Refutation graphs

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ARTIFICIAL INTELLIGENCE

Refutation Graphs' Robert E. Shostak 2 Harvard University, Cambridge, Moss., U.S.A. Recommended by W. W. Bledsoe

ABSTRACT A graph-theoretic characterization o f truth-functional consistency is introduced, providing a clear perspective on some resolution-bused systems' for deciding formulas in propositional and first-order logic. Various resolution strategies are analyzed in te~wts of"walks" about specially defined graphs. A new procedure---called Graph Construction~is presented that improves on the Model Elimination and SL strategies,

Introduction This article presents a new methodology that both generalizes the resolution p~-inciple, and casts new light on the effectiveness of its variot, s refinements. The key to this new approach lies in the characterization of the logical notion of truth-functional inconsistency in terms of the topolog~ :al properties of certain graphs. In particular, structures called clause grapl ; : 'e defined for arbitrary sets of clauses, and a set S is shown to be truth-fi ~c",ona~ly incon.~;iy~tentif and only if it has a clause graph containing no cy~ o This relation•:,hip leads t 0 a characterization of resolution derivatic, ns ir ms o' "walks "' about loop-free clause graphs, and o£ restrictions of res~ : , . terms of limitations on the manner in which such graphs can be traversed. The characterization is exploited to provide an intuitive expos~ of some faults in the Model Elimination [6] and SL procedures [3]. An improved version of these procedm'es is described which corrects these faults without introducing new ones.

Because most of the following deals on the propositional level, terms such as "clause" and "strategy" will be used in the truth-functional sense unless otherwise noted. Proofs not given here can be found in [10]. 1 Work was supported in part by ARPA, NSF, and IBM. Currently at Stanford R.csearch Institute, Menlo Park, Calif., U.S.A. Artificial Intelligence "i (1976), 51-64 Copyright © 1976 by North-Holland Publishing Company

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I. C-lL ee Grains In this section, ~'aph-like structures called clause graphs are informally defined and investigated. As the examples of Fig. I suggest, clause graphs are not graphs at all in the conventional sense. Each node represents a clause and is drawn as a chain of contiguous cells labeled with the ]iterals of the clause. The "edges" in clause graphs are called bridges. Each bridge connects two sets of cells in the graph. Each such set is called a shore of the bridge. As shown in Fig. I, bridges are drawn as solid lines that fan out at either end into dotted lines called romps. The ramps at one end of a bridge attach to all

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the cells on the shore at that end. When a shore of a bridge contains only one c~ll, the dotted line is omitted and the end of the bridge attaches directly to the celi. A clause graph for a finite set S of clauses is a finite, non-empty ~et of nodes together with a set of bridges connectin~ sets of cells in those nodes such that: (!) Each node represents z clause in S. (2) Each cell of each n'~de lie.- !:n exactly one shore. (3) Cells in the sam,: shore are marked with identical literals, and cells in opposite shores are marked with complementary literals. Fig. 1 shows clause graphs for several sets of clauses. Note that a clause graph may have more than one node representing a given clause. Note also that many clause graphs generally exist for a given set of clauses. Two nodes are said to be connected by a bridge if they contain cells on opposite shores of the bridge. A path through a clause graph is a sequence of nodes Nl, N 2 , . . . , N~ and a sequence of distinct bridges bl, b a . . . , bt-i such that k >1 I, and for I ~< i ~< k - 1 , b~ connects N~ to N~+~. Two nodes are accessible to each other if a path exists between them. A graph is said to be pad:wise-connected if each two of its nodes are mutually accessible. A loop is a path that begins and ends at the same node, and that uses one or more bridges. A graph containing no loop is said to be loop-free. Note, for example, that the two nodes i'or {P} in Fig. l(c) are not accessible to each other since getting from one to ~.heother would require crossing the same bridge twice. The graphs of l(a) and l(d) are pathwise-connected and those of l(a), l(b) and I(c) are loop-free. The following theoren, characterizes truth-functional inconsistency in terms of clause graphs. THEOREM. ,~ set of clauses is truth-functionally inconsistent i.f and only if it has a loop-free clause graph. Proof. (~=) To see that no trut.h-functionally con3is~nt set S has a loop-free clause graph, it suffices to show that any chmse graph for S must have a !oop. Let G be a clause graph for S, and let ~ ' be a verifying truth assignment for S. Starting with any node of G, choose a path through G, always exiting a node through a cell marked by a literal true in.~', but never crossing the same bridge twice. Since the graph is finite, one of the following must eventually occur: either some node will be visited twice, giving a loop, or some node/v will be reached from which it is impossible to exit. Since.4{ verifies S, N must have a cell C marked with a literal L that is true in .4{. Moreover, since N cannot be exited, the bridge b attached to C must have been crossed earlier. Because L is true in ~ the crossing n~ust have been from a node N' in the same shore ofb as C to a node M in the opposite shore. A path was thus traced from M to N that did not use b, and so b completes a loop. Artificial Intelligence 7 (1976), 51-64

