Synchronizing Automata Preserving a Chain of Partial Orders
|
|
- Madlyn Flowers
- 6 years ago
- Views:
Transcription
1 Synchronizing Automata Preserving a Chain of Partial Orders M. V. Volkov Ural State University, Ekaterinburg, Russia CIAA 2007, Prague, Finland, p.1/25
2 Synchronizing automata We consider DFA:. CIAA 2007, Prague, Finland, p.2/25
3 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, p.2/25
4 ) ( $ " & # #!! 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, p.2/25
5 10 /. * * 4 / = -3 < 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 ; Any word with this property is said to be a reset word for the automaton. CIAA 2007, Prague, Finland, p.2/25
6 ?? >? Synchronizing automata >? 3 > > CIAA 2007, Prague, Finland, p.3/25
7 A A A A A Synchronizing automata 1 2 A reset sequence of actions is. Applying it at any state brings the automaton to the state 2. CIAA 2007, Prague, Finland, p.3/25
8 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) ) though implicitly it had been studied since CIAA 2007, Prague, Finland, p.4/25
9 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) ) though implicitly it had been studied since 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, p.4/25
10 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) ) though implicitly it had been studied since 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. Think of a satellite which loops around the Moon and cannot be controlled from the Earth while behind the Moon (Černý s original motivation). CIAA 2007, Prague, Finland, p.4/25
11 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). CIAA 2007, Prague, Finland, p.5/25
12 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, p.5/25
13 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: Such parts arrive at manufacturing sites in boxes and they need to be sorted and oriented before assembly. CIAA 2007, Prague, Finland, p.5/25
14 Engineering Applications Assume that only four initial orientations of the part shown above are possible, namely, the following ones: CIAA 2007, Prague, Finland, p.6/25
15 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, p.6/25
16 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, p.7/25
17 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, p.7/25
18 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, p.7/25
19 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, p.7/25
20 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. CIAA 2007, Prague, Finland, p.8/25
21 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, p.8/25
22 Engineering Applications We met this picture a few slides ago: CIAA 2007, Prague, Finland, p.9/25
23 L L K L L L L K L L L K K Engineering Applications We met this picture a few slides ago: 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, p.9/25
24 N N M N N N N M N N N M M Engineering Applications We met this picture a few slides ago: 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, p.9/25
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) ; DNA molecule provides a computing machine with both data and fuel, Proc. National Acad. Sci. USA 100 (2003) , etc). CIAA 2007, Prague, Finland, p.10/25
26 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) ; DNA molecule provides a computing machine with both data and fuel, Proc. National Acad. Sci. USA 100 (2003) , etc). They have produced a solution containing identical DNA-based automata per l. OQP CIAA 2007, Prague, Finland, p.10/25
27 \[ 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) ; DNA molecule provides a computing machine with both data and fuel, Proc. National Acad. Sci. USA 100 (2003) , 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, p.10/25
28 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) ; DNA molecule provides a computing machine with both data and fuel, Proc. National Acad. Sci. USA 100 (2003) , 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, p.10/25 ^Q_
29 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, p.11/25
30 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, p.11/25
31 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, p.11/25
32 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, p.11/25
33 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, p.11/25
34 The Černý conjecture The best upper bound known so far is (J.-E. Pin, 1983). z ~} y{z CIAA 2007, Prague, Finland, p.12/25
35 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, p.12/25
36 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, p.12/25
37 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, p.12/25
38 Aperiodic Automata Some progress has been achieved for various restricted classes of synchronizing automata. CIAA 2007, Prague, Finland, p.13/25
39 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, p.13/25
40 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, p.13/25
41 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, p.13/25
42 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, p.13/25
43 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, p.13/25
44 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, p.14/25
45 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, p.14/25
46 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, p.14/25
47 ! 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, p.14/25
48 # (' % $ ) - #,* + * # 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, p.14/25
49 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, p.15/25
50 ; < ; 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).? CDFE CIAA 2007, Prague, Finland, p.15/25
51 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, p.15/25
52 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, p.15/25
