O mputational gic L Dependency Pairs and Polynomial Path Orders - PowerPoint PPT Presentation

C O mputational gic L Dependency Pairs and Polynomial Path Orders Martin Avanzini and Georg Moser Computational Logic Faculty of Computer Science, University of Innsbruck RTA ’09

Introduction Automatic Complexity Analysis Goal ◮ purely automatic complexity analysis Approach ◮ employ term rewriting as model of computation 1. proof termination 2. extract complexity certificates from termination proof MA (ICS @ UIBK) Dependency Pairs and POP ∗ 2/17

Introduction Automatic Complexity Analysis Goal ◮ purely automatic complexity analysis Approach ◮ employ term rewriting as model of computation 1. proof termination 2. extract complexity certificates from termination proof Problem ◮ to detect feasible computation, restrictions on termination technique usually inevitable MA (ICS @ UIBK) Dependency Pairs and POP ∗ 2/17

Introduction . . . direct techniques Polynomial Path (Automatic) Orders Complexity Analysis . . . by Rewriting Weak transformation Dependency techniques Pairs . . . . . . MA (ICS @ UIBK) Dependency Pairs and POP ∗ 3/17

Introduction . . . direct techniques Polynomial Path (Automatic) Orders Complexity Analysis . . . by Rewriting Weak transformation Dependency techniques Pairs . . . . . . MA (ICS @ UIBK) Dependency Pairs and POP ∗ 3/17

Introduction Term Rewriting TRS R bits half(0) → 0 bits(0) → 0 half(s(0)) → 0 bits(s(0)) → s(0) half(s(s( x ))) → s(half( x )) bits(s(s( x ))) → s(bits(s(half( x )))) MA (ICS @ UIBK) Dependency Pairs and POP ∗ 4/17

Introduction Term Rewriting TRS R bits half(0) → 0 bits(0) → 0 half(s(0)) → 0 bits(s(0)) → s(0) half(s(s( x ))) → s(half( x )) bits(s(s( x ))) → s(bits(s(half( x )))) bits(s(s(0))) MA (ICS @ UIBK) Dependency Pairs and POP ∗ 4/17

Introduction Term Rewriting TRS R bits half(0) → 0 bits(0) → 0 half(s(0)) → 0 bits(s(0)) → s(0) half(s(s( x ))) → s(half( x )) bits(s(s( x ))) → s(bits(s(half( x )))) bits(s(s(0))) − → R s(bits(s(half(0)))) MA (ICS @ UIBK) Dependency Pairs and POP ∗ 4/17

Introduction Term Rewriting TRS R bits half(0) → 0 bits(0) → 0 half(s(0)) → 0 bits(s(0)) → s(0) half(s(s( x ))) → s(half( x )) bits(s(s( x ))) → s(bits(s(half( x )))) bits(s(s(0))) − → R s(bits(s(half(0)))) − → R s(bits(s(0))) MA (ICS @ UIBK) Dependency Pairs and POP ∗ 4/17

Introduction Term Rewriting TRS R bits half(0) → 0 bits(0) → 0 half(s(0)) → 0 bits(s(0)) → s(0) half(s(s( x ))) → s(half( x )) bits(s(s( x ))) → s(bits(s(half( x )))) bits(s(s(0))) − → R s(bits(s(half(0)))) − → R s(bits(s(0))) − → R s(s(0)) MA (ICS @ UIBK) Dependency Pairs and POP ∗ 4/17

Introduction Term Rewriting TRS R bits half(0) → 0 bits(0) → 0 half(s(0)) → 0 bits(s(0)) → s(0) half(s(s( x ))) → s(half( x )) bits(s(s( x ))) → s(bits(s(half( x )))) bits(s(s(0))) − → R s(bits(s(half(0)))) − → R s(bits(s(0))) − → R s(s(0)) → ! bits( n ) = v ⇐ ⇒ bits( � n � ) − R � v � computation MA (ICS @ UIBK) Dependency Pairs and POP ∗ 4/17