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It. E. SHOSTAK

( ~ ) Let S be a truth-functionaUy inconsistent set of clauses. It will be shown by induction on the number n of distinct atomic formulas occurring in clauses in S that a loop-free clause graph exists for S. (I) Basis. n - O. Then the empty clause is in S and the one-node graph containing a node with no cells is a loop-free graph for S. (I1) Induction step. n > 0. Let L be a literal occurring in S. Let SL be the subset of S not having occurrences of E and St that subset not having occurrences of L. Let DL be obtained from Sc by removing all occurrences of L from its clauses, and define Dr symmetrically. Now DL and Dt must both be inconsistent, since, for example, if . ~ verified D,, then the truth assignment that falsified L but agreed with ~'. on all other atoms in S would verify S. By hypothesis, loopfree clause graphs exist for both DL and DL. If the graph for DL has no nodes representing clauses from which L has been removed, that graph is also a loop-free graph for So, and thus for S. Similarly, if the graph for D L has no nodes representing clauses from which Ehas been removed, it is also a graph for S. Otherwise, a clause graph for S can be constructed by adding back cells marked L and Eto the appropriate nodes of the graphs for DL and Dr, respectively, and connecting the two graphs together with 2 new bridge with all the cells marked L on one shore, and all those marked/:on the other shore. The new graph contains no loops, since the new bridge connects graphs that are themselves loop-free and that are otherwise disconnected from each other. For example, since the graph of Fig. l(a) is loop-free, the set S -- {TQR, TO.R, TR, ~ , RO} must be inconsistent. A careful examination of the inductive construction in the above proof will ~;erifythat the graph obtained is pathwise-connected as well as loop-free. Such graphs are called refutationgraphs. The graph of Fig. l(a), for example, is a refutation graph. It is also easy to show that the construction always produces a refutation graph having no more than 2~ nodes, where N is the number of distinct atoms occurring in clauses in S. Thus, any exhaustive search for refutation graphs with no more than 2N nodes provides a decision procedure for ground sets of clauses. Such searches may quite naturally take the form of deductions, in which refutation graphs are built up one step at a time, each step adding a bridge or a new node, or extending an existing shore. Not surprisingly, such a deduction system can be lifted to the general level to furnish a semi-decision procedure for predicate calculus. On the general level, adding a bridge entails unifying the literals that mark cells on either end of the proposed new bridge, and extending a shore involves factoring the literals that mark cells on the extended shore. Artificial Intelligence 7 (1976), 51-64

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REFUTATION GRAPHS

One can show that every refutation graph has a separating bridge, i.e., a bridge whose removal from the graph (together with the removal of the cells on both shores) divides the graph into two smaller refutation graphs. For example, removal of the R - ~ bridge of Fig. l(a) results in the two components of l(b). The two component graphs can each be separated into still smaller refutation graphs, and so on. One can formalize this idea by means of decomposition orderings. A partial ordering ~< on the bridges of a clause graph G is a decomposition orderingfor G if for each bridge b of G, b separates .the graph obtained from G by removing all bridges b' such that b' < b. Fig. 2 shows a refutation graph and a decomposition ordering for it. Such orderings will prove useful in the next section.