53 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, p.15/25
54 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, p.15/25
55 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, p.16/25
56 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, p.16/25
57 º ¹ ½ ¼» Á ¹» ¼ ¹» Á ¼  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 ûœÃ Monotonic automata are aperiodic (known and easy). CIAA 2007, Prague, Finland, p.16/25
58 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, p.16/25
59 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, p.16/25
60 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, p.16/25
61 # # $ ' $,. - ) / 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, p.16/25
62 : 9 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 : >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, p.16/25
63 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, p.17/25
64 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 CIAA 2007, Prague, Finland, p.17/25
65 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 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, p.17/25
66 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 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, p.17/25
67 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 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, p.17/25
68 ž Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ ž ž ž ž ž Ÿ Monotonicity ž CIAA 2007, Prague, Finland, p.18/25
69 µ Monotonicity This 0-monotonic automaton is the first in Ananichev s series that yields the lower bound. ª «² ³ ± CIAA 2007, Prague, Finland, p.18/25
70 ¹ º ¹ Å Ç Æ É À È Ì Â Monotonicity 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, p.18/25
71 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, p.19/25
72 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, p.19/25
73 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 CIAA 2007, Prague, Finland, p.19/25
74 %! " '! " ' & # ' #! % ) *!! ) ' ' - - ' ' ) - - 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 ( # +, "$# CIAA 2007, Prague, Finland, p.19/25
75 / = : 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 < CD 3 < : G CD 6 D CD4 ; ;$< ;CD 7 6 D 6 D CIAA 2007, Prague, Finland, p.19/25
76 K J SR Q Y V P O X V W [ V T R W [ R Z Q X [ Y P P ] ^ V V ] a NO a L Q ] ] P P d ^b a O V ]M [ V T R ^ ^T [ V P [ [ [ e [ e e [ [ ] e e e Generalized Monotonicity A binary relation on the state set of a DFA is a stable if implies for all and. A congruence is a stable equivalence. For being a congruence, is the -class containing the state The quotient is the DFA where and the function is defined by the rule. LM T U P NOP \ X _` O X V c _` R ` 3 _`P W M 4 T U P U P W$X W_` 3,4 S R ` R ` ,2 CIAA 2007, Prague, Finland, p.19/25
77 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, p.20/25
78 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, p.20/25
79 { 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, p.20/25
80 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, p.20/25
81 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, p.20/25
82 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 Á à Áº m¹ ¹ ¹ ½ ¾ µºp»»»à À À µâ à µâ Á Á à Á à CIAA 2007, Prague, Finland, p.20/25
83 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, p.21/25
84 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, p.21/25
85 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, p.21/25
86 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, p.21/25
87 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, p.21/25
88 Generalized Monotonicity 1. The hierarchy of generalized monotonic automata is strict: there are automata of each level. CIAA 2007, Prague, Finland, p.22/25
89 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, p.22/25
90 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. CIAA 2007, Prague, Finland, p.22/25
91 ! " # 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, p.22/25
92 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, p.23/25
93 * +.. 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:,-,/ ,2 ( ) CIAA 2007, Prague, Finland, p.23/25
94 = 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; : 91; ; ; 7< 3 4 CIAA 2007, Prague, Finland, p.23/25
95 @ 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, p.23/25
96 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, p.23/25
97 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, p.23/25
98 Yet Another Generalization Examples: CIAA 2007, Prague, Finland, p.24/25
99 Yet Another Generalization Examples: every aperiodic automaton is weakly monotonic; CIAA 2007, Prague, Finland, p.24/25
100 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, p.24/25
101 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, p.24/25
102 Yet Another Generalization Main results: CIAA 2007, Prague, Finland, p.25/25
103 ƒ Yet Another Generalization Main results: Every weakly monotonic automaton with a strongly connected underlying digraph is synchronizing. CIAA 2007, Prague, Finland, p.25/25
104 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.) CIAA 2007, Prague, Finland, p.25/25
105 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, p.25/25
106 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, p.25/25
An Example file... log.txt
# ' ' Start of fie & %$ " 1 - : 5? ;., B - ( * * B - ( * * F I / 0. )- +, * ( ) 8 8 7 /. 6 )- +, 5 5 3 2( 7 7 +, 6 6 9( 3 5( ) 7-0 +, => - +< ( ) )- +, 7 / +, 5 9 (. 6 )- 0 * D>. C )- +, (A :, C 0 )- +,
More informationCERNY CONJECTURE FOR DFA ACCEPTING STAR-FREE LANGUAGES
CERNY CONJECTURE FOR DFA ACCEPTING STAR-FREE LANGUAGES A.N. Trahtman? Bar-Ilan University, Dep. of Math. and St., 52900, Ramat Gan, Israel ICALP, Workshop synchr. autom., Turku, Finland, 2004 Abstract.
