St Andrews, September 59, 2006 Interpreting graphs in 0-simple - PowerPoint PPT Presentation

❫ ❨ ❬ ❭ ❨ ❴ ❨ ❲ ❫ ❪ ❪ ❞ ❴ ❵ ✐ ❞ ❪ ❫ ❣ ❴ ❨ ❵ ❞ ❪ ❫ ❣ ❴ ❵ ❵ ❵ ❭ ❲ ❳ ❨ ❩ ❵ ❨ ❬ ❭ ❴ ❲ ❫ ❪ ❜ ❨ ❦ ❪ ❫ ❲ ❴ ❨ ❵ ❛ ❪ ❫ ❥ ❴ ❨ Piecewise testable languages More precisely, a language is called piecewise testable of height if can be recognized by a hydra automaton with heads. Let [resp. ❲❱❛ ] denote the family of all piecewise testable languages [of height ] over a fixed alphabet . Simon’s hierarchy of piecewise testable languages: ❲❝❜ ❲❝❡ ❲❝❢ ❞❤❣ ❲❧❛ St Andrews 2006 – p.8/32

♠ ♥ ♦ ♣ ♠ Piecewise testable languages Question 1. Given a language , how to decide whether or not is piecewise testable? St Andrews 2006 – p.9/32

✉ ✇ ✇ t s r q ✈ q q ② ③ t s r q ✈ Piecewise testable languages Question 1. Given a language , how to decide whether or not is piecewise testable? Question 2. Given a piecewise testable language , how to determine its height ( the least such that belongs to q❧① but not to q❧① )? St Andrews 2006 – p.9/32

⑩ ⑤ ⑩ ④ ❷ ❸ ⑧ ⑥ ⑦ ⑥ ④ ⑨ ❻ ⑥ ④ ⑦ ⑥ ⑦ ⑥ ⑤ ④ ⑨ Piecewise testable languages Question 1. Given a language , how to decide whether or not is piecewise testable? Question 2. Given a piecewise testable language , how to determine its height ( the least such that belongs to ④❧❶ but not to ④❧❶ )? Exercise. Is the language ⑦❺❹ ❻❾❽ ( ) piecewise ❹❧❼ testable? St Andrews 2006 – p.9/32

➅ ➂ ➈ ➁ ➅ ➄ ➊ ❿ ➁ ➂ ➃ ➁ ➁ ➇ ➀ ❿ ➍ ➍ ❿ ➍ ➍ ➂ ➁ ➀ ❿ ➄ Piecewise testable languages Question 1. Given a language , how to decide whether or not is piecewise testable? Question 2. Given a piecewise testable language , how to determine its height ( the least such that belongs to ❿❧➆ but not to ❿❧➆ )? Exercise. Is the language ➂❺➉ ➊❾➌ ( ) piecewise ➉❧➋ testable? Yes No Don’t know It depends St Andrews 2006 – p.9/32

➑ ➙ ➎ ➙ ↕ ➔ ➣ ↔ → ➐ ➙ ↕ ↕ → ➣ ➓ ➒ ➔ ➔ → ➓ ➒ ➎ ➑ ➐ ➏ ➎ ➛➜ Syntactic monoids For a language its syntactic congruence is defined by if, for any ➎➞➝ ➣❧↔ Thus, and occur in in the same contexts. St Andrews 2006 – p.10/32

➭ ➫ ➦ ➫ ➭ ➟ ➯➲ ➨ ➧ ➡ ➩ ➟ ➵ ➦ ➟ ➧ ➢ ➟ ➡ ➨ ➢ ➤ ➥ ➟ ➠ ➡ ➢ ➟ ➤ ➥ ➤ ➡ ➦ ➤ ➥ ➧ ➼ ➢ ➫ ➭ ➥ Syntactic monoids For a language its syntactic congruence is defined by if, for any ➟➞➳ ➨❧➩ Thus, and occur in in the same contexts. One can check that is the largest congruence on for which is a union of classes. The quotient monoid ➸➻➺ is called the syntactic monoid of the language . St Andrews 2006 – p.10/32

