Decidable Theories 1. Linear order. p.1/9 Decidable Theories 1. - PowerPoint PPT Presentation

Decidable Theories 1. Linear order. – p.1/9

Decidable Theories 1. Linear order. 2. Presburger arithmetic: (Natural with +) – p.1/9

Decidable Theories 1. Linear order. 2. Presburger arithmetic: (Natural with +) 3. Real Arithmetic (Reals +, , <) � – p.1/9

Decidable Theories 1. Linear order. 2. Presburger arithmetic: (Natural with +) 3. Real Arithmetic (Reals +, , <) � 4. Elementary Geometry – p.1/9

Decidable Theories 1. Linear order. 2. Presburger arithmetic: (Natural with +) 3. Real Arithmetic (Reals +, , <) � 4. Elementary Geometry 5. Linear orders with monadic predicates. – p.1/9

Decidable Theories ✂ ✆ 1. Linear order. ✁ ✄ ☎ 2. Presburger arithmetic: (Natural with +) 3. Real Arithmetic (Reals +, , <) � 4. Elementary Geometry 5. Linear orders with monadic predicates. – p.1/9

Decidable Theories ✂ ✆ 1. Linear order. ✁ ✄ ☎ ✞ ✝ 2. Presburger arithmetic: (Natural with +) ✂ ✆ ✁ ✄ 3. Real Arithmetic (Reals +, , <) � 4. Elementary Geometry 5. Linear orders with monadic predicates. – p.1/9

Decidable Theories ✂ ✆ 1. Linear order. ✁ ✄ ☎ ✞ ✝ 2. Presburger arithmetic: (Natural with +) ✂ ✆ ✁ ✄ ✝ ✞ 3. Real Arithmetic (Reals +, , <) ✂ ✆ ✁ ✄ � 4. Elementary Geometry 5. Linear orders with monadic predicates. – p.1/9

Decidable Theories ✂ ✆ 1. Linear order. ✁ ✄ ☎ ✞ ✝ 2. Presburger arithmetic: (Natural with +) ✂ ✆ ✁ ✄ ✝ ✞ 3. Real Arithmetic (Reals +, , <) ✂ ✆ ✁ ✄ � 4. Elementary Geometry 5. Linear orders with monadic predicates. tower of 2: ✟ ✝ heights n ✄ ✠ ✠ ✠ – p.1/9

Validity problem over finite structures Input: a formula ✡ Question: Is true over all finite structures? ✡ – p.2/9

Validity problem over finite structures Input: a formula ✡ Question: Is true over all finite structures? ✡ Theorem (Trakhtenbrot) There is no procedure for checking validity over finite structures. – p.2/9

Validity problem over finite structures Input: a formula ✡ Question: Is true over all finite structures? ✡ Theorem (Trakhtenbrot) There is no procedure for checking validity over finite structures. The theory of finite structures is very different from the the- ory of arbitrary structures – p.2/9

Hilbert Calculus Axioms a ✂ ✆ Ax1 ☛ ☛ ✌ ☞ ☞ ✂ ✂ ✆ ✆ ✂ ✂ ✆ ✂ ✆ ✆ ☛ ✍ ☛ ☛ ✍ Ax2 ✌ ✌ ☞ ☞ ☞ ☞ ☞ ☞ ✂ ✆ ✂ ✂ ✆ ✆ ☛ ☛ Ax3 ✌ ✌ ✌ ☞ ☞ ☞ ☞ ✎ ✎ ✎ ✂ ✂ ✆ ✆ ✒ ✔ ✕ , where is a term. Ax4 ✏ ☛ ☛ ✓ ✓ ☞ ✑ ✑ ✑ ✂ ✂ ✆ ✆ ✂ ✆ , where is not free in . Ax5 ✏ ☛ ✌ ☛ ✏ ✌ ☛ ☞ ☞ ☞ ✑ ✑ ✑ Inference Rules MP Derive from and . ✌ ☛ ☛ ✌ ☞ Gen Derive from . ✏ ☛ ☛ ✑ a We do not distinguish between formulas with the same skeleton – p.3/9

Completeness Theorem ✗ ☛ iff ☛ . Theorem (Completeness) ✖ ✖ ✣ ✘ ✛ ✢ ✤ ✥ ✜ ✙ ✚ – p.4/9

Completeness Theorem ✗ ☛ iff ☛ . Theorem (Completeness) ✖ ✖ ✣ ✘ ✛ ✢ ✤ ✥ ✜ ✙ ✚ is consis- Theorem (Completeness - Satisfiability version) ✖ tent iff holds in a structure. ✖ – p.4/9

Completeness Theorem – p.5/9

Completeness Theorem Theorem (Completeness - Satisfiability version) If is ✖ consistent, then is satisfiable (holds in a structure for ✖ FO). – p.5/9

Completeness Theorem Theorem (Completeness - Satisfiability version) If is ✖ consistent, then is satisfiable (holds in a structure for ✖ FO). Propositional Calculus – p.5/9

Completeness Theorem Theorem (Completeness - Satisfiability version) If is ✖ consistent, then is satisfiable (holds in a structure for ✖ FO). Propositional Calculus Theorem Every consistent set of formulas is a subset of a maximal consistent set of formulas. Theorem Every maximal consistent set of formulas is satisfiable. – p.5/9

Completeness Theorem Theorem (Completeness - Satisfiability version) If is ✖ consistent, then is satisfiable (holds in a structure for ✖ FO). Propositional Calculus Theorem Every consistent set of formulas is a subset of a maximal consistent set of formulas. Theorem Every maximal consistent set of formulas is satisfiable. Predicate Calculus – p.5/9