More informationOptimal Control of PDEs
Optimal Control of PDEs Suzanne Lenhart University of Tennessee, Knoville Department of Mathematics Lecture1 p.1/36 Outline 1. Idea of diffusion PDE 2. Motivating Eample 3. Big picture of optimal control
More informationLA PRISE DE CALAIS. çoys, çoys, har - dis. çoys, dis. tons, mantz, tons, Gas. c est. à ce. C est à ce. coup, c est à ce
> ƒ? @ Z [ \ _ ' µ `. l 1 2 3 z Æ Ñ 6 = Ð l sl (~131 1606) rn % & +, l r s s, r 7 nr ss r r s s s, r s, r! " # $ s s ( ) r * s, / 0 s, r 4 r r 9;: < 10 r mnz, rz, r ns, 1 s ; j;k ns, q r s { } ~ l r mnz,
More information! " # $! % & '! , ) ( + - (. ) ( ) * + / 0 1 2 3 0 / 4 5 / 6 0 ; 8 7 < = 7 > 8 7 8 9 : Œ Š ž P P h ˆ Š ˆ Œ ˆ Š ˆ Ž Ž Ý Ü Ý Ü Ý Ž Ý ê ç è ± ¹ ¼ ¹ ä ± ¹ w ç ¹ è ¼ è Œ ¹ ± ¹ è ¹ è ä ç w ¹ ã ¼ ¹ ä ¹ ¼ ¹ ±
More informationConstructive Decision Theory
Constructive Decision Theory Joe Halpern Cornell University Joint work with Larry Blume and David Easley Economics Cornell Constructive Decision Theory p. 1/2 Savage s Approach Savage s approach to decision
More informationLoop parallelization using compiler analysis
Loop parallelization using compiler analysis Which of these loops is parallel? How can we determine this automatically using compiler analysis? Organization of a Modern Compiler Source Program Front-end
More informationRedoing the Foundations of Decision Theory
Redoing the Foundations of Decision Theory Joe Halpern Cornell University Joint work with Larry Blume and David Easley Economics Cornell Redoing the Foundations of Decision Theory p. 1/21 Decision Making:
More informationGeneral Neoclassical Closure Theory: Diagonalizing the Drift Kinetic Operator
General Neoclassical Closure Theory: Diagonalizing the Drift Kinetic Operator E. D. Held eheld@cc.usu.edu Utah State University General Neoclassical Closure Theory:Diagonalizing the Drift Kinetic Operator
More information" #$ P UTS W U X [ZY \ Z _ `a \ dfe ih j mlk n p q sr t u s q e ps s t x q s y i_z { U U z W } y ~ y x t i e l US T { d ƒ ƒ ƒ j s q e uˆ ps i ˆ p q y
" #$ +. 0. + 4 6 4 : + 4 ; 6 4 < = =@ = = =@ = =@ " #$ P UTS W U X [ZY \ Z _ `a \ dfe ih j mlk n p q sr t u s q e ps s t x q s y i_z { U U z W } y ~ y x t i e l US T { d ƒ ƒ ƒ j s q e uˆ ps i ˆ p q y h
More informationFramework for functional tree simulation applied to 'golden delicious' apple trees
Purdue University Purdue e-pubs Open Access Theses Theses and Dissertations Spring 2015 Framework for functional tree simulation applied to 'golden delicious' apple trees Marek Fiser Purdue University
More informationThe University of Bath School of Management is one of the oldest established management schools in Britain. It enjoys an international reputation for
The University of Bath School of Management is one of the oldest established management schools in Britain. It enjoys an international reputation for the quality of its teaching and research. Its mission
More informationFront-end. Organization of a Modern Compiler. Middle1. Middle2. Back-end. converted to control flow) Representation
register allocation instruction selection Code Low-level intermediate Representation Back-end Assembly array references converted into low level operations, loops converted to control flow Middle2 Low-level
More information4.3 Laplace Transform in Linear System Analysis
4.3 Laplace Transform in Linear System Analysis The main goal in analysis of any dynamic system is to find its response to a given input. The system response in general has two components: zero-state response
More informationObtain from theory/experiment a value for the absorbtion cross-section. Calculate the number of atoms/ions/molecules giving rise to the line.
Line Emission Observations of spectral lines offer the possibility of determining a wealth of information about an astronomical source. A usual plan of attack is :- Identify the line. Obtain from theory/experiment
More informationarxiv: v3 [cs.fl] 2 Jul 2018
COMPLEXITY OF PREIMAGE PROBLEMS FOR DETERMINISTIC FINITE AUTOMATA MIKHAIL V. BERLINKOV arxiv:1704.08233v3 [cs.fl] 2 Jul 2018 Institute of Natural Sciences and Mathematics, Ural Federal University, Ekaterinburg,
More informationOptimizing Stresses for Testing DRAM Cell Defects Using Electrical Simulation
Optimizing Stresses for Testing RAM Cell efects Using Electrical Simulation Zaid Al-Ars Ad J. van de Goor Faculty of Information Technology and Systems Section of Computer Engineering elft University of
More informationVectors. Teaching Learning Point. Ç, where OP. l m n
Vectors 9 Teaching Learning Point l A quantity that has magnitude as well as direction is called is called a vector. l A directed line segment represents a vector and is denoted y AB Å or a Æ. l Position
More informationMatrices and Determinants
Matrices and Determinants Teaching-Learning Points A matri is an ordered rectanguar arra (arrangement) of numbers and encosed b capita bracket [ ]. These numbers are caed eements of the matri. Matri is
More informationALTER TABLE Employee ADD ( Mname VARCHAR2(20), Birthday DATE );
!! "# $ % & '( ) # * +, - # $ "# $ % & '( ) # *.! / 0 "# "1 "& # 2 3 & 4 4 "# $ % & '( ) # *!! "# $ % & # * 1 3 - "# 1 * #! ) & 3 / 5 6 7 8 9 ALTER ; ? @ A B C D E F G H I A = @ A J > K L ; ?
