Compositional Methods Alex Rabinovich Department of Computer - PowerPoint PPT Presentation

Compositional Methods Alex Rabinovich Department of Computer Science Tel-Aviv University – p.1/30

Composition Theorem s Composition Theorems are tools which reduce sentences about some compound structures to sentences about their parts. – p.2/30

Composition Theorem s Composition Theorems are tools which reduce sentences about some compound structures to sentences about their parts. Reduce verification of on FO structure ✂ ✟ to ✁ ✄ ✄ � ☎ ✞ ✆ ✝ ✝ ✝ verification of on . . . on . ✄ ✄ � � ☎ ☎ ✞ ✞ – p.2/30

Composition Theorem s Composition Theorems are tools which reduce sentences about some compound structures to sentences about their parts. Reduce verification of on FO structure ✂ ✟ to ✁ ✄ ✄ � ☎ ✞ ✆ ✝ ✝ ✝ verification of on . . . on . ✄ ✄ � � ☎ ☎ ✞ ✞ Composition theorems (1)Provide a very powerful technique for decidability/definability. – p.2/30

Composition Theorem s Composition Theorems are tools which reduce sentences about some compound structures to sentences about their parts. Reduce verification of on FO structure ✂ ✟ to ✁ ✄ ✄ � ☎ ✞ ✆ ✝ ✝ ✝ verification of on . . . on . ✄ ✄ � � ☎ ☎ ✞ ✞ Composition theorems (1)Provide a very powerful technique for decidability/definability. (2) Among few fundamental theorems which hold in finite models – p.2/30

Landmark papers Mostowski 1952 - Direct products and power – p.3/30

Landmark papers Mostowski 1952 - Direct products and power Feferman - Vaught 1959 -generalized product – p.3/30

Landmark papers Mostowski 1952 - Direct products and power Feferman - Vaught 1959 -generalized product Shelah 1975 -generalized sum. – p.3/30

Plan Generalized Product (Feferman Vaught 1959). Generalized Sum (Shelah 1975, Rabinovich 1997). Applications to decidability and definability. Recursively (inductively?) Defined Types. – p.4/30

The Aim of this talk: 1. explain the definitions and 2. state the composition theorems. – p.5/30

Generalized Product - Two examples Example 1 Cartesian Product. Let ✠ ( ) be a family of structures for . ✌ ✍ ☛ ☞ ✎ ✡ ✒ ✏ ✑ - Cartesian product. ✠ ✔ ✠ ✓ ✡ ✖ ✡ ✕ – p.6/30

Generalized Product - Two examples Example 1 Cartesian Product. Let ✠ ( ) be a family of structures for . ✌ ✍ ☛ ☞ ✎ ✡ ✒ ✏ ✑ - Cartesian product. ✠ ✔ ✠ ✓ ✡ ✖ ✡ ✕ Universe: all functions with domain such that ✂ ✟ . ✗ ✗ ✠ ✌ ☛ ☞ ✡ – p.6/30

Generalized Product - Two examples Example 1 Cartesian Product. Let ✠ ( ) be a family of structures for . ✌ ✍ ☛ ☞ ✎ ✡ ✒ ✏ ✑ - Cartesian product. ✠ ✔ ✠ ✓ ✡ ✖ ✡ ✕ Universe: all functions with domain such that ✂ ✟ . ✗ ✗ ✠ ✌ ☛ ☞ ✡ Interpretation of the relations: e.g. binary symbol, ✘ ✙ ✙ ✂ ✟ iff ✂ ✂ ✟ ✂ ✟ ✟ for every . ✗ ✗ ✘ ✘ ☛ ☛ ☛ ✌ ✛ ☞ ✚ ✚ ✆ ✆ – p.6/30

Generalized Product - Two examples Example 2 Ordinal Product. a linear order and ✠ ( ) a family of linear orders. ✌ ✢ ✌ ✢ ☛ ☞ ✜ ✜ ✡ is a structure for . ✠ ✔ ✠ ✤ ✓ ✡ ✖ ✣ ✡ ✕ ✞ – p.7/30

Generalized Product - Two examples Example 2 Ordinal Product. a linear order and ✠ ( ) a family of linear orders. ✌ ✢ ✌ ✢ ☛ ☞ ✜ ✜ ✡ is a structure for . ✠ ✔ ✠ ✤ ✓ ✡ ✖ ✣ ✡ ✕ ✞ Universe: like before, all functions with domain such ✗ ✌ that ✂ ✟ . ✗ ✠ ☛ ☞ ✡ – p.7/30

Generalized Product - Two examples Example 2 Ordinal Product. a linear order and ✠ ( ) a family of linear orders. ✌ ✢ ✌ ✢ ☛ ☞ ✜ ✜ ✡ is a structure for . ✠ ✔ ✠ ✤ ✓ ✡ ✖ ✣ ✡ ✕ ✞ Universe: like before, all functions with domain such ✗ ✌ that ✂ ✟ . ✗ ✠ ☛ ☞ ✡ Interpretation of (lexicographical order): ✤ ✂ ✟ ✂ ✟ iff there is such that and ✗ ✌ ✢ ✗ ✥ ✥ ✥ ✤ ☞ ✤ ✚ ✜ ✚ ✙ ✙ ✦ ✂ ✟ ✂ ✟ for all . ✗ ☛ ☛ ☛ ✥ ✤ ✚ ✓ ✣ ✖ ✞ – p.7/30

