On Confluence of Innermost Terminating Term Rewriting Systems - PowerPoint PPT Presentation

On Confluence of Innermost Terminating Term Rewriting Systems Masahiko Sakai (Nagoya University) joint work with Sayaka Ishizuki, Michio Oyamaguchi Sep. 9, 2017, IFIP WG 1.6, Oxford 1

This talk • Background: - Open problems on confluence for inner- most terminating TRSs • Tackling the open problem in negative direc- tion - A simple proof of undecidability of ground confluence for terminating TRSs • Sufficient conditions of confluence for inner- most terminating TRSs 2

Confluence for terminating TRSs • Well-known result of [KB’70] : for SN( → R ) CP R ⊆ ∗ → R · ∗ ← R iff CR( → R ) • Notations: SN( → ) : → is terminating CR( → ) : → is Church-Rosser ( ∗ → ⊆ ∗ → · ∗ ← ← ) Conf( → ) : → is confluent ( ∗ ← · ∗ → ⊆ ∗ → · ∗ ← ) NF( → ) : the set of normal forms CP R : the set of critical pairs of R 3

Critical pairs • Ex. R = { h ( k ( x )) → f ( x ) , k ( c ) → d } h ( d ) ← R h ( k ( c )) → R f ( c ) CP R = { ( h ( d ) , f ( c )) } • Formally, for ℓ → r, ℓ ′ → r ′ ∈ R (no common variables), let θ = mgu( ℓ | p , ℓ ′ ) for some pos. p , i.e. ℓθ [ r ′ θ ] p ← R ℓθ [ ℓ ′ θ ] p = ℓθ → R rθ Then ( ℓθ [ r ′ θ ] p , rθ ) is a critical pair 4

Confluence for terminating TRSs • Well-known result of [KB’70] : for SN( → R ) CP R ⊆ ∗ → R · ∗ ← R iff CR( → R ) • How about on innermost terminating TRSs? - Innermost rewrite step → rewrites a redex i having no redex strictly below 5

Confluence for innermost terminating TRSs • An open problem [Ohlebusch’01] : for SN( → i R ) CP R ⊆ ∗ i R · ∗ → ← i R iff CR( → R ) • A partial result for the problem [Gramlich’95] Th. The claim holds if OV( R ) Proof: by showing SN( → R ) • R is overlay OV( R ) : CPs are all produced from root-overlaps 6

The open prob. is negatively solved • Counter example [presented at IWC’16] : R = { f ( x ) → x, h ( k ( x )) → f ( x ) , k ( c ) → d, g ( x ) → x, f ( c ) → g ( c ) , g ( x ) → h ( k ( x )) } CP R = { ( x, h ( k ( x ))) , ( h ( d ) , f ( c )) , ( c, g ( c )) } i R ) and CP R ⊆ ∗ i R · ∗ • SN( → → ← i R , but is not con- fluent x f ( x ) h ( k ( x )) c f ( c ) h ( k ( c )) h ( d ) i i i ¬ i i i i i i i g ( x ) g ( c ) 7

Characterization of confluence [presented at IWC’16] • Instances of CPs by normalized substitutions iCP : { ( uσ, vσ ) | ( u, v ) ∈ CP , σ : normalized subst. } Note that iCP ⊇ CP • Th. For SN( → i ) , iCP ⊆ ∗ i · ∗ → ← CR( → ) iff i • New open problem Is confluence of innermost terminating TRSs decidable? 8

This talk • Background: - Open problems on confluence for inner- most terminating TRSs • Tackling the open problem in negative direc- tion - A simple proof of undecidability of ground confluence for terminating TRSs • Sufficient conditions of confluence for inner- most terminating TRSs 9

Tackling the new open problem • Tried to show undecidability of confluence in vain, and have shown undecidability of ground confluence for terminating TRSs [Ka- pur’87] ) • Normal technique to show undecidability: re- ducing an undecidable problem (here PCP) into confluence problem We adopt the policy that an instance has a solution iff non-confluent - Difficulties: necessary to design innermost terminating TRSs 10

