CHURCHS SYNTHESIS PROBLEM and its EXTENSIONS Alexander Rabinovich - PowerPoint PPT Presentation

CHURCH’S SYNTHESIS PROBLEM and its EXTENSIONS Alexander Rabinovich Tel-Aviv University, Israel http://www.tau.ac.il/ ∼ rabinoa 4-th ISLA January, 2012

Plan of the Course 1 The Church problem - logic and automata. 2 Games - basic notions. 3 Memoryless determinacy. 4 Finite memory determinacy 5 Applications and trends.

Sources 1 D. Perrin, J.E. Pin. Infinite Words, Elsevier, Amsterdam 2004. 2 E. Gr¨ adel, W. Thomas, Th. Wilke (Eds). Automata, Logics, and Infinite Games, Springer LNCS 2500 (2002). 3 Publications in “Game training network”.

Sources 1 D. Perrin, J.E. Pin. Infinite Words, Elsevier, Amsterdam 2004. 2 E. Gr¨ adel, W. Thomas, Th. Wilke (Eds). Automata, Logics, and Infinite Games, Springer LNCS 2500 (2002). 3 Publications in “Game training network”.

Synthesis Problem Input: A specification S ( I, O ) Task: Find a program P which implements S , i.e., ∀ I ( S ( I, P ( I )) .

Synthesis Problem Input: A specification S ( I, O ) Task: Find a program P which implements S , i.e., ∀ I ( S ( I, P ( I )) . Formal and Expressive Specification and Implementation languages.

Church’s Problem Consider a bit by bit transformation of bit streams: …I t …I 3 I 2 I 1 …O t …O 3 O 2 O 1 F Church’s Problem: For a given I-O specification fill the box.

Church’s Problem Consider a bit by bit transformation of bit streams: …I t …I 3 I 2 I 1 …O t …O 3 O 2 O 1 F Church’s Problem: For a given I-O specification fill the box. Given a logical specification of the input-output relation R find a causal mapping (implementation) F : I → F ( I ) such that ( I, F ( I )) ∈ R for all I .

Church’s Problem Consider a bit by bit transformation of bit streams: …I t …I 3 I 2 I 1 …O t …O 3 O 2 O 1 F Church’s Problem: For a given I-O specification fill the box. Given a logical specification of the input-output relation R find a causal mapping (implementation) F : I → F ( I ) such that ( I, F ( I )) ∈ R for all I . Causal-operator - the output bit O t at moment t depends only on I 1 I 2 . . . I t .

Church’s Problem Consider a bit by bit transformation of bit streams: …I t …I 3 I 2 I 1 …O t …O 3 O 2 O 1 F Church’s Problem: For a given I-O specification fill the box. Given a logical specification of the input-output relation R find a causal mapping (implementation) F : I → F ( I ) such that ( I, F ( I )) ∈ R for all I . Causal-operator - the output bit O t at moment t depends only on I 1 I 2 . . . I t . Synthesis ∼ games; Causal operators ∼ strategies.

Example …I t …I 3 I 2 I 1 …O t …O 3 O 2 O 1 F Consider R defined by If all I ( t ) = 0 then all O ( t ) = 0; otherwise all O ( t ) = 1.

Example …I t …I 3 I 2 I 1 …O t …O 3 O 2 O 1 F Consider R defined by If all I ( t ) = 0 then all O ( t ) = 0; otherwise all O ( t ) = 1. It is impossible to implement this R by a causal operator.

