Unique Normal Forms in Infinitary Weakly Orthogonal Term Rewriting - PowerPoint PPT Presentation

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Unique Normal Forms in Infinitary Weakly Orthogonal Term Rewriting Jörg Endrullis ⋆ Clemens Grabmayer △ Dimitri Hendriks ⋆ Jan Willem Klop ⋆ Vincent van Oostrom △ △ ) Universiteit Utrecht ⋆ ) Vrije Universiteit Amsterdam RTA 2010, Edinburgh, UK July 11–13, 2010 UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal vs. orthogonal Weakly orthogonal (first-/higher-order) rewrite systems: ◮ definition: ‘harmless’ weakening of orthogonality ◮ for finitary TRSs: most ‘nice’ properties of orthogonal systems are preserved ◮ but: new concepts, and non-trivial adaptations are needed In this paper we: ◮ investigate infinitary weakly orthogonal rewrite systems ◮ show that uniqueness of infinitary normal forms fails in contrast to orthogonal systems ◮ explain how this failure can be repaired UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal vs. orthogonal Weakly orthogonal (first-/higher-order) rewrite systems: ◮ definition: ‘harmless’ weakening of orthogonality ◮ for finitary TRSs: most ‘nice’ properties of orthogonal systems are preserved ◮ but: new concepts, and non-trivial adaptations are needed In this paper we: ◮ investigate infinitary weakly orthogonal rewrite systems ◮ show that uniqueness of infinitary normal forms fails in contrast to orthogonal systems ◮ explain how this failure can be repaired UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Overview ◮ Definitions: weakly orthogonal, UN ∞ ◮ Counterexample to UN ∞ for weakly orthogonal TRSs ◮ Counterexample to UN ∞ for λ ∞ βη ◮ Restoring infinitary confluence ◮ Diamond and triangle properties for developments UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal Weakly orthogonal (first-/higher-order) systems: ◮ left-linear ◮ all critical pairs are trivial. Examples. ◮ Successor/Predecessor TRS: P ( S ( x )) → x S ( P ( x )) → x with critical pairs: S ( x ) ← S ( P ( S ( x ))) → S ( x ) P ( x ) ← P ( S ( P ( x ))) → P ( x ) ◮ Parallel-Or TRS (‘almost orthogonal’): por ( true , x ) → true por ( x , true ) → true por ( false , false ) → false UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal Weakly orthogonal (first-/higher-order) systems: ◮ left-linear ◮ all critical pairs are trivial. Examples. ◮ Successor/Predecessor TRS: P ( S ( x )) → x S ( P ( x )) → x with critical pairs: S ( x ) ← S ( P ( S ( x ))) → S ( x ) P ( x ) ← P ( S ( P ( x ))) → P ( x ) ◮ Parallel-Or TRS (‘almost orthogonal’): por ( true , x ) → true por ( x , true ) → true por ( false , false ) → false UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal Weakly orthogonal (first-/higher-order) systems: ◮ left-linear ◮ all critical pairs are trivial. Examples. ◮ Successor/Predecessor TRS: P ( S ( x )) → x S ( P ( x )) → x with critical pairs: S ( x ) ← S ( P ( S ( x ))) → S ( x ) P ( x ) ← P ( S ( P ( x ))) → P ( x ) ◮ Parallel-Or TRS (‘almost orthogonal’): por ( true , x ) → true por ( x , true ) → true por ( false , false ) → false UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal Weakly orthogonal (first-/higher-order) systems: ◮ left-linear ◮ all critical