Introduction Term Rewriting TRS R bits half(0) → 0 bits(0) → 0 half(s(0)) → 0 bits(s(0)) → s(0) half(s(s( x ))) → s(half( x )) bits(s(s( x ))) → s(bits(s(half( x )))) confluent and terminating bits(s(s(0))) − → R s(bits(s(half(0)))) − → R s(bits(s(0))) − → R s(s(0)) constructor TRS → ! bits( n ) = v ⇐ ⇒ bits( � n � ) − R � v � computation MA (ICS @ UIBK) Dependency Pairs and POP ∗ 4/17

Introduction Term Rewriting TRS R bits half(0) → 0 bits(0) → 0 half(s(0)) → 0 bits(s(0)) → s(0) half(s(s( x ))) → s(half( x )) bits(s(s( x ))) → s(bits(s(half( x )))) confluent and terminating bits(s(s(0))) − → R s(bits(s(half(0)))) − → R s(bits(s(0))) − → R s(s(0)) constructor TRS → ! bits( n ) = v ⇐ ⇒ bits( � n � ) − R � v � = � log( n + 1) � computation MA (ICS @ UIBK) Dependency Pairs and POP ∗ 4/17

Introduction Runtime Complexity of TRSs ◮ derivation length dl( t , → ) = max { n | ∃ s . t → n s } dl( n , T , → ) = max { dl( t , → ) | t ∈ T and | t | � n } MA (ICS @ UIBK) Dependency Pairs and POP ∗ 5/17

Introduction Runtime Complexity of TRSs ◮ derivation length dl( t , → ) = max { n | ∃ s . t → n s } dl( n , T , → ) = max { dl( t , → ) | t ∈ T and | t | � n } ◮ derivational complexity dc R ( n ) = dl( n , T , − → R ) MA (ICS @ UIBK) Dependency Pairs and POP ∗ 5/17

Introduction Runtime Complexity of TRSs ◮ derivation length dl( t , → ) = max { n | ∃ s . t → n s } dl( n , T , → ) = max { dl( t , → ) | t ∈ T and | t | � n } ◮ derivational complexity dc R ( n ) = dl( n , T , − → R ) capture complexity of computed functions ◮ runtime complexity rc R ( n ) = dl( n , B , − → R ) B := { f ( v 1 , . . . , v n ) | f defined, v i build from constructors } MA (ICS @ UIBK) Dependency Pairs and POP ∗ 5/17

Introduction Runtime Complexity of TRSs ◮ derivation length dl( t , → ) = max { n | ∃ s . t → n s } dl( n , T , → ) = max { dl( t , → ) | t ∈ T and | t | � n } ◮ derivational complexity dc R ( n ) = dl( n , T , − → R ) avoid duplication of redexes ◮ innermost runtime complexity i rc i R ( n ) = dl( n , B , → R ) − B := { f ( v 1 , . . . , v n ) | f defined, v i build from constructors } MA (ICS @ UIBK) Dependency Pairs and POP ∗ 5/17

Polynomial Path Orders Polynomial Path Orders > pop ∗ ◮ > pop ∗ ⊆ > mpo Theorem (A, Moser 2008) R ⊆ > pop ∗ ⇒ rc i R polynomially bounded MA (ICS @ UIBK) Dependency Pairs and POP ∗ 6/17

Polynomial Path Orders Polynomial Path Orders > pop ∗ ◮ > pop ∗ ⊆ > mpo ◮ > pop ∗ ≈ > mpo ∩ predicative recursion Predicative Recursion [Bellantoni, Cook 1992] f ( ε,� x ; � y ) = g ( � x ; � y ) f ( z · i ,� x ; � y ) = h i ( z ,� x ; � y , f ( z ,� x ; � y )) , i ∈ { 0 , 1 } f ( x 1 , . . . , x m ; y 1 , . . . , y n ) � �� � � �� � normal safe Theorem (A, Moser 2008) R ⊆ > pop ∗ ⇒ rc i R polynomially bounded MA (ICS @ UIBK) Dependency Pairs and POP ∗ 6/17