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FIG. 2. 2. St-Linear, SL, and ME Walks It was suggested earlier that resolution derivations can be viewed as walks about loop-free clause graphs. In this section, the propositional versions of the t-linear and SL strategies of Kcwalsk. dnd Kuehner and the Mode[ Elimination procedure of Loveland are viewed in this light. For clarity, a slightly simplified version of the t-linear strategy is considered. Simplified t-linear (st-linear) is a linear strategy in which ancestor resolution Artificial bztelligence 7 (1976), ~! -64

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can occur only with particular ancestor clauses called A-ancestors. An ancestor Cj of a clause C~ is an A-ancestor if Cj+1 was obtained by input resolution and if all of the iiterals of C~ except the one resolved upon in obtaining Cj+t appear in all resoh, ents from Cj+t down to Cl. Furthermore, ancestor resolution is compulsory when it is possible, and if a given resolvent has more than one A-ancestor, the top one must be resolved against first. Fig. 2(c) shows an st-linear refutation for the clauses of Fig. 2(a). Kowalski and Kuehner have shown that the st-linear strategy always admits, for a given inconsistent set S and set of support for S, a refutation as simple as any non-linear proof for S. "Simple", here, is defined relative to the rm-size metric described in [3]. Owing to the compulsory ancestor resolution restriction, an st-linear refutation is completely determined by the sequence of input clauses starting with the top clause. For example, the st-linear sequence associated with the deduction of Fig. 2 is:

N~, MQN, L~T!.,LO, LP, RPN, ~£, T. Now, given a refutation graph G and a decomposition ordering <~ for G, construct a sequence of nodes No, N ~ , . . . , Nt, called an st-linear walk for (7 as follows: First, choose any node No as starting node. Among all the bridges adjacent to No, choose one that is no lower in ~< than any other. Then choose N~ from among all the nodes on the opposite shore of the chosen bridge from No. In general, after N~ has been found, consider all of the bridges connecting only between nodes that have already been chosen and nodes that have yet to be chosen, and pick one such bridge not lower than the others in ~. Then select N~+I from among the yet-to-be-chosen nodes adjacent to this bridge. The walk ends when all nodes of the graph have been chosen. The nodes of the graph in Fig. 2 have been labeled to exemplify an st-linear walk, starting with the node representing NT and using the decomposition ordering diagrammed to the right of the graph. The sequence of clauses represented by the nodes in the walk is the st-linear sequence associated with deduction of Fig. 2. One can show that if G is a refutation graph, (i) G t, as an st-linear walk starting at any node. (ii) Each st-linear walk about G generates a t-linear sequence. (iii) Each st-linear sequence is generated by exactly one st-linear walk about exactly one refutation graph. Because several decomposition orderings may exist for a given graph and because the st-linear walk algorithm involves making choices, a given graph will usually yield several st-linear walks. Thus, the set of st-linear refutations for a given set S of clauses partitions into equivalence classes, each class containing derivations corresponding to walks about the same graph. Now Arti,6ciai Intelligence 7 (1976), 51-64

REFUTATION GRAPHS

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since the refutations in a given class differ, in a strong sense, only in the order in which their input parents are used, the size of the classes can be. taken as a measure of redundancy. The egregious redundancy of st'linear resolution is exemplified in Fig. 3, which shows that portion of the search space for the Kowalski clauses deriving simplest derivations with top clause PQ. Each of the refutations falls into one of but two classes.

