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CO3 a COnverter for proving COnfluence of COnditional term - PowerPoint PPT Presentation

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CO3 a COnverter for proving COnfluence of COnditional term rewriting systems Ver. 1.1 Naoki Nishida Makishi Yanagisawa Karl Gmeiner Nagoya University UAS Technikum Wien CoCo 2014 Vienna, July 13, 2014 Target and Main

  1. CO3 a COnverter for proving COnfluence of COnditional term rewriting systems Ver. 1.1 Naoki Nishida † Makishi Yanagisawa † Karl Gmeiner ‡ † Nagoya University ‡ UAS Technikum Wien CoCo 2014 Vienna, July 13, 2014

  2. Target and Main Function Convert a normal 1-CTRS to a TRS by using the simultaneous unraveling U [Marchiori, 96][Ohlebusch, 02][Gmeiner et al, 13] the SR transformation SR [S ¸erb˘ anut ¸˘ a & Ro¸ su, 06] ◮ if the input CTRS is constructor-based, then the special bracket symbol and its rewrite rules are not introduced, i.e., the result is the same as by [Antoy et al, 03] Theorem (theoretical background) R is confluent if R is weakly left-linear (WLL) and U ( R ) is confluent, [Gmeiner et al, 13] or SR ( R ) is confluent [Nishida et al, today (before lunch)] 1 /4

  3. Example   even(0) → true   R = even(s( x )) → true ⇐ even( x ) ։ false even(s( x )) → false ⇐ even( x ) ։ true     even(0) → true     even(s( x )) → U 1 (even( x ) , x )   U ( R ) = U 1 (false , x ) → true     U 1 (true , x ) → false    even(0 , z ) → true      even(s( x ) , ⊥ ) → even(s( x ) , even( x ))   SR ( R ) = even(s( x ) , false) → true     even(s( x ) , true) → false   U ( R ) and SR ( R ) are orthogonal, and thus, R is confluent 2 /4

  4. How to Prove/Disprove Confluence of R Implemented Criteria for Confluence Orthogonality of U ( R ) or SR ( R ) if R is WLL Termination and CP-joinability of U ( R ) or SR ( R ) if R is WLL ◮ the emptiness of the union of the SCCs in EDG ◮ the simplest reduction pair ⋆ s ≥ t if | s | ≥ | t | and ∀ x ∈ V . | s | x ≥ | t | x ⋆ s > t if | s | > | t | and ∀ x ∈ V . | s | x ≥ | t | x Implemented Criterion for Non-confluence Existence of unconditional CP ( s , t ) such that s and t are ground irreducible on R u (= { l → r | l → r ⇐ c ∈ R} ), or CAP ( s ) and CAP ( t ) is not unifiable 3 /4

  5. Remark The feature of CO3 is syntactic analysis ◮ the power of proving termination and confluence of TRSs is weak ◮ CO3 will rely on other tools CO3 website: http://www.trs.cm.is.nagoya-u.ac.jp/co3/ ◮ Ver. 1.0 is now available ◮ Updated to Ver. 1.1 soon 4 /4

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