PCP (Post’s correspondence problem) • Instance: a set P of pair of strings Ex. P = { ( aa, a ) , ( b, ab ) } • Answer: whether there exist non-empty se- quence ( u i , v i ) ∈ P ( 1 ≤ i ≤ n ) s.t. u 1 · · · u n = v 1 · · · v n (called solution) For ex. above, it is YES because sequence ( aa, a ) , ( b, ab ) gives a solution aab • Known as undecidable whether an instance has a solution or not 11

TRS construction for confluence • Unary representation a ( a ( b ( e ))) of string aab (abbreviated as aab ( e ) ) • For an instance { ( aa, a ) , ( b, ab ) } , R = { f ( x, x, y ) → c, f ( aa ( x ) , a ( y ) , 1( z )) ↔ f ( x, y, z ) , f ( aa ( e ) , a ( e ) , 1( e )) → c ′ , f ( b ( x ) , ab ( y ) , 2( z )) ↔ f ( x, y, z ) , f ( b ( e ) , ab ( e ) , 2( e )) → c ′ } f f | | c ′ aab ( e ) b ( e ) c i i i aab ( e ) ab ( e ) 1(2( e )) 2( e ) 12

TRS construction for confluence • For an instance { ( aa, a ) , ( b, ab ) } , R = { f ( x, x, y ) → c, f ( aa ( x ) , a ( y ) , 1( z )) ↔ f ( x, y, z ) , f ( aa ( e ) , a ( e ) , 1( e )) → c ′ , f ( b ( x ) , ab ( y ) , 2( z )) ↔ f ( x, y, z ) , f ( b ( e ) , ab ( e ) , 2( e )) → c ′ } f f | | e aa ( e ) c i i a ( e ) aa ( e ) 2( e ) 1(2( e )) 13

Innermost-terminating TRS constr. • Idea: introduce an argument that control in- nermost or not R = { d → e, f ( x, x, y, z ) → c, f ( aa ( x ) , a ( y ) , 1( z ) , d ) ← f ( x, y, z, d ) , f ( aa ( x ) , a ( y ) , 1( z ) , e ) → f ( x, y, z, d ) , f ( aa ( e ) , a ( e ) , e, z ′ ) → c ′ , · · · } ¬ i f f f | | | aab ( e ) b ( e ) aab ( e ) c ′ c aab ( e ) ab ( e ) aab ( e ) i i 1(2( e )) 2( e ) 1(2( e )) e d d 14

Innermost-terminating TRS constr. • Rules avoiding unexpected normal forms caused by non-solution (complement rules) R = { d → e, f ( x, x, y, z ′ ) → c, f ( aa ( x ) , a ( y ) , 1( z ) , e ) → f ( x, y, z, d ) , f ( b ( x ) , ab ( y ) , 2( z ) , e ) → f ( x, y, z, d ) , f ( e, y, z, z ′ ) → c, f ( b ( x ) , y, 1( z ) , z ′ ) → c,, · · · } ¬ i f f f f | | | | e aa ( e ) e aa ( e ) c a ( e ) a ( e ) aa ( e ) aa ( e ) i i i 2( e ) 2( e ) 1(2( e )) 1(2( e )) e d e d i 15

Problem on non-ground terms • Unexpected normal forms cannot be re- moved due to variables. Construction failed. R = { d → e, f ( x, x, y, z ′ ) → c, f ( aa ( x ) , a ( y ) , 1( z ) , e ) → f ( x, y, z, d ) , f ( b ( x ) , ab ( y ) , 2( z ) , e ) → f ( x, y, z, d ) , f ( e, y, z, z ′ ) → c, f ( b ( x ) , y, 1( z ) , z ′ ) → c, · · · } ¬ i f f f f | | | | aa ( x ) x x aa ( x ) c a ( x ) a ( x ) aa ( x ) aa ( x ) i i i 2( e ) 2( e ) 1(2( e )) 1(2( e )) e d e d • Obtained undecidability of ground confluence 16

