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 SOME NEW RESULTS ON RESOLUTION IN AUTOMATED THEOREM PROVING 
 
 by 
 Lawrence J. Henschen 
 
 May 1, 1968 
 
 THE UBttAiir 8r 
 
 km ^^' 
 
 UHNEKI1Y Of ILUMWS 
 
Report No. 26l 
 
 SOME NEW RESULTS ON RESOLUTION IN AUTOMATED THEOREM PROVING* 
 
 by 
 Lawrence J. Henschen 
 
 May 1, 1968 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 This work was submitted in partial fulfillment of the Master of 
 Arts degree in Mathematics and was supported in part by the 
 National Science Foundation under Grant No. NSF-GP-U636, 
 
Ill 
 
 TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION. . «... 1 
 
 2. PRELIMINARY DEFINITIONS AND RESULTS k 
 
 3. ALTERNATE DEFINITION OF RESOLUTION 13 
 
 k. SETS OF SUPPORT IT 
 
 LIST OF REFERENCES 23 
 
1. INTRODUCTION 
 
 The idea of proof in logic is well known „ A proof for some 
 theorem in logic is a finite sequence of statements each of which is 
 derived from a basic set of statements, called axioms and hypotheses, 
 or from earlier statements in the sequence, by applying specific rules 
 of inference. The last statement is usually the conclusion of the 
 theorem, although it may be a statement which is obviously false in 
 case of a proof by contradiction , The sequence of statements is a 
 proof of the last statement or a proof of the theorem with the given 
 hypotheses. 
 
 Closely related to the idea of proof is the concept of 
 completeness . Suppose, in a particular logical system, the set of 
 statements is partitioned by some property into two subsets, a set S 
 of statements which have the property and its complement „ The system 
 is said to be complete with respect to that property if S is precisely 
 the set of statements provable in the system. For example, propositional 
 calculus is complete with respect to the property of a statement being 
 a truth-functionally true formula [3], 
 
 Many classes of mathematical systems can each be represented 
 in the first order predicate calculus. Axioms of the system are 
 represented by a particular set of statements, and theorems are 
 represented by other sets of statements. Much progress has been made 
 in the area of automated theorem proving in the first order predicate 
 calculus [2, k t 5? 6], A decision procedure for a logical system is an 
 algorithm which decides in a finite number of steps if an arbitrary 
 
 * Numbers in brackets refer to list of references, 
 
2 
 
 proposition is a theorem or not. It has been proved by Church [l] that 
 there does not exist a decision procedure for the first order predicate 
 calculus „ However , there are many processes, called semi-de cl sion 
 procedures , which decide in a finite number of steps if a formula is a 
 theorem, but which may never terminate if it is not „ 
 
 In 196k, J, A, Robinson [5] presented a new form of first 
 order predicate calculus based on a single rule of inference, called the 
 resolution principle „ In the first order predicate calculus, the 
 statements, called clauses, can be written in prenex normal form in 
 which all variables are 'universally quantified and all quantifiers 
 occur at the beginning of each clause. Hence, the quantifiers need not 
 be written , In handling a mathematical system via the first order 
 predicate calculus, the axioms, and possibly some other useful results, 
 are written as clauses in the above form without quantifiers „ To prove 
 a theorem, its denial is expressed as one or more clauses, and an 
 attempt is made to show that the sentence S consisting of all the 
 clauses mentioned above is always false. This is done by asking 
 whether, in all possible instances in which the variables may be 
 functionally related, S remains false. In Robinson's system, the 
 method is to process S itself using the resolution principle, rather 
 than enumerate instances of S. He shows the processing terminates if, 
 and only if, the theorem embodied in S is true, and hence his system 
 satisfies a completeness property. 
 
 Direct application of the resolution principle leads to a 
 rather inefficient method of constructing proofs. One reason is that 
 in processing S, unrelated theorems may be proved as well, A method 
 
3 
 of applying the resolution principle is needed which will omit many 
 of the unrelated theorems, but not omit a proof for S if one exists. 
 This paper deals with one such method, proposed in [k]^ called the 
 set of support strategy „ 
 
2, PRELIMINARY DEFINITIONS AND RESULTS 
 
 This section is an outline of the definitions and principal 
 results in [5] together with some added description to clarify the 
 material. For proofs of the lemmas and the theorem,, the reader is 
 referred to [5] . 
 
 2.1 Variables , The following symbols , ordered as shown., are 
 variables : 
 
 u,v,w,x,y,z,u 1 ,v 1 ,w 1 , x 1 ,y 1 ,z i ,w 1 , ....,etc. 
 
