J. Ferrante and C. W. Racko#. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of automatic structures for which the rstorder theory becomes elementary decidable. In this paper we will present such a subclass, namely automatic structures of bounded degree, where the bounded degree property refers to the Gaifman graph of the structure. Using a method of Ferrante and Racko [9] (see Section 3) we show in Section 4 that for every automatic structure of bounded degree the rst order theory can be decided in triply exponential alternating time with a linear number of alternations (Theorem 3) We are currently not able to match this upper bound by a sharp lower bound. But ....

.... structures of bounded degree is closed under operations like for instance disjoint union or direct product [3] 3 The method of Ferrante and Racko In order to prove that the rst order theory of an automatic structure of bounded degree is elementary, we have to introduce a general method from [9]. Let us x a structure A with universe A. Roughly speaking, Gaifman s Theorem [11] states that rst order logic only allows to express local properties of structures, see [7] for a recent account of this result. For our use, the following weaker statement is sucient, which is an immediate ....

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J. Ferrante and C. Racko. The Computational Complexity of Logical Theories. Number 718 in Lecture Notes in Mathematics. Springer, 1979.

....theory of term algebras in which all function symbols have arity 1. The rst order theory of such term algebras is complete in LATIME(2 ) the class of problems solvable by alternating Turing machines running in exponential time but only with a linear number of alternations, for details see [14, 7, 32, 31, 34]. The range restricted fragment of rstorder logic over lists is PSPACE complete, so there is also a gap between the complexities of the full and the rangerestricted fragment. The next two theorems characterize the data complexity. As usual in the case of the data complexity, we restrict ....

J. Ferrante and C. W. Racko. The computational complexity of logical theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

....with respect to Mazurkiewicz s trace equivalence (in fact, it is a generalization of this quotient) We show that the FO theory of a factorized unfolding can be reduced to the FO theory of the underlying structure (Theorem 5. 7) The proof of this result uses techniques of Ferrante and Racko [13] and a thorough analysis of factorized unfoldings using ideas from the theory of Mazurkiewicz traces [8] From this result on factorized unfoldings, we obtain the closure under graph products similarly to the closure under free products. Our results on FO theories of Cayley graphs should be also ....

....ut In the complete version of this extended abstract [17] we prove that every FOsentence is equivalent in C (G; to the same sentence but with all quanti ers restricted to spheres around the unit of at most exponential diameter. This proof uses techniques developed by Ferrante and Racko [13]. In addition to the above result, it provides a tight relationship between the word problem of G and FOTh(C (G; in terms of complexity: the space complexity of FOTh(C (G; is bounded exponentially in the space complexity of the word problem of G [17] 5 Factorized unfoldings In [31] ....

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J. Ferrante and C. Racko. The Computational Complexity of Logical Theories, number 718 of Lecture Notes in Mathematics. Springer, 1979.

....decision procedures in automated theorem proving. The complexity of the decidability problem for term powers is nonelementary because term powers extend term algebras. The non elementary bound applies to term algebras as a consequence of the lower bound on the theory of pairing functions [14], see also [11] Previous Quantifier Elimination Results. We show our decidability result using quantifier elimination. Quantifier elimination [20, Section 2.7] is a fruitful technique that has been used to show decidability and classification of boolean algebras [40, 44] Presburger arithmetic ....

J. Ferrante and C. W. Rackoff. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

....arising in other quantifier elimination procedures [31, 11, 30] Our terminology also borrows from congruence closure graphs like those of [39, 38] although we are not primarily concerned with e#ciency of the algorithm described. Term algebra is an example of a theory of pairing functions, and [15] shows that non empty family of theories of pairing functions as nonelementary lower bound on time complexity. 3.4.1 Term Algebra in Selector Language To facilitate quantifier elimination we use a selector language Sel(#) for term algebra [22, Page 61] We define term algebra in selector ....

....functions seems to preserve most of the properties of two valued logic and appears to agree with the way partial functions are used in informal mathematical practice. An alternative direction for proving decidability of structural subtyping would be to use Ehrenfeucht Fraisse games [53, Page 405] [15] uses techniques based on games to study both the decidability and the computational complexity of theories. The complexity of our the decidability for structural subtyping non recursive types is non elementary and is a consequence of the non elementary complexity of the term algebra, whose ....

