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Layer Systems for Confluence Formalized 1 Bertram Felgenhauer , Franziska Rapp University of Innsbruck, Allgemeines Rechenzentrum Innsbruck ICTAC, Stellenbosch 2018-10-16 1 supported by FWF project P27528 Motivation Big Picture TRS yes

  1. Layer Systems for Confluence — Formalized 1 Bertram Felgenhauer , Franziska Rapp University of Innsbruck, Allgemeines Rechenzentrum Innsbruck ICTAC, Stellenbosch 2018-10-16 1 supported by FWF project P27528

  2. Motivation Big Picture TRS yes implement no Literature CSI maybe • automated confluence of first-order term rewrite systems Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 2/22

  3. Motivation Big Picture TRS yes implement no Literature CSI maybe proof formalize yes generate no IsaFoR Ce T A maybe • certified automated confluence of first-order term rewrite systems Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 2/22

  4. Motivation Big Picture TRS yes implement no Literature CSI maybe proof formalize yes generate no IsaFoR Ce T A maybe • certified automated confluence of first-order term rewrite systems • here: formalizing layer systems Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 2/22

  5. Motivation Term Rewriting +( x, 0 ) → x +( x, S ( y )) → S (+( x, y )) +( 0 , +( 0 , S ( S ( 0 )))) Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 3/22

  6. Motivation Term Rewriting +( x, 0 ) → x +( x, S ( y )) → S (+( x, y )) +( 0 , +( 0 , S ( S ( 0 )))) +( 0 , S (+( 0 , S ( 0 )))) Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 3/22

  7. Motivation Term Rewriting +( x, 0 ) → x +( x, S ( y )) → S (+( x, y )) +( 0 , +( 0 , S ( S ( 0 )))) +( 0 , S (+( 0 , S ( 0 )))) +( 0 , S ( S (+( 0 , 0 )))) S (+( 0 , +( 0 , S ( 0 )))) Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 3/22

  8. Motivation Term Rewriting +( x, 0 ) → x +( x, S ( y )) → S (+( x, y )) +( 0 , +( 0 , S ( S ( 0 )))) +( 0 , S (+( 0 , S ( 0 )))) +( 0 , S ( S (+( 0 , 0 )))) +( 0 , S ( S ( 0 ))) S (+( 0 , S (+( 0 , 0 )))) S (+( 0 , +( 0 , S ( 0 )))) S (+( 0 , S ( 0 ))) S ( S (+( 0 , +( 0 , 0 )))) S ( S (+( 0 , 0 ))) S ( S ( 0 )) Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 3/22

  9. Motivation Confluence s ∗ ∗ u t ∗ ∗ v Definition • ∗ ← · → ∗ ⊆ → ∗ · ∗ ← Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 4/22

  10. Motivation Confluence s ∗ ∗ u t ∗ ∗ v Definition • ∗ ← · → ∗ ⊆ → ∗ · ∗ ← Criteria for TRSs • orthogonality: left-linear, no critical pairs • Knuth-Bendix: terminating, joinable critical pairs • . . . Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 4/22

  11. Motivation Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ Orthogonal? • not left-linear � • no critical pairs � Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 5/22

  12. Motivation Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ Orthogonal? • not left-linear � • no critical pairs � Knuth-Bendix? • non-terminating � @(@(@( S , I ) , I ) , @(@( S , I ) , I )) → + @(@(@( S , I ) , I ) , @(@( S , I ) , I )) where I = @(@( S , K ) , K ) • joinable critical pairs � Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 5/22

  13. Motivation Modularity Theorem Let R 1 , R 2 be TRSs over disjoint signatures. Then CR ( R 1 ∪ R 2 ) ⇐ ⇒ CR ( R 1 ) ∧ CR ( R 2 ) Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

  14. Motivation Modularity Theorem Let R 1 , R 2 be TRSs over disjoint signatures. Then CR ( R 1 ∪ R 2 ) ⇐ ⇒ CR ( R 1 ) ∧ CR ( R 2 ) Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

  15. Motivation Modularity Theorem Let R 1 , R 2 be TRSs over disjoint signatures. Then CR ( R 1 ∪ R 2 ) ⇐ ⇒ CR ( R 1 ) ∧ CR ( R 2 ) Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ • first two rules are orthogonal � Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