 2.2 Functio n Symbols, The following symbols, ordered as 
 
 shown, are function symbols of degree n, for n = blank,, 1, 2, , „, : 
 
 n , n n ,n nn n , n n , n 
 a ,b ,c ,d ,e ,f } % ,h ,a ,1^, . . .,etc . 
 
 Function symbols with no superscripts are constants , 
 
 2.3 Predicate Symbols , The following symbols, ordered as 
 
 shown, are predicate symbols of degree n, for n = blank, 1, 2, ,,, : 
 
 n n n n n n n 
 P ,q ,r ,P 1 ,q 1 ,r 1 ,p 2 _, ...,etc. 
 
 2,U The Negation Symbol , "~" is the negation symbol, 
 
 2,5 Alphabetical Order of Symbols , The set of symbols is 
 well ordered by the ordering above and the conventions that variables 
 precede function symbols, function symbols precede predicate symbols, 
 predicate symbols precede "~", function symbols of lower degree 
 (blank = 0) precede function symbols of higher degree, and predicate 
 symbols of lower degree ( blank = 0) precede predicate symbols of 
 higher degree. 
 
2.6 Terms. A variable is a term. A string of symbols 
 starting with a function symbol of degree n followed by n terms is a 
 term. 
 
 2.7 Atomic Formu las . A string of symbols starting with a 
 predicate symbol of degree n followed by n terms is called an atomic 
 formula . 
 
 2.8 Literals . An atomic formula is a literal; if A is an 
 atomic formula, then ~A is a literal. 
 
 2.9 Complements . If A is an atomic formula, then A and 
 ~A are said to be complementary literals. 
 
 2.10 Clauses . A finite unordered set of literals is called 
 a clause. The empty clause is denoted by O. 
 
 As an example of how the above definitions can be interpreted, 
 
 suppose we are studying Abelian group theory. The variables represent 
 
 2 
 elements of the group. We choose one binary function symbol, say b , 
 
 2 
 to represent the group operation +, and another, say c , to represent 
 
 the solution of u + x = v. Finally, we choose a binary predicate, 
 
 2 
 
 say p , to represent equality. Then, some of the axioms would take 
 
 the following form: 
 
 2/2 2 
 p (b (u,v), b (v,u)) means u+v=v+u; 
 
 2 2 
 p (b (u,a),u) means existence of an identity element; 
 
 2 2 2 
 p (b (u,c (u,v)),v) means existence of a right solution of 
 
 u + x = V, 
 
 2 2 2 2 2 
 
 p (b (b (u,v),w), b (u,b (v,w))) means associativity is 
 
 satisfied. 
 
6 
 
 2.11 Well-formed Expressions . Terms and literals are 
 well-formed expressions. 
 
 2 . 12 Lexical Order of Well-formed Expressions . The set 
 of well -formed expressions is well ordered by the conventions that 
 
 A precedes B if A has fewer symbols than B or, if A and B have the same 
 number of symbols, if A has the alphabetically earlier symbol in the 
 first position in which A and B differ. 
 
 2.1.3 Ground Terms, Ground Literals, Ground Glauses . The 
 adjective "ground" appearing before term, literal, or clause means 
 that no variables appear in the term, literal, or clause. In 
 particular, G is a ground clause. 
 
 2.1^ Substitution Components . A substitution component is 
 any expression of the form T/v, where V is any variable and T is any 
 term different from V. T is called the term of the component and V 
 the variable of the component. 
 
 2.15 Substitutions . A substitution is any set 
 
 {T /V , ..., T /v } of substitution components such that V. / V, if 
 
 -L J- K. K. 1 J 
 
 i/j, 1 < i, j < k. Substitutions are denoted by lower case Greek 
 letters. In particular, e is the empty substitution. 
 
 2.16 Instantiation . If E is any finite string of symbols 
 and w = {T,/V, , ..., T. /v } is any substitution, then the instantiation 
 
 -L J- K. K 
 
 of E by u) is the operation of replacing each occurrence of the variable 
 
 V. , 1 < i < k, in E by an occurrence of the term T. , all variables 
 
 being replaced simultaneously. The resulting string, denoted by Ew, 
 
 is called the instance of E by w. Thus, if E is the string 
 
 E n V. E, ... V. E , then Ew is the string EJT. E n ... T. E . 
 i n 1 in' to i, 1 in 
 
 In In 
 
7 
 
 Here, none of the substrings E. of E contain occurrences of the 
 
 variables V , . ,., V , some of the E. are possibly null, n is 
 Ik. j 
 
 possibly 0, and each V. is an occurrence of one of the variables 
 
 v x , ...,v k . 
 
