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 UIUCDCS-R-79-978 
 
 UILU-ENG 79 1726 
 
 Complete Problems in the First-Order 
 Predicate Calculus 
 
 by 
 
 David A. Plaisted 
 
 July 1979 
 
Complete Problems in the First-Order 
 Predicate Calculus 
 
 by 
 
 David A. Plaisted 
 Department of Computer Science 
 University of Illinois 
 Digital Computer Lab 
 Urbana, IL 61801 
 
 Key words: Satisfiability, predicate calculus, resolution theorem proving, 
 complete problems, computational complexity. 
 
 This research was partially supported by the National Science Foundation 
 under grant MCS 77-22830. 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/completeproblems978plai 
 
Abstract 
 
 We present a survey of recent results concerning the complexity 
 of deciding if a first-order predicate calculus formula in Schonfinkel- 
 Bernays form is satisfiable. We also present results concerning the 
 complexity of deciding the existence of resolution proofs of restricted 
 depths and sizes. In this way we obtain natural problems complete for 
 the classes NP, PSPACE, deterministic and nondeterministic exponential 
 time, and deterministic and nondeterministic double exponential time. 
 The results concerning the existence of resolution proofs of restricted 
 depths and sizes seem to have implications for the design and analysis 
 of resolution theorem proving programs. This report mainly lists results 
 without proof. 
 
-2- 
 
 We exhibit problems involving first-order predicate calculus 
 formulae that are complete for the classes NLOGSPACE, P, NP, 
 PSPACE, DEXPTIME, and NEXPTIME. These problems involve satisfiability 
 and resolution proof depth for formulae in Schdnfinkel-Bernays form 
 (that is, the prefix is of the form 33..3W-- \/) • Most of these results 
 are obtained by direct encoding of Turing computations, using an 
 economical representation of the successor relation on integers. Some 
 of the results are obtained by different methods. This work 
 was originally motivated by a problem involving relational databases 
 posed by Yehoshua Sagiv. It was then realized that this problem was 
 related to recent work of Harry Lewis [ 7] involving decidable subclasses 
 of the first-order predicate calculus. We improved his lower bound of 
 nondeterministic time c for the Schonfinkel-Bernays class to nondeterministic 
 time c n/ ogn . Harry Lewis has improved this to c 11 since then [8]. 
 
 We have also obtained some results which precisely 
 characterize the difficulty of resolution theorem proving in the first- 
 order predicate calculus (not restricted to Schonfinkel-Bernays form). 
 In particular, one of these results states that it is complete for 
 nondeterministic exponential time to decide if a depth d proof of a 
 particular clause exists from a set W of clauses, if d is given in 
 unary notation. Other results deal with the special case in which W 
 consists entirely of 2-literal clauses. Also, it is NP-complete to 
 determine if a size d resolution proof of a particular clause from a set 
 W of propositional clauses exists, if d is given in unary. (This last 
 result is not just a restatement of the NP-completeness of satisfiability 
 of Boolean expressions.) 
 
-3- 
 
 Some of the results for deterministic and nondeterministic 
 exponential time completeness are obtained by encoding Turing computations 
 using statements of the form "The symbol on tape square i at time j is 
 s" or "The Turing machine at time j is at tape square i in state t." 
 In this way we encode deterministic and nondeterministic Turing com- 
 putations. Some of the results for polynomial space completeness are 
 obtained by letting P(x Q , x, , ..., x ) represent a configuration 
 XqX-,...x of a Turing machine, specifying the tape symbols, the 
 current state, and the position of the read-write head. By cycling 
 the variables to the left or the right, we get the effect of moving 
 the read-write head to the right or to the left. 
 
 We first consider formulae of the classical first-order predicate 
 calculus without the identity sign or function signs. In particular, 
 an atom is a predicate symbol followed by a list of variables (and 
 constants). A formula is said to be in Schbnfinkel-Bernays form 
 if it is of the form 3y,3y 2 - . .3y Vx,Vx 2 . . -Vx A where A is a well- 
 formed expression containing atoms and the Boolean connectives 
 a( conjunction), v(disjunction) , =(equivalence) , and l(negation). 
 We will assume that formulae in Schb'nfinkel-Bernays form have no free 
 variables. The Skolemized (or functional) form of A is obtained by 
 replacing each y. in A by a distinct constant symbol, and by deleting 
 all quantifiers. For example, 3y-|3y 2 Vx.|\/x 2 (P(x.. , y,)vP(x„, yj) is a 
 formula in Schbnfinkel-Bernays form and its Skolemized form is 
 P(x,, c-, ) v P(x 2 , c 2 ). It is known that a formula F is satisfiable 
 iff its Skolemized form is satisfiable. We will assume all formulae 
 