➴ ➘ ➽ ✃❐ ➷ ➴ ➮ ➱ ➽ ➹ ➶ ➮ Ð ➪ ➽ ➚ ➽ ➶ ➹ ❮ ➚ ➱ ➘ ➽ ➘ ➽ ➾ ➚ ➪ ➽ ➶ ➹ ❰ ➽ ➶ ➷ ➹ ➴ ❮ ➽ ➮ ➱ ➚ ➪ ➬ ➪ Syntactic monoids For a language its syntactic congruence is defined by if, for any ➽➞❒ ➷❧➬ Thus, and occur in in the same contexts. One can check that is the largest congruence on for which is a union of classes. The quotient monoid ❰➻Ï is called the syntactic monoid of the language . For a regular language , the syntactic monoid can be also defined as the transition monoid of the minimal automaton of . St Andrews 2006 – p.10/32

Syntactic monoids Rather than formal definitions from the previous slide, the following crucial ideas are to be understood: St Andrews 2006 – p.11/32

Ò Ñ Ñ Ó Syntactic monoids Rather than formal definitions from the previous slide, the following crucial ideas are to be understood: For a regular language , its syntactic monoid is always finite (and vice versa) — this is Myhill’s form of Kleene’s theorem. St Andrews 2006 – p.11/32

Ö Ô Õ Ô Ö Ô Õ Ô Syntactic monoids Rather than formal definitions from the previous slide, the following crucial ideas are to be understood: For a regular language , its syntactic monoid is always finite (and vice versa) — this is Myhill’s form of Kleene’s theorem. The syntactic monoid can be efficiently calculated whenever is efficiently presented — say, by a regular expression or by a finite automaton. St Andrews 2006 – p.11/32

Ù Ø Ù × × Ø Ø × Ù × × × Syntactic monoids Rather than formal definitions from the previous slide, the following crucial ideas are to be understood: For a regular language , its syntactic monoid is always finite (and vice versa) — this is Myhill’s form of Kleene’s theorem. The syntactic monoid can be efficiently calculated whenever is efficiently presented — say, by a regular expression or by a finite automaton. Thus, whenever is “given”, so is . St Andrews 2006 – p.11/32

Ú Û Ü Û Ý Ú Û Simon’s theorem A monoid is said to be - trivial if every principal ideal of has a unique generator: ÜßÞ St Andrews 2006 – p.12/32

è ã â à à é â á à ã á â á à Simon’s theorem A monoid is said to be - trivial if every principal ideal of has a unique generator: âßä In different terms, being -trivial amounts to saying that the ( bilateral ) divisibility relation åçæ is an order relation on . St Andrews 2006 – p.12/32

ô í ñ í ó ê ë ê ë ì ì ë ê ò ì õ ó ê Simon’s theorem A monoid is said to be - trivial if every principal ideal of has a unique generator: ìßî In different terms, being -trivial amounts to saying that the ( bilateral ) divisibility relation ïçð is an order relation on . Theorem. (Imre Simon, 1972) A language is piecewise testable if and only if its syntactic monoid is -trivial. St Andrews 2006 – p.12/32

ù ø ø ÷ ü ÷ ù ÷ ÷ ù û ÷ ø ö ø ø ö ù ú ù ö ö ö ÷ ù ö ø ù ÷ ö ø Simon’s theorem 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 The monoid St Andrews 2006 – p.13/32

þ þ � ý ý � ÿ � ý � ÿ � þ þ � ÿ ÿ ý ÿ ý ý ✠ ý þ ÿ � ÿ ✞ ✆ þ ☎ ✄ ✂ ✁ ý þ ý ÿ Simon’s theorem 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 ✝✟✞ St Andrews 2006 – p.14/32

✌ ☛ ✍ ☞ ✡ ☛ ✏ ☛ ✌ ☛ ✎ ✡ ☞ ✍ ☞ ✡ ☞ ☞ ✡ ✌ ☛ ✡ ✎ ✌ ✔ ✕ ✡ ☛ ☞ ✌ ✌ ✔ ☞ ☛ ✡ ✔ ✡ ✒ ✑ ✏ ✡ ☛ ☞ ✌ ✌ Simon’s theorem 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 ✓✟✔ St Andrews 2006 – p.15/32