 2„17 Herbrand Universe „ With any set S of clauses there is 
 associated a set of ground terms, called the Herbrand universe H(S) of 
 S, as follows; Let F be the set of all function symbols which occur 
 in So If F contains any constants, the functional vocabulary of S is 
 F; otherwise, the single constant "a" is adjoined to F, and the 
 functional vocabulary is {a'J U F The Herbrand universe of S is then 
 the set of all ground terms in which there occur only symbols in the 
 functional vocabulary of S„ Unless F consists exclusively of constants, 
 H(S) is an infinite set„ 
 
 2. 18 Saturation,, If S is any set of clauses and P is any 
 set of ground terms, then P(s) , the saturation of S over P, is the set 
 of all ground clauses obtained from members of S by instantiation with 
 members of P 
 
 2.19 Mode Is „ A set of ground literals which does not 
 include a complementary pair is called a model, M is a model of a 
 set of ground clauses S if every clause C in S contains a member of M 
 Then, in general, if S is any set of clauses and H(s) is the Herbrand 
 universe of S, M is a model of S if M is a model of H(s)„ 
 
 2.20 S a t is fi. a b_i 1 i t y « A set S of clauses is satisfiable if 
 there is a model of S; otherwise S is unsatisfiable. 
 
8 
 One interpretation of the above definition is that each 
 clause is the disjunction of its literals, S is the conjunction of its 
 clauses, ~ is the complement symbol, and functions are mappings from 
 the space of variables into itself. Then a set S of clauses is 
 unsatisfiable if there is no way of assigning values 0, 1 to the 
 predicate symbols for each set of arguments such that S takes on the 
 value 1 under the normal rules of Boolean algebra „ In essence, this 
 says that every situation to which Sis applicable is a counterexample 
 to S, so the original proposition is always true. 
 
 2.21 Standardiza tions „ If C is any finite set of strings 
 
 and V , <,„„, V are all the distinct variables, in alphabetical order, 
 
 which occur in strings in C, then the x-standardization of C, denoted 
 
 by £«, is the substitution (x./V , <,„„, x /V | and the y-standardization 
 *-> linn 
 
 of C, denoted by t^, is the substitution ly'/V , „ „. 5 y /V ). 
 u linn 
 
 2.22 Composition of Sub stitutions . If w = (T /v , . *., \/\) 
 and A. are any two substitutions, then the set w f (j \> } where A.' is the 
 set of all components of A. whose variables are not among V , „„„, V 
 
 1 K 
 
 and ^ v is the set of all components T. A./V., 1 < i < k, such that 
 T. is different from V., is called the composition of w and A, and is 
 denoted by w A. 
 
 Lemma 2.1; Composition of substitutions obeys the following 
 laws ; 
 
 1) (Aa)p =A(ap) for all strings A; 
 
 2) (ap)>. = a(pk) for all substitutions a, 0, and K} 
 
 3) (A U B)a = A3 U B3 for all strings A and B and all 
 substitutions a. 
 
on 
 
 9 
 
 2o23 Disagreement Set s, Let A consist of two or more 
 well-formed expressions which all begin with a common sequence of k 
 symbols, k > o The disagreement set of A is the set of well-formed 
 subexpressions beginning at the (k + l)st position of each expressi 
 of A s 
 
 •3 2 ^ 1 3 
 
 Example: A = [p (x,h (x,y),y),p (x,k (y),y),p (x,a,b)}, 
 
 2 . \ 1 v 
 disagreement set of A = {h (x,y),k (y),a} „ 
 
 2„2H Unification o If A is any set of well-formed expressions 
 and \ is a substitution such that A A. is a singleton, then A. is said to 
 unify A, or to be a unifier of A, and A is said to be unifiable 
 
 2„25 Unification Algor ithm,, The following process, applicable 
 to any finite nonempty set A of well-formed expressions, is called the 
 Unification Algorithm: 
 
 Step 1„ Set A.-, = t and k = 1, and go to Step 2„ 
 
 Step 2 If A K is not a singleton, go to Step 3° Otherwise, 
 set A.. - A», and terminate, having unified A 
 
 Pi K. 
 