Example Consider R defined by the conjunction of three conditions on the input-output stream (I, O): 1 ∀ t ( I ( t ) = 1 → O ( t ) = 1) 2 never O ( t ) = O ( t + 1) = 0 3 If infinitely often I ( t ) = 0 then infinitely often O ( t ) = 0

Example Consider R defined by the conjunction of three conditions on the input-output stream (I, O): 1 ∀ t ( I ( t ) = 1 → O ( t ) = 1) 2 never O ( t ) = O ( t + 1) = 0 3 If infinitely often I ( t ) = 0 then infinitely often O ( t ) = 0 Common-Sense Solution 1 for input 1 produce output 1 2 for input 0 produce output 1 if last output was 0 output 0 if last output was 1

Example Consider R defined by the conjunction of three conditions on the input-output stream (I, O): 1 ∀ t ( I ( t ) = 1 → O ( t ) = 1) 2 never O ( t ) = O ( t + 1) = 0 3 If infinitely often I ( t ) = 0 then infinitely often O ( t ) = 0 Common-Sense Solution 1 for input 1 produce output 1 1/1 1/1 2 for input 0 produce Last 1 Last 0 0/1 output 1 if last output was 0 output 0 if last output 0/0 was 1 Can be described by a finite state automaton with output.

B¨ uchi-Landweber Theorem In the examples the input-output specification R ( I, O ) can be formalized in the Monadic second-order logic of order (MLO).

B¨ uchi-Landweber Theorem In the examples the input-output specification R ( I, O ) can be formalized in the Monadic second-order logic of order (MLO). B¨ uchi-Landweber(69) proved that the Church synthesis problem is computable for MLO specification. Theorem. For every MLO formula ψ ( X, Y ) it is decidable whether there is causal operator F which implements ψ , i.e. Nat | = ∀ Xψ ( X, F ( X ))

B¨ uchi-Landweber Theorem In the examples the input-output specification R ( I, O ) can be formalized in the Monadic second-order logic of order (MLO). B¨ uchi-Landweber(69) proved that the Church synthesis problem is computable for MLO specification. Theorem. For every MLO formula ψ ( X, Y ) it is decidable whether there is causal operator F which implements ψ , i.e. Nat | = ∀ Xψ ( X, F ( X )) If such an operator exists then there is a finite state operator which implements ψ . Moreover, this finite state operator is computable from ψ .

Techniques Rich interplay of 1 Mathematical logic - Monadic Second-Order Logics 2 Automata theory - automata on infinite objects . 3 Games of infinite length.

The language for specifying temporal behavior MLO (Monadic second-order Logic of Order ) 1st-order variables x,y,z,. . . ranging over elements 2nd-order monadic variables X,Y,Z,. . . ranging over sets of elements Formulas x < y x ∈ X x ∈ P a φ ∧ φ ′ ¬ φ ∃ xφ ∃ Xφ FOMLO No second-order quantifiers ∃ Xφ

The language for specifying temporal behavior MLO (Monadic second-order Logic of Order ) 1st-order variables x,y,z,. . . ranging over elements 2nd-order monadic variables X,Y,Z,. . . ranging over sets of elements Formulas x < y x ∈ X x ∈ P a φ ∧ φ ′ ¬ φ ∃ xφ ∃ Xφ FOMLO No second-order quantifiers ∃ Xφ Models - Discrete Linear Time: T = ( N , < )

The language for specifying temporal behavior MLO (Monadic second-order Logic of Order ) 1st-order variables x,y,z,. . . ranging over elements 2nd-order monadic variables X,Y,Z,. . . ranging over sets of elements Formulas x < y x ∈ X x ∈ P a φ ∧ φ ′ ¬ φ ∃ xφ ∃ Xφ FOMLO No second-order quantifiers ∃ Xφ Models - Discrete Linear Time: T = ( N , < ) Other models - Rationals, Reals, Tree order, etc. A monadic predicate P on Nat - an ω -sequence over { 0 , 1 } . A formula ψ ( X, Y ) defines a binary relation - on ω -sequences.