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Jeanne Ferrante and Charles W. Racko#. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979. 1, 3.4, 8, 8

....arising in other quanti er elimination procedures [31, 11, 30] Our terminology also borrows from congruence closure graphs like those of [39, 38] although we are not primarily concerned with eciency of the algorithm described. Term algebra is an example of a theory of pairing functions, and [15] shows that non empty family of theories of pairing functions as nonelementary lower bound on time complexity. 3.4.1 Term Algebra in Selector Language To facilitate quanti er elimination we use a selector language Sel( for term algebra [22, Page 61] We de ne term algebra in selector ....

....functions seems to preserve most of the properties of two valued logic and appears to agree with the way partial functions are used in informal mathematical practice. An alternative direction for proving decidability of structural subtyping would be to use Ehrenfeucht Fraisse games [53, Page 405] [15] uses techniques based on games to study both the decidability and the computational complexity of theories. 48 The complexity of our the decidability for structural subtyping non recursive types is non elementary and is a consequence of the non elementary complexity of the term algebra, whose ....

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Jeanne Ferrante and Charles W. Racko. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

.... fx 7 ng j= for some n 2 N In particular, a formula is satis able i there is an assignment such that j= It is well known that satis ability of Presburger arithmetic is decidable in doubly exponential space. For complexity results of corresponding decision procedures, we refer to [9, 8, 2]. Note that the set of vectors v satisfying a Presburger formula with free variables is a semi linear set which can be e ectively computed [10, 11] Recall that a semi linear set is a nite union of linear sets, i.e. sets of the form f c k i=0 x i p i j x i 2 Ng where c and the p i ....

J. Ferrante and C.W. Racko. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer Verlag, 1979.

....: such that i 1 (x) j Phi( i (x) x) Then we can simply substitute 0 (x) in Phi(X; x) to get 1 (x) then substitute 1 (x) in Phi(X; x) to get 2 (x) etc. But if X occurs at least twice in Phi(X; x) the size of such formulas i (x) grows exponentially. Ferrante and Rackoff [13] and Solovay [unpublished] have devised techniques which produce formulas i (x) of polynomial size. 4 Results 4.1 For a formula J(x) and a sentence , we shall denote by J the sentence obtained by restricting the quantifiers to the domain J(x) The following is a strengthening of the ....

J. Ferrante and Ch. W. Rackoff. The Computational Complexity of Logical Theories. Lecture Notes in Mathematics #718. Springer-Verlag, Berlin, 1979.

....Arithmetic are decidable. 2.5.1 Related Results We just mention the two fundamental results on the complexity of Presburger Arithmetic: Theorem 4 ( FR74] The non deterministic time complexity of deciding validity of formulas in APr has lower bound 2 2 cn for some constant c. Theorem 5 ( FR79] There is an algorithm for deciding validity of formulas in APr running in deterministic space O(2 2 cn ) and hence in deterministic time O(2 2 2 cn ) for some constant c. 2.6 Weak Quanti er Elimination The theory FT does not allow quanti er elimination since there is no quanti erfree ....

Jeanne Ferrante and Charles W. Racko. The computational complexity of logical theories. Number 718 in Lecture Notes in Mathematics. Springer-Verlag, 1979.

....relational signature b Sigma = We present the elementary decision procedure in Section 8. 7 Ferrante Rackoff Games for Decidability In this section we briefly present a smart complexity tailored refinement of the Ehrenfeucht Fraiss e games due to Ferrante and Rackoff, following Chapter 2 of [6]. We discuss only a small fragment of the general techniques, which is needed to decide a theory of a single structure. The classical Ehrenfeucht Fraiss e method [5, 7] gives criteria, in terms of partial isomorphisms or back and forth games, of indistinguishability of two structures by ....

....we consider are the models of finite purely relational signatures. ut We discuss the needed modifications for infinite signatures in Section 8. This is necessary because even though Sigma is finite, the companion relational signature b Sigma = is always infinite. Definition14 (Boundedness, [6]) Suppose A is a model and H : 3 is a function. Let for every n; k; m 2 , every a k 2 A k such that a k m, and every formula Phi(x k 1 ) of quantifier depth n the following be true: if A j= 9x k 1 Phi(a k ; x k 1 ) then for some a k 1 H(n; k; m) one has A j= Phi(a k ; a k 1 ....