-4- 
 
 are in Skolemized form. Also, we will assume all formulae are expressed 
 as a conjunction of disjunctions of literals, where a literal L is 
 an expression of the form B or "IB for some atom B. In the former 
 case, we say L is a positive literal, and in the latter case we say 
 L is a negative literal. A clause is a disjunction of literals. 
 Hence we will assume all formulae are conjunctions of clauses. We say 
 a problem S is complete for a class of problems if S is in the class and 
 if e^ery other problem in the class can be reduced to it by deterministic 
 log tape reductions. Completeness for DLOGSPACE is defined by using 
 deterministic one-way log-tape reductions as defined in [2]. Note that 
 a problem may be complete for DEXPTIME under this definition even if it can 
 
 be solved in time c n/logn . 
 
 We now give a list of results. We need more terminology. 
 
 A Horn clause is a clause that has at most one positive literal. The 
 positive literal of a Horn clause (if it exists) is called the consequent 
 of the clause. The negative literals (if they exist) are called 
 antecedents of the clause. We say two literals LI and L2 are unifiable 
 if they have a common instance. For Schbnfinkel-Bernays form formulae, this 
 means that there is some literal L that can be obtained from both LI and L2 
 by replacing variables by constant symbols. Thus P(x, c) and P(d, x) are 
 unifiable because they have P(d, c) as a common instance. However, P(x, x) 
 and P(c, d) are not unifiable, nor are P(c, x) and P(d, y). 
 
 We use the notation #P < c to indicate that the number of 
 predicate symbols in a class of formulae is bounded. Also, HORN 
 indicates that all formulae in the class are conjunctions of HORN clauses. 
 The notation DET means that no two consequents of distinct Horn clauses 
 in the formula are unifiable, and that ewery (universally quantified) 
 variable appearing in an antecedent must also appear in the consequent. 
 
-5- 
 
 In particular, a clause having no consequent must have no universally 
 quantified variables. This condition implies the determinism of a certain kind 
 of program expressed as a set of Horn clauses, hence the notation DET. 
 
 Furthermore, 3LIT means that all formulae are conjunctions 
 of clauses having three or fewer literals, and 2LIT means all clauses 
 have two or fewer literals. The symbol E refers to the number of 
 existential quantifiers and U refers to the number of universal quantifiers. 
 With each combination of conditions, we indicate which deterministic 
 or nondeterministic time or tape complexity class the problem is complete 
 for. Note that E = 2 is equivalent to E >_ 2 for completeness results. 
 
 1. Complexity of the satisfiability problem for formulae 
 in Schonfinkel-Bernays form with E = 2 and #P <_ c. 
 
 Restrictions Complete for 
 
 a) 3LIT NEXPTIME 
 
 b) 3LIT HORN DEXPTIME 
 
 c) 3LIT HORN, DET PSPACE 
 
 d) 2LIT PSPACE 
 
 e) 2LIT HORN PSPACE 
 
 f) 2LIT HORN, DET PSPACE 
 
 Comments: la) is due to Lewis [7]. Id) can be obtained from the function 
 generation problem of Kozen [6], but we have a simple direct reduction 
 from Turing computations. 
 
 2. Complexity of the satisfiability problem for formulae in 
 Schonfinkel-Bernays form with E = 2 and #P <c and U < log 2 |W| where |W| 
 is the length of the formula W. 
 
-6- 
 
 a) 3LIT NP 
 
 b) 3LIT HORN P 
 
 c) 3LIT HORN, DET NLOGSPACE 
 
 d) 2LIT CoNLOGSPACE 
 
 e) 2LIT HORN CoNLOGSPACE 
 
 f) 2LIT HORN, DET DLOGSPACE 
 Comments: All of these results can be obtained for E = 2, #P <_ c, 
 and U = by encoding n predicate symbols as sequences of logn bits 
 and using the results in Part 3. 
 