The language for specifying temporal behavior MLO (Monadic second-order Logic of Order ) 1st-order variables x,y,z,. . . ranging over elements 2nd-order monadic variables X,Y,Z,. . . ranging over sets of elements Formulas x < y x ∈ X x ∈ P a φ ∧ φ ′ ¬ φ ∃ xφ ∃ Xφ FOMLO No second-order quantifiers ∃ Xφ Models - Discrete Linear Time: T = ( N , < ) Other models - Rationals, Reals, Tree order, etc. A monadic predicate P on Nat - an ω -sequence over { 0 , 1 } . A formula ψ ( X, Y ) defines a binary relation - on ω -sequences. Fundamental connection between MLO and automata theory - B¨ uchii, Trakhtenbrot, Rabin.

Examples - Formalization over ( N , < ) 1 X is infinite: Inf ( X ) := ∀ t ∃ t ′ ( t ′ > t ∧ X ( t ′ ))

Examples - Formalization over ( N , < ) 1 X is infinite: Inf ( X ) := ∀ t ∃ t ′ ( t ′ > t ∧ X ( t ′ )) 2 t 2 is a successor of t 1 : ϕ ( t 1 , t 2 ) := t 1 < t 2 ∧ ¬∃ t 3 ( t 1 < t 3 < t 2 )

Examples - Formalization over ( N , < ) 1 X is infinite: Inf ( X ) := ∀ t ∃ t ′ ( t ′ > t ∧ X ( t ′ )) 2 t 2 is a successor of t 1 : ϕ ( t 1 , t 2 ) := t 1 < t 2 ∧ ¬∃ t 3 ( t 1 < t 3 < t 2 ) 3 X is the set of even numbers X (0) ∧ ∀ tX ( t ) ↔ ¬ X ( t + 1)

Examples - Formalization over ( N , < ) 1 X is infinite: Inf ( X ) := ∀ t ∃ t ′ ( t ′ > t ∧ X ( t ′ )) 2 t 2 is a successor of t 1 : ϕ ( t 1 , t 2 ) := t 1 < t 2 ∧ ¬∃ t 3 ( t 1 < t 3 < t 2 ) 3 X is the set of even numbers X (0) ∧ ∀ tX ( t ) ↔ ¬ X ( t + 1) 4 t 1 is an even number: Even ( t 1 ) := ∃ X ( X ( t 1 ) ∧ X (0) ∧ ∀ tX ( t ) ↔ ¬ X ( t + 1))

Examples - Formalization over ( N , < ) 1 X is infinite: Inf ( X ) := ∀ t ∃ t ′ ( t ′ > t ∧ X ( t ′ )) 2 t 2 is a successor of t 1 : ϕ ( t 1 , t 2 ) := t 1 < t 2 ∧ ¬∃ t 3 ( t 1 < t 3 < t 2 ) 3 X is the set of even numbers X (0) ∧ ∀ tX ( t ) ↔ ¬ X ( t + 1) 4 t 1 is an even number: Even ( t 1 ) := ∃ X ( X ( t 1 ) ∧ X (0) ∧ ∀ tX ( t ) ↔ ¬ X ( t + 1)) 5 After every occurrence of X there is an occurrence of Y ϕ ( X, Y ) := ∀ tX ( t ) → ∃ t 1 ( t 1 > t ∧ Y ( t 1 ))

Examples - Formalization over ( N , < ) 1 X is infinite: Inf ( X ) := ∀ t ∃ t ′ ( t ′ > t ∧ X ( t ′ )) 2 t 2 is a successor of t 1 : ϕ ( t 1 , t 2 ) := t 1 < t 2 ∧ ¬∃ t 3 ( t 1 < t 3 < t 2 ) 3 X is the set of even numbers X (0) ∧ ∀ tX ( t ) ↔ ¬ X ( t + 1) 4 t 1 is an even number: Even ( t 1 ) := ∃ X ( X ( t 1 ) ∧ X (0) ∧ ∀ tX ( t ) ↔ ¬ X ( t + 1)) 5 After every occurrence of X there is an occurrence of Y ϕ ( X, Y ) := ∀ tX ( t ) → ∃ t 1 ( t 1 > t ∧ Y ( t 1 )) S1S - Second order theory of One Successor is expressive equivalent to MLO over ( N , < ).