : 9 8 > 6 3 8 9 6 8 > 9? ; 6 A 9? 4 F H G I C J Monotonicity A DFA order ; <= 4 is monotonic if admits a linear such that, for, the transformation of preserves : 234 01 >B ; @ 4 8&@ Monotonic automata are aperiodic (known and easy). E CD E CD E C D contradiction! E C D CIAA 2007, Prague, Finland, 16.07.07 p.16/25

K L P M 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. LON CIAA 2007, Prague, Finland, 16.07.07 p.17/25

Q R U S 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. ROT A much more difficult case of so-called 0-monotonic automata was analyzed by Ananichev in 2005. CIAA 2007, Prague, Finland, 16.07.07 p.17/25

V W Z X ba ` _ g fe ^ h ^d c c m a l g fe ^ c 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. WOY 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. [\ ]^_ i jk _ CIAA 2007, Prague, Finland, 16.07.07 p.17/25

n o r p zy x w ~} v v { { y ~} v { 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. ooq 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. Clearly, 0-monotonic automata are in a 1-1 correspondence with incomplete monotonic automata. st uvw ƒ w CIAA 2007, Prague, Finland, 16.07.07 p.17/25

Š ˆ Ž Ž œ Ž 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. O 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. 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. Œ Ž š CIAA 2007, Prague, Finland, 16.07.07 p.17/25

ž Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ ž ž ž ž ž Ÿ Monotonicity 1 2 3 4 5 6 0 ž CIAA 2007, Prague, Finland, 16.07.07 p.18/25

µ Monotonicity 1 2 3 4 5 6 0 This 0-monotonic automaton is the first in Ananichev s series that yields the lower bound. ª «² ³ ± CIAA 2007, Prague, Finland, 16.07.07 p.18/25

¹ º ¹ Å Ç Æ É À È Ì Â Monotonicity 1 2 3 4 5 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. Ã Ä ÅËÊ ¼ ½ ¾» ¼ Ã Ä À Á Â CIAA 2007, Prague, Finland, 16.07.07 p.18/25

Î Í ÖÕ Ô Ý Ù Ó Õ Þ Ô Û ß Ò Û Ù Ú ß Ù Õ Ú ß Ý Ó Ó Generalized Monotonicity A binary relation on the state set is a stable if for all ÏÐ ËØ Ó ÑÒÓ à Û ËØ Ó Ø Ó ÚÜÛ of a DFA implies and. CIAA 2007, Prague, Finland, 16.07.07 p.19/25

â á êé è ð í ç î ò í ë é î ò é ñ è ï ò æ ï í ð ç ç ô õ í í ô Generalized Monotonicity A binary relation on the state set of a DFA is a stable if implies for all and. A congruence is a stable equivalence. For being a congruence, is the -class containing the state ãä ëëì ç åæç ó ï öø ëëì ç ì ç îüï. CIAA 2007, Prague, Finland, 16.07.07 p.19/25

ú ù þ ÿ ÿ Generalized Monotonicity A binary relation on the state set of a DFA is a stable if implies for all and. A congruence is a stable equivalence. For being a congruence, is the -class containing the state ûü ÿ ýþÿ ÿ ÿ ÿ. 3 4 1 2 CIAA 2007, Prague, Finland, 16.07.07 p.19/25

%! " '! " ' & # ' #! % ) *!! ) ' ' - - ' ' ) - - Generalized Monotonicity A binary relation on the state set of a DFA is a stable if implies for all and. A congruence is a stable equivalence. For being a congruence, is the -class containing the state ( # +, "$#. 3 4 1 2 CIAA 2007, Prague, Finland, 16.07.07 p.19/25

/. 76 5 = : 4 3 < : ;? : 8 6 ;? 6 > 5 <? = 4 4 A B : : A E 23 E 0 5 A A 4 4 H B F E 3 : A1? : 8 6 B B 8? : 4?? I I?? A I I Generalized Monotonicity A binary relation on the state set of a DFA is a stable if implies for all and. A congruence is a stable equivalence. For being a congruence, is the -class containing the state The quotient is the DFA where and the function is defined by the rule. 01 8 9 4 234 @ < CD 3 < : G CD 6 D CD4 ; 1 8 9 4 9 4 ;$< ;CD 7 6 D 6 D. 3 4 1 2 CIAA 2007, Prague, Finland, 16.07.07 p.19/25