✥ ✛ ✗ ✥ ✖ ✛ ✣ ✢ ✜ ✚ ✘ ✖ ✜ ✛ ✚ ✜ ✗ ✙ ✥ ✥ ✙ ✢ ✘ ✥ ✗ ✥ ✖ ✦ ✣ ✜ ✙ ✛ ✚ ✗ ✜ ✛ ✚ ✚ ✦ ✙ ✙ ✖ ✗ ✖ ✘ ✖ ✖ ✗ ✘ ✖ ✗ ✙ ✙ ✘ ✗ ✗ ✙ ✘ ✘ ✘ ✗ ✖ ✘ ✖ ✙ Simon’s theorem 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 ✤✟✥ Similarly one can verify that ✤✟✥ whence the monoid is not -trivial. St Andrews 2006 – p.15/32

Simon’s theorem Example : solution to the above Exercise. St Andrews 2006 – p.16/32

✧ ✫ ✧ ★ ✫ ✮ ✧ Simon’s theorem Example : solution to the above Exercise. Exercise. Is the language ★✪✩ ( ) piecewise ✩✭✬ testable? St Andrews 2006 – p.16/32

✯ ✴ ✵ ✯ ✵ ✲ ✯ ✰ ✵ ✵ ✲ Simon’s theorem Example : solution to the above Exercise. Exercise. Is the language ✰✪✱ ( ) piecewise ✱✭✳ testable? Yes No Don’t know It depends St Andrews 2006 – p.16/32

✼ ✼ ✼ ✼ ✼ ✽ ✼ ✼ ✶ ✻ ✹ ✷ ✶ ✹ ✶ Simon’s theorem Example : solution to the above Exercise. Exercise. Is the language ✷✪✸ ( ) piecewise ✸✭✺ testable? Yes Yes No No Don’t know Don’t know It depends ! St Andrews 2006 – p.16/32

❄ ❈ ❅ ❄ ✾ ❄ ❆ ❄ ❇ ❄ ❁ ✾ ❄ ❃ ❁ ✾ ❁ ✿ ✾ ❁ ✾ ✾ ✿ ❄ Simon’s theorem Example : solution to the above Exercise. Exercise. Is the language ✿✪❀ ( ) piecewise ❀✭❂ testable? Yes Yes No No Don’t know Don’t know It depends ! If , the language is piecewise testable. ✿✪❀ ❀✭❂ St Andrews 2006 – p.16/32

❖ ❏ ❊ ▼ ❑ ❉ ❏ ● ❏ ▼ ❏ ◆ ❏ ● ❏ ❉ ❉ ▲ ❏ ● ❉ ❉ ■ ● ❊ ◆ ❊ ❉ ● ● ❉ ❉ ❉ Simon’s theorem Example : solution to the above Exercise. Exercise. Is the language ❊✪❋ ( ) piecewise ❋✭❍ testable? Yes Yes No No Don’t know Don’t know It depends ! If , the language is piecewise testable. ❊✪❋ ❋✭❍ If , the language ❊✪❋ is not piecewise ❋✭❍ testable. St Andrews 2006 – p.16/32

❩ P ❱ ❭ ❬ ❬ ❬ ❬ ❩ ❙ ❩ ❱ ❨ ❱ ❭ ❙ ❨ P ◗ ❳ ❙ ❲ ❱ ❱ ◗ P ❭ Simon’s theorem ❘❚❙✭❯ If , the minimal automaton of looks as follows: ❙✭❯ ❭❫❪ ❭❫❪ ❭❫❪ St Andrews 2006 – p.17/32

♥ ❞ ❴ ❧ ❢ ♠ ❵ ❛ ❵ ❝ ♦ ❝ ❢ ❞ ❝ ❜ ❜ ❡ ♠ ❜ ♣ ❵ ❢ ❞ ♣ ❵ ❞ ❝ ❜ ❞ ❵ ❞ ❞ ❧ ❞ ❴ ❵ ❣ ❞ ❡ ❴ ❢ ❵ ❴ ❣ ❜ ❞ ❴ ❣ ❤ ❣ ❤ ❤ ✐ ✐ ✐ ✐ ❥ ❣ ❥ ❴ ❥ ❜ ❞ ❜ ♥ Simon’s theorem ❛❚❜✭❝ If , the minimal automaton of looks as follows: ❜✭❝ ❥❫❦ ❥❫❦ ❥❫❦ Using this, one readily calculates that ❜✭❝ subject to the relations and that ❜✭❝ is -trivial. In fact, . St Andrews 2006 – p.17/32