More informationAn Introduction to Optimal Control Applied to Disease Models
An Introduction to Optimal Control Applied to Disease Models Suzanne Lenhart University of Tennessee, Knoxville Departments of Mathematics Lecture1 p.1/37 Example Number of cancer cells at time (exponential
More informationQUESTIONS ON QUARKONIUM PRODUCTION IN NUCLEAR COLLISIONS
International Workshop Quarkonium Working Group QUESTIONS ON QUARKONIUM PRODUCTION IN NUCLEAR COLLISIONS ALBERTO POLLERI TU München and ECT* Trento CERN - November 2002 Outline What do we know for sure?
More informationAN IDENTIFICATION ALGORITHM FOR ARMAX SYSTEMS
AN IDENTIFICATION ALGORITHM FOR ARMAX SYSTEMS First the X, then the AR, finally the MA Jan C. Willems, K.U. Leuven Workshop on Observation and Estimation Ben Gurion University, July 3, 2004 p./2 Joint
More informationOC330C. Wiring Diagram. Recommended PKH- P35 / P50 GALH PKA- RP35 / RP50. Remarks (Drawing No.) No. Parts No. Parts Name Specifications
G G " # $ % & " ' ( ) $ * " # $ % & " ( + ) $ * " # C % " ' ( ) $ * C " # C % " ( + ) $ * C D ; E @ F @ 9 = H I J ; @ = : @ A > B ; : K 9 L 9 M N O D K P D N O Q P D R S > T ; U V > = : W X Y J > E ; Z
More informationComplex Analysis. PH 503 Course TM. Charudatt Kadolkar Indian Institute of Technology, Guwahati
Complex Analysis PH 503 Course TM Charudatt Kadolkar Indian Institute of Technology, Guwahati ii Copyright 2000 by Charudatt Kadolkar Preface Preface Head These notes were prepared during the lectures
More informationVector analysis. 1 Scalars and vectors. Fields. Coordinate systems 1. 2 The operator The gradient, divergence, curl, and Laplacian...
Vector analysis Abstract These notes present some background material on vector analysis. Except for the material related to proving vector identities (including Einstein s summation convention and the
More informationMax. Input Power (W) Input Current (Arms) Dimming. Enclosure
Product Overview XI025100V036NM1M Input Voltage (Vac) Output Power (W) Output Voltage Range (V) Output urrent (A) Efficiency@ Max Load and 70 ase Max ase Temp. ( ) Input urrent (Arms) Max. Input Power
More informationImproving the Upper Bound on the Length of the Shortest Reset Words
Improving the Upper Bound on the Length of the Shortest Reset Words Marek Szykuła Institute of Computer Science, University of Wrocław, Wrocław, Poland msz@csuniwrocpl Abstract We improve the best known
More informationPlanning for Reactive Behaviors in Hide and Seek
University of Pennsylvania ScholarlyCommons Center for Human Modeling and Simulation Department of Computer & Information Science May 1995 Planning for Reactive Behaviors in Hide and Seek Michael B. Moore
More informationA REPRESENTATION THEORETIC APPROACH TO SYNCHRONIZING AUTOMATA
A REPRESENTATION THEORETIC APPROACH TO SYNCHRONIZING AUTOMATA FREDRICK ARNOLD AND BENJAMIN STEINBERG Abstract. This paper is a first attempt to apply the techniques of representation theory to synchronizing
More information$%! & (, -3 / 0 4, 5 6/ 6 +7, 6 8 9/ 5 :/ 5 A BDC EF G H I EJ KL N G H I. ] ^ _ ` _ ^ a b=c o e f p a q i h f i a j k e i l _ ^ m=c n ^
! #" $%! & ' ( ) ) (, -. / ( 0 1#2 ' ( ) ) (, -3 / 0 4, 5 6/ 6 7, 6 8 9/ 5 :/ 5 ;=? @ A BDC EF G H I EJ KL M @C N G H I OPQ ;=R F L EI E G H A S T U S V@C N G H IDW G Q G XYU Z A [ H R C \ G ] ^ _ `
More informationA Functional Quantum Programming Language
A Functional Quantum Programming Language Thorsten Altenkirch University of Nottingham based on joint work with Jonathan Grattage and discussions with V.P. Belavkin supported by EPSRC grant GR/S30818/01
More informationConnection equations with stream variables are generated in a model when using the # $ % () operator or the & ' %
7 9 9 7 The two basic variable types in a connector potential (or across) variable and flow (or through) variable are not sufficient to describe in a numerically sound way the bi-directional flow of matter
More informationPrincipal Secretary to Government Haryana, Town & Country Planning Department, Haryana, Chandigarh.
1 From To Principal Secretary to Government Haryana, Town & Country Planning Department, Haryana, Chandigarh. The Director General, Town & Country Planning Department, Haryana, Chandigarh. Memo No. Misc-2339
More informationExamination paper for TFY4240 Electromagnetic theory
Department of Physics Examination paper for TFY4240 Electromagnetic theory Academic contact during examination: Associate Professor John Ove Fjærestad Phone: 97 94 00 36 Examination date: 16 December 2015
More informationDISSIPATIVE SYSTEMS. Lectures by. Jan C. Willems. K.U. Leuven
DISSIPATIVE SYSTEMS Lectures by Jan C. Willems K.U. Leuven p.1/26 LYAPUNOV FUNCTIONS Consider the classical dynamical system, the flow with, the state, and Denote the set of solutions, the vector-field.