Terminating TRS construction for grand confluence • NON-terminating const. (already shown) R = { f ( x, x, z ) → c, f ( aa ( x ) , a ( y ) , 1( z )) ↔ f ( x, y, z ) , f ( aa ( e ) , a ( e ) , 1( e )) → c ′ , f ( b ( x ) , ab ( y ) , 2( z )) ↔ f ( x, y, z ) , f ( b ( e ) , ab ( e ) , 2( e )) → c ′ } f f | | c ′ aab ( e ) b ( e ) c aab ( e ) ab ( e ) 1(2( e )) 2( e ) 17

Terminating TRS construction for grand confluence • terminating const. R = { f ( x, x, z ) → c, f ( aa ( x ) , a ( y ) , 1( z )) → f ( x, y, z ) , f ( aa ( e ) , a ( e ) , 1( e )) → c ′ , f ( b ( x ) , ab ( y ) , 2( z )) → f ( x, y, z ) , f ( b ( e ) , ab ( e ) , 2( e )) → c ′ } ∪ complement rules f f | | c ′ aab ( e ) b ( e ) c aab ( e ) ab ( e ) 1(2( e )) 2( e ) 18

Terminating TRS construction for grand confluence • terminating const. R = { f ( x, x, z ) → c, f ( aa ( x ) , a ( y ) , 1( z )) → f ( x, y, z ) , f ( aa ( e ) , a ( e ) , 1( e )) → c ′ , f ( b ( x ) , ab ( y ) , 2( z )) → f ( x, y, z ) , f ( b ( e ) , ab ( e ) , 2( e )) → c ′ } ∪ complement rules f f | | e aa ( e ) c a ( e ) aa ( e ) 2( e ) 1(2( e )) 19

This talk • Background: - Open problems on confluence for inner- most terminating TRSs • Tackling the open problem in negative direc- tion - A simple proof of undecidability of ground confluence for terminating TRSs • Sufficient conditions of confluence for inner- most terminating TRSs 20

Characterization of confluence [presented at IWC’16] • Th. For SN( → i ) , iCP ⊆ ∗ i · ∗ → ← iff CR( → ) i where iCP = { ( uσ, vσ ) | ( u, v ) ∈ CP , σ : normal. subst. } Note that iCP ⊇ CP 21

Sufficient conditions • Ground innermost rewrite step s → gi t : s → i t and the redex is a ground term • → gi rewrite sequences are stable under substitutions • Th.[presented at IWC’16] For SN( → i ) , CP ⊆ ∗ gi · ∗ → ← = ⇒ CR( → ) gi 22

Sufficient conditions • Bidirectional parallel step s ← → t : � s = s [ s 1 , . . . , s n ] , t = s [ t 1 , . . . , t n ] , and s i ↔ t i • Th. For SN( → i ) , CP ⊆ ∗ → · ∗ → gi · ← ← = ⇒ CR( → ) � gi Proven by Lem. in the next slide • An example: R = { k ( c ) → d, h ( k ( x ) , x ) → g ( x ) , g ( c ) → h ( k ( c ) , c ) } CP R = { ( h ( d, c ) , g ( c )) } gi h ( d, c ) h ( k ( c ) , c ) g ( c ) gi ¬ i 23

Sufficient conditions • Lem. For SN( → i ) , iCP ⊆ ∗ → · ∗ → i · ← ← iff CR( → ) � i → t implies s ∗ i · ∗ Proof ( ⇒ ): Show that s ← → ← i t � by Noetherian induction on { s, t } wrt multiset extension of → i ∪ ⊲ ( ⊲ : proper subterm rel.) → ε< f ( t 1 , . . . , t n ) = t Case s = f ( s 1 , . . . , s n ) ← � s ⊲ s i , t ⊲ t i and s i ← → t i for each i � By IH s i ∗ i · ∗ → ← i t i Hence, s ∗ i · ∗ → ← i t Case s = ℓσ → i rσ = t : it is trivial 24