Generalized Product Ingredients: a structure for ; ✢ ✌ ✍ ✜ ✡ ✣ ✞ – p.8/30

Generalized Product Ingredients: a structure for ; ✢ ✌ ✍ ✜ ✡ ✣ ✞ A family ✠ ( ✟ of structures for ; ✌ ✢ ✍ ☛ ☞ ✜ ✎ ✡ ✒ ✏ ✑ – p.8/30

Generalized Product Ingredients: a structure for ; ✢ ✌ ✍ ✜ ✡ ✣ ✞ A family ✠ ( ✟ of structures for ; ✌ ✢ ✍ ☛ ☞ ✜ ✎ ✡ ✒ ✏ ✑ A structure ✠ for to be defined. ✍ ✧ ★ ✩ – p.8/30

Generalized Product Ingredients: a structure for ; ✢ ✌ ✍ ✜ ✡ ✣ ✞ A family ✠ ( ✟ of structures for ; ✌ ✢ ✍ ☛ ☞ ✜ ✎ ✡ ✒ ✏ ✑ A structure ✠ for to be defined. ✍ ✧ ★ ✩ Universe of the product: ✠ . ✔ ✣ ✡ ✖ ✡ ✕ ✞ – p.8/30

Generalized Product Ingredients: a structure for ; ✢ ✌ ✍ ✜ ✡ ✣ ✞ A family ✠ ( ✟ of structures for ; ✌ ✢ ✍ ☛ ☞ ✜ ✎ ✡ ✒ ✏ ✑ A structure ✠ for to be defined. ✍ ✧ ★ ✩ Universe of the product: ✠ . ✔ ✣ ✡ ✖ ✡ ✕ ✞ The interpretation of the relational symbols in will be ✍ ✧ ★ ✩ defined by “Conditions”. – p.8/30

Conditions -ary Condition - Syntax ✜ – p.9/30

Conditions -ary Condition - Syntax ✜ 1. ✂ ✟ ✂ ✟ ✂ ✟ formulas in ✫ ✫ ✫ ✫ ✫ ✫ ✪ ✪ ✪ ☎ ☎ ✬ ☎ ✭ ☎ ✞ ✞ ✞ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ . ✍ ✎ ✒ ✏ ✑ – p.9/30

Conditions -ary Condition - Syntax ✜ 1. ✂ ✟ ✂ ✟ ✂ ✟ formulas in ✫ ✫ ✫ ✫ ✫ ✫ ✪ ✪ ✪ ☎ ☎ ✬ ☎ ✭ ☎ ✞ ✞ ✞ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ . ✍ ✎ ✒ ✏ ✑ 2. ✂ ✟ a formula in the second-order monadic ✮ ✯ ✯ ☎ ✭ ✆ ✝ ✝ ✝ language for . ✍ ✖ ✣ ✞ – p.9/30

Conditions -ary Condition - Syntax ✜ 1. ✂ ✟ ✂ ✟ ✂ ✟ formulas in ✫ ✫ ✫ ✫ ✫ ✫ ✪ ✪ ✪ ☎ ☎ ✬ ☎ ✭ ☎ ✞ ✞ ✞ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ . ✍ ✎ ✒ ✏ ✑ 2. ✂ ✟ a formula in the second-order monadic ✮ ✯ ✯ ☎ ✭ ✆ ✝ ✝ ✝ language for . ✍ ✖ ✣ ✞ -ary Condition - Semantics ✜ – p.9/30

Conditions -ary Condition - Syntax ✜ 1. ✂ ✟ ✂ ✟ ✂ ✟ formulas in ✫ ✫ ✫ ✫ ✫ ✫ ✪ ✪ ✪ ☎ ☎ ✬ ☎ ✭ ☎ ✞ ✞ ✞ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ . ✍ ✎ ✒ ✏ ✑ 2. ✂ ✟ a formula in the second-order monadic ✮ ✯ ✯ ☎ ✭ ✆ ✝ ✝ ✝ language for . ✍ ✖ ✣ ✞ -ary Condition - Semantics ✜ We have to define when satisfies the condition. ✗ ✗ ☎ ✞ ✆ ✝ ✝ ✝ – p.9/30

Conditions -ary Condition - Syntax ✜ 1. ✂ ✟ ✂ ✟ ✂ ✟ formulas in ✫ ✫ ✫ ✫ ✫ ✫ ✪ ✪ ✪ ☎ ☎ ✬ ☎ ✭ ☎ ✞ ✞ ✞ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ ✆ ✝ ✝ ✝ ✆ ✆ ✝ ✝ ✝ . ✍ ✎ ✒ ✏ ✑ 2. ✂ ✟ a formula in the second-order monadic ✮ ✯ ✯ ☎ ✭ ✆ ✝ ✝ ✝ language for . ✍ ✖ ✣ ✞ -ary Condition - Semantics ✜ We have to define when satisfies the condition. ✗ ✗ ☎ ✞ ✆ ✝ ✝ ✝ Let ✱ ✂ ✟ ✂ ✟ ✳ ✂ ✟ ✴ ✢ ✠ ✗ ✗ ✰ ☛ ✌ ☛ ☛ ✫ ✫ ☞ ✜ ✲ ✪ ✓ ✓ ☎ ✡ ☎ ☎ ☎ ✞ ✞ ✆ ✆ ✝ ✝ ✝ ✆ ✝ ✝ ✝ . . . ✱ ✂ ✟ ✂ ✟ ✳ ✂ ✟ ✴ . ✰ ✌ ✢ ✠ ✗ ✗ ☛ ☛ ☛ ✫ ✫ ☞ ✜ ✪ ✲ ✓ ✓ ✭ ✭ ✡ ☎ ☎ ✞ ✞ ✆ ✆ ✝ ✝ ✝ ✆ ✝ ✝ ✝ – p.9/30