  16. Motivation Modularity Theorem Let R 1 , R 2 be TRSs over disjoint signatures. Then CR ( R 1 ∪ R 2 ) ⇐ ⇒ CR ( R 1 ) ∧ CR ( R 2 ) Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ • first two rules are orthogonal � • last rule is terminating, and has no critical pairs � Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

  17. Motivation Modularity Theorem Let R 1 , R 2 be TRSs over disjoint signatures. Then CR ( R 1 ∪ R 2 ) ⇐ ⇒ CR ( R 1 ) ∧ CR ( R 2 ) Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ • first two rules are orthogonal � • last rule is terminating, and has no critical pairs � ⇒ confluent by modularity � • disjoint signatures = Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

  18. Layer Systems Table of Contents Motivation Layer Systems Formalization Implementation Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 7/22

  19. Layer Systems Proving Modularity History • Toyama 1987 • Klop et al. 1994 • van Oostrom 2008 • . . . Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 8/22

  20. Layer Systems Proving Modularity History • Toyama 1987 • Klop et al. 1994 • van Oostrom 2008 • . . . Proof idea • = ⇒ is easy (homogeneous terms are closed under rewriting) Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 8/22

  21. Layer Systems Proving Modularity History • Toyama 1987 • Klop et al. 1994 • van Oostrom 2008 • . . . Proof idea • = ⇒ is easy (homogeneous terms are closed under rewriting) • decompose terms into maximal top and aliens recursively Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 8/22

  22. Layer Systems Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ • e (@( x, e ( S , x )) , K ) e @ K e x x S Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 9/22

  23. Layer Systems Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ • e (@( x, e ( S , x )) , K ) e e @ K @ K e e x x x x S S Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 9/22

  24. Layer Systems Example @(@( K , x ) , y ) → x @(@(@( S , x ) , y ) , z ) → @(@( x, z ) , @( y, z )) e ( x, x ) → ⊤ • e (@( x, e ( S , x )) , K ) e e @ K @ K e e x x x x S S • max-top e ( � , � ) , aliens @( x, e ( S , x )) and K, rank 4 Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 9/22

  25. Layer Systems Proving Modularity History • Toyama 1987 • Klop et al. 1994 • van Oostrom 2008 • . . . Proof idea • = ⇒ is easy (homogeneous terms are closed under rewriting) • decompose terms into maximal top and aliens recursively • use induction on rank Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 10/22

  26. Layer Systems Proving Modularity History • Toyama 1987 • Klop et al. 1994 • van Oostrom 2008 • . . . Proof idea • = ⇒ is easy (homogeneous terms are closed under rewriting) • decompose terms into maximal top and aliens recursively • use induction on rank • . . . details are complicated Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 10/22

  27. Layer Systems Related Results Results • persistence (Aoto and Toyama 1997) • layer preservation (Ohlebusch 1994) • currying (Kahrs 1995) • . . . Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 11/22

  28. Layer Systems Related Results Results • persistence (Aoto and Toyama 1997) • layer preservation (Ohlebusch 1994) • currying (Kahrs 1995) • . . . Proof idea • similar to modularity • different decomposition into max-top and aliens Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 11/22

  29. Layer Systems Layer Systems in a Nutshell Idea • layer system L : set of admissible tops Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

  30. Layer Systems Layer Systems in a Nutshell Idea • layer system L : set of admissible tops • theorem: if R is confluent on L then R is confluent Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

  31. Layer Systems Layer Systems in a Nutshell Idea • layer system L : set of admissible tops • theorem: if R is confluent on L then R is confluent • adapt modularity proof Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

  32. Layer Systems Layer Systems in a Nutshell Idea • layer system L : set of admissible tops • theorem: if R is confluent on L then R is confluent • adapt modularity proof Complications • max-tops must be unique Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

  33. Layer Systems Layer Systems in a Nutshell Idea • layer system L : set of admissible tops • theorem: if R is confluent on L then R is confluent • adapt modularity proof Complications • max-tops must be unique • rewriting must not increase rank Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

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