Monadic second order logic as a model companion Sam van Gool - PowerPoint PPT Presentation

Monadic second order logic as a model companion Sam van Gool samvangool@me.com University of Utrecht (until 31 Aug) / Paris (starting 1 Sept) Logic Colloquium, Prague 13 August 2019

Automata and logic: example ◮ A programming problem: given a natural number in binary, w ∈ { 0 , 1 } ∗ , determine if w is congruent to 1 modulo 3. 1 / 9

Automata and logic: example ◮ A programming problem: given a natural number in binary, w ∈ { 0 , 1 } ∗ , determine if w is congruent to 1 modulo 3. ◮ Solution 1: a (deterministic) automaton A : 0 1 1 0 q 0 q 1 q 2 1 0 Answer yes iff A accepts w . 1 / 9

Automata and logic: example ◮ A programming problem: given a natural number in binary, w ∈ { 0 , 1 } ∗ , determine if w is congruent to 1 modulo 3. ◮ Solution 1: a (deterministic) automaton A : 0 1 1 0 q 0 q 1 q 2 1 0 Answer yes iff A accepts w . ◮ Solution 2: a monadic second order formula ϕ ( W 0 , W 1 ) : ∃ Q 0 ∃ Q 1 ∃ Q 2 ( Q 0 ( first ) ∧ Q 1 ( last ) ∧ Partition ( Q 0 , Q 1 , Q 2 ) ∧ ∀ x ([ W 0 ( x ) ∧ Q 0 ( x ) → Q 0 ( S x )] ∧ [ W 1 ( x ) ∧ Q 0 ( x ) → Q 1 ( S x )] ∧ . . . )) Answer yes iff w = ( W 0 , W 1 ) makes ϕ true. 1 / 9

Regular languages Regular languages over a finite alphabet Σ are subsets L ⊆ Σ ω which are ... ◮ recognizable by a finite automaton; or, equivalently, ◮ definable by a formula of S1S, the monadic second order logic of one successor. Büchi 1960 2 / 9

A model complete theory A functional language L : Boolean algebra operations ( ⊥ , ∪ , − ), two unary functions, X and F , and a constant I . 3 / 9

A model complete theory A functional language L : Boolean algebra operations ( ⊥ , ∪ , − ), two unary functions, X and F , and a constant I . The Boolean algebra P ( ω ) is an L -structure with: ◮ X a := { t ∈ ω | t + 1 ∈ a } , ◮ F a := { t ∈ ω | ∃ t ′ ≥ t : t ′ ∈ a } , ◮ I := { 0 } . 3 / 9

A model complete theory A functional language L : Boolean algebra operations ( ⊥ , ∪ , − ), two unary functions, X and F , and a constant I . The Boolean algebra P ( ω ) is an L -structure with: ◮ X a := { t ∈ ω | t + 1 ∈ a } , ◮ F a := { t ∈ ω | ∃ t ′ ≥ t : t ′ ∈ a } , ◮ I := { 0 } . Theorem The first order L -theory of P ( ω ) is model complete. A theory T ∗ is model complete iff every formula is T ∗ -equivalent to an existential formula. Ghilardi, G. JSL 2017 3 / 9

Proving model completeness with automata L -theory of P ( ω ) 4 / 9

Proving model completeness with automata S1S “standard translation” L -theory of P ( ω ) 4 / 9

Proving model completeness with automata S1S “standard translation” Büchi’s Theorem L -theory of P ( ω ) Word automaton 4 / 9

Proving model completeness with automata S1S “standard translation” Büchi’s Theorem L -theory of P ( ω ) Word automaton existential L -description 4 / 9

Proving model completeness with automata S1S “standard translation” Büchi’s Theorem L -theory of P ( ω ) Word automaton existential L -description existential L -description 4 / 9

An existential L -description of a word automaton ◮ Let A = ( Q , Σ , δ, q 0 , F ) be a word automaton over a finite alphabet Σ , i.e., a function δ : Q × Σ → P ( Q ) , an initial state q 0 ∈ Q and a subset F ⊆ Q of final states. 5 / 9

An existential L -description of a word automaton ◮ Let A = ( Q , Σ , δ, q 0 , F ) be a word automaton over a finite alphabet Σ , i.e., a function δ : Q × Σ → P ( Q ) , an initial state q 0 ∈ Q and a subset F ⊆ Q of final states. ◮ Write Σ = { 0 , . . . , s } , Q = { 0 , . . . , m } , q 0 = 0. ◮ A word W : ω → Σ is a partition ( W 0 , . . . , W s ) of ω ; W j = W − 1 ( j ) . 5 / 9

An existential L -description of a word automaton ◮ Let A = ( Q , Σ , δ, q 0 , F ) be a word automaton over a finite alphabet Σ , i.e., a function δ : Q × Σ → P ( Q ) , an initial state q 0 ∈ Q and a subset F ⊆ Q of final states. ◮ Write Σ = { 0 , . . . , s } , Q = { 0 , . . . , m } , q 0 = 0. ◮ A word W : ω → Σ is a partition ( W 0 , . . . , W s ) of ω ; W j = W − 1 ( j ) . Key Observation. The automaton A accepts a word W : ω → Σ iff P ( ω ) , [ w i �→ W i ] | = α ( w 0 , . . . , w s ) , where α is the ∃ L -formula: 5 / 9

