Expressive Completeness over Nat and Finite orders - PowerPoint PPT Presentation

Expressive Completeness over Nat and Finite orders MLO=Automata=regular expressions (over finite orders). – p.1/12

Expressive Completeness over Nat and Finite orders MLO=Automata=regular expressions (over finite orders). MLO= -Automata= -regular expressions (over Nat). � � – p.1/12

Expressive Completeness over Nat and Finite orders MLO=Automata=regular expressions (over finite orders). MLO= -Automata= -regular expressions (over Nat). � � FOMLO=TL(U,S) (over Dedekind complete orders) – p.1/12

Expressive Completeness over Nat and Finite orders MLO=Automata=regular expressions (over finite orders). MLO= -Automata= -regular expressions (over Nat). � � FOMLO=TL(U,S) (over Dedekind complete orders) FOMLO= star free regular expressions (over finite orders) – p.1/12

Expressive Completeness over Nat and Finite orders MLO=Automata=regular expressions (over finite orders). MLO= -Automata= -regular expressions (over Nat). � � FOMLO=TL(U,S) (over Dedekind complete orders) FOMLO= star free regular expressions (over finite orders) FOMLO = Counter-free automata (over finite orders) – p.1/12

Counter-free automata Def. A sequence of states (for ) in an ✠ ✟ ✁ ✁ ✁ ✞ ✂ ☎ ✝ ✄ ✄ ✆ ✆ ✆ automaton is a counter for a string if ✌ ✎ ✡ ☞ ☛ ✁ ☛ ✁ ✏ ✍ ✍ ☎ ✑ ✄ where by convention . ✁ ✁ ✏ ✂ ☎ ✑ ✝ – p.2/12

Counter-free automata Def. A sequence of states (for ) in an ✠ ✟ ✁ ✁ ✁ ✞ ✂ ☎ ✝ ✄ ✄ ✆ ✆ ✆ automaton is a counter for a string if ✌ ✎ ✡ ☞ ☛ ✁ ☛ ✁ ✏ ✍ ✍ ☎ ✑ ✄ where by convention . ✁ ✁ ✏ ✂ ☎ ✑ ✝ Def. An automaton is counter-free iff it does not have a counter. – p.2/12

Counter-free automata Def. A sequence of states (for ) in an ✠ ✟ ✁ ✁ ✁ ✞ ✂ ☎ ✝ ✄ ✄ ✆ ✆ ✆ automaton is a counter for a string if ✌ ✎ ✡ ☞ ☛ ✁ ☛ ✁ ✏ ✍ ✍ ☎ ✑ ✄ where by convention . ✁ ✁ ✏ ✂ ☎ ✑ ✝ Def. An automaton is counter-free iff it does not have a counter. Theorem (MacNaughton) A language is definable by FOMLO formula iff it is accepted by a deterministic counter-free au- tomaton iff it is definable by a star free regular expression. – p.2/12

The complexity of TL(U) over Nat Theorem The satisfiability problem for TL(U) over Nat is in PSPACE. – p.3/12

The complexity of TL(U) over Nat Theorem The satisfiability problem for TL(U) over Nat is in PSPACE. Lemma (Small Model property) If is satisfiable then it is ✒ satisfiable on a quasi-periodic model with small ✔ ☛ ✓ ☛ ✓ ✄ ✗ ✗ ✘ ( ✌ ✚ ✚ ✎ ✕ ✖ ✒ ✙ – p.3/12

The complexity of TL(U) over Nat Theorem The satisfiability problem for TL(U) over Nat is in PSPACE. Lemma (Small Model property) If is satisfiable then it is ✒ satisfiable on a quasi-periodic model with small ✔ ☛ ✓ ☛ ✓ ✄ ✗ ✗ ✘ ( ✌ ✚ ✚ ✎ ✕ ✖ ✒ ✙ Lemma The satisfiability of over small model can be ✒ checked in NPSPACE. – p.3/12

The complexity of TL(U) over Nat Theorem The satisfiability problem for TL(U) over Nat is in PSPACE. Lemma (Small Model property) If is satisfiable then it is ✒ satisfiable on a quasi-periodic model with small ✔ ☛ ✓ ☛ ✓ ✄ ✗ ✗ ✘ ✌ ( ✚ ✚ ✎ ✕ ✖ ✒ ✙ Lemma The satisfiability of over small model can be ✒ checked in NPSPACE. Homework: Prove PSPACE lower bound for the satifiability problem – p.3/12

