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Clearing Restarting Automata

Clearing Restarting Automata. Peter Černo František MráZ. About. We propose a new restricted version of restarting automata called C learing R estarting A utomata . About. We propose a new restricted version of restarting automata called C learing R estarting A utomata .

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Clearing Restarting Automata

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  1. Clearing Restarting Automata Peter Černo František MráZ

  2. About We propose a new restricted versionof restarting automata called ClearingRestartingAutomata.

  3. About We propose a new restricted versionof restarting automata called ClearingRestartingAutomata. The newmodel can be learned very efficiently from positive examples and its stronger version enablesto learn effectively a large class of languages.

  4. About We propose a new restricted versionof restarting automata called ClearingRestartingAutomata. The newmodel can be learned very efficiently from positive examples and its stronger version enablesto learn effectively a large class of languages. We relate the class of languages recognized byclearing restarting automata to the Chomsky hierarchy.

  5. Definition Let kbe a positiveinteger.

  6. Definition Let kbe a positiveinteger. k-clearingrestartingautomaton(k-cl-RA-automaton for short) is a couple M = (Σ, I), where:

  7. Definition • Let kbe a positiveinteger. • k-clearingrestartingautomaton(k-cl-RA-automaton for short) is a couple M = (Σ, I), where: • Σ is a finite nonempty alphabet, ¢, $ ∉ Σ.

  8. Definition • Let kbe a positiveinteger. • k-clearingrestartingautomaton(k-cl-RA-automaton for short) is a couple M = (Σ, I), where: • Σ is a finite nonempty alphabet, ¢, $ ∉ Σ. • I is a finite set of instructions(x, z, y), x∊ LCk, y ∊ RCk, z ∊ Σ+, • left context LCk = Σk∪¢.Σ≤k-1 • right context RCk = Σk∪Σ≤k-1.$

  9. Definition • Let kbe a positiveinteger. • k-clearingrestartingautomaton(k-cl-RA-automaton for short) is a couple M = (Σ, I), where: • Σ is a finite nonempty alphabet, ¢, $ ∉ Σ. • I is a finite set of instructions(x, z, y), x∊ LCk, y ∊ RCk, z ∊ Σ+, • left context LCk = Σk∪¢.Σ≤k-1 • right context RCk = Σk∪Σ≤k-1.$ • The special symbols: ¢ and $ are called sentinels.

  10. Definition • Let kbe a positiveinteger. • k-clearingrestartingautomaton(k-cl-RA-automaton for short) is a couple M = (Σ, I), where: • Σ is a finite nonempty alphabet, ¢, $ ∉ Σ. • I is a finite set of instructions(x, z, y), x∊ LCk, y ∊ RCk, z ∊ Σ+, • left context LCk = Σk∪¢.Σ≤k-1 • right context RCk = Σk∪Σ≤k-1.$ • The special symbols: ¢ and $ are called sentinels. • The width of the instructioni = (x, z, y) is |i| = |xzy|.

  11. Definition • A word w = uzvcan be rewritten to uv (uzv⊢Muv) if and only if there exist an instruction i = (x, z, y) ∊ I such that: • x⊒ ¢.u(x is a suffix of ¢.u) • y⊑ v.$ (y is a prefix of v.$)

  12. Definition • A word w = uzvcan be rewritten to uv (uzv⊢Muv) if and only if there exist an instruction i = (x, z, y) ∊ I such that: • x⊒ ¢.u(x is a suffix of ¢.u) • y⊑ v.$ (y is a prefix of v.$) • A word w is accepted if and only if w ⊢*M λwhere ⊢*M is reflexive and transitive closure of ⊢M.

  13. Definition • A word w = uzvcan be rewritten to uv (uzv⊢Muv) if and only if there exist an instruction i = (x, z, y) ∊ I such that: • x⊒ ¢.u(x is a suffix of ¢.u) • y⊑ v.$ (y is a prefix of v.$) • A word w is accepted if and only if w ⊢*M λwhere ⊢*M is reflexive and transitive closure of ⊢M . • The k-cl-RA-automatonMrecognizes the language L(M) = {w ∊ Σ* | M accepts w}.

  14. Definition By cl-RA we will denote the class of all clearingrestarting automata.

  15. Definition By cl-RA we will denote the class of all clearingrestarting automata. ℒ(k-cl-RA) denotes the class of all languages accepted by k-cl-RA-automata.