Polynomial Path Orders Polynomial Path Orders > pop ∗ ◮ > pop ∗ ⊆ > mpo ◮ > pop ∗ ≈ > mpo ∩ predicative recursion ◮ R ⊆ > pop ∗ and s − → R t �⇒ s > pop ∗ t Theorem (A, Moser 2008) R ⊆ > pop ∗ ⇒ rc i R polynomially bounded MA (ICS @ UIBK) Dependency Pairs and POP ∗ 6/17

Polynomial Path Orders Polynomial Path Orders > pop ∗ ◮ > pop ∗ ⊆ > mpo ◮ > pop ∗ ≈ > mpo ∩ predicative recursion ◮ R ⊆ > pop ∗ and s − → R t �⇒ s > pop ∗ t Lemma if R ⊆ > pop ∗ then there exists I : T ( F , V ) → N satisfying i 1. s − → R t ⇒ I ( s ) > I ( t ) Theorem (A, Moser 2008) R ⊆ > pop ∗ ⇒ rc i R polynomially bounded MA (ICS @ UIBK) Dependency Pairs and POP ∗ 6/17

Polynomial Path Orders Polynomial Path Orders > pop ∗ ◮ > pop ∗ ⊆ > mpo ◮ > pop ∗ ≈ > mpo ∩ predicative recursion ◮ R ⊆ > pop ∗ and s − → R t �⇒ s > pop ∗ t Lemma if R ⊆ > pop ∗ then there exists I : T ( F , V ) → N satisfying i 1. s − → R t ⇒ I ( s ) > I ( t ) 2. I ( t ) polynomially bounded (in the size of t) for basic terms t Theorem (A, Moser 2008) R ⊆ > pop ∗ ⇒ rc i R polynomially bounded MA (ICS @ UIBK) Dependency Pairs and POP ∗ 6/17

❘ Polynomial Path Orders Polynomial Path Orders > pop ∗ (continued) Observation > pop ∗ ⊆ > mpo ◮ application of polynomial path orders restricted to MPO-terminating TRS’s MA (ICS @ UIBK) Dependency Pairs and POP ∗ 7/17

Polynomial Path Orders Polynomial Path Orders > pop ∗ (continued) Observation > pop ∗ ⊆ > mpo ◮ application of polynomial path orders restricted to MPO-terminating TRS’s ❘ dependency pairs for complexity analysis ◮ reduction pairs, argument filterings, usable rules, dependency graphs, subterm criterion . . . MA (ICS @ UIBK) Dependency Pairs and POP ∗ 7/17

Polynomial Path Orders Polynomial Path Orders > pop ∗ (continued) Observation > pop ∗ ⊆ > mpo ◮ application of polynomial path orders restricted to MPO-terminating TRS’s ❘ dependency pairs for complexity analysis ◮ reduction pairs, argument filterings, usable rules, dependency graphs, subterm criterion . . . MA (ICS @ UIBK) Dependency Pairs and POP ∗ 7/17

Dependency Pairs for Complexity Analysis Dependency Pairs for Complexity Analysis � t if t a variable t ♯ = f ♯ ( t 1 , . . . , t n ) if t = f ( t 1 , . . . , t n ) WIDP( R ) = { l ♯ → c( r ♯ 1 , . . . r ♯ n ) | l → C [ r 1 , . . . , r n ] ∈ R} ◮ C maximal context built from constructors and variables ◮ c fresh compound symbol (but we set c( t ) = t ) MA (ICS @ UIBK) Dependency Pairs and POP ∗ 8/17

Dependency Pairs for Complexity Analysis Dependency Pairs for Complexity Analysis � t if t a variable t ♯ = f ♯ ( t 1 , . . . , t n ) if t = f ( t 1 , . . . , t n ) WIDP( R ) = { l ♯ → c( r ♯ 1 , . . . r ♯ n ) | l → C [ r 1 , . . . , r n ] ∈ R} ◮ C maximal context built from constructors and variables ◮ c fresh compound symbol (but we set c( t ) = t ) TRS R WIDP( R ) f ♯ (0) → c 1 f(0) → 0 f ♯ (s( x )) → c 2 (f ♯ ( x ) , f ♯ ( x )) f(s( x )) → d(f( x ) , f( x )) MA (ICS @ UIBK) Dependency Pairs and POP ∗ 8/17