The complexity of TL(U) over Nat Theorem The satisfiability problem for TL(U) over Nat is in PSPACE. Lemma (Small Model property) If is satisfiable then it is ✒ satisfiable on a quasi-periodic model with small ✔ ☛ ✓ ☛ ✓ ✄ ✗ ✗ ✘ ( ✌ ✚ ✚ ✎ ✕ ✖ ✒ ✙ Lemma The satisfiability of over small model can be ✒ checked in NPSPACE. Homework: Prove PSPACE lower bound for the satifiability problem Hint: For every PSPACE TM and a word construct a ☛ formula which is satisfiable iff accepts . ✒ ☛ ✛ ✢ – p.3/12 ✜

Proof of a small model property Notations: Sub( ) - the set of subformulas of ✒ ✒ – p.4/12

Proof of a small model property Notations: Sub( ) - the set of subformulas of ✒ ✒ Example ✌ ✌ ✌ ✎ ✎ ✎ ✒ ✣ ✤ ✥ ✤ ✦ ✣ ✧ ✏ ★ The Subformulas of ✒ ✩ ✌ ✎ ✌ ✎ ✣ ✣ ✥ ✥ ✦ ✦ ✥ ✤ ✦ ✣ ✥ ✤ ✦ ✣ ✦ ✧ ✧ ✧ ★ ★ ★ ★ ★ ★ ✆ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✌ ✎ ✪ ✩ ✪ ✣ ✦ ✣ ✒ ✧ ✫ ★ ★ ★ ✄ – p.4/12

Proof of a small model property Notations: Sub( ) - the set of subformulas of ✒ ✒ Example ✌ ✌ ✌ ✎ ✎ ✎ ✒ ✣ ✤ ✥ ✤ ✦ ✣ ✧ ✏ ★ The Subformulas of ✒ ✩ ✌ ✎ ✌ ✎ ✣ ✣ ✥ ✥ ✦ ✦ ✥ ✤ ✦ ✣ ✥ ✤ ✦ ✣ ✦ ✧ ✧ ✧ ★ ★ ★ ★ ★ ★ ✆ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✌ ✎ ✪ ✩ ✪ ✣ ✦ ✣ ✒ ✧ ✫ ★ ★ ★ ✄ ✌ ✚ ✚ ✎ Number of subformulas - ✕ ✒ – p.4/12

Proof of a small model property Notations: Sub( ) - the set of subformulas of ✒ ✒ Example ✌ ✌ ✌ ✎ ✎ ✎ ✒ ✣ ✤ ✥ ✤ ✦ ✣ ✧ ✏ ★ The Subformulas of ✒ ✩ ✌ ✎ ✌ ✎ ✣ ✣ ✥ ✥ ✦ ✦ ✥ ✤ ✦ ✣ ✥ ✤ ✦ ✣ ✦ ✧ ✧ ✧ ★ ★ ★ ★ ★ ★ ✆ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✌ ✎ ✪ ✩ ✪ ✣ ✦ ✣ ✒ ✧ ✫ ★ ★ ★ ✄ ✌ ✚ ✚ ✎ Number of subformulas - ✕ ✒ Def (Type) Let be a formula be a linear order with ✒ ✡ monadic predicates and an element of . ✬ ✡ ✘ ✌ ✎ ✩ ✌ ✎✵ ✚ ✪ ✬ ✲ ✴ ✬ ✒ ✡ ✬ ✲ ✭ ✳ ✮ ✯ ✰ ☛ ✏ ✏ ✱ ✄ – p.4/12

Proof of a small model property ✘ ✘ Assume ✌ ✎ ✌ ✎ ✬ ✭ ✭ ✮ ✯ ✰ ✹ ✮ ✯ ✰ ✏ ✱ ✱ ✱ ✱ ✱ ✱ ✑ ✑ ✑ ✑ ✶ ✷ ✸ ✶ ✷ ✸ A 1 A 2 A 3 a b A 1 A 3 a Then – p.5/12

– p.5/12 Proof of a small model property ✎ ✺ ✌ ✱ ✸ ✎ ✑ ✬ ✌ A 3 A 3 ✱ ✶ ✘ ✱ ✸ ✰ ✑ ✯ ✮ ✱ ✷ ✭ ✑ ✏ b ✱ ✶ ✘ ✰ ✎ ✺ ✯ ✌ ✮ ✭ ✱ ✸ A 2 ✑ ✏ ✱ ✷ ✎ ✑ ✹ a a ✌ ✱ ✶ ✘ ✱ ✸ ✰ ✻ ✡ ✯ ✑ ✮ ✱ ✷ A 1 A 1 ✭ ✳ ✑ ✺ ✱ ✶ ✘ 1. For every ✰ ✯ ✮ ✭ Assume Then

