Type Introduction for Runtime Complexity Analysis Scienza e Ingegneria Università degli Studi di Bologna University of Innsbruck Workshop on Termination, October 27, 2014 Martin Avanzini 1 (Joint work with Bertram Felgenhauer 2 ) 1 Dipartimento di Informatica 2 Institute for Computer Science
; 2. runtime-complexity function basic term v are values Runtime Complexity of TRSs [Hirokawa & Moser, 2008] [Hofbauer & Lautemann, 1989] 1. derivational-complexity function • derivation height dh ( t , → ) := max { n | ∃ s . t → n s } dc R ( n ) = max { dh ( f ( ⃗ → R ) | f ( ⃗ s ) , − s ) of size up to n } rc R ( n ) = max { dh ( f ( ⃗ v ) , − → R ) | f ( ⃗ v ) and ⃗ of size up to n } � �� �
; 2. runtime-complexity function basic term v are values Runtime Complexity of TRSs [Hirokawa & Moser, 2008] [Hofbauer & Lautemann, 1989] 1. derivational-complexity function • derivation height dh ( t , → ) := max { n | ∃ s . t → n s } dc R ( n ) = max { dh ( f ( ⃗ → R ) | f ( ⃗ s ) , − s ) of size up to n } rc R ( n ) = max { dh ( f ( ⃗ v ) , − → R ) | f ( ⃗ v ) and ⃗ of size up to n } � �� �
; 2. runtime-complexity function basic term Runtime Complexity of TRSs [Hirokawa & Moser, 2008] [Hofbauer & Lautemann, 1989] 1. derivational-complexity function • derivation height dh ( t , → ) := max { n | ∃ s . t → n s } dc R ( n ) = max { dh ( f ( ⃗ → R ) | f ( ⃗ s ) , − s ) of size up to n } rc R ( n ) = max { dh ( f ( ⃗ v ) , − → R ) | f ( ⃗ v ) and ⃗ of size up to n } v are values � �� �
; 2. runtime-complexity function basic term v are values Runtime Complexity of TRSs [Hirokawa & Moser, 2008] [Hofbauer & Lautemann, 1989] 1. derivational-complexity function • derivation height dh ( t , → ) := max { n | ∃ s . t → n s } dc R ( n ) = max { dh ( f ( ⃗ → R ) | f ( ⃗ s ) , − s ) of size up to n } rc R ( n ) = max { dh ( f ( ⃗ v ) , − → R ) | f ( ⃗ v ) and ⃗ of size up to n } � �� �
; v are values 1. derivational-complexity function [Hofbauer & Lautemann, 1989] 2. innermost runtime-complexity function [Hirokawa & Moser, 2008] rc i Runtime Complexity of TRSs basic term i • derivation height dh ( t , → ) := max { n | ∃ s . t → n s } dc R ( n ) = max { dh ( f ( ⃗ → R ) | f ( ⃗ s ) , − s ) of size up to n } R ( n ) = max { dh ( f ( ⃗ v ) , → R ) | f ( ⃗ − v ) and ⃗ of size up to n } � �� �
; C – theory recently reformulated in context of many-sorted TRSs – fast and powerful – automated, amortised cost analysis of functional programs http://raml.tcs.ifi.lmu.de/prototype – RAML prototype http://cl-informatik.uibk.ac.at/software/tct T – T Automated Complexity Analysis of TRSs http://dfa.imn.htwk-leipzig.de/matchbox – Matchbox/Poly http://cl-informatik.uibk.ac.at/software/cat – Cat http://aprove.informatik.rwth-aachen.de – AProVE [Hofmann and Moser, 2014] • part of annual termination competition since 2008 • Tools
; C – theory recently reformulated in context of many-sorted TRSs – fast and powerful – automated, amortised cost analysis of functional programs http://raml.tcs.ifi.lmu.de/prototype – RAML prototype http://cl-informatik.uibk.ac.at/software/tct T – T Automated Complexity Analysis of TRSs http://dfa.imn.htwk-leipzig.de/matchbox – Matchbox/Poly http://cl-informatik.uibk.ac.at/software/cat – Cat http://aprove.informatik.rwth-aachen.de – AProVE [Hofmann and Moser, 2014] • part of annual termination competition since 2008 • Tools
; C – theory recently reformulated in context of many-sorted TRSs – fast and powerful – automated, amortised cost analysis of functional programs http://raml.tcs.ifi.lmu.de/prototype – RAML prototype http://cl-informatik.uibk.ac.at/software/tct T – T Automated Complexity Analysis of TRSs http://dfa.imn.htwk-leipzig.de/matchbox – Matchbox/Poly http://cl-informatik.uibk.ac.at/software/cat – Cat http://aprove.informatik.rwth-aachen.de – AProVE [Hofmann and Moser, 2014] • part of annual termination competition since 2008 • Tools
; Persistency 3. innermost termination – … [Iwama, 2003; 2004] locally confluent overlay TRSs – non-overlapping or [Aoto, 1998] – all variables of same sort [Zantema,1994] non-duplicating TRSs – non-collapsing or 2. termination 1. confluence Definition [Iwama, 2004] [Aoto and Toyama, 1997] property P is persistent if for all many-sorted TRS R P ( R ) ⇐ ⇒ P (Θ( R )) • Θ( R ) denotes un(i)sorted TRS underlying R ✔ ✗ ✔ ✔ ✔ ✔