More informationCSC Metric Embeddings Lecture 8: Sparsest Cut and Embedding to
CSC21 - Metric Embeddings Lecture 8: Sparsest Cut and Embedding to Notes taken by Nilesh ansal revised by Hamed Hatami Summary: Sparsest Cut (SC) is an important problem ith various applications, including
More informationWords guaranteeing minimal image
Words guaranteeing minimal image S.W. Margolis J.-E. Pin M.V. Volkov Abstract Given a positive integer n and a finite alphabet A, a word w over A is said to guarantee minimal image if, for every homomorphism
More informationNew BaBar Results on Rare Leptonic B Decays
New BaBar Results on Rare Leptonic B Decays Valerie Halyo Stanford Linear Accelerator Center (SLAC) FPCP 22 valerieh@slac.stanford.edu 1 Table of Contents Motivation to look for and Analysis Strategy Measurement
More informationQueues, Stack Modules, and Abstract Data Types. CS2023 Winter 2004
Queues Stack Modules and Abstact Data Types CS2023 Wnte 2004 Outcomes: Queues Stack Modules and Abstact Data Types C fo Java Pogammes Chapte 11 (11.5) and C Pogammng - a Moden Appoach Chapte 19 Afte the
More informationDamping Ring Requirements for 3 TeV CLIC
Damping Ring Requirements for 3 TeV CLIC Quantity Symbol Value Bunch population N b 9 4.1 10 No. of bunches/train k bt 154 Repetition frequency f r 100 Hz Horizontal emittance γε x 7 4.5 10 m Vertical
More informationTELEMATICS LINK LEADS
EEAICS I EADS UI CD PHOE VOICE AV PREIU I EADS REQ E E A + A + I A + I E B + E + I B + E + I B + E + H B + I D + UI CD PHOE VOICE AV PREIU I EADS REQ D + D + D + I C + C + C + C + I G G + I G + I G + H
More informationSymbols and dingbats. A 41 Α a 61 α À K cb ➋ à esc. Á g e7 á esc. Â e e5 â. Ã L cc ➌ ã esc ~ Ä esc : ä esc : Å esc * å esc *
Note: Although every effort ws tken to get complete nd ccurte tble, the uhtor cn not be held responsible for ny errors. Vrious sources hd to be consulted nd MIF hd to be exmined to get s much informtion
More informationUnit 3. Digital encoding
Unit 3. Digital encoding Digital Electronic Circuits (Circuitos Electrónicos Digitales) E.T.S.I. Informática Universidad de Sevilla 9/2012 Jorge Juan 2010, 2011, 2012 You are free to
More informationSubrings and Ideals 2.1 INTRODUCTION 2.2 SUBRING
Subrings and Ideals Chapter 2 2.1 INTRODUCTION In this chapter, we discuss, subrings, sub fields. Ideals and quotient ring. We begin our study by defining a subring. If (R, +, ) is a ring and S is a non-empty
More informationThe Fourier series are applicable to periodic signals. They were discovered by the
3.1 Fourier Series The Fourier series are applicable to periodic signals. They were discovered by the famous French mathematician Joseph Fourier in 1807. By using the Fourier series, a periodic signal
More informationPART IV LIVESTOCK, POULTRY AND FISH PRODUCTION
! " $#%(' ) PART IV LIVSTOCK, POULTRY AND FISH PRODUCTION Table (93) MAIN GROUPS OF ANIMAL PRODUCTS Production 1000 M.T Numbers 1000 Head Type 2012 Numbers Cattle 54164.5 53434.6 Buffaloes 4304.51 4292.51
More informationPermutation distance measures for memetic algorithms with population management
MIC2005: The Sixth Metaheuristics International Conference??-1 Permutation distance measures for memetic algorithms with population management Marc Sevaux Kenneth Sörensen University of Valenciennes, CNRS,
More informationETIKA V PROFESII PSYCHOLÓGA
P r a ž s k á v y s o k á š k o l a p s y c h o s o c i á l n í c h s t u d i í ETIKA V PROFESII PSYCHOLÓGA N a t á l i a S l o b o d n í k o v á v e d ú c i p r á c e : P h D r. M a r t i n S t r o u
More informationSYNTACTIC SEMIGROUP PROBLEM FOR THE SEMIGROUP REDUCTS OF AFFINE NEAR-SEMIRINGS OVER BRANDT SEMIGROUPS
SYNTACTIC SEMIGROUP PROBLEM FOR THE SEMIGROUP REDUCTS OF AFFINE NEAR-SEMIRINGS OVER BRANDT SEMIGROUPS JITENDER KUMAR AND K. V. KRISHNA Abstract. The syntactic semigroup problem is to decide whether a given
More information3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 REDCLIFF MUNICIPAL PLANNING COMMISSION FOR COMMENT/DISCUSSION DATE: TOPIC: April 27 th, 2018 Bylaw 1860/2018, proposed amendments to the Land Use Bylaw regarding cannabis