An existential L -description of a word automaton ◮ Let A = ( Q , Σ , δ, q 0 , F ) be a word automaton over a finite alphabet Σ , i.e., a function δ : Q × Σ → P ( Q ) , an initial state q 0 ∈ Q and a subset F ⊆ Q of final states. ◮ Write Σ = { 0 , . . . , s } , Q = { 0 , . . . , m } , q 0 = 0. ◮ A word W : ω → Σ is a partition ( W 0 , . . . , W s ) of ω ; W j = W − 1 ( j ) . Key Observation. The automaton A accepts a word W : ω → Σ iff P ( ω ) , [ w i �→ W i ] | = α ( w 0 , . . . , w s ) , where α is the ∃ L -formula:   � � ∃ q 0 , . . . , q m ( “the q i partition ω ” ∧  q i ∩ w j ⊆ X q k  0 ≤ i ≤ m k ∈ δ ( i , j ) 0 ≤ j ≤ s 5 / 9

An existential L -description of a word automaton ◮ Let A = ( Q , Σ , δ, q 0 , F ) be a word automaton over a finite alphabet Σ , i.e., a function δ : Q × Σ → P ( Q ) , an initial state q 0 ∈ Q and a subset F ⊆ Q of final states. ◮ Write Σ = { 0 , . . . , s } , Q = { 0 , . . . , m } , q 0 = 0. ◮ A word W : ω → Σ is a partition ( W 0 , . . . , W s ) of ω ; W j = W − 1 ( j ) . Key Observation. The automaton A accepts a word W : ω → Σ iff P ( ω ) , [ w i �→ W i ] | = α ( w 0 , . . . , w s ) , where α is the ∃ L -formula:   � � ∃ q 0 , . . . , q m ( “the q i partition ω ” ∧  q i ∩ w j ⊆ X q k  0 ≤ i ≤ m k ∈ δ ( i , j ) 0 ≤ j ≤ s �� � ∧ I ⊆ q 0 ∧ F = ⊤ ) . i ∈ F q i 5 / 9

The theory is a model companion A theory T ∗ is a model companion of a theory T iff T ∗ is model complete, and T and T ∗ have the same universal consequences. Theorem The L -theory of P ( ω ) is the model companion of the theory of L -structures axiomatized by the following universal sentences: 6 / 9

The theory is a model companion A theory T ∗ is a model companion of a theory T iff T ∗ is model complete, and T and T ∗ have the same universal consequences. Theorem The L -theory of P ( ω ) is the model companion of the theory of L -structures axiomatized by the following universal sentences: ◮ equations for Boolean algebras; ◮ X is a Boolean homomorphism; ◮ F a is the least fixed point of x �→ a ∨ X x ; ◮ I is an atom which is below F a for any a � = ⊥ , and XI = ⊥ . Ghilardi, G. JSL 2017 6 / 9

Binary trees The full binary tree is 2 ∗ , finite sequences of 0’s and 1’s. 7 / 9

Binary trees The full binary tree is 2 ∗ , finite sequences of 0’s and 1’s. To obtain a model complete theory, more complex operators are needed to express acceptance by a tree automaton. 7 / 9

Binary trees The full binary tree is 2 ∗ , finite sequences of 0’s and 1’s. To obtain a model complete theory, more complex operators are needed to express acceptance by a tree automaton. A functional language L 2 : Boolean algebra operations ( ⊥ , ∪ , − ), constant I , unary operations X 0 , X 1 , binary operations EU and AF . 7 / 9

Binary trees The full binary tree is 2 ∗ , finite sequences of 0’s and 1’s. To obtain a model complete theory, more complex operators are needed to express acceptance by a tree automaton. A functional language L 2 : Boolean algebra operations ( ⊥ , ∪ , − ), constant I , unary operations X 0 , X 1 , binary operations EU and AF . The Boolean algebra P ( 2 ∗ ) is an L 2 -structure with ◮ I := { ǫ } , ◮ X i a := { t ∈ ω | t · i ∈ a } for i = 0 , 1, ◮ t ∈ EU ( a , b ) iff there E xists a path t = t 0 , . . . , t n such that, for i < n , t i ∈ a , and ( U ntil) t n ∈ b , 7 / 9

Binary trees The full binary tree is 2 ∗ , finite sequences of 0’s and 1’s. To obtain a model complete theory, more complex operators are needed to express acceptance by a tree automaton. A functional language L 2 : Boolean algebra operations ( ⊥ , ∪ , − ), constant I , unary operations X 0 , X 1 , binary operations EU and AF . The Boolean algebra P ( 2 ∗ ) is an L 2 -structure with ◮ I := { ǫ } , ◮ X i a := { t ∈ ω | t · i ∈ a } for i = 0 , 1, ◮ t ∈ EU ( a , b ) iff there E xists a path t = t 0 , . . . , t n such that, for i < n , t i ∈ a , and ( U ntil) t n ∈ b , ◮ t ∈ AF ( a , − b ) iff for A ll infinite paths t = t 0 , t 1 , . . . there is a ( F uture) t i ∈ a , provided that t j ∈ b for infinitely many j . 7 / 9

Model companion for binary trees Theorem The L 2 -theory of P ( 2 ∗ ) is model complete, and is in fact the model companion of an L 2 -theory with a finite universal axiomatization. Ghilardi, G. LICS 2016 8 / 9

Model companion for binary trees Theorem The L 2 -theory of P ( 2 ∗ ) is model complete, and is in fact the model companion of an L 2 -theory with a finite universal axiomatization. Ghilardi, G. LICS 2016 ◮ Proving model completeness crucially uses tree automata originally developed for deciding S2S (Rabin 1969). ◮ We obtain an analogous result for ‘bisimulation-invariant’ MSO, i.e., the modal µ -calculus (Janin-Walukiewicz 1996). 8 / 9