 Step 3° Let V be the earliest and U the next earliest term, 
 
 k k 
 
 k 
 
 in lexical ordering,, of the disagreement set of A A. n _„ 
 
 If V, is a variable and does not occur in U n , set 
 k k 
 
 A. = A. [U Jv ) , add 1 to k, and return to Step 2„ 
 k+J_ k. k k 
 
 Otherwise, terminate since unification can be shown 
 to be impossible in this case, 
 
 2.26 Most General Unifiers „ If A is a finite nonempty set 
 of well-formed expressions for which the Unification Algorithm 
 terminates in Step 2, the substitution A, is called the most general 
 unifier of A A. is denoted as MGU(A) 
 
10 
 Lemma 2 ,2; If w unifies A, there exists a X => M3U(A), and 
 w = \ a for some substitution a 
 
 This lemma shows that for any unifier u of A, A w is an 
 instance of A X, hence the name most general unifier „ 
 
 2 27 Key Triples . The ordered triple (L,M,N) of finite sets 
 of literals is said to be a key triple of the ordered pair (C,D) of 
 clauses if the following conditions are satisfied ; 
 
 1) L and M are nonempty, and L c C ; M c Dj 
 
 2) N is the set of atomic formulas which are members, or 
 complements of members, of the set L£ U Mr)-* i.e<, } N is formed from 
 L£ f , U Mr; by removing all ~ symbols; 
 
 3) N is unifiable with most general unifier \° f 
 
 h) The sets LL^X and Mtj-X are singletons whose members 
 are complements „ 
 
 2»28 Resolution „ A resolvent of two clauses C and D is 
 any clause of the form (C-lHJv U (D-M)tj_X where (L,M,N) is a key 
 triple for (C,D) and \ = MGU('N) If S is any set of clauses, the 
 resolution of S,, written R (s) or R(s),, is the set of all clauses 
 which are either in S or are resolvents of pairs of clauses in S„ 
 Then for n > 1, the nth resolution of S,, written R (s), is defined 
 as R(R n_1 (s)) 
 
 We continue the interpretation of this system in which 
 each clause is the disjunction of its literals „ Then resolving two 
 clauses is a generalization of the simple deduction; If A => B and 
 B => C, then A => C, An example will show this„ 
 
11 
 
 2 1 1 P ~\ 
 
 Let C = {p (x,y),q (y)] and D ^{~p (u,v),r (y)} . 
 
 2 1 2 1 
 
 Then C£ c = {p (x^x^q (x 2 )} and Dt^ -{~p (y^y^r (y )) . 
 
 If L = (p 2 (x ; y)} and M = {~p 2 (u,v)}, 
 
 then (L,M,N) is a key triple for \C,D) where 
 
 N = (p 2 (x 1 ,x 2 ),p 2 (y 1 ,y 2 )} A. = MGU(n) = {y^x^y^x^ . 
 
 In the above interpretation, 
 
 2 1 • 
 
 l) Ci n k is {p (y,,y_) v q (y D )} 
 
 \>Jn> v ^ wp 
 
 1 2 - 
 which is equivalent to ~q (y p ) => p (y-,.,y p ) 
 
 2 1 
 
 2) Dti d X is {~p (y 1 ,y 2 ) v r (y )) 
 
 p 1 
 
 which is equivalent to p (y ,y_) => r (y ) „ 
 
 Then (l) and (2) imply 
 
 3) ~q {y 2 ) => r (y 3 ) 
 
 which is equivalent to q (y ) v r (y ) by the simple deductic 
 and (3) corresponds to the resolvent of C and D on L and M 
 
 The device is to make the most general substitition which 
 will allow an inference from C and D based on L and M„ The use of 
 the x- and y-standardization is to insure that the most general 
 inference is made,, If in the above example, resolution were performed 
 without first standardizing, the result would have been 
 h) {q (v),r (v)}, which is a special case of (3)0 
 
 Theorem 2.1 ; A set S of clauses is unsatisfiable if, and 
 only if, R (s) contains □ for some finite n 
 
 Theorem 2,1 is the completeness theorem for this system 
 of logic under the property of unsatisf lability of a set S of clauses „ 
 A proof for S consists of a sequence of clauses, A , A } 000 , A , 
 
 such that n is finite and each A is either in S or is the resolvent 
 
 1 
 
12 
 
 of two clauses A. and A. j,k < i, and A = □„ The method of 
 j k ' ' ' n 
 
 1 2 
 searching for a proof consists of generating the sets R (S), R (S), , 
 
 until □ is found o Theorem 2„1 guarantees that □ will be found after 
 
 a finite number of steps if S is unsatisfiable, while D will never 
 
 be found and the process continues indefinitely if S is satisfiable 
 
13 
 
 3. ALTERNATE DEFINITION OF RESOLUTION 
 
 In this section, it will be shown that the definition of 
 resolution given by J. A, Robinson is somewhat restrictive in requiring 
 x- and y-standardizations „ An alternate definition will now be given, 
 and the two definitions will be shown to be equivalent. This is done 
 to simplify the proofs of the major results of this paper. 
 