More informationSCOTT PLUMMER ASHTON ANTONETTI
2742 A E E E, UE, AAAMA 3802 231 EE AEA 38,000-cre dversfed federl cmus 41,000 emloyees wth 72+ dfferent gences UE roosed 80-cre mster-lnned develoment 20 home stes 3,000F of vllgestyle retl 100,000 E
More informationA Study on the Analysis of Measurement Errors of Specific Gravity Meter
HWAHAK KONGHAK Vol. 40, No. 6, December, 2002, pp. 676-680 (2001 7 2, 2002 8 5 ) A Study on the Analysis of Measurement Errors of Specific Gravity Meter Kang-Jin Lee, Jae-Young Her, Young-Cheol Ha, Seung-Hee
More informationPharmacological and genomic profiling identifies NF-κB targeted treatment strategies for mantle cell lymphoma
CORRECTION NOTICE Nat. Med. 0, 87 9 (014) Pharmacoogica and genomic profiing identifies NF-κB targeted treatment strategies for mante ce ymphoma Rami Raha, Mareie Fric, Rodrigo Romero, Joshua M Korn, Robert
More informationChapter 2: Finite Automata
Chapter 2: Finite Automata 2.1 States, State Diagrams, and Transitions Finite automaton is the simplest acceptor or recognizer for language specification. It is also the simplest model of a computer. A
More informationAlgebraic Approach to Automata Theory
Algebraic Approach to Automata Theory Deepak D Souza Department of Computer Science and Automation Indian Institute of Science, Bangalore. 20 September 2016 Outline 1 Overview 2 Recognition via monoid
More informationSurfaces of Section I
Surfaces of Section I Consider a system with n = 2 degrees of freedom (e.g., planar motion), and with a Hamiltonian H( x, p) = 1 2 (p2 x + p2 y ) + Φ(x, y) Conservation of energy, E = H, restricts the
More informationProbabilistic testing coverage
Probabilistic testing coverage NICOLAE GOGA Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven THE NETHERLANDS Abstract: This paper describes a way to compute the coverage for an on-the-fly
More informationA Language for Task Orchestration and its Semantic Properties
DEPARTMENT OF COMPUTER SCIENCES A Language for Task Orchestration and its Semantic Properties David Kitchin, William Cook and Jayadev Misra Department of Computer Science University of Texas at Austin
More informationRarefied Gas FlowThroughan Orifice at Finite Pressure Ratio
Rarefied Gas FlowThroughan Orifice at Finite Pressure Ratio Felix Sharipov Departamento de Física, Universidade Federal do Paraná Caixa Postal 90, 85-990 Curitiba, Brazil Email: sharipov@fisica.ufpr.br;
More informationA Study on Dental Health Awareness of High School Students
Journal of Dental Hygiene Science Vol. 3, No. 1,. 23~ 31 (2003), 1 Su-Min Yoo and Geum-Sun Ahn 1 Deartment of Dental Hygiene, Dong-u College 1 Deartment of Dental Hygiene, Kyung Bok College In this research,
More informationTowards a High Level Quantum Programming Language
MGS Xmas 04 p.1/?? Towards a High Level Quantum Programming Language Thorsten Altenkirch University of Nottingham based on joint work with Jonathan Grattage and discussions with V.P. Belavkin supported
More informationA General Procedure to Design Good Codes at a Target BER
A General Procedure to Design Good odes at a Target BER Speaker: Xiao Ma 1 maxiao@mail.sysu.edu.cn Joint work with: hulong Liang 1, Qiutao Zhuang 1, and Baoming Bai 2 1 Dept. Electronics and omm. Eng.,
More informationAlso, course highlights the importance of discrete structures towards simulation of a problem in computer science and engineering.
The objectives of Discrete Mathematical Structures are: To introduce a number of Discrete Mathematical Structures (DMS) found to be serving as tools even today in the development of theoretical computer
More informationDefinitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch
Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary
More informationBooks. Book Collection Editor. Editor. Name Name Company. Title "SA" A tree pattern. A database instance
"! # #%$ $ $ & & & # ( # ) $ + $, -. 0 1 1 1 2430 5 6 78 9 =?