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Kowalski and Kuehner's 5L resolution is a ~'estriction of the ,~t-linear resolution procedure that attempts to deal with the t-linear redundancy problem. Briefly, SL is obtained from st-linear by adding (among others) a restriction calling for a single literal to be chosen from each non-empty cla ~e C in a derivation. This selected literal must be most recently introduced (i.e., must have been a member of the last input parent that contributed literals to C), and is the only literal that can be resolved upon when C is used as the near parent in input resolution. The new restriction thus cuts down Artlfu:ial Intelligence 7 (1976), 51-64

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enormously on the number of possibilities for input clauses to be resolved with C. Unfortanately, the added selectivity gained by the new restriction is bought at the expense of simplicity of derivations. Note first that while any SL derivation is also an st-linear derivation, the converse is false. Consider once again the derivation of Fig. 2 and its corresponding walk. Up to the third line where TQL is derived, this derivation could well be an SL derivation in which ~ is selected from .~Tand M is selected from TQM. But whereas Q is the next literal resolved upon in the st-linear proof, L, which is most recently introduced, must be selected in the SL version. In terms of the walk, bridge b4 must now be crossed in the SL version rather than b3. Since in any decomposition ordering of the graph b~ will be greater than b4, choosing b4 at this point is not compatible with any st-linear walk of the graph. A generalization of this argument shows that no st-linear sequence corresponding to a walk about the graph is also an SL sequence. The SL restriction thus trims the number of walks about the graph to zero! Consequently, the possibility arises that the clause graphs for this set of clauses that do correspond to SL derivations have greater complexity. That possibility is indeed realized in this example. Fig. 4(a) shows a simplest SL proof for the clauses of Fig. 2 using NT as the top clause; 4(b) shows the c.orresponding derived graph. Note that the size of this refutation is significantly greater than that of the st-linear refutation. Now compare the two refutation graphs. The SL graph is similar to the st-linear graph, except that the two-cell shore on the right-hand side of the bridge b4 in Fig. 2 has been split into two one-cell shores, each connected to its own private copy of that portion of the graph lying on the left-hand side of the bridge. Intuitively, this splitting phenomenon might be viewed as the resultant of two opposing forces--one dictating that b4 be crossed before b3 in accordance with the SL restriction, and the other dictating that bridges with higher decomposition precedence be crossed before ones with lower decomposition precedence, in accordance with st-linear walks. The increased complexity of the SL graph obviously results from the redundant structure created by the split. In the next ~¢ction, a new procedure is intro,~,uced that retains the advantages of SL resolution without sacrificing simplicity of' derivations. The new procedure is described in terms of the chainformat, which was devised by Loveland for Model Elimination and later adopted by Kowalski. Loveland's Model Elimination procedure differs from the chain format version of SL resolution only in that merging operations are disallowed. Model Elimination is therefore not, strictly speaking, a restriction of st-linear resolution. It is easy to show, however, that ME refutations can be viewed as walks about refutation graphs, and that the splitting problem occurs in the ME context as Artificial Intelligence 7 (1976), 51--64

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well as in the SL context. In any case, Loveland's implementation [1 ] of firstorder ME provides for factoring (the first-order analog of merging), thus eliminating the distinction. 3, The Graph Construction (GC) Preeedure In this section a modified version of the ME and SL procedures is introduced that retains the selectivity of these strategies, while solving the splitting problem. The new procedure has an added mechanism that saves and later applies information about truncated A-literals. The memory element takes Artificial Intelligence 7 (1976), 51--64

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the form of the C-literal, which is indicated by enclosui'e in a circle. C-literals may be used for reduction just as A-literals, and are truncated as are A-literals. C-literals are created as follows. Associated with each A-literal is a position to its left in the chain, called its C-point. Whenever an A-literal is truncated from the chain, that literal is complemented, circled, and placed in the chain at its C-point. Truncated A-literals are thus reincarnated as C-literals. When C-literals are truncated, however, they are gone forever. The C-point of an A-literal is always initially at the extreme left end of the chain. Whenever reduction occurs using either an A-literal or C-literal, say R, the C-point of e°~h A-literal L to the right of R whose C-point is (before the reduction) to the left of R is moved to the position just to the right of R in the chain. Unlike SL resolution, GC has no merging operation. Aside from the introduction of C-literals, then, it is exactly like Model Elimination. OBTAINED BY : o)

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Fig. 5 displays a GC refutation using the same top clause and selectioa function as the SL deduction of Fig. 4. The arrows indicate C-points of those A-literals having C-points other than at the extreme left end of the chain. At step (5), for example, reduction has occurred using the A-fiteral [~. Since the tN-~lies between the A-literals ['MI, I'Ll, and ]PI and their C-poin'~ (all at the ex'-treme left end of the chain), those C-points--are all moved just to the right of IN__J,