 3. Complexity of the satisfiability problem for propositional 
 calculus formulae. This corresponds to Schb'nfinkel-Bernays form 
 with E = or E = 1. These results are all known or follow easily 
 from known results. 
 
 a) 3LIT NP 
 
 b) 3LIT HORN P 
 
 c) 3LIT HORN, DET NLOGSPACE 
 
 d) 2LIT CoNLOGSPACE 
 
 e) 2LIT HORN CoNLOGSPACE 
 
 f) 2LIT HORN, DET DLOGSPACE 
 
 g) #P <_ c Regular set 
 Comments: 3a) is of course Cook's result [1]. 3b) is essentially the 
 complement of the GEN problem [4]. 3c) can be obtained from the graph 
 accessibility problem GAP of [3]. 3d) is essentially from [3]. 3e) can 
 be obtained from the complement of GAP of [3]. 3f) can be obtained from 
 the GAP! problem of [2]. 
 
-7- 
 
 4. Complexity of deciding the existence of various types of 
 resolution proofs having depths or sizes restricted in various ways. 
 The size of a resolution proof is the number of resolutions in it. 
 Assume we are given an integer n and a formula W in Schbnfinkel-Bernays 
 form with E = 2 and #P <_ c, and are looking for a resolution proof of NIL 
 (the empty clause) from W. An all-positive resolution proof is a resolution 
 proof in which one of the parent clauses in each resolution consists 
 entirely of positive literals. In a unit resolution proof, one of the 
 parents in each resolution must consist of a single literal. Note that 
 all-positive resolution and hyper-resolution are identical for 2-literal 
 Horn clauses. 
 
 a) 3LIT HORN, Complete for DEXPTIME to decide if an 
 n in binary (all-positive, unit, hyper-resolution) 
 
 proof of (depth, size) n exists. 
 
 b) 3LIT HORN PSPACE - complete to decide if an (all- 
 
 n in unary positive, unit, hyper-resolution) proof of 
 depth n exists. 
 
 .IP-complete to determine if an (all- 
 positive, unit, hyper-resolution) proof of 
 size n exists. (See 4f . ) 
 
 c) 3LIT HORN, DET PSPACE - complete to decide if a hyper- 
 n in binary resolution proof of depth n exists. 
 
 PSPACE - hard to decide if an (all-positive, 
 unit) proof of depth n exists. 
 
 d) 3LIT HORN, DET CoNP - complete to decide if a hyper- 
 n unary resolution proof of depth <_ n exists. 
 
 CoNP - hard to decide if an (all-positive, 
 unit) proof of depth <_ n exists. 
 
 e) 2LIT HORN PSPACE - complete to decide if an (all- 
 n binary positive, unit) proof of (depth, size) 
 
 n exists. 
 
 f) 2LIT HORN NP-complete to decide if an (all-positive, 
 n unary unit) proof of (depth, size) n exists. 
 
 PSPACE - complete to decide if an arbitrary 
 resolution proof of depth n exists. 
 
 g) 2LIT HORN, DET PSPACE- complete to decide if an (all- 
 n binary positive, unit) proof of (depth, size) 
 
 n exists. 
 
-8- 
 
 h) 2LIT HORN, DET P - complete to decide if an (all-positive, 
 n unary unit) proof of (depth, size) n exists. 
 
 PSPACE - complete to decide if an arbitrary 
 resolution proof of depth n exists. 
 
 Comments: 4b) is obtained by reduction from satisfiability of quantified 
 
 Boolean formulae [11]. 4f), first part, is obtained by reduction from 
 
 conjunctive normal form satisfiability. We do not know the complexity 
 
 of 4d) if we require the proof to have depth equal to n. Many of these 
 
 results can be extended to the problem of determining if there exists an 
 
 inconsistent set of n ground instances of the clauses in W. 
 
 5. Complexity of deciding the existence of various types of 
 
 resolution proofs of NIL from a set of clauses in the proposi tional 
 
 calculus. (This corresponds to Schonfinkel-Bernays form formulae with 
 
 E = or E = 1 . ) Assume we are given a set W of clauses and an integer n. 
 
 a) arbitrary set of clauses, P-complete to decide if a unit 
 
 n unary resolution proof of depth n exists. 
 
 NP-complete to decide if an (all- 
 positive, hyper-resolution, arbitrary) 
 proof of size n exists. 
 NP-hard to decide if an (all-positive, 
 arbitrary) proof of depth n exists. 
 These results still hold if 3LIT is 
 required. 
 
 b) 3LIT HORN, NP-complete to decide if an (all- 
 
 n unary negative, arbitrary) proof of size 
 
 n exists. 
 