pairs are trivial. Examples. ◮ Successor/Predecessor TRS: P ( S ( x )) → x S ( P ( x )) → x with critical pairs: S ( x ) ← S ( P ( S ( x ))) → S ( x ) P ( x ) ← P ( S ( P ( x ))) → P ( x ) ◮ Parallel-Or TRS (‘almost orthogonal’): por ( true , x ) → true por ( x , true ) → true por ( false , false ) → false UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal Weakly orthogonal (first-/higher-order) systems: ◮ left-linear ◮ all critical pairs are trivial. Examples. ◮ Successor/Predecessor TRS: P ( S ( x )) → x S ( P ( x )) → x with critical pairs: S ( x ) ← S ( P ( S ( x ))) → S ( x ) P ( x ) ← P ( S ( P ( x ))) → P ( x ) ◮ Parallel-Or TRS (‘almost orthogonal’): por ( true , x ) → true por ( x , true ) → true por ( false , false ) → false UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal Weakly orthogonal (first-/higher-order) systems: ◮ left-linear ◮ all critical pairs are trivial. Examples. ◮ Successor/Predecessor TRS: P ( S ( x )) → x S ( P ( x )) → x with critical pairs: S ( x ) ← S ( P ( S ( x ))) → S ( x ) P ( x ) ← P ( S ( P ( x ))) → P ( x ) ◮ Parallel-Or TRS (‘almost orthogonal’): por ( true , x ) → true por ( x , true ) → true por ( false , false ) → false UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary Weakly orthogonal Weakly orthogonal (first-/higher-order) systems: ◮ left-linear ◮ all critical pairs are trivial. Examples. ◮ Successor/Predecessor TRS: P ( S ( x )) → x S ( P ( x )) → x with critical pairs: S ( x ) ← S ( P ( S ( x ))) → S ( x ) P ( x ) ← P ( S ( P ( x ))) → P ( x ) ◮ Parallel-Or TRS (‘almost orthogonal’): por ( true , x ) → true por ( x , true ) → true por ( false , false ) → false UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary CR ∞ en UN ∞ (definitions). Situation in OTRSs ◮ CR ∞ : t 1 և և t ։ ։ t 2 = ⇒ ∃ s . t 1 ։ ։ s և և t 2 ◮ UN ∞ : ։ t 2 ∧ t 1 , t 2 normal forms = ⇒ t 1 = t 2 t 1 և և t ։ ◮ SN ∞ : all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known): ◮ SN ∞ = ⇒ CR ∞ , and CR ∞ = ⇒ UN ∞ . ◮ CR ∞ fails (Kennaway). ◮ But for non-collapsing TRSs: CR ∞ holds. ◮ UN ∞ holds (Kennaway/Klop). UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary CR ∞ en UN ∞ (definitions). Situation in OTRSs ◮ CR ∞ : t 1 և և t ։ ։ t 2 = ⇒ ∃ s . t 1 ։ ։ s և և t 2 ◮ UN ∞ : ։ t 2 ∧ t 1 , t 2 normal forms = ⇒ t 1 = t 2 t 1 և և t ։ ◮ SN ∞ : all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known): ◮ SN ∞ = ⇒ CR ∞ , and CR ∞ = ⇒ UN ∞ . ◮ CR ∞ fails (Kennaway). ◮ But for non-collapsing TRSs: CR ∞ holds. ◮ UN ∞ holds (Kennaway/Klop). UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom

Counterexample in λ ∞ βη Introduction Counterexample in w-o iTRSs Restoring infinitary confluence Diamond and triangle properties Summary CR ∞ en UN ∞ (definitions). Situation in OTRSs ◮ CR ∞ : t 1 և և t ։ ։ t 2 = ⇒ ∃ s . t 1 ։ ։ s և և t 2 ◮ UN ∞ : ։ t 2 ∧ t 1 , t 2 normal forms = ⇒ t 1 = t 2 t 1 և և t ։ ◮ SN ∞ : all infinite rewrite sequences are progressive (str. conv.) In orthogonal TRSs (well-known): ◮ SN ∞ = ⇒ CR ∞ , and CR ∞ = ⇒ UN ∞ . ◮ CR ∞ fails (Kennaway). ◮ But for non-collapsing TRSs: CR ∞ holds. ◮ UN ∞ holds (Kennaway/Klop). UN ∞ in weakly-orthogonal iTRSs Endrullis, Grabmayer, Hendriks, Klop, van Oostrom