⑤ ⑥ ④ ④ t ⑤ ⑤ ⑤ ✇ ✈ ⑥ ✉ ⑥ ✈ ✈ ✉ ④ ③ t ① q r ✇ ✈ ✉ ✇ ⑧ q ② r q ③ t ✈ ✇ Simon’s theorem s❚t✭✉ If , the minimal automaton of only slightly changes: t✭✉ ⑥❫⑦ ⑥❫⑦ ⑥❫⑦ St Andrews 2006 – p.18/32

❷ ➄ ❺ ❷ ➈ ❹ ❸ ❺ ⑩ ❹ ➂ ❺ ❻ ➃ ❼ ⑩ ❹ ❶ ➅ ❸ ➆ ❸ ➄ ❹ ❸ ❺ ❸ ❹ ❷ ➃ ❸ ➉ ➇ ❹ ⑨ ⑩ ❼ ❹ ❸ ❺ ❻ ➊ ❼ ⑩ ⑨ ❽ ❷ ⑨ ➀ ❽ ➉ ❾ ❾ ❾ ❿ ❿ ❿ ❿ ➀ ➈ ➀ ➊ ⑩ Simon’s theorem ❶❚❷✭❸ If , the minimal automaton of only slightly changes: ❷✭❸ ➀❫➁ ➀❫➁ ➀❫➁ One gets ❷✭❸ and we already know that is not -trivial. Thus, the language is not piecewise testable. St Andrews 2006 – p.18/32

Simon’s theorem Nice: relates a very natural combinatorial property to a very natural semigroup-theoretic property. St Andrews 2006 – p.19/32

➋ ➌➍ ➍ ➎ ➏ Simon’s theorem Nice: relates a very natural combinatorial property to a very natural semigroup-theoretic property. Efficient: given a monoid (by its Cayley table, say), one can easily (in time ) verify whether or not is -trivial. St Andrews 2006 – p.19/32

→ ➑ ➔ → ➔ ➓ ➒ ➑➒ ➐ Simon’s theorem Nice: relates a very natural combinatorial property to a very natural semigroup-theoretic property. Efficient: given a monoid (by its Cayley table, say), one can easily (in time ) verify whether or not is -trivial. Very efficient: There are polynomial time algorithms to verify if the syntactic monoid is -trivial when presented the minimal automaton of . St Andrews 2006 – p.19/32

➛ ↔ ➞ ➝ ➝ ➛ ➣ ↔↕ ↕ ➙ ➛ ➜ ↔ ➜ ➜ Simon’s theorem Nice: relates a very natural combinatorial property to a very natural semigroup-theoretic property. Efficient: given a monoid (by its Cayley table, say), one can easily (in time ) verify whether or not is -trivial. Very efficient: There are polynomial time algorithms to verify if the syntactic monoid is -trivial when presented the minimal automaton of . Such a description of is much more compact than the Cayley table — recall that the transition monoid of an automaton with states may consist of as many as elements! St Andrews 2006 – p.19/32

Simon’s theorem Deep: a crossing where many ideas meet. St Andrews 2006 – p.20/32

Simon’s theorem Deep: a crossing where many ideas meet. Proofs come from: St Andrews 2006 – p.20/32

Simon’s theorem Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; St Andrews 2006 – p.20/32

Simon’s theorem Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; Model theory — Stern, 1985; St Andrews 2006 – p.20/32

Simon’s theorem Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; Model theory — Stern, 1985; Ordered monoids — Straubing and Thérien, 1988; St Andrews 2006 – p.20/32

Simon’s theorem Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; Model theory — Stern, 1985; Ordered monoids — Straubing and Thérien, 1988; Profinite topology — Almeida, 1990; St Andrews 2006 – p.20/32

Simon’s theorem Deep: a crossing where many ideas meet. Proofs come from: Combinatorics on words — Simon’s original proof, 1972, 1975; Model theory — Stern, 1985; Ordered monoids — Straubing and Thérien, 1988; Profinite topology — Almeida, 1990; Transformation semigroups — Higgins, 1997. St Andrews 2006 – p.20/32

Recognizing height A pseudovariety of finite monoids is a class of finite monoids closed under taking submonoids, morphic images and finite direct products. St Andrews 2006 – p.21/32