More informationAutomata Specialities:
Applied Mathematical Sciences, Vol. 1, 018, no. 0, 959-974 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ams.018.8794 Automata Specialities: Černý Automata David Casas Departamento de Matemáticas
More informationTowards a numerical solution of the "1/2 vs. 3/2" puzzle
Towards a numerical solution of the /2 s 3/2 puzzle Benoît Blossier DESY Zeuthen Motiations Hea Quark Effectie Theor Theoretical expectations Experimental measurements umerical computation of Conclusion
More informationLower and Upper Bound Results for Hard Problems Related to Finite Automata
Lower and Upper Bound Results for Hard Problems Related to Finite Automata Henning Fernau Universität Trier, Germany fernau@informatik.uni-trier.de Andreas Krebs Universität Tübingen, Germany krebs@informatik.uni-tuebingen.de
More informationF O R SOCI AL WORK RESE ARCH
7 TH EUROPE AN CONFERENCE F O R SOCI AL WORK RESE ARCH C h a l l e n g e s i n s o c i a l w o r k r e s e a r c h c o n f l i c t s, b a r r i e r s a n d p o s s i b i l i t i e s i n r e l a t i o n
More informationI118 Graphs and Automata
I8 Graphs and Automata Takako Nemoto http://www.jaist.ac.jp/ t-nemoto/teaching/203--.html April 23 0. Û. Û ÒÈÓ 2. Ø ÈÌ (a) ÏÛ Í (b) Ø Ó Ë (c) ÒÑ ÈÌ (d) ÒÌ (e) É Ö ÈÌ 3. ÈÌ (a) Î ÎÖ Í (b) ÒÌ . Û Ñ ÐÒ f
More informationNew method for solving nonlinear sum of ratios problem based on simplicial bisection
V Ù â ð f 33 3 Vol33, No3 2013 3 Systems Engineering Theory & Practice Mar, 2013 : 1000-6788(2013)03-0742-06 : O2112!"#$%&')(*)+),-))/0)1)23)45 : A 687:9 1, ;:= 2 (1?@ACBEDCFHCFEIJKLCFFM, NCO 453007;
More informationTesting SUSY Dark Matter
Testing SUSY Dark Matter Wi de Boer, Markus Horn, Christian Sander Institut für Experientelle Kernphysik Universität Karlsruhe Wi.de.Boer@cern.ch http://hoe.cern.ch/ deboerw SPACE Part Elba, May 7, CMSSM
More information1. Allstate Group Critical Illness Claim Form: When filing Critical Illness claim, please be sure to include the following:
Dear Policyholder, Per your recent request, enclosed you will find the following forms: 1. Allstate Group Critical Illness Claim Form: When filing Critical Illness claim, please be sure to include the
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2017 http://cseweb.ucsd.edu/classes/sp17/cse105-ab/ Today's learning goals Sipser Ch 1.4 Explain the limits of the class of regular languages Justify why the Pumping
More informationcse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska
cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 6 CHAPTER 2 FINITE AUTOMATA 2. Nondeterministic Finite Automata NFA 3. Finite Automata and Regular Expressions 4. Languages
More informationSOLAR MORTEC INDUSTRIES.
SOLAR MORTEC INDUSTRIES www.mortecindustries.com.au Wind Regions Region A Callytharra Springs Gascoyne Junction Green Head Kununurra Lord Howe Island Morawa Toowoomba Wittanoom Bourke Region B Adelaide
More informationComputer Systems Organization. Plan for Today
Computer Systems Organization "!#%$&(' *),+-/. 021 345$&7634 098 :;!/.:?$&4@344A$&76 Plan for Today 1 ! ILP Limit Study #"%$'&)('*+-,.&/ 0 1+-*2 435&)$'768&9,:+8&.* &:>@7ABAC&ED(
More informationThe Separating Words Problem
The Separating Words Problem Jeffrey Shallit School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada shallit@cs.uwaterloo.ca https://www.cs.uwaterloo.ca/~shallit 1/54 The Simplest
More informationhal , version 1-27 Mar 2014
Author manuscript, published in "2nd Multidisciplinary International Conference on Scheduling : Theory and Applications (MISTA 2005), New York, NY. : United States (2005)" 2 More formally, we denote by
More informationA Beamforming Method for Blind Calibration of Time-Interleaved A/D Converters
A Beamforming Method for Blind Calibration of Time-nterleaved A/D Converters Bernard C. Levy University of California, Davis Joint wor with Steve Huang September 30, 200 Outline 1 Motivation Problem Formulation
More informationLecture 16: Modern Classification (I) - Separating Hyperplanes
Lecture 16: Modern Classification (I) - Separating Hyperplanes Outline 1 2 Separating Hyperplane Binary SVM for Separable Case Bayes Rule for Binary Problems Consider the simplest case: two classes are
More informationA Theory of Universal AI
A Theory of Universa AI Literature Marcus Hutter Kircherr, Li, and Vitanyi Presenter Prashant J. Doshi CS594: Optima Decision Maing A Theory of Universa Artificia Inteigence p.1/18 Roadmap Caim Bacground
More informationRemarks on Separating Words
Remarks on Separating Words Erik D. Demaine, Sarah Eisenstat, Jeffrey Shallit 2, and David A. Wilson MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA 239, USA,