– p.5/12 Proof of a small model property ✎ ✎ ✺ ✺ ✌ ✌ ✱ ✸ ✱ ✸ ✎ ✑ ✑ ✬ ✌ A 3 A 3 ✱ ✶ ✱ ✶ ✘ ✘ ✱ ✸ ✰ ✰ ✑ ✯ ✯ ✮ ✮ ✱ ✷ ✭ ✭ ✑ ✏ ✏ b ✱ ✶ ✘ ✰ ✎ ✎ ✺ ✺ ✯ ✌ ✌ ✮ ✭ ✱ ✸ ✱ ✸ A 2 ✑ ✑ ✏ ✱ ✷ ✱ ✷ ✎ ✑ ✑ ✹ a a ✌ ✱ ✶ ✱ ✶ ✘ ✘ ✱ ✸ ✰ ✰ ✻ ☎ ✡ ✡ ✯ ✯ ✑ ✮ ✮ ✱ ✷ A 1 A 1 ✭ ✭ ✳ ✳ ✑ ✺ ✺ ✱ ✶ ✘ 1. For every 2. For every ✰ ✯ ✮ ✭ Assume Then

Proof of a small model property Additional transformations Image of a point A 1 A 2 A 3 c a b A 2 A 2 A 3 A 1 b b a c’ c’’ ✘ ✘ Assume ✌ ✎ ✌ ✎ ✬ ✭ ✭ ✮ ✯ ✰ ✹ ✮ ✯ ✰ ✏ ✱ ✱ ✱ ✱ ✱ ✱ ✑ ✑ ✑ ✑ ✶ ✷ ✸ ✶ ✷ ✸ Then – p.6/12

Proof of a small model property Additional transformations Image of a point A 1 A 2 A 3 c a b A 2 A 2 A 3 A 1 b b a c’ c’’ ✘ ✘ Assume ✌ ✎ ✌ ✎ ✬ ✭ ✭ ✮ ✯ ✰ ✹ ✮ ✯ ✰ ✏ ✱ ✱ ✱ ✱ ✱ ✱ ✑ ✑ ✑ ✑ ✶ ✷ ✸ ✶ ✷ ✸ Then For every and its image ✼ ✺ ✘ ✘ ✌ ✎ ✌ ✎ ✼ ✭ ✭ ✮ ✯ ✰ ✺ ✮ ✯ ✰ ✏ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✑ ✑ ✑ ✑ ✑ ✶ ✷ ✸ ✶ ✷ ✷ ✸ – p.6/12

Proof of a small model property A 1 A 2 A 3 c b 3 b 1 a b 2 A 2 A 2 A 2 A 1 Assume that is an unbounded increasing sequence and ✬ ✍ ✘ ✘ ✌ ✎ ✌ ✎ for ✬ ✬ ✾ ✿ ❀ ✭ ✭ ✭ ✳ ✮ ✯ ✰ ✮ ✯ ✰ ✹ ✏ ✍ ✽ ✱ ✱ ✄ – p.7/12

Proof of a small model property A 1 A 2 A 3 c b 3 b 1 a b 2 A 2 A 2 A 2 A 1 Assume that is an unbounded increasing sequence and ✬ ✍ ✘ ✘ ✌ ✎ ✌ ✎ for ✬ ✬ ✾ ✿ ❀ ✭ ✭ ✭ ✳ ✮ ✯ ✰ ✮ ✯ ✰ ✹ ✏ ✍ ✽ ✱ ✱ ✄ ✘ For ✌ ✎ there is such that ✒ ✒ ✬ ✡ ✤ ✭ ✳ ✳ ✮ ✯ ✰ ✺ ☎ ❁ ☎ ❁ ✱ ✚ ✡ . ✒ ✺ ✏ ❁ ✄ – p.7/12

Proof of a small model property A 1 A 2 A 3 c b 3 b 1 a b 2 A 2 A 2 A 2 A 1 Assume that is an unbounded increasing sequence and ✬ ✍ ✘ ✘ ✌ ✎ ✌ ✎ for ✬ ✬ ✾ ✿ ❀ ✭ ✭ ✭ ✳ ✮ ✯ ✰ ✮ ✯ ✰ ✹ ✏ ✍ ✽ ✱ ✱ ✄ ✘ For ✌ ✎ there is such that ✒ ✒ ✬ ✡ ✤ ✭ ✳ ✳ ✮ ✯ ✰ ✺ ☎ ❁ ☎ ❁ ✱ ✚ ✡ . ✒ ✺ ✏ ❁ ✄ – p.7/12