Recognizing height A pseudovariety of finite monoids is a class of finite monoids closed under taking submonoids, morphic images and finite direct products. Fact (Eilenberg). Each pseudovariety is generated by syntactic monoids it contains. St Andrews 2006 – p.21/32

➟ Recognizing height A pseudovariety of finite monoids is a class of finite monoids closed under taking submonoids, morphic images and finite direct products. Fact (Eilenberg). Each pseudovariety is generated by syntactic monoids it contains. Simon’s theorem means that the pseudovariety of all finite -trivial monoids is generated by syntactic monoids of piecewise testable languages. St Andrews 2006 – p.21/32

➡ ➦ ➳ ➲ ➠ ➨ ➤ ➥ ➭ ➭ ➨ ➵ ➨ Recognizing height Let ➠➢➡ denote the pseudovariety of finite monoids generated by the syntactic monoids of piecewise testable languages of height . We have ➠➧➦ ➠➧➩ ➠➧➫ ➠➢➡ ➨➯➭ St Andrews 2006 – p.22/32

➾ ➶ ➼ ➽ ➴ ➺ ➻ ➼ ➘ ➹ ➾ ➸ ➾ ➶ Recognizing height Let ➸➢➺ denote the pseudovariety of finite monoids generated by the syntactic monoids of piecewise testable languages of height . We have ➸➧➽ ➸➧➚ ➸➧➪ ➸➢➺ ➾➯➶ Thus, the algebraic counterpart of Question 2 is: Question 3. Given a finite monoid and a number , how to determine whether or not belongs to ➸➷➺ ? St Andrews 2006 – p.22/32

Ð Ï ✃ ❐ Ò ➮ ➱ ✃ Ñ ➬ ❒ ❒ ❒ Ï Recognizing height Let ➬➢➮ denote the pseudovariety of finite monoids generated by the syntactic monoids of piecewise testable languages of height . We have ➬➧❐ ➬➧❮ ➬➧❰ ➬➢➮ ❒➯Ï Thus, the algebraic counterpart of Question 2 is: Question 3. Given a finite monoid and a number , how to determine whether or not belongs to ➬➷➮ ? This is a typical instance of the PMP (Pseudovariety Membership Problem). The PMP has proved to systematically arise whenever one translates a “real world” (computer science) question into algebra. St Andrews 2006 – p.22/32

Ú à ß Û á ×Ø Ö Ó Ô Õ Ô Ô Straubing’s theorem — the monoid of all reflexive binary relations on a set with elements. It can be thought of as the monoid of all matrices whose diagonal entries ÜÞÝ are 1 over the boolean semiring Ù✟Ú . St Andrews 2006 – p.23/32

â â ï î ã é í ã ä ã ê â å æç Straubing’s theorem — the monoid of all reflexive binary relations on a set with elements. It can be thought of as the monoid of all matrices whose diagonal entries ëÞì are 1 over the boolean semiring è✟é . — the submonoid of consisting of upper triangular matrices. St Andrews 2006 – p.23/32

û ñ ø ü ôõ ó ý ð ò ð ñ ð þ ñ ð ñ ÷ Straubing’s theorem — the monoid of all reflexive binary relations on a set with elements. It can be thought of as the monoid of all matrices whose diagonal entries ùÞú are 1 over the boolean semiring ö✟÷ . — the submonoid of consisting of upper triangular matrices. — the monoid of all order preserving and extensive transformations of a chain with elements. St Andrews 2006 – p.23/32

✄ ✒ ÿ ✏ ✕ ✎ ÿ ✔ ✓ � ✏ ÿ ✏ ✔ ÿ ✑ ✝ ✒ ✍ ✓ ✓ ✝ ✓ ✒ ✑ ✗ � ✓ ✏ � ✁ � ✓ ✍ ✂ ✄☎ ✑ ✟ ✓ ☞ ✝ ✌ ✒ Straubing’s theorem — the monoid of all reflexive binary relations on a set with elements. It can be thought of as the monoid of all matrices whose diagonal entries ✠☛✡ are 1 over the boolean semiring ✆✞✝ . — the submonoid of consisting of upper triangular matrices. — the monoid of all order preserving and extensive transformations of a chain with elements. A transformation of a chain is order preserving if implies for all ✓✖✕ ✔✘✗ and extensive if for every . ✓✖✕ St Andrews 2006 – p.23/32