 301 Separated Variables , Two clauses A and B have separated 
 variables if no variable occurring in A occurs in B„ 
 
 302 Equivalent Clauses . Two clauses A and B are equivalent 
 if there exist two substitutions Oi and B such that A = BOL and B = A(3. 
 
 Example: A and A£. are equivalent where a = £ A = fx n /V. ,„„„, x /V 1 
 
 K A A 1 1 ' ' n n 
 
 and B = [V./x., ..., V /x ). 
 
 11' ' n n 
 
 Suppose A and B are equivalent and A = Ba, B = A(3 Then 
 A differs from B only by a change of variable names. For, suppose not. 
 Then there is a substitution component T/V of Ot where V is a variable 
 occurring in B and T is a term beginning with a function symbol. 
 B --■ A B and A - B a together imply B - B OL (3 . Then V occurs in some 
 position in B on the left while T|3 occurs in the corresponding position 
 on the right, TB again begins with a function symbol and hence can't 
 equal V. This contradicts B = B a B, and so A differs from B only by a 
 change of variable names. Note, also, that the change is reversible. 
 Hence A is identical to B except for a relabeling of variable names. 
 Then if variables represent arbitrary elements of the underlying 
 space of whatever branch of mathematics is 'under consideration, two 
 equivalent clauses would represent the same statement. 
 
14 
 
 3-3 Resolution Sets . The set N is a resolution set for 
 two subsets of literals L and M if 
 
 1) N is the set of atomic formulas which are members or 
 iplements of members oi L U M, 
 
 2) N is unifiable, 
 
 3) if X = MGU(N), L X and M X are complementary literal. . 
 3-4 H-Resolvents . Let A = L U C, B=MUDbe two clauses 
 
 with separated variables where L,M,C, and D are disjoint subsets of 
 literals, L,M nonempty. A and B are H-resolvable on L and M if there 
 exists a resolution set N for L and M. Then, if \ = MGU(N), the 
 Jl-resolvent of A and B on L and M is defined to be C X U D A,. In case 
 the variables are not separated in A and B, the two clauses are 
 H-resolvable if there exist two clauses A 1 and B* which have separated 
 variables, are equivalent to A and B respectively, and are H-resolvable 
 on L' and M' . The H-resolvent of A and B is then defined to be the 
 H-resolvent of A" and B 1 . 
 
 According to this definition, if A and B do not have 
 separateri variables, they may have an infinite number of H-resolvents 
 on a a'/ n pair L and M. The following lemma shows that these are 
 all equi'al-nt. 
 
 Notation . The following symbolism will be used to denote an 
 
 H-resolvent i< of two clauses C and D on L and M: F = (C,D) T ... Here \ 
 
 N ' L,M 
 
 will always denote the MGU(N) where N is the resolution set for L and M. 
 
 Lemma 3.1 : Let A = L U C, B=MUDbe two clauses with 
 separated variables, and suppose A and B are H-resolvable on L and M. 
 Let A' be a clause equivalent to A and B' a clause equivalent to B such 
 
 that A' and B' also have separated variables. Then A 1 and B' are H- 
 
 X " X 
 
 resolvable on L' and M" , and (A',B') T ', M , and (A,B) T m are equivalent. 
 
15 
 
 Proof ; Let u be the substitution such that A = A'u and u p 
 
 such that B = B'u p . Let u. = u U u . Since A' and B' have separated 
 variables, A'u = A'(u U ji ? ) = A'p. = A since u. can't affect A. 
 Similarly, B'u = B'u p = B. Let N be the resolution set for L and M, 
 and let N' be the corresponding set for L' and M' . Then N = N'u. Hence, 
 N'u X = N \ is a singleton, and N' is unifiable. By Lemma 2.2 there is 
 
 a substitution X' = MGU(N' ) and a substitution GC such that u X = X*Ct. 
 Further, L'\' = ~(M'\'), i.e., N' is a resolution set for L' and M' . 
 
 X ' 
 Then A' and B' are H-resolvable on L' and M', and F' = (A',B') , , is 
 
 the H-resolvent. If F =■ (A,B) , then clearly F = F'a. But now the 
 roles of A,B and A', B' can be interchanged, and F' = F 0L X for some 
 substitution a'. Hence F and F' are equivalent. 
 