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J. Ferrante and C. W. Rackoff. The computational complexity of logical theories, volume 718 of Lect. Notes Math. Springer-Verlag, 1979.

....2n ary relation S to describe the successor function limited to binary numbers of size n bits. That is, we want S( o) to be true if and only if the binary number o is the successor of , in shorthand 1 = o. The successor function can be expressed by a sentence of size O(n) as described in [FR79]. Similarly, we use the 2n 1 ary relation M to describe the next tape position after the machine moves one tape cell in direction m. That is we want M( n) be true for each 0 2 n , and we want M( o; r) to be true for each 0 2 n and o successor of , and we want M( o; ....

J. Ferrante, C.W. Rackoff. The Computational Complexity of Logical Theories, Springer-Verlag, 1979.

....We note that the two equivalent formalisms considered in above theorem are extremely distant in succinctness. Indeed, the complexity of the emptiness problem T Sig 6= is in PTIME (Corollary 4. 5) while the complexity of the analogous problem Mod( 6= is known to be non elementary [14]. Acknowledgments. I wish to thank Jerzy Tiuryn for introducing me to the theory of fixed points. I am greatful to Andrzej W. Mostowski, Andr e Arnold, Bruno Courcelle and Wolfgang Thomas for enlightening discussions and for encouragement. I am also indebted to the referees for corrections and ....

J. Ferrante and C.W. Rackoff, The computational complexity of logical theories, Lecture Notes in Mathematics Vol. 718 (Springer, Berlin, 1979).

....no products or reciprocals of quanti ed variables. Quanti er elimination methods for valued elds have been extensively investigated in the past, cf. Weispfenning (1984) and the references there. The procedures given there are primitive recursive, but far beyond feasibility. Based on ideas of Ferrante and Racko (1979) for decision problems, quanti er elimination by elimination sets containing test terms has been introduced for linear formulas by Weispfenning (1988) This technique is very attractive due to its comparatively low complexity a q O(c) where a is the number of atomic formulas, q is the number ....

Ferrante, J., Racko, C. W. (1979). The Computational Complexity of Logical Theories. Number 718 in Lecture Notes in Mathematics. Springer-Verlag, Berlin.

....then the bounded theory of trees can be decided within polynomial space and is PSPACE complete if Sigma contains 2 constants. ut 4 Ferrante Rackoff s Games for Complexity Analysis In the next section we prove our Main Theorem by applying Ferrante Rackoff s games described in Section 2 of [3]. We have to spend additional effort to make these games applicable to infinite signatures. This is necessary because companion relational signatures (Definition 8) are always infinite, whereas original Ferrante Rackoff s games apply to finite signatures only. We attain the needed generalization ....

....original Ferrante Rackoff s games apply to finite signatures only. We attain the needed generalization by relativizing Ferrante Rackoff s boundedness conditions to finite subsignatures and by proving that the games carry over with this modification. Ferrante Rackoff s complexity tailored games [3] refine Ehrenfeucht Fraiss egames [2, 4] by additional boundedness analysis in the back and forth conditions. Boundedness means that whenever a formula of the form 9x Phi(x) is true, one can always find a small witness for Phi(x) from a finite subset of a model. Contrapositively, if there are no ....

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J. Ferrante and C. W. Rackoff. The computational complexity of logical theories, volume 718 of Lect. Notes Math. Springer-Verlag, 1979.

....24 6.2 Normal programs In this section we show that the complexity of the SUCCESS problem for nonrecursive logic programs with negation and function symbols of arity at most one is complete for a complexity class intermediate between NEXPTIME and EXPSPACE. The following key theorem is due to (Ferrante Rackoff 1979, Chapters 4 and 9) Theorem 6.5 Th(TA(1; 2; 0) can be decided in DSPACE(2 O(n) and is NTIME(2 O(n) hard w.r.t. loglin reducibility 2 . 2 (Volger 1983b, Volger 1983a) improved it 3 to Theorem 6.6 Th(TA(1; 2; 0) is LATIME(2 O(n) complete. 2 LATIME(2 O(n) is a class of ....