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Fig. 6 shows the evolution of the corresponding refutation graph. Note that the graph is modified only by extension and reduction operations. Truncation and the associated creation of C-literals are ch~ply implemented bookkeeping operations not contributing to the development (or complexity) Artiftclal Intelligence "1 (1976), 51-64

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of the graph. The finished graph is that of Fig. 2, corresponding to the simple st-linear deduction. The critical action differentiating the GC reduction from the SL proof occurs at step 10. Whereas extension takes place at this point in the SL proof, a reduction using C-literal E occurs in the GC proof. Referring to the graph (Fig. 6), the effect of this reduction is to extend the L-shore of the L-Ebridge to the node for LO. The C-literal T. thus "recalls" the development of the graph structure on the left-hand side of that bridge. The effect of the C-point mechanism is to ensure that such shore extension operations preserve loopfreeness in the graph. The informal description of the propositional version of the GC strategy given above can be made precise as follows. Define a chain as a list of literals, each of which has the status of A-literal, B-literal, or C-literal. Given a clause C, a chain C* representing C is an all-B-literal chain listing all the literals of C in arbitrary order. For a given set S of clauses, define an input set S* for S as exactly containing one chain representing each clause in S. If So is a subset of ~S, let Sf be that subset of S* representing clauses in So. Two B-literals in a chain not separated by a non-B-literal are said to belong t~ the same ceil. The rJohtmost cell of a chain consist~ of all the literals to the r ~ u of the rklt~,tmost non-B-literal if a non-B-literal exists, and all the literals i~t the chain other'wise. A selection function ¢~ is a mapping taking any chain with a non-empty rightmost cell to a literal in that cell. For a given set of clauses S, support set So, and selection function • a GC derivation for S is a sequence of chains C*, C * , . . . , C*, satisfying conditions (i) to (3) below: (1) Cf ¢ Sf where S* is any input set for S. (2) Each Cr+l is obtained from C? by one of extension, reduction, or truncation defined below. (3) Cf+~ must be obtained from Cf by reduction, if reduction is possible. Furthermore, no two non-B-literals in C* may have identical atoms. Cr+ I is obtained by extension with input chain B* ¢ S* if: (a) Cf has a non-empty rightmost cell, and the selected literal L q) (C*) is the complement of a literal K in B*. (b) C~'+t is obtained by deleting L from the rightmost cell of C~ and K from B*, then appending the two chains together with B* on the right with a boxed copy of L sandwiched in between. The new boxed literal is an A-literal, and. the position just to the left of the ieftmost literal in the chain is given as its C-point. All the iiterals descending from Cf retain their statuses. The literals from B* are B-literals and make up the rightmost cell. C?+l is obtained from C* by reduction if: Arlificial Intelligence7 (1976), 51-64

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(a) A B-literal L in the right.most cell of Ct' is the complement of an Aor C-literal K. (b) C~+t is obtained by deleting L from the right most cell. All iiterals retain their A, B, or C status, and each A-fiteral to the right of the K whose C-point in C~ is to the left of the K has as its C-point in C~÷t the position just to the right of the K. C~'+t is obtained from C* by truncation if: (a) The rightmost literal L in C~ is a non-B-literal. (b) C*+ t is obtained from C* by first deleting the L, and then, if the L was an A-literal, complementing it, circling it, and placing it in the chain at its C-point, t!ms creating a C-literal. The status of all the other titerals in the chain remains unchanged. The GC procedure lifts in a straightforward manner to the first-order level. It is not, strictly speaking, a refinement of resolution, and so requires a soundness as well as a completeness result. Both of these arguments are given in [10]. 4. Relation to Other Work