 NP-hard to decide if an (all-negative, 
 arbitrary) proof of depth n exists. 
 P-complete to decide if an (all- 
 positive, unit, hyper-resolution) 
 proof of (depth, size) n exists. 
 
 c) 3LIT HORN, DET, CoNLOGSPACE - complete to determine 
 
 n unary if a hyper-resolution proof of depth <_ 
 
 n exists. 
 
 CoNLOGSPACE - hard to determine if 
 an (all-positive, unit) proof of 
 depth <_ n exists. 
 
-9- 
 
 d) 2LIT HORN, NLOGSPACE - complete to determine if 
 n unary an (all-positive, unit) proof of 
 
 (depth, size) n exists. 
 
 e) 2LIT HORN, DET, DLOGSPACE - complete to determine if 
 n unary an (all-positive, unit) proof of 
 
 (depth, size) n exists. 
 
 f) arbitrary set of clauses, Regular set to determine if an (all- 
 #P £ c, positive, all -negative, unit, 
 
 n unary hyper - resolution, arbitrary) proof 
 
 of depth n exists. 
 
 g) arbitrary set of clauses, Regular set to determine if an (all- 
 #P 1 c, positive, all -negative, unit, hyper- 
 n binary resolution, arbitrary) proof of depth 
 
 n exists. 
 
 Comments: There is a polynomial time algorithm for the first part of a) 
 even though the straightforward search method may take exponential time. 
 See [4] for a related result. Also, the second part of a) is obtained 
 by reduction from the hitting set problem [5]. Furthermore, results a) 
 through e) can also be obtained for E = 2 and #P <_ c and U = 0, and for 
 
 E = 2 and IP < c and U <_ 1 og^ I W | . This is done by encoding n predicate 
 symbols as sequences of logn bits. 
 
 Note that the results in Part 5 give a precise meaning to the 
 statements "unit resolution is easier than general resolution" and 
 "all-positive resolution for Horn sets is easier than all-negative 
 resolution for Horn sets". 
 
■10- 
 
 6- Complexity of deciding the existence of a depth n 
 resolution proof of NIL (the empty clause) from a set W of clauses, 
 not necessarily in Schbnfinkel-Bernays form. Let |x| be the length 
 of the input x consisting of W and n. The following results all 
 still hold if the function symbols of W are restricted to a fixed 
 finite set of at least two unary function symbols and at least one constant. 
 
 ? C I X I 
 
 a) n binary Complete for nondeterministic time 2 
 
 Still true if 3LIT HORN is required, and if 
 we restrict the proofs to be all -positive, 
 unit, or hyper-resolution proofs. 
 
 clxl 
 
 b) n unary Complete for nondeterministic time 2 ' '. 
 
 Still true if 3LIT HORN is required, et cetera 
 as in part a). 
 
 2 c|x| 
 
 c) 2LIT, n binary Complete for nondeterministic time 2 
 
 Still true if HORN is required. 
 Complete for NEXPTIME if we restrict the 
 proofs to be all -positive or unit proofs, 
 even if HORN is required. 
 
 d) 2LIT, n unary Complete for NEXPTIME (i.e., nondeterministic time 2 
 
 Still true if HORN is required. 
 NP-complete if we restrict the proofs to be 
 all-positive or unit resolution proofs, even 
 if HORN is required. 
 
 n) 3LIT HORN, DET, Complete for CoHEXPTIME to decide if a 
 n binary hyper-resolution proof of depth n exists. 
 
 Hard for CoNEXPTIME to decide if an (all- 
 positive, unit) proof of depth n exists. 
 
 f) 3LIT HORN, DET CoNP - complete to decide if a hyper-resolution 
 n unary proof of depth n exists. 
 
 CoNP hard to decide if an (all-positive, 
 unit) proof of depth n exists. 
 
 clxl 
 
 g) 2LIT HORN, DET Complete for deterministic time 2 2 
 
 n binary to determine if an arbitrary resolution proof 
 
 of depth n exists. 
 
 Complete for DEXPTIME to determine if an (all- 
 positive, unit) proof of depth n exists. 
 
 c|x| 
 
11- 
 
 h) 2LIT HORN, DET Complete for DEXPTIME to determine if an 
 
 n unary arbitrary resolution proof of depth n exists. 
 