K J SR Q Y V P O X V W [ V T R W [ R Z Q X [ Y P P ] ^ V V ] a NO a L Q ] ] P P d ^b a O V ]M [ V T R ^ ^T [ V P [ [ [ e [ e e [ [ ] e e e Generalized Monotonicity A binary relation on the state set of a DFA is a stable if implies for all and. A congruence is a stable equivalence. For being a congruence, is the -class containing the state The quotient is the DFA where and the function is defined by the rule. LM T U P NOP \ X _` O X V c _` R ` 3 _`P W M 4 T U P U P W$X W_` 3,4 S R ` R `. 1 2 1,2 CIAA 2007, Prague, Finland, 16.07.07 p.19/25

f g j j Generalized Monotonicity We call a DFA generalized monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: hi hk jml l l hnpo CIAA 2007, Prague, Finland, 16.07.07 p.20/25

q r u u z Generalized Monotonicity We call a DFA generalized monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality; st st sv umw w w sxpy CIAA 2007, Prague, Finland, 16.07.07 p.20/25

{ Ž Ž Generalized Monotonicity We call a DFA generalized monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality; for each, the congruence generated by is contained in and the relation is a linear order on each -class; }~ } }~ } Š Œ m } pƒ ƒ ˆ ƒ } Œ } Š Š Œ Š Œ CIAA 2007, Prague, Finland, 16.07.07 p.20/25

Generalized Monotonicity We call a DFA generalized monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality; for each, the congruence generated by is contained in and the relation is a linear order on each -class; is the universal relation. ž ž Ÿ m š p œ œ œ ž Ÿ ž ž ž Ÿ ž Ÿ CIAA 2007, Prague, Finland, 16.07.07 p.20/25

ª ª ² ² ª Generalized Monotonicity We call a DFA generalized monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality; for each, the congruence generated by is contained in and the relation is a linear order on each -class; is the universal relation. Monotonic automata are precisely generalized monotonic automata of level 1. ± m «p ± ± ± CIAA 2007, Prague, Finland, 16.07.07 p.20/25

³ ¼ ¼ Ä Ä ¼ Generalized Monotonicity We call a DFA generalized monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality; for each, the congruence generated by is contained in and the relation is a linear order on each -class; is the universal relation. Monotonic automata are precisely generalized monotonic automata of level 1. The automaton in the previous example is a generalized monotonic automaton of level 2. µ µâ µ µ ÁÂ Ã Áº m¹ ¹ ¹ ½ ¾ µºp»»»à À À µâ Ã µâ ÁÂ ÁÂ Ã ÁÂ Ã CIAA 2007, Prague, Finland, 16.07.07 p.20/25

Å Å Å Æ Æ Æ Å Å Å Æ Æ Æ Generalized Monotonicity 3 4 3,4 ÇÈ 1 2 1,2 CIAA 2007, Prague, Finland, 16.07.07 p.21/25

É É É Ê Ê Ê É É É Ê Ê Ê Í Ò Ì Ô Ì Generalized Monotonicity 3 4 3,4 ËÌ 1 2 1,2 Endowing with the order such that, we get a linear order on each -class. Ó Ð Î Ì ËÌ ÏÑÐ and CIAA 2007, Prague, Finland, 16.07.07 p.21/25

Õ Õ Õ Ö Ö Ö Õ Õ Õ Ö Ö Ö Ù Ý Ø ß Ø á á à Ù ä Generalized Monotonicity 3 4 3,4 Ø 1 2 1,2 Endowing with the order such that and, we get a linear order on each -class. If we order by letting, the quotient automaton becomes monotonic. Þ Ü Ø ã Ý Ü Ûâ Ú Ø ã ß Þ â Ø ÛÑÜ CIAA 2007, Prague, Finland, 16.07.07 p.21/25

å å å æ æ æ å å å æ æ æ é í è ï è ñ ñ ð é ô Generalized Monotonicity 3 4 3,4 çè 1 2 1,2 Endowing with the order such that and, we get a linear order on each -class. If we order by letting, the quotient automaton becomes monotonic. It can be shown that the automaton is not monotonic. î ì çè ó í ì ëò ê è ó ï î ò çè ëñì CIAA 2007, Prague, Finland, 16.07.07 p.21/25

