Reasoning by Analogy via Abstraction Reasoning by Analogy via Abstraction

theoremsnametheoremlenabs ag+ ag ag& ag"lemmasass ]x ] (y ] z) = (x ] y) ] z150.71 0.65 0.65 0.78com ]x ] y = y ] x230.92 0.79 0.3{lemc ]com2 ] x ] (y ] z) = y ] (x ] z)150.71 0.65{{lemc ]lemc ] ? ] (y ] x) = y ] ( ? ] x)130.76 0.48 0.40{even ]e(x) e(x ] y)290.82 1.00{{dbl+, len+, dbl , sum& , prod&, rem&a ]a(x ] y) a(x) ] a(y)180.81, 0.78, 0.64, 0.78,0.78, 0.47Table 4: theorems proved by abstraction (abstract theory).ground theoremsnametheoremlen ap lemmasic tcass+x+ (y + z) = (x+ y) + z21 ass ]7 3.0assx (y z) = (x y) z23 ass ] distr17 3.0ass&x&(y& z) = (x& y)& z23 ass ]12 1.7ass"zx y = (zy)x19 ass ] z(x+y) = zx zy15 2.2com+x+ y = y + x25 com ] lemc+22 2.4comx y = y x29 com ] lemc22 2.2com&len(x &y;) = len(y &x;)38 com ] lemc&25 1.9com2+x+ (y + z) = y + (x+ z)21 com2 ] lemc+10 2.1com2x (y z) = y (x z)23 com2 ] brick2, distr27 2.2lemc+s(y + x) = y + s(x)17 lemc ]13 2.7lemcx y + x = x s(y)27 lemc ] ass+, com+18 1.8lemc&len([Z, x &y;]) = len(x & [W, y])32 lemc ]18 1.5even+e(x) ^ e(y) e(x+ y)35 even ]24 1.9evene(x) e(x y)29 even ] leme16 3.0dbl+2 (x + y) = 2 (x) + 2 (y)22a ] assoc+ LS20 2.8len+len(x& y) = len(x) + len(y)23a ]20 2.5dbl2 (2 (x y)) = 2 (x) + 2 (y)28a ] com+ dbl+ ls d0 18 1.8sum&sum(x& y) = sum(x) + sum(y)23a ] assoc+16 1.8prod&prod(x& y) = prod(x) + prod(y)23a ] assoc16 1.8rem& rem(n; x& y) = rem(n; x) + rem(n; y) 38a ]21 1.4Table 5: theorems proved by abstraction (ground theory).ABSFOL:: alle ripplewave Y Z;1 (? # Y) # Z = ? # (Y # Z)The command line written above tells ABSFOL to ap-ply 8E to the axiom ripplewave, substituting the vari-ables universally quanti ed in ripplewave with Y and Z.The commands that implement inference rules in ABSFOLhave as name a string identifying a logical connective (inthis case all for 8), su xed by i or e for the introduc-tion or the elimination rule respectively. ABSFOL reactsto the command by printing the proof line obtained byapplying the rule speci ed by the user. A proof line inABSFOL is composed of a label, a formula, and a set ofthe proof lines on which the line depends (in this casethe empty set). The step case is p(X) imp p(? # X),that is,(X # Y) # Z = X # (Y # Z) imp((? # X) # Y) # Z = (? # X) # (Y # Z)This formula is proven by assuming the antecedent ofthe implication written above, and then by rewriting itwith the wave and the rippling axioms.ABSFOL:: assume (X # Y) # Z = X # (Y # Z);2 (X # Y) # Z = X # (Y # Z) (2)ABSFOL:: alle wave (X # Y) # Z X # (Y # Z);3 (X # Y) # Z = X # (Y # Z) imp? # ((X # Y) # Z) = ? # (X # (Y # Z))ABSFOL:: impe 2 3;4 ? # ((X # Y) # Z) = ? # (X # (Y # Z)) (2)ABSFOL:: alle ripplewave X (Y # Z);5 ((? # X) # (Y # Z)) = (? # (X # (Y # Z)))ABSFOL:: subst 4 5 right;6 (? # ((X # Y) # Z)) = ((? # X) # (Y # Z)) (2)...alle stands for 8E, impe for E and subst for thesubstitution rule. The arguments of these commandsare the proof lines or the name of the axioms to whichthe rule has to be applied, and additional informationlike the term that has to be substituted for a universallyquanti ed variable or the \direction" of a substitution.Once the abstract goal has been proven, the abstractproof is mapped back to the target formal theory, gen-erating a schematic outline of the target proof. ABSFOLrepresents schematic outlines as trees of schematic for-mulas: ABSFOL:: mapback proof by fpa;I am switching from the current context to: pa1.0 (? # Y) # Z = ? # (Y # Z)ALLE ripplewave2.0 (X # Y) # Z = X # (Y # Z)ASSUMPTION...15.0 forall y z x. (x # y) # z=x # (y # z)ALLI 14.0Each line of the schematic outline has a label to identifyit, a formula, and a justi cation saying what rule hasbeen applied in the abstract formal system to infer theformula. The formulae of a schematic outline are para-metric: thus | for instance | line 1.0 represents allthe ground formulas which are mapped by fpa to:(? # Y) # Z = ? # (Y # Z)The instantiation of the schematic outline can be per-formed in ABSFOL line by line. The commands for instan-tiating a line