Straubing’s theorem Theorem. (Howard Straubing, 1980) For a finite monoid the following are equivalent: St Andrews 2006 – p.24/32

Straubing’s theorem Theorem. (Howard Straubing, 1980) For a finite monoid the following are equivalent: (i) is -trivial ; St Andrews 2006 – p.24/32

✙ ✚ Straubing’s theorem Theorem. (Howard Straubing, 1980) For a finite monoid the following are equivalent: (i) is -trivial ; (ii) divides (is a morphic image of a submonoid of) for some ; St Andrews 2006 – p.24/32

✛ ✛ ✜ ✜ Straubing’s theorem Theorem. (Howard Straubing, 1980) For a finite monoid the following are equivalent: (i) is -trivial ; (ii) divides (is a morphic image of a submonoid of) for some ; (iii) divides for some ; St Andrews 2006 – p.24/32

✣ ✤ ✢ ✢ ✣ ✣ ✢ Straubing’s theorem Theorem. (Howard Straubing, 1980) For a finite monoid the following are equivalent: (i) is -trivial ; (ii) divides (is a morphic image of a submonoid of) for some ; (iii) divides for some ; (iv) divides for some . St Andrews 2006 – p.24/32

✥ ✥ ✦ ✥ ✧ ✦ ✦ Straubing’s theorem Theorem. (Howard Straubing, 1980) For a finite monoid the following are equivalent: (i) is -trivial ; (ii) divides (is a morphic image of a submonoid of) for some ; (iii) divides for some ; (iv) divides for some . This looks as a quite innocent Cayley-type theorem but in fact the proof heavily depends on Simon’s theorem, and moreover, it can be shown relatively easily that the two theorems are equivalent. St Andrews 2006 – p.24/32

✪ ✱ ✬ ✯✱ ✱ ✩ ✫ ★ ✪ ✰ ✩ ★ ✲ ✪ ✩ ★ ✭ Straubing’s theorem Corollary. Each of the three sequences , and ( ✮✞✯ ) generates the pseudovariety of all finite -trivial monoids. St Andrews 2006 – p.25/32

✽ ✷ ✽ ❃ ❃ ❀ ✽ ✸ ✼ ✼ ✺✼ ✻ ❅ ✸ ✾ ❀ ✵ ✴ ✶ ✳ ✽ ✵ ✴ ✳ ❆ ✵ ✴ ✳ ❀ Straubing’s theorem Corollary. Each of the three sequences , and ( ✹✞✺ ) generates the pseudovariety of all finite -trivial monoids. We thus have four stratifications for : ✽✿✾ ✽✿❁ ✽✿❂ ❀❄❃ ❆❈❇ St Andrews 2006 – p.25/32

❩ ❪ ❘ ❭ ❬ ❩ ❨ ❳ ❲ P ❫ ◗ ❩ ● ❭ ❬ ❭ ❨ ◗ ❭ ❬ ❩ ❨ ❨ ❬ ❭ ❙ ❳ ❲ ❖ ❱ ❊ ❚ ■ ◆ P ❙ ❭ ❬ ❫ ❪ ❘ ● ❭ ❬ ❩ ❨ ❘ ◗ ❬ ❩ ❋ ❭ ▼ ▼ ❑▼ ▲ ● ■ ❍ ❊ ❊ ❬ ● ❉ ❪ ❋ ❊ ❉ ❫ ❋ ❊ ❉ ◆ ❩ ❩ ◆ ❨ ❳ ❲ ❯ ◆ ❖ ❊ ❚ ■ P ❨ ❙ ❙ ❨ ❩ P ❬ P ❭ ❊ ◆ ❨ Straubing’s theorem Corollary. Each of the three sequences , and ( ❏✞❑ ) generates the pseudovariety of all finite -trivial monoids. We thus have four stratifications for : ◆✿❖ ◆✿◗ ◆✿❘ P❄❙ ❯❈❱ P❄❙ St Andrews 2006 – p.25/32

Straubing’s theorem: a refinement Surprisingly enough, the four stratifications coincide: St Andrews 2006 – p.26/32