 Complete for P (polynomial time) to determine 
 if an (all-positive, unit) proof of depth n 
 exists. 
 
 Comments: To obtain the upper bounds, we represent literals as 
 directed acyclic graphs as in [10]. If L is such a literal, let |L| 
 denote the number of nodes in L. The number of edges in L will be 
 proportional to |L| if we have a fixed number of function symbols of 
 
 fixed arities. In the worst case, the number of edges will still be 
 
 2 2 
 
 0(|L| ). It may take 0(|L| log|L|) space to represent L in a Turing 
 
 machine since each node may require 0(log|L|) bits. We obtain the 
 
 upper bounds by observing that if substitution a is a most general 
 
 unifier of literals A and B, then |Aa| < |A| + |B| and |Ba| < |A| + |B|. 
 
 Also, if C is a resolvent of CI and C2 then |C| < |C1| + |C2|. The 
 
 lower bounds are obtained by encoding Turing computations using two 
 
 stacks. These results show that resolution theorem proving is inherently 
 
 difficult, even when the depth is restricted. Also, these results suggest 
 
 that resolution theorem provers should concentrate more on the "size" 
 
 of a proof than on its depth, since proofs of small depth can have a 
 
 large size. By size we mean in this context the sum of the sizes of the 
 
 clauses in the proof. We know that finding proofs of small size is no 
 
 worse than NP-complete. 
 
 Encoding the Successor Relation 
 
 Many of the completeness results for Schonfinkel-Bernays form 
 are based on an economical encoding of the successor relation. We represent 
 
-12- 
 
 integers in binary notation as sequences of zeroes and ones. Assume 
 constants of zero and one and other constants are available. (We have 
 restricted E to 2 so there are strictly speaking only two constants, but 
 other constants can be encoded as sequences of these two.) Let I(c Q , c, , 
 ..., c ) be the integer c '2 n + c^"" 1 + ... + c n _-|' 2 + c n - We construct a 
 formula A(P1, P2) in Schonfinkel-Bernays form which represents the 
 following assertion: 
 
 \/x ] \/x 2 . . . vx m [(0< I(c , c 1 , . . . , c n ) < 2 -1 and 
 I(d Q , d ]s . .., d n ) = I(c Q , Cp ..., c n ) + 1) implies 
 
 • I \ C«i C-jj • • • » C n > ^1 > ^? > •••> X J = 'M^f)) 1' •••» n ' *1 ' *2 ' " 
 
 Thus we can count up or down by one by going from PI to P2 or from P2 
 to PI. Also, the formula A(P1, P2) consists of 2-literal Horn clauses 
 and satisfies the restriction DET. The formula A(P1 , P2) makes use of 
 auxiliary predicate symbols Ql and Q2, and is constructed as follows: 
 Let c be a constant symbol other than and 1. Let x be an abbreviation 
 for x,, x 2> ..., x , y be an abbreviation for y, , y 2 , ..., y , and z 
 be an abbreviation for z-, , z ? , ..., z . 
 
 1. Pl(y, x) d Ql(c, y, x) Insert c 
 
 2. Ql(z, 1, x) 3 Q1(0, z, x) Change rightmost ones to zeroes 
 
 3. Ql(z, 0, x) ^ Q2(l, z, x) Change rightmost zero to one 
 
 4. Q2(z, 1, x) ^ Q2(l, z, x) Cycle past remaining bits 
 
 5. Q2(z, 0, x) 3 Q2(0, z, x) Cycle past remaining bits 
 
 6. Q2(z, c, x) d P2(z, x) Delete c 
 
 Example: Pl(Ollx) 3 Ql(cOllx) 3 Ql(OcOlx) 3 Ql(OOcOx) 3 02(100cx) 3 P2(100x). 
 
13- 
 
 We represent Wl D W2 by ~IW1 v W2 and thereby obtain 2-literal 
 Horn clauses. The above six clauses add one in going from PI to P2. 
 There are six similar clauses which subtract one in going from P2 to 
 PI. The formula A(P1, P2) is the conjunction of these twelve clauses. 
 Note that adding one always requires a constant number of steps, 
 regardless of the number involved. This is useful for establishing 
 bounds concerning sizes of resolution proofs. 
 