Completeness Theorem Theorem (Completeness - Satisfiability version) If is ✖ consistent, then is satisfiable (holds in a structure for ✖ FO). Propositional Calculus Theorem Every consistent set of formulas is a subset of a maximal consistent set of formulas. Theorem Every maximal consistent set of formulas is satisfiable. Predicate Calculus Theorem Every consistent set of formulas is a subset of a Complete Henkin consistent set of formulas. Theorem Every Complete Henkin consistent set of formulas holds (in a Herbrand Structure). – p.5/9

Complete Set of Formulas – p.6/9

Complete Set of Formulas is complete if for every sentence in the Definition ✖ ✦ ✡ signature , either or (sign. of ). ✦ ✖ ✖ ✖ ✦ ★ ✧ ✧ ✡ ✡ ✎ – p.6/9

Complete Set of Formulas is complete if for every sentence in the Definition ✖ ✦ ✡ signature , either or (sign. of ). ✦ ✖ ✖ ✖ ✦ ★ ✧ ✧ ✡ ✡ ✎ Theorem If is consistent then there is such that ✩ ✩ ✖ ✖ ✖ ✖ ★ is consistent and complete. ✦ – p.6/9

Complete Set of Formulas is complete if for every sentence in the Definition ✖ ✦ ✡ signature , either or (sign. of ). ✦ ✖ ✖ ✖ ✦ ★ ✧ ✧ ✡ ✡ ✎ Theorem If is consistent then there is such that ✩ ✩ ✖ ✖ ✖ ✖ ★ is consistent and complete. ✦ Proof Define a sequence of set of formulas as follows: ✖ ☎ – p.6/9

Complete Set of Formulas is complete if for every sentence in the Definition ✖ ✦ ✡ signature , either or (sign. of ). ✦ ✖ ✖ ✖ ✦ ★ ✧ ✧ ✡ ✡ ✎ Theorem If is consistent then there is such that ✩ ✩ ✖ ✖ ✖ ✖ ★ is consistent and complete. ✦ Proof Define a sequence of set of formulas as follows: ✖ ☎ Take an enumeration of all sentences. ☛ ☛ ✦ ✪ ☎ ✫ ✬ ✬ ✬ ✫ ✬ ✬ ✬ – p.6/9

Complete Set of Formulas is complete if for every sentence in the Definition ✖ ✦ ✡ signature , either or (sign. of ). ✦ ✖ ✖ ✖ ✦ ★ ✧ ✧ ✡ ✡ ✎ Theorem If is consistent then there is such that ✩ ✩ ✖ ✖ ✖ ✖ ★ is consistent and complete. ✦ Proof Define a sequence of set of formulas as follows: ✖ ☎ Take an enumeration of all sentences. ☛ ☛ ✦ ✪ ☎ ✫ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✖ ✖ ✘ ✭ ✒ ✕ if ✒ ✕ is consistent; ✖ ☛ ✖ ☛ ✯ ✯ ✪ ✪ ✮ ✮ ☎ ☎ ☎ ☎ ✖ ✘ ✪ ✮ ☎ ✒ ✕ otherwise. ☛ ✖ ✯ ✎ ✪ ✮ ☎ ☎ – p.6/9

Complete Set of Formulas is complete if for every sentence in the Definition ✖ ✦ ✡ signature , either or (sign. of ). ✦ ✖ ✖ ✖ ✦ ★ ✧ ✧ ✡ ✡ ✎ Theorem If is consistent then there is such that ✩ ✩ ✖ ✖ ✖ ✖ ★ is consistent and complete. ✦ Proof Define a sequence of set of formulas as follows: ✖ ☎ Take an enumeration of all sentences. ☛ ☛ ✦ ✪ ☎ ✫ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✖ ✖ ✘ ✭ ✒ ✕ if ✒ ✕ is consistent; ✖ ☛ ✖ ☛ ✯ ✯ ✪ ✪ ✮ ✮ ☎ ☎ ☎ ☎ ✖ ✘ ✪ ✮ ☎ ✒ ✕ otherwise. ☛ ✖ ✯ ✎ ✪ ✮ ☎ ☎ Show that is consistent by induction on . ✖ ✰ ☎ – p.6/9

Complete Set of Formulas is complete if for every sentence in the Definition ✖ ✦ ✡ signature , either or (sign. of ). ✦ ✖ ✖ ✖ ✦ ★ ✧ ✧ ✡ ✡ ✎ Theorem If is consistent then there is such that ✩ ✩ ✖ ✖ ✖ ✖ ★ is consistent and complete. ✦ Proof Define a sequence of set of formulas as follows: ✖ ☎ Take an enumeration of all sentences. ☛ ☛ ✦ ✪ ☎ ✫ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✖ ✖ ✘ ✭ ✒ ✕ if ✒ ✕ is consistent; ✖ ☛ ✖ ☛ ✯ ✯ ✪ ✪ ✮ ✮ ☎ ☎ ☎ ☎ ✖ ✘ ✪ ✮ ☎ ✒ ✕ otherwise. ☛ ✖ ✯ ✎ ✪ ✮ ☎ ☎ Show that is consistent by induction on . ✖ ✰ ☎ Show that is a consistent and -complete. ✖ ✦ ✯ ☎ ☎ – p.6/9

Henkin Sets of Formulas has Henkin property for if for every sentence Definition ✖ ✦ of the form in the signature there is a ✱ ✖ ✏ ✦ ✧ ✑ ✡ ✎ constant such that ✒ ✔ ✕ (sign. of ). ✖ ✖ ✦ ★ ✧ ✲ ✡ ✲ ✑ ✎ – p.7/9