  16. Definition By cl-RA we will denote the class of all clearingrestarting automata. ℒ(k-cl-RA) denotes the class of all languages accepted by k-cl-RA-automata. Similarly ℒ(cl-RA) denotes the class of all languages accepted by cl-RA-automata.

  17. Definition By cl-RA we will denote the class of all clearingrestarting automata. ℒ(k-cl-RA) denotes the class of all languages accepted by k-cl-RA-automata. Similarly ℒ(cl-RA) denotes the class of all languages accepted by cl-RA-automata. ℒ(cl-RA) = ⋃k≥1ℒ(k-cl-RA).

  18. Definition By cl-RA we will denote the class of all clearingrestarting automata. ℒ(k-cl-RA) denotes the class of all languages accepted by k-cl-RA-automata. Similarly ℒ(cl-RA) denotes the class of all languages accepted by cl-RA-automata. ℒ(cl-RA) = ⋃k≥1ℒ(k-cl-RA). Note: For every cl-RA M: λ ⊢*M λhence λ∊ L(M). If we say that cl-RA Mrecognizes a language L, we mean that L(M) = L ∪ {λ}.

  19. Motivation • This model was inspired by the Associative Language Descriptions (ALD) model: • By Alessandra Cherubini, Stefano Crespi-Reghizzi, Matteo Pradella, Pierluigi San Pietro. • See: http://home.dei.polimi.it/sanpietr/ALD/ALD.html

  20. Motivation • This model was inspired by the Associative Language Descriptions (ALD) model: • By Alessandra Cherubini, Stefano Crespi-Reghizzi, Matteo Pradella, Pierluigi San Pietro. • See: http://home.dei.polimi.it/sanpietr/ALD/ALD.html • The simplicity of cl-RA model implies that the investigation of its properties is not so difficult and also the learning of languages is easy.

  21. Motivation • This model was inspired by the Associative Language Descriptions (ALD) model: • By Alessandra Cherubini, Stefano Crespi-Reghizzi, Matteo Pradella, Pierluigi San Pietro. • See: http://home.dei.polimi.it/sanpietr/ALD/ALD.html • The simplicity of cl-RA model implies that the investigation of its properties is not so difficult and also the learning of languages is easy. • Another important advantage of this model is that the instructions are human readable.

  22. Example Language L = {anbn | n ≥ 0}.

  23. Example • Language L = {anbn | n ≥ 0}. • Can be recognized by the 1-cl-RA M = ({a, b}, I), where the instructions I are: • R1 = (a, ab, b) • R2 = (¢, ab, $)

  24. Example • Language L = {anbn | n ≥ 0}. • Can be recognized by the 1-cl-RA M = ({a, b}, I), where the instructions I are: • R1 = (a, ab, b) • R2 = (¢, ab, $) • For instance: • aaaabbbb ⊢R1aaabbb

  25. Example • Language L = {anbn | n ≥ 0}. • Can be recognized by the 1-cl-RA M = ({a, b}, I), where the instructions I are: • R1 = (a, ab, b) • R2 = (¢, ab, $) • For instance: • aaaabbbb ⊢R1aaabbb⊢R1aabb

  26. Example • Language L = {anbn | n ≥ 0}. • Can be recognized by the 1-cl-RA M = ({a, b}, I), where the instructions I are: • R1 = (a, ab, b) • R2 = (¢, ab, $) • For instance: • aaaabbbb ⊢R1aaabbb⊢R1aabb⊢R1ab

  27. Example • Language L = {anbn | n ≥ 0}. • Can be recognized by the 1-cl-RA M = ({a, b}, I), where the instructions I are: • R1 = (a, ab, b) • R2 = (¢, ab, $) • For instance: • aaaabbbb ⊢R1aaabbb⊢R1aabb⊢R1ab⊢R2λ. • Now we see that the word aaaabbbb is accepted because aaaabbbb ⊢*Mλ.