 Lemma 3«1 shows that, in the definition of H-resolvent, the 
 
 particular way in which the variables are separated in A and B is 
 irrelevant, and the H-resolvent of A and B on L and M is unique up to a 
 change of variable names. 
 
 3-5 H-Resolution . Let S be a set of clauses. An H-resolu- 
 tion of S is a set of clauses which are either in S or H-resolvents of 
 pairs of clauses in S with the understanding that only one H-resolvent 
 of two clauses on a given pair of subsets of literals is included. 
 For n > 1, an nth H-resolution of S is an H-resolution of an (n-l)th 
 H-resolution of S. 
 
 Since any two clauses may have an infinite number of 
 H-resolvents on two subsets of literals, there are an infinite number 
 of H-resolutions . However, Lemma 3«1 shows that any two H-resolutions 
 of S are equivalent sets of clauses. By induction, the same is true 
 of any two nth H-resolutions of S. Robinson's resolution of S is a 
 particular H-resolution of S. Clearly D is the only clause equivalent 
 
l6 
 
 to Do Combining this with Theorem 2,1, we have the following 
 completeness theorems 
 
 Theorem 3°1 ° S is unsatisfiable if, and only if, D is a 
 member of any nth H-resolution of S for some finite n„ 
 
 We have defined resolution in a way independent of particular 
 variable names to prove later, in a compact way, a theorem on resolutions 
 affecting three or more clauses „ Robinson's formulation involving x- and 
 y-standardizatlons leads to proofs in which the concepts used are obscured 
 by notation,, R(s) is now used to denote any H-resolution of S, 
 
 In preparation for the major theorems to be proved in the 
 next section, the following lemma will be proved. 
 
 Lemma 3° 2; Let A be a set of literals which is unifiable, and 
 let K = MGU(A), Then no variable appears in any component of X that 
 does not appear in A, 
 
 Proof; X is a product of substitutions \ = (U, /v.) ,,, (U /V ) 
 
 ■ r 1 1 n n - 
 
 given in the Unification Algorithm, By induction, each substitution 
 [U./v. ] involves only variables occurring in A, Hence, X also contains 
 only variables occurring in A, 
 
17 
 h. SETS OF SUPPORT 
 
 H-resolution provides a very good method for automatic 
 theorem proving. For a set S of clauses, the method consists of 
 generating the sets R(s), P. (S), R (S), ... . S will be unsatisfiable 
 if, and only if, D € R (s) for some finite n. If S is satisfiable, 
 the method will continue generating sets R (s) forever . Thus using 
 resolution, proof is by contradiction. One problem is that the sets 
 R (s) may contain many clauses which are irrelevant. For example, 
 suppose S = S, U S U S where S is a set of clauses representing 
 the axioms of some consistent system (and hence, known to be 
 satisfiable), S p is a set of clauses representing the hypotheses of 
 some theorem, and S_ is a set of clauses representing the denial of 
 that theorem. It may be possible to H-resolve pairs of clauses in S, 
 and produce a consequence of the axioms which may have no relevance 
 to the particular theorem embodied in S. A strategy for applying 
 H-resolution to S is needed which does not enumerate many of the 
 non-essential clauses, but which still finds a proof for S if one 
 exists. One strategy proposed in [h] is the set of support strategy. 
 
 k-.l Set of Support Strategy . Let T be a subset of So 
 
 Let T, = T, and let T. . be the set of clauses which are either 
 1 i+I 
 
 in T. or an H-resolvent of a clause in T. with a clause in T. US with 
 l 11 
 
 the understanding that only one H-resolvent of a pair of clauses 
 on a given pair of subsets of literals is included. A clause A has 
 T support or is derived from T if A is in T. for some i. A proof 
 of S has T support if every clause in the proof has T support or is 
 in S itself. 
 
18 
 Again, for each i, there are infinitely many sets T., 
 all equivalent, and the definition leads to one representative 
 
 set T. for each i. 
 
 1 
 
 The set of support strategy has been shown to work if a 
 particular class of subsets T is used [6]. The major result of 
 this paper is to extend this class. 
 
 Theorem h.l i Let A = I U K U D, B = J U L U E, and C = M U F 
 be three clauses. Suppose A and B are H-resolvable on I and J with 
 
 G = (A,B)t t = (K U L)X U DX U EX , 
 
 and suppose further that G and C are H-resolvable on (K U L)\ and M 
 with 
 
 H= (G ' C) (K U L)\,M = ^ UE ^ U ^ * 
 
 Then there is a clause H derivable from A, B, and C such that H 
 
 -* 
 has C support and H = H w for some substitution co« 
 
 Proof ; By Lemma 3-1, it can be assumed without loss of 
 generality that the variables of A, B, and C are all separated. Then 
 by Lemma 3-2, A, does not involve variables in C. Hence CX - C, and 
 CXX = Ck so that 
 
 H = DXX U EXX U FXX = (D U E U F)XX . 
 