....) hard for nonrecursive normal programs. The upper bound appears to be of the same kind: Theorem 6.9 SUCCESS( 2; 0) is in LATIME(2 O(n) for nonrecursive logic programs with negation. 2 Consequently, NEXPTIME hard w.r.t. polynomial reducibility, cf. Johnson 1990) 3 At the time when (Ferrante Rackoff 1979) was written, the complexity classes defined simultaneously in terms of time, space, and alternations were not yet well known. They first appeared in (Berman 1977, Bruss Meyer 1980, Berman 1980) 4 (Johnson 1990) calls this class TA(2 O(n) n) which clashes with our usage of TA( for term ....

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Ferrante, J. & Rackoff, C. W. (1979), The computational complexity of logical theories, Vol. 718 of Lecture Notes in Mathematics, Springer-Verlag.

....c 0, and thus can be considered a useful practical alternative to the usual (unbounded) nonelementary recursive theory of finite trees. 2 Our basic decision and complexity analysis techniques are model theoretic games. More specifically, we use Ferrante Rackoff s complexity tailored games [FR79], which refine Ehrenfeucht Fraiss e games [Ehr61, Hod93] by additional boundedness analysis in the back and forth conditions. Boundedness means that whenever a formula of the form 9x Phi(x) is true, one can always find a small witness for Phi(x) from a finite subset of a model. Contrapositively, ....

....relational signature b Sigma = We present the elementary decision procedure in Section 7. 9 5 Ferrante Rackoff Games for Decidability In this section we briefly survey a complexity tailored refinement of the Ehrenfeucht Fraiss e games due to Ferrante and Rackoff, following Chapter 2 of [FR79]. We discuss only a small fragment of their general techniques, which is needed to decide a theory of a single structure. The classical Ehrenfeucht Fraiss e method, see [Ehr61, Hod93] gives criteria, in terms of partial isomorphisms or back and forth games, of indistinguishability of two ....

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J. Ferrante and C. W. Rackoff. The computational complexity of logical theories, volume 718 of Lect. Notes Math. Springer-Verlag, 1979.

....a class of problems solvable by alternating Turing machines in time 2 O(n) with linear number of alternations 2 . By [15] LATIME(2 O(n) DSPACE(2 O(n) Also, obviously, NTIME(2 O(n) LATIME(2 O(n) Both inclusions are presumably proper. The following key theorem is due to [24]. Theorem 6.2 Th(TA(1; 2; 0) is NTIME(2 O(n) hard w.r.t. loglin reducibility 3 , and can be decided in DSPACE(2 O(n) 59, 58] improved it 4 to Theorem 6.3 Th(TA(1; 2; 0) is LATIME(2 O(n) complete. Theorem 6.3 and our Theorem 3.1 imply Corollary 6.4 SUCCESS(1; 2; 0) for ....

....6.5 SUCCESS( 2; for nonrecursive normal programs is LATIME(2 O(n) hard. 2 The upper bound appears to be of the same kind: Theorem 6. 6 SUCCESS( 2; 0) for nonrecursive logic programs with negation is in the class LATIME(2 O(n) 2 The proof may be found in [62] We use the result of [24, 59] that the theory Th(TA(1; 2; 0) of two successors is LATIME(2 O(n) complete. In [62] we show how the results of [24, 59] generalize to any number of successors greater than 1. Summarizing, the complexity of the SUCCESS problem in unary signatures is as follows: Theorem 6.7 The SUCCESS( ....

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J. Ferrante and C. W. Rackoff. The computational complexity of logical theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

....to express by short formulas of a theory all models, up to a certain size, of a unique binary relation. The larger is the size, the higher is the respective lower bound. Technically, the method of Compton and Henson has a great advantage over the pioneer methods of Meyer, Stockmeyer, Fisher, Rabin [16, 23, 9, 8], since it allows one to avoid tedious encodings of Turing machines. To estimate the size of a binary relation representable by short formulas of a theory is much simpler than to prove that all Turing machine computations up to a certain length are encodable in a theory. One of the (numerous) ....

....of the same subformula (but with different parameters) as an equivalent formula containing just one occurrence of the subformula, and only fixed number of variables. This standard abbreviation trick is described in full detail by Stockmeyer [22] pp. 189 190, also by Ferrante and Rackoff [8], Chapter 7, pp. 153 161, and further 8 Consider Pn 1 (x; y) 9z(Pn(x; z) Pn(z;y) growing as 2 n . by Compton and Henson in Section 3 of [6] This trick allows one for the iterative definitions as above, to keep the size of Pn (x) proportional to n. In Section 8 we proceed semi formally ....