  28. Some Theorems • Error preserving property: Let M = (Σ, I)be a cl-RA-automaton and u, v be two words from Σ*. If u ⊢*M v and u ∉ L(M), then v ∉ L(M). • Proof. v ∊ L(M) ⇒ v ⊢*M λ ⇒ u ⊢*M v ⊢*M λ ⇒ u ∊ L(M). ∎

  29. Some Theorems • Error preserving property: Let M = (Σ, I)be a cl-RA-automaton and u, v be two words from Σ*. If u ⊢*M v and u ∉ L(M), then v ∉ L(M). • Proof. v ∊ L(M) ⇒ v ⊢*M λ ⇒ u ⊢*M v ⊢*M λ ⇒ u ∊ L(M). ∎ • Observation: For each finite L ⊆ Σ*there exist 1-cl-RA-automaton M such that L(M) = L ∪ {λ}. • Proof. Suppose L = {w1, …, wn}. Consider I = {(¢, w1, $), …, (¢, wn, $)}. ∎

  30. Some Theorems • Theorem: ℒ(k-cl-RA) ⊂ ℒ((k+1)-cl-RA), for all k ≥ 1. • Note: The following language: { (ckack)n(ckbck)n | n ≥ 0 } belongs to ℒ((k+1)-cl-RA) - ℒ(k-cl-RA).

  31. Some Theorems • Theorem: ℒ(k-cl-RA) ⊂ ℒ((k+1)-cl-RA), for all k ≥ 1. • Note: The following language: { (ckack)n(ckbck)n | n ≥ 0 } belongs to ℒ((k+1)-cl-RA) - ℒ(k-cl-RA). • Theorem: For each regular language L ⊆ Σ* there exist a k-cl-RA-automatonM : L(M) = L ∪ {λ}.

  32. Some Theorems • Theorem: ℒ(k-cl-RA) ⊂ ℒ((k+1)-cl-RA), for all k ≥ 1. • Note: The following language: { (ckack)n(ckbck)n | n ≥ 0 } belongs to ℒ((k+1)-cl-RA) - ℒ(k-cl-RA). • Theorem: For each regular language L ⊆ Σ* there exist a k-cl-RA-automatonM : L(M) = L ∪ {λ}. • Proof. Based on pumping lemma for regular languages.

  33. Some Theorems • Theorem: ℒ(k-cl-RA) ⊂ ℒ((k+1)-cl-RA), for all k ≥ 1. • Note: The following language: { (ckack)n(ckbck)n | n ≥ 0 } belongs to ℒ((k+1)-cl-RA) - ℒ(k-cl-RA). • Theorem: For each regular language L ⊆ Σ* there exist a k-cl-RA-automatonM : L(M) = L ∪ {λ}. • Proof. Based on pumping lemma for regular languages. • For each z ∊ Σ*, |z|=nthere exist u, v, wsuch that |v|≥1and δ(q0, uv) = δ(q0, u); the word v can be crossed out.

  34. Some Theorems • Theorem: ℒ(k-cl-RA) ⊂ ℒ((k+1)-cl-RA), for all k ≥ 1. • Note: The following language: { (ckack)n(ckbck)n | n ≥ 0 } belongs to ℒ((k+1)-cl-RA) - ℒ(k-cl-RA). • Theorem: For each regular language L ⊆ Σ* there exist a k-cl-RA-automatonM : L(M) = L ∪ {λ}. • Proof. Based on pumping lemma for regular languages. • For each z ∊ Σ*, |z|=nthere exist u, v, wsuch that |v|≥1and δ(q0, uv) = δ(q0, u); the word v can be crossed out. • We add corresponding instruction iz = (¢.u, v, w).

  35. Some Theorems • Theorem: ℒ(k-cl-RA) ⊂ ℒ((k+1)-cl-RA), for all k ≥ 1. • Note: The following language: { (ckack)n(ckbck)n | n ≥ 0 } belongs to ℒ((k+1)-cl-RA) - ℒ(k-cl-RA). • Theorem: For each regular language L ⊆ Σ* there exist a k-cl-RA-automatonM : L(M) = L ∪ {λ}. • Proof. Based on pumping lemma for regular languages. • For each z ∊ Σ*, |z|=nthere exist u, v, wsuch that |v|≥1and δ(q0, uv) = δ(q0, u); the word v can be crossed out. • We add corresponding instruction iz = (¢.u, v, w). • For each accepted z ∊ Σ<n - {λ} we add instruction iz = (¢, z, $).

  36. Some Theorems Theorem: The language L1 = {ancbn | n ≥ 0} ∪ {λ} is not recognized by any cl-RA-automaton.

  37. Some Theorems • Theorem: The language L1 = {ancbn | n ≥ 0} ∪ {λ} is not recognized by any cl-RA-automaton. • Note: L1 can be recognized by a simple RRWW-automaton. Moreover L1 is a context-free language, thus we get the following corollary: • Corollary: • ℒ(cl-RA) ⊂ ℒ(RRWW). • CFL - ℒ(cl-RA) ≠ ∅.

