A remark on higher order RUE-resolution with EXTRUE

AGS 2004 A remark on higher order R U E-resolution with E R U E Christoph Benzm üller F a h b erei h Informatik Univ ersität des Saarlandes, Saarbrü k en, German y hrisags.uni- sb.de SEKI Rep ort SR0205 SEKI -REPORT ISSN 1437-4447 UNIVERSIT Ä T DES SAARLANDES F A CHRICHTUNG INF ORMA TIK D66123 SAARBRÜCKEN GERMANY WWW: http://www.ags.uni-sb.d e/ This SEKI Rep ort w as in ternally review ed b y: Claus-P eter Wirth E-mail: pags.uni- sb.de WWW: http://www.ags.uni- sb.de/~p/welome.html Editor of SEKI series: Claus-P eter Wirth FR Informatik, Univ ersität des Saarlandes, D66123 Saarbrü k en, German y E-mail: pags.uni- sb.de WWW: http://www.ags.uni- sb.de/~p/welome.html A remark on higher order R U E-resolution with E R U E Christoph Benzm üller F a h b erei h Informatik Univ ersität des Saarlandes, Saarbrü k en, German y hrisags.uni- sb.de 2002 Abstrat W e sho w that a prominen t oun terexample for the ompleteness of rst order R U E-resolution do es not apply to the higher order R U E-resolution approa h E R U E. Bonaina sho ws in [BH92℄ that the rst order R U E-N R F resolution approa h as in tro dued in [ Dig79 , Dig81, DH86℄ is not omplete. The oun terexample onsists in the follo wing set of rst order lauses: { g ( f ( a )) = a, f ( g ( X )) 6 = X } Here X is a v ariable and f , g are unary funtion sym b ols. It is illustrated in [BH92 ℄ that this ob viously inonsisten t lause set annot b e refuted in the rst order R U E-resolution approa h of Digrioli. The extensional higher order R U E-resolution v arian t E R U E has b een prop osed in [ Ben99b , Ben99a ℄ and ompleteness is analyzed in [Ben99a ℄. An in teresting question is whether the ab o v e example is also a oun terexample to the ompleteness of E R U E. The t w o E R U E refutations presen ted b elo w illustrate that this is not the ase. W e do not presen t the E R U E alulus here and instead refer to [ Ben99b , Ben99a ℄. In the follo wing w e onsider ( A ⇔ B ) as shorthand for ( A ∧ B ) ∨ ( ¬ A ∧ ¬ B ) . W e furthermore use the [ . . . ] T and [ . . . ] F -notation of [Ben99a ℄ to denote p ositiv e and negativ e literals. T erms are presen ted in the usual rst order st yle notation, i.e. w e write g ( f ( a )) instead of ( g ( f a )) as done in [Ben99b , Ben99a ℄. The deomp osition rule emplo y ed in the refutations b elo w is C ∨ [ hA n = hV n ] F C ∨ [ A 1 = V 1 ] F ∨ . . . ∨ [ A n = V n ] F D ec The reader migh t b e more used to this form of deomp osition than to the one emplo y ed in [ Ben99b , Ben99a ℄. Compared to the latter the ab o v e rule D ec also shortens the presen tation. The deom- p osition rule emplo y ed in [Ben99b , Ben99a ℄ is more general, i.e. rule D ec ab o v e is deriv able in alulus E R U E. The rst refutation in E R U E presen ted b elo w (whi h has b een suggested b y Chad Bro wn) emplo ys a ex-rigid uniation step ( F lexRig ) in the v ery b eginning. In this k ey step v ariable X is b ound to an imitation binding that in tro dues f at head p osition. The rest of the refutation is then straigh t forw ard. 1 2 The seond refutation sho ws that there are alternativ es to the ex-rigid uniation step for v ariable X at the b eginning. The k ey idea no w is to deriv e the p ositiv e reexivit y literal [ f ( a ) = f ( a )] F in lause C 18 . While p ositiv e reexivit y literals annot b e deriv ed in rst order R U E- resolution, our example sho ws that this is (theoretially) p ossible in E R U E for some sym b ols and terms o uring in the giv en lause on text, lik e f ( a ) in our ase. W e no w presen t b oth E R U E-refutations in detail. f and g are still unary funtion sym b ols, while X is a v ariable. H and Y are freshly in tro dued v ariables. Refutation I C 1 : [ g ( f ( a )) = a ] T C 2 : [ f ( g ( X )) = X ] F F l exRig ( C 2 ) : C 