❤ ❡ ❤ ✐ ❣ ❢ ❢ ❣ ❡ ❝❡ ❣ ❞ ❤ ❛ ❵ ❥ ❴ ❢ ❢ Straubing’s theorem: a refinement Surprisingly enough, the four stratifications coincide: Theorem. ( , 2004) For every , ❜✞❝ each of the monoids , , generates the pseudovariety . St Andrews 2006 – p.26/32

t r s s r t t s r ✈ q ✉ q ♦q ♣ ❧ ♠ ❧ ✈ ❦ r r Straubing’s theorem: a refinement Surprisingly enough, the four stratifications coincide: Theorem. ( , 2004) For every , ♥✞♦ each of the monoids , , generates the pseudovariety . Thus, for each the pseudovariety is generated by a single finite monoid. It easily follows from some basic universal algebra that the PMP for a (pseudo)variety generated by a single finite algebra is always decidable. St Andrews 2006 – p.26/32

❶ ⑨ ❶ ⑦ ① ⑨ ⑧ ⑦ ⑩ ② ⑨ ⑧ ⑦ ⑤ ⑧ ① ⑦ ④⑥ ⑥ ⑥ ⑥ ④⑥ ⑤ ⑥ ② ① ❶ ✇ ⑦ ⑦ Straubing’s theorem: a refinement Surprisingly enough, the four stratifications coincide: Theorem. ( , 2004) For every , ③✞④ each of the monoids , , generates the pseudovariety . Thus, for each the pseudovariety is generated by a single finite monoid. It easily follows from some basic universal algebra that the PMP for a (pseudo)variety generated by a single finite algebra is always decidable. Corollary. (Jean-Eric Pin, 1984) For each ③✞④ , the membership problem for the pseudovariety is decidable, and hence, given a piecewise testable language, its height can be algorithmically determined. St Andrews 2006 – p.26/32

Recognizing height Is this an efficient solution? St Andrews 2006 – p.27/32

Recognizing height Is this an efficient solution? It doesn’t seem so — even if we allow ourselves the luxury of the language being presented by its syntactic monoid rather than by its minimal automaton. St Andrews 2006 – p.27/32

➄ ❹ ➃ ❺ ❻ ❸ ➀ ❿ ❾ ❽ ❷ ❻ ❷ ❼ ❷ ❺ ❷ ❸ ❻ ❹ ❷ ❷ Recognizing height Is this an efficient solution? It doesn’t seem so — even if we allow ourselves the luxury of the language being presented by its syntactic monoid rather than by its minimal automaton. Indeed, if and , then the only known time bound for the algorithm that recognizes whether or ➁✘➂ not belongs to is — so requires doubly exponential time (as a function of ). St Andrews 2006 – p.27/32

➉ ➅ ➍ ➎ ➆ ➈ ➊ ➇ ➅ ➉ ➑ ➋ ➈ ➇ ➅ ➆ ➅ ➒ ➅ ➉ ➅ ➌ Recognizing height Is this an efficient solution? It doesn’t seem so — even if we allow ourselves the luxury of the language being presented by its syntactic monoid rather than by its minimal automaton. Indeed, if and , then the only known time bound for the algorithm that recognizes whether or ➏✘➐ not belongs to is — so requires doubly exponential time (as a function of ). Can we do better? St Andrews 2006 – p.27/32

Recognizing height via identities Lemma. (Eilenberg-Sch¨ utzenberger, 1976) Every pseudovariety generated by a single finite monoid is equational, that is, it consists precisely of finite monoids satisfying a certain system of usual monoid identities. St Andrews 2006 – p.28/32

Recognizing height via identities Lemma. (Eilenberg-Sch¨ utzenberger, 1976) Every pseudovariety generated by a single finite monoid is equational, that is, it consists precisely of finite monoids satisfying a certain system of usual monoid identities. A monoid is said to be finitely based if all identities holding in follow from a finite set of such identities (an identity basis of ). St Andrews 2006 – p.28/32

↔ ➔ ➓ ➣ → Recognizing height via identities Lemma. (Eilenberg-Sch¨ utzenberger, 1976) Every pseudovariety generated by a single finite monoid is equational, that is, it consists precisely of finite monoids satisfying a certain system of usual monoid identities. A monoid is said to be finitely based if all identities holding in follow from a finite set of such identities (an identity basis of ). If we know a finite identity basis of a monoid then we can use it to efficiently decide the membership in . St Andrews 2006 – p.28/32