 Let N be the resolution set for (K U L)X and M. Let N 
 be the set of atomic formulas which are members or complements of members 
 of K U M, and let N p be the set of atomic formulas which are members 
 or complements of members of L U M. Since MX = M, N,X and N ? X are 
 subsets of N. Hence X unifies both N,X and N ? X. By Lemma 2.2, 
 there exists a X, = MGU(N, ) and a substitution a such that 
 XX = Xa . Similarly, there exists a X ? = MGU(N ) and a substitution 
 
19 
 OL such that KK = XJX p . Since KX\ and MXA are complements, KX, 
 and MX. must "be complements also. Similarly, IA p and M\ p are 
 complements. Thus, N is a resolution set for K and M, and N is a 
 resolution set for L and M. Then A and C are H-resolvable on K 
 and M with 
 
 G .l = (A ' C) K,M = 1X 1 U DX 1 U **1 ' 
 and B and C are H-resolvable on L and M with 
 
 G 2 = ( B > C ) L M = ^2 U EX 2 U FX 2 ' 
 
 Clearly G, and G p have C support; however, they may not have separated 
 
 variables. In order to show that G, and G p are H-resolvable, the 
 
 variables of G-, are relabeled to produce an equivalent formula G, 
 
 whose variables are distinct from the variable names occurring in 
 
 G p , H, \ , and a. . (This is done even if the variables were already 
 
 separated. Lemma 3-1 allows G, to be used, instead of G, , to 
 
 H-resolve with G„ . ) If the variables of G, are V, . . ... V . then 
 2 1 1' } n f 
 
 * 
 
 G-, = G, u where 
 
 = {V V 1' •••' V V n) 
 
 and no V. occurs in G , H, X p , or Q^ . Since a may not change all 
 the variables in G. , let the variable names be chosen so that 
 
 a i = {w ■••' V v r} • 
 
 Let 
 
 a ' = -|t /v, , . . . , T /V , V , /V , , . . . , V /V \ 
 
 1 [_ 1' 1 r" r r+i r+1 r/ nj 
 
20 
 
 Then clearly, 
 
 G i a i = ^^l = G i a i = ( x u D u F )^ • 
 
 ' * 
 Now consider the substitution OCJDC . Since G contains 
 
 * -* 
 no variables occurring in a p , G,CC = G and hence 
 
 G i a 2 a i = G i a i = ( X u D u F )^ • 
 
 i _ i 
 
 In particular (l\ n u)a o a. = IW. Next G a_a_ = G„a: since V. does 
 v 1 2 1 22122 l 
 
 t _ 
 
 not occur in G p , a , and hence, G ? CC . Thus (JX )a a = JX p CC = JTvA. 
 
 But JXA = ~ (IXA) since J.\ = ~ (IX.) by hypothesis (A and B were 
 
 assumed H-resolvable on I and j). 
 
 Hence, there is a resolution set N for IX, u = I and JA, p . 
 
 If \_ - MGU(N ), then by Lemma 2.2, OCJX = X_co for some substitution 
 
 co since OLJX unifies N~. Thus, G, and G p are H resolvable on I and 
 
 J\ with 
 
 H* = (G*,G 2 ) I J JX = D^uA U FX ± [x\ U EX^ U FX^ . 
 
 Since G and G p bc.th have C support, H has C support. Finally, since 
 
 G X co = G ] a 2 a i = (l U D U F)\A and 
 
 G 2 \ u - G 2 a 2 a jL = G 2 a 2 = (J U E U F)W . 
 
 * _ 
 
 Then H w - (D U E U F)AA = H, and the theorem is proved. 
 
 Theorem h.2. i Let A = L U C, B = M U D be two clauses, and let 
 co be some substitution. Suppose Aco and B are H-resolvable on Lw and 
 M with F = (A.co,B) T ... Then A and B are H-resolvable on L and M, and if 
 
21 
 G = (A,B) L „, then F = GCC for some substitution a. 
 
 Proof ; Again, it can be assumed that the variables in B 
 are distinct from those occurring in A, Aw, and w. Let N be the 
 resolution set for Lu> and M. Then X = MGU(N). Let N be the set of 
 atomic formulas or complements of atomic formulas in L U M. Since M 
 contains no variable occurring in 10, Mu = M, and hence N, w = N. 
 Then (N,cu)\ = N (uA.) is a singleton, and there exists a X = MGU(N-). 
 Further uX = XQi for some substitution a. Finally, LA, = ~ (MX) 
 since LiuX and MX are complements. Then A and B are H-resolvable on 
 L and M, and G = (A,B)£ = CA U DX. 
 Finally, Ga - CXCC U DXDC = CwX U DcoX = CwX U DX = F, since Doo = D. 
 