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J. Ferrante and C. W. Rackoff. The computational complexity of logical theories, volume 718 of Lect. Notes Math. Springer-Verlag, 1979.

....will be considered, such as the structure of number theory, hN; 6; Thetai, and structures over larger vocabulary, but we will see that they are generally meaningless for constraint databases. Note that all theories of the previous structures are decidable and have hyper exponential complexity [FR79,Ren92]. The structure Q satisfies the theory of dense order without endpoints that is known to be complete [CK73] Moreover, it admits the elimination of quantifiers, which as we shall see in the following is the fundamental property of constraint databases. The structure R satisfies the theory of ....

....representable linear constraint databases. Details follow in the sequel about finitely representable databases. For richer languages, there is a serious gap of data complexity. This gap exists already for the complexity of the decision problem of the logical theories, Presburger arithmetic [Pre29,FR79] versus number theory [End72] Theorem 3.11 First order queries in the contexts of the natural numbers, hN; 6; Thetai, the integers, hZ; 6; Thetai, and the rationals hQ; 6; Thetai define mappings in the arithmetical hierarchy. Proof: It is clear in the context of the natural numbers ....

J. Ferrante and C. W. Rackoff. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

.... equations, integer feasibility of systems of (parametric) linear constraints, integer programming, and certain problems in program description and verification (compare [G85, SJ80] This motivated an extensive study of the complexity of the decision problem for fragments and extensions of PA [Ber77, Ber80, Coo72, FR75, FR79, FiR74, F82, GS78, G87, GI79, GI81, L78, O73, RL78, Sc84, Sh77, V83]. While the decision problem is concerned with closed (i.e. parameter free) formulas, the quantifier elimination problem deals with simplification of formulas involving free variables to a form that is easy to evaluate. It has turned out that for applications a quantifier elimination procedure is ....

J. Ferrante, Ch. Rackoff, The computational complexity of logical theories, Springer Lecture Notes in Mathmatics, 718 (1979).

....Because the first of the above theories is a subtheory of Presburger arithmetic and the second is a subtheory of real addition with order, our results must be of independent interest to the theoretical computer science community. Previous results on the same line of research include [FR74, FR75, FR79, FG77, RL78] ffl Using the results on quantifier elimination and decision, we study the complexity of query evaluation in temporal constraint databases and indefinite temporal constraint databases with constraints from the above theories (chapter 7) We show that the complexity of query ....

....is Sigma p k complete. Since diPC is a subtheory of Presburger arithmetic and dePC is a subtheory of real addition with order these results must be of independent interest to the theoretical computer science community. Previous results on the same line of research include [FR74, FR75, FR79, FG77, RL78] This chapter is organized as follows. The next section gives some preliminary definitions and results. In section 6.2 we present naive algorithms based on standard quantifier elimination techniques. Then we develop more efficient quantifier elimination algorithms in sections 6.3, ....

J. Ferrante and C. Rackoff. The Computational Complexity of Logical Theories. Lecture Notes in Mathematics. Springer Verlag, 1979.

....is used in the proofs below. We have reduced the problem of establishing the k variable property for Sigma to checking the condition of Corollary 4(i) This will done using Ehrenfeucht Fraisse games [3, 5] Ehrenfeucht Fraisse games have been used widely in theoretical computer science; see e.g. [4, 6, 8, 10, 12, 13, 17, 18]. Here we use a modified version in which the number of pebbles is finite [9, 14, 10] Definition 5 Let A; B be structures for L and (u; v) a k configuration. We call (u; v) a local isomorphism if the map u(x) 7 v(x) x 2 u, is well defined and extends to an isomorphism of the substructures of ....

J. Ferrante and C. Rackoff, The Computational Complexity of Logical Theories, Springer Lect. Notes in Math. 718, 1979.

....: with rational order and any inflationary Datalog : with equality over an infinite set of constants query can be also evaluated in PTIME data complexity. The latter case is also considered in [2, 11, 19, 25] However, the case of Datalog queries combined with the theory of integer order [10] was left as an open problem. There are other attempts to combine some form of integer constraint solving with existing relational database languages. For example, Kabanza, Stevenne and Wolper [15] examined a way of combining Relational Calculus with a limited form of recursion called linear ....