3 : [ f ( g ( X )) = X ] F ∨ [ X = f ( H ( X ))] F S olv e ( C 3 ) : C 4 : [ f ( g ( X )) = f ( H ( X ))] F D ec ( C 4 ) : C 5 : [ g ( X ) = H ( X )] F Res ( C 1 , C 5 ) : C 6 : [( g ( f ( a )) = a ) = ( g ( X ) = H ( X ))] F D ec ( C 6 ) : C 7 : [ g ( f ( a )) = g ( X )] F ∨ [ a = H ( X )] F D ec ( C 7 ) : C 8 : [ f ( a ) = X ] F ∨ [ a = H ( X )] F S olv e ( C 8 ) : C 9 : [ f ( a ) = f ( a )] F ∨ [ a = H ( f ( a ))] F T r iv ( C 9 ) : C 10 : [ a = H ( f ( a ))] F F l exRig ( C 10 ) : C 11 : [ a = H ( f ( a ))] F ∨ [ h = λY a ] F S olv e ( C 11 ) : C 12 : [ a = a ] F T r iv ( C 12 ) : [] Refutation I I C 1 : [ g ( f ( a )) = a ] T C 2 : [ f ( g ( X )) = X ] F Res ( C 1 , C 2 ) : C 3 : [( g ( f ( a )) = a ) = ( f ( g ( X )) = X )] F E q uiv ( C 3 ) : C 4 : [( g ( f ( a )) = a ) ⇔ ( f ( g ( X )) = X )] F n × C nf ( C 4 ) : C 5 : [ g ( f ( a )) = a ] T ∨ [ f ( g ( X )) = X ] T C 6 : [ g ( f ( a )) = a ] F ∨ [ f ( g ( X )) = X ] F Res ( C 6 , C 1 ) : C 7 : [( g ( f ( a )) = a ) = ( g ( f ( a )) = a )] F ∨ [ f ( g ( X )) = X ] F D ec ( C 7 ) : C 8 : [ f ( a ) = f ( a )] F ∨ [ a = a ] F ∨ [ f ( g ( X )) = X ] F T r iv ( C 8 ) : C 9 : [ f ( a ) = f ( a )] F ∨ [ f ( g ( X )) = X ] F F ac ( C 9 ) : C 10 : [ f ( a ) = f ( a )] F ∨ [( f ( a ) = f ( a )) = ( f ( g ( X )) = X )] F T r iv ( C 10 ) : C 11 : [( f ( a ) = f ( a )) = ( f ( g ( X )) = X )] F E q uiv ( C 11 ) C 12 : [( f ( a ) = f ( a )) ⇔ ( f ( g ( X )) = X )] F n × C nf ( C 12 ) : C 13 : [ f ( a ) = f ( a )] T ∨ [ f ( g ( X )) = X ] T C 14 : [ f ( a ) = f ( a )] F ∨ [ f ( g ( X )) = X ] F Res ( C 13 , C 2 ) : C 15 : [ f ( a ) = f ( a )] T ∨ [( f ( g ( X )) = X ) = ( f ( g ( X ′ )) = X ′ )] F D ec ( C 15 ) : C 16 : [ f ( a ) = f ( a )] T ∨ [ f ( g ( X )) = f ( g ( X ′ ))] F ∨ [ X = X ′ ] F S olv e ( C 16 ) : C 17 : [ f ( a ) = f ( a )] T ∨ [ f ( g ( X ′ )) = f ( g ( X ′ ))] F T r iv ( C 17 ) : C 18 : [ f ( a ) = f ( a )] T Res ( C 2 , C 18 ) : C 19 : [( f ( g ( X )) = X ) = ( f ( a ) = f ( a ))] F D ec ( C 19 ) : C 20 : [ f ( g ( X )) = f ( a )] F ∨ [ X = f ( a )] F S olv e ( C 20 ) : C 21 : [ f ( g ( f ( a ))) = f ( a )] F D ec ( C 21 ) : C 22 : [ g ( f ( a )) = a ] F Res ( C 22 , C 1 ) : C 23 : [( g ( f ( a )) = a ) = ( g ( f ( a )) = a )] F T r iv ( C 23 ) : C 24 : [] 3 The ab o v e refutations are admittedly non-trivial. F or this partiular kind of problems paramo d- ulation therefore seems to b e a more appropriate approa h. Ho w ev er, w e suggest a more thorough analysis to suien tly larify this question for the higher order ase. A  kno wledgmen t: I thank Chad Bro wn, CMU, Pittsburgh, USA, for his ommen ts and on- tribution. Referenes [Ben99a℄ C. Benzm üller. Equality and Extensionality in Higher-Or der The or em Pr oving . PhD thesis, Departmen t of Computer Siene, Saarland Univ ersit y , 1999. [Ben99b℄ C. Benzmüller. Extensional higher-order paramo dulation and R UE-resolution. In H. Ganzinger, editor, Pr o  e e dings of the 16th International Confer en e on Automate d De dution (CADE-16) , n um b er 1632 in LNCS, pages 399413, T ren to, Italy , 1999. Springer. [BH92℄ M. P . Bonaina and Jieh Hsiang. Inompleteness of the R UE/NRF inferene systems. Newsletter of the Asso iation for Automated Reasoning, No. 20, pages 912, Ma y 1992. [DH86℄ V. J. Digrioli and M.C. Harrison. Equalit y-based binary resolution. Journal of the A CM , 33(2):254289, 1986. [Dig79℄ V. J. Digrioli. Resolution b y uniation and equalit y . In W. H. Jo yner, editor, Pr o . of the 4th W orkshop on A utomate d De dution , Austin, T exas, USA, 1979. [Dig81℄ V. J. Digrioli. The eay of rue resolution, exp erimen tal results and heuristi theory . In A. Dri- nan, editor, Pr o . of the 7th International Joint Confer en e on A rtiial Intel ligen e (IJCAI81) , pages 539547, V anouv er, Canada, 1981. Morgan Kaufmann, San Mateo, California, USA.