of a schematic outline are insts, instt,and instw. These commands take as arguments the listof lines to be instantiated, a pair of symbols , terms , orformulas , and a list of the occurences to be instantiated.The following are examples of instantiations used duringthe re nement of ass+, ass , ass& :9.0 s(X) # Y = s(X # Y)ABSFOL:: insts 9.0 #: + all by fpa;9.0 s(X) + Y = s(X + Y)ALLE ripplewave9.0 s(X) * Y = ? # (X # Y)ABSFOL:: instt 9.0 ? # (X # Y): X * Y + Yall by fpa;9.0 s(X) * Y = (X * Y) + YALLE ripplewave9.0 cons(T, X) # Y = cons(T, X && Y)ABSFOL:: insts 9.0 #: && all by absappend;9.0 cons(T, X) && Y = cons(T, X && Y)ALLE ripplewaveThe outline is transformed to a proof by adding thoseinference steps which are abstracted away by the ab-straction function. For instance, the proof of the basecase | which is just one step in the abstract proof |requires eight steps in the ground proof:ABSFOL:: tryalle brick Y + Z;0.1 O + (Y + Z) = Y + ZALLE brick...ABSFOL:: trysubst 0.6 0.1 right;0.7 (O + Y) + Z = O + (Y + Z)SUBST 0.6 0.1ABSFOL:: matchtry 0.7 0.1;1.0 has been bound to 0.7.Commands referring to schematic outlines are pre xedby try; the matchtry command states that the re ne-ment in a subtree is completed (in the example, that thetry 1.0 can be derived by applying the inference rulespeci ed by the try 0.7).Finally, the outline must be proof checked , that is,ABSFOL has to verify whether all the nodes in the outlineare a consequence of a valid application of an inferencerule to previous nodes of the outline. The proof checkingis invoked by the ol2prf command:ABSFOL:: ol2prf;1 O + (Y + Z) = Y + Z2 O + Y = Y...15 forall y z x. (((x + y) + z) = (x + (y + z)))5 ConclusionsWe described how abstraction can be used to model someforms of analogical reasoning: abstraction functions areused to collapse di erences between proofs, so that asingle abstract proof can be used to guide the proof ofdistinct ground proofs. Since outlines are generally sim-pler to be proven than the corresponding ground proofs,the computational e ort of analogical reuse is paid backby 1) a search space in the abstract theory signi cantlysmaller and 2) by using the same abstract proof as aguide to distinct ground proofs. The validity of the ap-proach has been shown by presenting various examplesof analogical reusing.References[Bundy et al., 1991] A. Bundy, F. Van Harlemen,J. Hesketh, and A. Smaill. Experiments with proofplans for induction. Journal of Automated Reason-ing, 7:303{324, 1991. Also DAI Research Paper 413,University of Edinburgh, 1988.[Bundy et al., 1993] A. Bundy, F. Giunchiglia, A. Vil-la orita, and T. Walsh. Godel's Incompleteness Theo-rem via Abstraction. Technical Report 9302-15, IRST,Trento, Italy, 1993. Also DAI Research Paper, Uni-versity of Edinburgh.[de La Tour and Kreitz, 1992] T. Boy de La Tour andCh. Kreitz. Building proofs by analogy via the CurryHoward isomorphism. In A. Voronkov, editor, Pro-ceedings of Logic Programming and Automated Rea-soning '92, volume 624 of Lecture Notes on Arti -cial Intelligence, pages 202{213, St. Petersburg, 1992.Springer Verlag.[Giunchiglia and Traverso, 1991] F. Giunchiglia andP. Traverso. GETFOL User Manual GETFOL version1. Manual 9109-09, IRST, Trento, Italy, 1991. AlsoDIST Technical Report 91-0005, DIST, University ofGenova.[Giunchiglia and Walsh, 1992a] F. Giunchiglia andT. Walsh. A Theory of Abstraction. Arti cial Intel-ligence, 56(2-3):323{390, 1992. Also IRST-TechnicalReport 9001-14, IRST, Trento, Italy.[Giunchiglia and Walsh, 1992b] F. Giunchiglia andT.Walsh. 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Simpson. Grazing: A Stand AloneTactic for Theoretical Inference. Master's thesis,Dept. of Arti cial Intelligence, University of Edin-burgh, 1988.[Villa orita, 1993] A. Villa orita. Progetto e sviluppodi un dimostratore per astrazione. Technical ReportThesis, DIST University of Genova, 1993.[Weyhrauch, 1977] R.W. Weyhrauch. A Users Manualfor FOL. Technical Report STAN-CS-77-432, Com-puter Science Department, Stanford University, 1977.