 Possible Extensions 
 
 These results can be extended in many ways. Some gaps in the 
 above tables of results still exist. We might look at other classes 
 of formulae, also. For example, we can consider the satisfiability 
 problem for formulae in which the connectives are chosen from some set 
 other than a, v, ~1. This is related to recent work of Lewis [9]. We 
 can consider other decidable subclasses of the first-order predicate 
 calculus investigated by Lewis [7] with the HORN, 2LIT, or DET restrictions 
 added. Also, we can consider decidable subclasses of the second order 
 predicate calculus. We have shown an analogue of 3-satisfiability to 
 be complete for nondeterministic exponential time. Are there analogues 
 of other NP-complete problems (such as the clique problem) that are 
 complete for nondeterministic exponential time? Many of our results 
 are based on encodings of deterministic and nondeterministic Turing 
 computations by predicate calculus formulae. Can parallel and alternating 
 Turing computations also be modeled in this way? Lewis [7] has done some 
 work along this line. Finally, we can look at inference rules other than 
 resolution and determine the complexity of searching for proofs at restricted 
 depths. 
 
■14- 
 
 References 
 
 1] Cook, S. A. The complexity of theorem proving procedures. 
 
 Proc . 3 rd Annual ACM Symp . on Theory of Computing (1971) 151-158. 
 
 2] Hartmanis, J., Immerman, N. and Mahaney, S. One-way log tape 
 
 reductions. Proc . 19th Annual Symp . on Found , of Computer Science 
 (1978) 65-71. 
 
 3] Jones, N. D. , Lien, Y. E. and Laaser, W. T. New problems complete 
 for nondeterministic log space. Math. Systems Theory 10 (1976) 1-18. 
 
 4] Jones, N. D. and Laaser, W. T. Complete problems for deterministic 
 polynomial time. Theoretical Computer Science 3 (1976) 105-117. 
 
 5] Karp, R. M. Reducibility among combinatorial problems, in Miller, 
 R. E. and Thatcher, J. W. (eds.), Complexity of Computer Computations 
 (Plenum Press, New York, 1972) 85-103. 
 
 6] Kozen, Dexter. Lower bounds for natural proof systems. Proc . 
 18^ Annua l Symp. on Found , of Computer Science (1977) 254-266. 
 
 7] Lewis, Harry R. Complexity of solvable cases of the decision 
 problem for the predicate calculus. Proc . 19th Annual Symp . 
 on Found , of Computer Science (1978) 35-47. 
 
 8] Lewis, Harry R. Personal communication. 
 
 9] Lewis, H. R. Satisfiability problems for propositional calculi, 
 unpublished manuscript (1978). 
 
 X.L. 
 
 10] Paterson, M. S. and Wegman, M. N. Linear unifications. Proc . 8 
 
 Annual ACM Symp . on Theory of Computing (1976) 181-186. 
 
 11] Stockmeyer, L. J. and Meyer, A. R. Word problems requiring 
 exponential time. Proc . 5th Annual ACM Symp . on Theory of 
 Computing (1973) 1-9. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-79-978 
 
 2. 
 
 3. Recipient's Accession No. 
 
 4. Title and Subt itlc 
 
 Complete Problems in the First-Order 
 Predicate Calculus 
 
 5- Report Date 
 
 July 1979 
 
 6. 
 
 7. Author(sl . . - __ . 
 
 David A. Plaisted 
 
 8. Performing Organization Rept. 
 No. 
 
 9. Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, IL 61801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 NSF MCS 77-22830 
 
 12. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D.C. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 We present a survey of recent results concerning the complexity of deciding 
 if a first-order predicate calculus formula in Schonfinkel-Bernays form is 
 satisfiable. We also present results concerning the complexity of deciding the 
 existence of resolution proofs of restricted depths and sizes. In this way we 
 obtain natural problems complete for the classes NP, PSPACE, deterministic and 
 nondeterministic exponential time, and deterministic and nondeterministic 
 double exponential time. The results concerning the existence of resolution 
 proofs of restricted depths and sizes seem to have implications for the design 
 and analysis of resolution theorem proving programs. This report mainly lists 
 results without proof. 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 Satisfiability, predicate calculus, resolution theorem proving, 
 complete problems, comptuational complexity. 
 
 17b. Identifiers /Open-Ended Terms 
 17c. COSATI Field/Group 
 
 18. Availability Statement 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 
 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 22. Price 
 
 FORM N TtS-35 I 10-70) 
 
 USCOMM-DC 40329-P71 
 
'•miKi 
 
FEB 
 
 2 1381