J. Ferrante, C.W. Rackoff. The Computational Complexity of Logical Theories, Springer-Verlag, 1979.

....theory of the structure Q can be decided in DSPACE(n 3 ) where n is the length of the input sentence (Theorem 7. 1) Since the theory of structure Q is a subtheory of real addition with order, this result is of independent interest and adds to the literature on the complexity of logical theories [18,16,40,17,15,4,5,26,27]. The results discussed here were originally presented in less detail in [29,33] This paper is organized as follows. The next section presents some examples of constraint languages and defines the relevant abstract concepts. In Section 3 we introduce the concepts of variable and quantifier ....

....8 to analyze the complexity of query evaluation in TCDB and ITCDB. 7 Quantifier Elimination in Theories of Temporal Constraints In this section we study the problems of decision and quantifier elimination for theories Th(Q) and Th(Z) Our techniques will be similar to the ones described in [16,17]. The main point of these techniques is that given a particular theory, one gives an elimination of quantifiers procedure, analyzes it to see how large constants can grow, and then uses this analysis ( to limit quantifiers to range over finite sets instead of an infinite domain [16] We will ....

[Article contains additional citation context not shown here]

J. Ferrante and C. Rackoff. The Computational Complexity of Logical Theories. Lecture Notes in Mathematics. Springer Verlag, 1979.

....2n ary relation S to describe the successor function limited to binary numbers of size n bits. That is, we want S ; o to be true if and only if the binary number o is the successor of , in shorthand 1 = o. The successor function can be expressed by a sentence of size O(n) as described in [FR79]. Similarly, we use the 2n 1 ary relation M to describe the next tape position after the machine moves one tape cell in direction m. That is we want M ; n be true for each 0 2 n , and we want M ; o; r to be true for each 0 2 n and o successor of , and we want M( o; l) ....

J. Ferrante, C.W. Rackoff. The Computational Complexity of Logical Theories, Springer-Verlag, 1979.

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J. Ferrante and C. W. Racko#. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

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Jeanne Ferrante and Charles W. Racko#. The Computational Complexity of Logical Theories. Lecture Notes in Mathematics. Springer, 1979.

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J. Ferrante and C.W. Racko#. The computational complexity of logical theories. Lecture Notes in Mathematics, 718, 1979.

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J. Ferrante and C. W. Racko#. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

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Jeanne Ferrante and Charles W. Racko#. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

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J. Ferrante and C. W. Racko#. The Computational Complexity of Logical Theories. Springer-Verlag, 1979.

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J. Ferrante and C. W. Racko. The Computational Complexity of Logical Theories. Springer-Verlag, 1979.

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J. Ferrante and C. W. Racko#. The Computational Complexity of Logical Theories. Springer-Verlag, 1979.

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J. Ferrante and C. W. Racko. The Computational Complexity of Logical Theories. Springer, 1979.

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Ferrante J., Rackoff C.W., The Computational Complexity of Logical Theories, Springer Lecture Notes in Mathematics 718, (1979).

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Ferrante J., Racko# C.W., The Computational Complexity of Logical Theories, Springer Lecture Notes in Mathematics 718, (1979).

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J. Ferrante and C. W. Rackoff. The Computational Complexity of Logical Theories, volume 718 of Lecture Notes in Mathematics. Springer-Verlag, 1979.

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J. Ferrante and C. Racko. The Computational Complexity of Logical Theories. Lecture Notes in Mathematics 718, Springer-Verlag, 1979.

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J. Ferrante and C. Rackoff. The Computational Complexity of Logical Theories. Lecture Notes in Mathematics 718, Springer-Verlag, 1979.

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Jeanne Ferrante and Charles W. Racko. The Computational Complexity of Logical Theories. Number 718 in Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1979.

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J. Ferrante and C. Rackoff. The Computational Complexity of Logical Theories. LNM 718. 1979.

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J. Ferrante and C. W. Rackoff. The computational complexity of logical theories. Number 718 in Lecture Notes in Mathematics. Spinger-Verlag, 1979.

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J. Ferrante, C.W. Rackoff. The Computational Complexity of Logical Theories, Springer-Verlag (No. 718), 1979.

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