õ õ õ ö ö ö õ õ õ ö ö ö ù ý ø ÿ ø ù Generalized Monotonicity 3 4 3,4 ø 1 2 1,2 Endowing with the order such that and, we get a linear order on each -class. If we order by letting, the quotient automaton becomes monotonic. It can be shown that the automaton is not monotonic. Moreover, it cannot be emulated by any monotonic automaton. þ ü ø ý ü û ú ø ÿ þ ø ûñü CIAA 2007, Prague, Finland, 16.07.07 p.21/25

Generalized Monotonicity 1. The hierarchy of generalized monotonic automata is strict: there are automata of each level. CIAA 2007, Prague, Finland, 16.07.07 p.22/25

Generalized Monotonicity 1. The hierarchy of generalized monotonic automata is strict: there are automata of each level. 2. Every generalized monotonic automaton is aperiodic. CIAA 2007, Prague, Finland, 16.07.07 p.22/25

! " # 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. "%$ CIAA 2007, Prague, Finland, 16.07.07 p.22/25

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. & ' CIAA 2007, Prague, Finland, 16.07.07 p.23/25

* +.. 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. We call a DFA weakly monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions:,-,/.10 0 0,2 ( ) CIAA 2007, Prague, Finland, 16.07.07 p.23/25

5 6 9 9 = 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. We call a DFA weakly monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality relation; 78 78 7: 91; ; ; 7< 3 4 CIAA 2007, Prague, Finland, 16.07.07 p.23/25

@ A D D H H S S T 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. We call a DFA weakly monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality relation; for each, the congruence generated by is contained in and the relation is a (partial) order on ; BC BC BE E BPRQ D1F F F IKJ BG A M L MN N N BP E OPRQ >? E OPRQ BP E OPRQ CIAA 2007, Prague, Finland, 16.07.07 p.23/25

W X [ [ h h i e^ _ 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. We call a DFA weakly monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality relation; for each, the congruence generated by is contained in and the relation is a (partial) order on ; is the universal relation. YZ YZ Y\ \ YfRg [1] ] ] `Ka Y^ X c b cd d d Yf \ efrg U V \ efrg Yf \ efrg CIAA 2007, Prague, Finland, 16.07.07 p.23/25

l m p p t t } } ~ zs t } } 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. We call a DFA weakly monotonic of level if it has a strictly increasing chain of stable binary relations satisfying the following conditions: is the equality relation; for each, the congruence generated by is contained in and the relation is a (partial) order on ; is the universal relation. Thus, we just dropped the restriction that the order is linear on each -class. no n{ no nq q n{r q z{r p1r r r ukv ns m x w xy y y n{ q z{r z{ q z{r j k q z{r n{ q z{r CIAA 2007, Prague, Finland, 16.07.07 p.23/25

Yet Another Generalization Examples: CIAA 2007, Prague, Finland, 16.07.07 p.24/25

Yet Another Generalization Examples: every aperiodic automaton is weakly monotonic; CIAA 2007, Prague, Finland, 16.07.07 p.24/25

Yet Another Generalization Examples: every aperiodic automaton is weakly monotonic; every automaton with a unique sink state is weakly monotonic (of level 1). CIAA 2007, Prague, Finland, 16.07.07 p.24/25

Yet Another Generalization Main results: CIAA 2007, Prague, Finland, 16.07.07 p.25/25

ƒ Yet Another Generalization Main results: Every weakly monotonic automaton with a strongly connected underlying digraph is synchronizing. CIAA 2007, Prague, Finland, 16.07.07 p.25/25

Yet Another Generalization Main results: Every weakly monotonic automaton with a strongly connected underlying digraph is synchronizing. (A non-trivial generalization of the corresponding result for aperiodic automata.) Every weakly monotonic automaton with a strongly connected underlying digraph and states has a reset word of length. Ž Œ ˆŠ CIAA 2007, Prague, Finland, 16.07.07 p.25/25

Yet Another Generalization Main results: Every weakly monotonic automaton with a strongly connected underlying digraph is synchronizing. (A non-trivial generalization of the corresponding result for aperiodic automata.) Every weakly monotonic automaton with a strongly connected underlying digraph and states has a reset word of length š Š. (This upper bound is new even for the aperiodic case recall that Trakhtman s bound was 3 times higher, namely,.) œ ž CIAA 2007, Prague, Finland, 16.07.07 p.25/25