  38. Some Theorems Let L2 = {anbn | n≥0} and L3 = {anb2n | n≥0} be two sample languages. Apparently both L2 and L3 are recognized by 1-cl-RA-automata.

  39. Some Theorems Let L2 = {anbn | n≥0} and L3 = {anb2n | n≥0} be two sample languages. Apparently both L2 and L3 are recognized by 1-cl-RA-automata. Theorem: Languages L2∪ L3 and L2 . L3 are not recognized by any cl-RA-automaton.

  40. Some Theorems • Let L2 = {anbn | n≥0} and L3 = {anb2n | n≥0} be two sample languages. Apparently both L2 and L3 are recognized by 1-cl-RA-automata. • Theorem: Languages L2∪ L3 and L2 . L3 are not recognized by any cl-RA-automaton. • Corollary: ℒ(cl-RA) is not closed under union, concatenation, and homomorphism. • For homomorphism use{anbn | n≥0} ∪ {cnd2n | n≥0} and homomorphism defined as: a ↦ a, b ↦ b, c ↦ a, d ↦ b. ∎

  41. Some Theorems • It is easy to see that each of the following languages: • L4 = {ancbn | n ≥ 0} ∪ {ambm | m ≥ 0} • L5 = {ancbm | n, m ≥ 0} ∪ {λ} • L6 = {ambm | m ≥ 0} can be recognized by a 1-cl-RA-automaton.

  42. Some Theorems • It is easy to see that each of the following languages: • L4 = {ancbn | n ≥ 0} ∪ {ambm | m ≥ 0} • L5 = {ancbm | n, m ≥ 0} ∪ {λ} • L6 = {ambm | m ≥ 0} can be recognized by a 1-cl-RA-automaton. • Corollary:ℒ(cl-RA) is not closed under: • intersection: L1 = L4 ∩ L5.

  43. Some Theorems • It is easy to see that each of the following languages: • L4 = {ancbn | n ≥ 0} ∪ {ambm | m ≥ 0} • L5 = {ancbm | n, m ≥ 0} ∪ {λ} • L6 = {ambm | m ≥ 0} can be recognized by a 1-cl-RA-automaton. • Corollary:ℒ(cl-RA) is not closed under: • intersection: L1 = L4 ∩ L5. • intersection with regular language: L5 is regular.

  44. Some Theorems • It is easy to see that each of the following languages: • L4 = {ancbn | n ≥ 0} ∪ {ambm | m ≥ 0} • L5 = {ancbm | n, m ≥ 0} ∪ {λ} • L6 = {ambm | m ≥ 0} can be recognized by a 1-cl-RA-automaton. • Corollary:ℒ(cl-RA) is not closed under: • intersection: L1 = L4 ∩ L5. • intersection with regular language: L5 is regular. • set difference: L1 = (L4- L6) ∪ {λ} .

  45. Parentheses The following instruction of 1-cl-RAM is enough for recognizing the language of correct parentheses: (λ, ( ), λ)

  46. Parentheses • The following instruction of 1-cl-RAM is enough for recognizing the language of correct parentheses: • (λ, ( ), λ) • Note: This instruction represents a set of instructions: • ({¢}∪Σ, ( ), Σ∪{$}), where Σ = {(, )} and • (A, w, B) = {(a, w, b) | a∊A, b∊B}.

  47. Parentheses • The following instruction of 1-cl-RAM is enough for recognizing the language of correct parentheses: • (λ, ( ), λ) • Note: This instruction represents a set of instructions: • ({¢}∪Σ, ( ), Σ∪{$}), where Σ = {(, )} and • (A, w, B) = {(a, w, b) | a∊A, b∊B}. • Note: We use the following notation for the (A, w, B): A B w

  48. Arithmetic Expressions • Suppose that we want to check correctness of arithmetic expressions over the alphabet Σ = {α, +, *, (, )}.

  49. Arithmetic Expressions • Suppose that we want to check correctness of arithmetic expressions over the alphabet Σ = {α, +, *, (, )}. • For example α+(α*α+α)is correct, α*+αis not.

  50. Arithmetic Expressions • Suppose that we want to check correctness of arithmetic expressions over the alphabet Σ = {α, +, *, (, )}. • For example α+(α*α+α)is correct, α*+αis not. • The priority of the operations is considered.

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