More informationUNIQUE FJORDS AND THE ROYAL CAPITALS UNIQUE FJORDS & THE NORTH CAPE & UNIQUE NORTHERN CAPITALS
Q J j,. Y j, q.. Q J & j,. & x x. Q x q. ø. 2019 :. q - j Q J & 11 Y j,.. j,, q j q. : 10 x. 3 x - 1..,,. 1-10 ( ). / 2-10. : 02-06.19-12.06.19 23.06.19-03.07.19 30.06.19-10.07.19 07.07.19-17.07.19 14.07.19-24.07.19
More informationThis document has been prepared by Sunder Kidambi with the blessings of
Ö À Ö Ñ Ø Ò Ñ ÒØ Ñ Ý Ò Ñ À Ö Ñ Ò Ú º Ò Ì ÝÊ À Å Ú Ø Å Ê ý Ú ÒØ º ÝÊ Ú Ý Ê Ñ º Å º ² ºÅ ý ý ý ý Ö Ð º Ñ ÒÜ Æ Å Ò Ñ Ú «Ä À ý ý This document has been prepared by Sunder Kidambi with the blessings of Ö º
More informationManifold Regularization
Manifold Regularization Vikas Sindhwani Department of Computer Science University of Chicago Joint Work with Mikhail Belkin and Partha Niyogi TTI-C Talk September 14, 24 p.1 The Problem of Learning is
More informationSample Exam 1: Chapters 1, 2, and 3
L L 1 ' ] ^, % ' ) 3 Sample Exam 1: Chapters 1, 2, and 3 #1) Consider the lineartime invariant system represented by Find the system response and its zerostate and zeroinput components What are the response
More informationMacLane s coherence theorem expressed as a word problem
MacLane s coherence theorem expressed as a word problem Paul-André Melliès Preuves, Programmes, Systèmes CNRS UMR-7126, Université Paris 7 ÑÐÐ ÔÔ ºÙ ÙºÖ DRAFT In this draft, we reduce the coherence theorem
More informationPeriodic monopoles and difference modules
Periodic monopoles and difference modules Takuro Mochizuki RIMS, Kyoto University 2018 February Introduction In complex geometry it is interesting to obtain a correspondence between objects in differential
More informationOpen Problems in Automata Theory: An Idiosyncratic View
Open Problems in Automata Theory: An Idiosyncratic View Jeffrey Shallit School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada shallit@cs.uwaterloo.ca http://www.cs.uwaterloo.ca/~shallit
More information(Refer Slide Time: 0:21)
Theory of Computation Prof. Somenath Biswas Department of Computer Science and Engineering Indian Institute of Technology Kanpur Lecture 7 A generalisation of pumping lemma, Non-deterministic finite automata
More informationAn algebraic view of topological -machines
An algebraic view of topological -machines Luke Grecki Graduate Group in Applied Mathematics lgrecki@math.ucdavis.edu June 8, 2010 1 Contents 1 Semigroups 3 2 Semigroups of automata 4 3 -machine words
More informationA QUADRATIC UPPER BOUND ON THE SIZE OF A SYNCHRONIZING WORD IN ONE-CLUSTER AUTOMATA
A QUADRATIC UPPER BOUND ON THE SIZE OF A SYNCHRONIZING WORD IN ONE-CLUSTER AUTOMATA MARIE-PIERRE BÉAL Université Paris-Est Laboratoire d informatique Gaspard-Monge, CNRS 77454 Marne-la-Vallée Cedex 2,
More informationApplications of Discrete Mathematics to the Analysis of Algorithms
Applications of Discrete Mathematics to the Analysis of Algorithms Conrado Martínez Univ. Politècnica de Catalunya, Spain May 2007 Goal Given some algorithm taking inputs from some set Á, we would like
More informationT T V e g em D e j ) a S D } a o "m ek j g ed b m "d mq m [ d, )
. ) 6 3 ; 6 ;, G E E W T S W X D ^ L J R Y [ _ ` E ) '" " " -, 7 4-4 4-4 ; ; 7 4 4 4 4 4 ;= : " B C CA BA " ) 3D H E V U T T V e g em D e j ) a S D } a o "m ek j g ed b m "d mq m [ d, ) W X 6 G.. 6 [ X
More informationIdeal Regular Languages and Strongly Connected Synchronizing Automata
Ideal Regular Languages and Strongly Connected Synchronizing Automata Rogério Reis a,b, Emanuele Rodaro a a Centro de Matemática, Universidade do Porto b DCC-Faculdade de Ciências, Universidade do Porto
More informationRobert Krzyzanowski Algebra (Dummit and Foote) Chapter 1.1. These problems implicitly make use of the following lemma (stated casually in the book).
Chapter 11 These problems implicitly make use of the following lemma (stated casually in the book) Lemma If akß æb is a group and L K is closed under æ then æ is associative in L Proof We know æ is associative
More informationMARKET AREA UPDATE Report as of: 1Q 2Q 3Q 4Q
Year: 2013 Market Area (Cit, State): Greater Portland, OR MARKET AREA UPDATE Report as of: 1Q 2Q 3Q 4Q Provided b (Compan / Companies): Coldwell Banker Seal, Archibald Relocation, Prudential Northwest
More informationOptimization of a parallel 3d-FFT with non-blocking collective operations
Optimization of a parallel 3d-FFT with non-blocking collective operations Chair of Computer Architecture Technical University of Chemnitz Département de Physique Théorique et Appliquée Commissariat à l
More information