 Theorem ^-.3 ° Let S be a set of clauses for which there 
 exists a proof using S as the set of support. Let T be a subset of 
 S. If T contains at least one clause, B, occurring in some proof P 
 of S, then the set of support strategy with T as the set of support 
 will produce a proof of S. 
 
 Proof ; B is involved in one or more H-resolutions . If at 
 some point, a clause with B support is H resolved with a clause F that 
 does not have T support and F^S-T, then F itself is an H-resolvent of 
 two clauses C and D, neither of which has T support. By Theorem ^.1, 
 F can be replaced by a clause F such that F is supported by B and 
 F = F w for some substitution w. Furthermore, Theorem 4.2 allows F 
 to be used in any H-resolution where F had previously been used. 
 Continuing in this fashion, P can be replaced by a proof with T support 
 
 Corollary ; If T is a subset of S such that S-T is a 
 consistent set of clauses, then the set of .support strategy with T 
 as set of support is complete. 
 
22 
 
 Proof ; If S-T is consistent, and hence satisfiable, then 
 no proof of S could consist of clauses of S-T alone. Thus, any proof 
 of S must contain a clause of T, and by the Theorem 4„3, the set of 
 support strategy with T as set of support is complete. 
 
 The corollary has been proven previously in [6], However, 
 Theorem 4.3 extends the corollary by not requiring S-T to be consistent. 
 Thus, Theorem 4.3 allows a larger class of subsets of S to be valid 
 sets of support. 
 
 Experience has shown that the set of support strategy 
 improves the speed of finding a proof considerably [4]. The reason 
 is that the cardinality of the sets T is much smaller than that of 
 the corresponding sets S . Thus while more sets T must be generated, 
 the time required is much less. 
 
 Theorem 4.3 has the disadvantage of requiring prior knowledge 
 of a proof of S, or at least some clause occurring in a proof. 
 However, this restriction is not too severe. For example, if 
 S = S, (J S p U S as at the beginning of this section, then any proof 
 of S must contain a clause in S p U S_ since S, , the set of axioms, is 
 assumed to be consistent. Other choices for a set of support may 
 also work. For example, a reasonable assumption is that at least 
 one clause in the denial of the conclusion of the theorem appears 
 in a proof. Thus, S^ alone may be a good choice for T» Efforts should 
 now be directed toward discovering which subsets give the best results. 
 Comparisons should be made of various rules for choosing T, e.g., 
 S U S_, S alone, etc. Another possibility is that the mathematician, 
 
 with his Insight into the particular problem, chooses the subset T. 
 Attempts should be made to understand how to choose optimum sets of 
 support. We are planning to write a computer program to compare 
 various methods for choosing sets of support. 
 
23 
 
 LIST OF REFERENCES 
 
 [1] Church, A., "An Unsolvable Problem in Number Theory," 
 American Journal of Mathematics , Vol. 58 (1936), pp< 
 
 3^5-363. 
 
 [2] Davis, M. , Putnam, H. , "A Computing Procedure for Quantification 
 Theory," Journal of the Association for Computing Machinery , 
 Vol. 7 (i960), pp. 201-215. 
 
 [3] Kleene, S., Introduction to Me tamat hematics , D. Van Nostrand 
 Company, Inc., Princeton, 1952. 
 
 [k] Robinson, G. A., Wos, L. T., Carson, D. F., "Some Theorem- 
 Proving Strategies and Their Implementation," Argonne National 
 Laboratory, Technical Memorandum No. 72, Argonne, Illinois, 
 1964. 
 
 [5] Robinson, J. A., "A Machine -Oriented Logic Based on the 
 
 Resolution Principle," Journal of the Association for Computing 
 Machinery , Vol. 12 (1965), pp. 23-i+l. ' ~~~ 
 
 [6] Wos, L. T., Robinson, G. A., Carson, D. F, , "Efficiency and 
 
 Completeness of the Set of Support Strategy in Theorem Proving, " 
 Journal of the Association for Computing Machinery , Vol . 12 
 (1965), pp. 536-5^1. 
 
AUG I 6 1SC8 
 

JUN 2 1969