Although the notion of graph is not new to the resolutior~ literature, the connection between this work and other efforts to make use of the concept is somewhat superficial. The resolution graphs of Yates et al. [l 1] bear what first seems to be a striking similarity to clause graphs. Both feature nodes representing clauses and edges that connect components of clauses representing the same atom; to this extent, they are quite similar. The notion of path in resolution graphs, however, is missing or at least different. Hence, these graphs fail to capture the crucial relationship between loop-freeness and truth-functional it~consiztency on which the results presented here depend. A much stronger connection exists between the C-literal mechanism of the GC procedure and the lemma accessory to Loveland's Model Elimination procedure, which has nothing to do with graphs. Lemmas are clauses (or more precisely, chains) that are formed as by-oroducts in the course of a Model Elimination derivation. These clauses can be added to the input set for use in extension. Lemmas are created at the time of A-literal reduction as are C-literals, and contain much the same kind of information about the chains from which they are derived as that represented ,by the C-literals. In the special case in which a C-liters', occurs at the left end of a chain, the tyro notions coincide, and the C-literal behaves like a unit lemma. While interesting from a theoretical standpoint, the lemma scheme has been found to have limited value in application. ME lemmas tend to be highly redundant--they are often subsumed by other lemmas and input chains. More important, those that are not subsumed are most often of little use in finding a Artificial Intelligence 7 (1976), 51-64

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proof, and so have the effect of cluttering the input set. In his summary of implementation results, Loveland [1 ] reports that the use of lemmas is usually detrimental, citing the lack of selection rules for them to be the principal cause. The C-literal mechanism is free of these difficulties. C-literals are totally irredundant, and are automatically selected via compulsory reduction. Moreover, C-literais do not have the effect of swelling the input set, because they are unit iiterals stored in the chain and are truncated when they are no longer relevant to the deduction. On the general level, the C-literal mechanism eliminates the need for factoring. (C-literal reduction, in fact, may be viewed as a kind of retrospective merging.)Thus, fruitlessdeductions i nwhich factors are formed but never resolved away are obviated. Although the GC procedure has yet to be implemented on the first-order level, its propositional version has been programmed (in the language EL1) for the DEC PDP-10, and tested against the SL and ME procedures on several dozen examples. The results of these tests confirm what the examples given here suggest--that the average amount of search required to find a first proof is substantially less for the GC procedure than for the others (typically 30 ~o fewer resolution and merging operations than for SL, and 40~o for ME). While it would be unjustified presently to extrapolate these findings to the first-order plain, even sharper gains on this level are likely. REFERENCES 1. Fleisi$,S., Lovelaod, D., Smile),,A. K. II! and Yarmt~h, D. L. An implementationof the model elimination proof procedure. J. ACM 21 (January 1974), 124-139. 2. KowahkJ,R. A. An improvedtheorem-provingsystemfor first-orderlogic. Mathematics Unit Memo 65, Universityof Edinburgh (1973), 3. Kowa~:~ki,R. A. and Kuehner, D. G. Linear resolution with selection ftngtion. Artificial Intellioence 2 (1971), 227-260. 4. Loveland,D. W. Mechanicaltheorem proving by modelelimination.J. ACM I$ (1968), 236-251. 5. Loveland,D. W. A linear format for resolution. Proc. IRIA Symposium on Automatic Demonstration (1968), 147-162. 6. Lovelan~ D. W. A simplified format for the model elimination theorem.pr0ving procedure. J. ACM 16 (1969), 349-363. 7. Lovelaud, D. W. Theorem provers combining model elimination and resolution. Machine Imelltoence 4 (1969), 73-86. 8. Loveland, D. W. A unifyingview of some liner herbraad proof procedures. J. ACM 19 (April 1972), 366-384. 9. Robinson, J. A. A machine-oriented logic based on the resolution principle. J. ACM 12 (January 1965),23--41. I0. Shostak, R. E. A graph-theoreticview of resolutiontheorem-proving. Ph.D. Thesis, Center for l~ese~rchin Computing Technology, Harvard University0974). II. Yates, R., Raphael, B. and Hart, T. Resolution graphs. ArtificialIntelligence1 0970), 224-239.

Received January 1974; accepted 21 July 1975 Art/fteio//~//~,m~ 7 (1970, 51--64