; Persistency 3. innermost termination – … [Iwama, 2003; 2004] locally confluent overlay TRSs – non-overlapping or [Aoto, 1998] – all variables of same sort [Zantema,1994] non-duplicating TRSs – non-collapsing or 2. termination 1. confluence Definition [Iwama, 2004] [Aoto and Toyama, 1997] property P is persistent if for all many-sorted TRS R P ( R ) ⇐ ⇒ P (Θ( R )) • Θ( R ) denotes un(i)sorted TRS underlying R ✔ ✗ ✔ ✔ ✔ ✔
; Persistency 3. innermost termination – … [Iwama, 2003; 2004] locally confluent overlay TRSs – non-overlapping or [Aoto, 1998] – all variables of same sort [Zantema,1994] non-duplicating TRSs – non-collapsing or 2. termination 1. confluence Definition [Iwama, 2004] [Aoto and Toyama, 1997] property P is persistent if for all many-sorted TRS R P ( R ) ⇐ ⇒ P (Θ( R )) • Θ( R ) denotes un(i)sorted TRS underlying R ✔ ✗ ✔ ✔ ✔ ✔
; Persistency 3. innermost termination – … [Iwama, 2003; 2004] locally confluent overlay TRSs – non-overlapping or [Aoto, 1998] – all variables of same sort [Zantema,1994] non-duplicating TRSs – non-collapsing or 2. termination 1. confluence Definition [Iwama, 2004] [Aoto and Toyama, 1997] property P is persistent if for all many-sorted TRS R P ( R ) ⇐ ⇒ P (Θ( R )) • Θ( R ) denotes un(i)sorted TRS underlying R ✔ ✗ ✔ ✔ ✔ ✔
; Persistency 3. innermost termination – … [Iwama, 2003; 2004] locally confluent overlay TRSs – non-overlapping or [Aoto, 1998] – all variables of same sort [Zantema,1994] non-duplicating TRSs – non-collapsing or 2. termination 1. confluence Definition [Iwama, 2004] [Aoto and Toyama, 1997] property P is persistent if for all many-sorted TRS R P ( R ) ⇐ ⇒ P (Θ( R )) • Θ( R ) denotes un(i)sorted TRS underlying R ✔ ✗ ✔ ✔ ✔ ✔
; Sorted Rewriting notations (i) 3. function symbols are equipped with sort declaration 4. term is well-sorted according to rules 1. finite set of sorts S 2. for each sort α ∈ S , set of sorted variables V α = { x α , y α , . . . } f :: ( α 1 , . . . , α k ) → α x ∈ V α t 1 :: α 1 . . . t k :: α k f :: ( α 1 , . . . , α k ) → α ( Var ) ( Fun ) x :: α f ( t 1 , . . . , t k ) :: α
; Sorted Rewriting notations (i) 3. function symbols are equipped with sort declaration 4. term is well-sorted according to rules 1. finite set of sorts S 2. for each sort α ∈ S , set of sorted variables V α = { x α , y α , . . . } f :: ( α 1 , . . . , α k ) → α x ∈ V α t 1 :: α 1 . . . t k :: α k f :: ( α 1 , . . . , α k ) → α ( Var ) ( Fun ) x :: α f ( t 1 , . . . , t k ) :: α
; Sorted Rewriting notations (ii) 5. many-sorted TRS R satisfies for each l → r ∈ R : l :: α and r :: α for some sort α ∈ S 6. rewrite relation − → R defined on well-sorted terms only
; Sorted Rewriting notations (ii) 5. many-sorted TRS R satisfies for each l → r ∈ R : l :: α and r :: α for some sort α ∈ S 6. rewrite relation − → R defined on well-sorted terms only
; Theorem proof outline. every reduction Main Result simulated stepwise by The (innermost) runtime-complexity functions of R and Θ( R ) coincide. • rc ( i ) Θ( R ) ( n ) ⩽ rc ( i ) R ( n ) : ( i ) ( i ) ( i ) f ( v 1 , . . . , v n ) − → Θ( R ) t 1 − → Θ( R ) t 2 − → Θ( R ) · · · ( i ) ( i ) ( i ) f ( v ′ 1 , . . . , v ′ t ′ t ′ n ) − → R − → R − → R · · · 1 2 where | v ′ i | ⩽ | v i | . • rc ( i ) Θ( R ) ( n ) ⩾ rc ( i ) R ( n ) : trivial
; Theorem proof outline. every reduction Main Result simulated stepwise by The (innermost) runtime-complexity functions of R and Θ( R ) coincide. • rc ( i ) Θ( R ) ( n ) ⩽ rc ( i ) R ( n ) : ( i ) ( i ) ( i ) f ( v 1 , . . . , v n ) − → Θ( R ) t 1 − → Θ( R ) t 2 − → Θ( R ) · · · ( i ) ( i ) ( i ) f ( v ′ 1 , . . . , v ′ t ′ t ′ n ) − → R − → R − → R · · · 1 2 where | v ′ i | ⩽ | v i | . • rc ( i ) Θ( R ) ( n ) ⩾ rc ( i ) R ( n ) : trivial
; Persistency Toyama’s example Example where TRS R T consists of f ( 0 , 1 , x ) → f ( x , x , x ) g ( y , z ) → y g ( y , z ) → z 0 :: • 1 :: • f :: ( • , • , • ) → • g :: ( • , • ) → ⋆ • uni-sorted variant Θ( R T ) gives cycle f ( 0 , 1 , g ( 0 , 1 )) − → Θ( R T ) f ( g ( 0 , 1 ) , g ( 0 , 1 ) , g ( 0 , 1 )) − → Θ( R T ) f ( 0 , g ( 0 , 1 ) , g ( 0 , 1 )) − → Θ( R T ) f ( 0 , 1 , g ( 0 , 1 )) • R T is terminating
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