N. Jones and W. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105--117, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....k ip pushdown automata languages is P complete. Proof. In both cases, the hardness results immediately follow from the inclusion L (CFL) L (FNPDA k ) for any k 0, and the LOG(CFL) completeness of xed membership for context free languages [18] and the P completeness for general membership [13]. For the upper bounds we argue as in the proof of Theorem 10. The main di erence in the proof is, that we can not guess a length k sequence of ip pushdown input reversals. Nevertheless, a deterministic logspace machine can enumerate all possible outcomes of length k sequences of ip pushdown ....

N. D. Jones and W. T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105-117, 1977.

....tation, an oracle Turing machine with a space bound on its worktape and an arbitrary number of oracle tapes is considered. Basic properties of the resulting reducibilities are examined. 1. Introduction Complexity bounded reducibilities between problems have been widely studied in recent years ([1, 2, 3, 5 8, 11, 12, 17, 19] and many others) There appear to be two major motivations for studies of this type. First, complexity bounded reducibilities are an extremely powerful tool for the classification of problems by their time and space complexities. There are many examples of problems whose absolute ....

....are an extremely powerful tool for the classification of problems by their time and space complexities. There are many examples of problems whose absolute complexities.are unknown, but which can be shown to be related to each other by very restrictive complexity bounded reducibilities ([3, 5, 6]) Classes of open problems are thus reduced to single open problems. Even if the absolute complexity of two problems were known, their relative complexity classification might still be interesting from the ioint of view of algorithm design; for example, complexity bounded reducibilities could be ....

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N. Jones and W. Laaser, Complete problems for deterministic polynomial time, Proc. 6th Ann. A CM Syrnp. Theory Cornput, (1974) 40-46. 39

....use arbitrary first order information on the input predicates from the underlying vocabulary. E HORN denotes the existential fragment of SO HORN, i.e. the set of SO HORN sentence where all second order quantifiers are existential. Example 2.13. The problem GEN is a well known P complete problem [52, 66]. It may be presented as the set of structures (A, f,a) in the vocabulary of one unary predicate , one binary function f and a constant a, such that a is contained in the closure of under f. Clearly, the complement of GEN is also P complete. It is defined by the following sentence of E HORN: ....

N. JONES AND W. LAASER, Complete problems for deterministic polynomial time, Theoretical Computer Science, 3 (1977), pp. 105-117.

....facts together with a set of simple ground implications (one atom as antecedent and one as consequent) This can be seen as a restricted form of a propositional Definite Horn Program, where queries are ground conjunctions of atoms. This proves that the derivation between CGs is polynomial in time [17, 18, 19, 20]. 2. CONCEPTUAL GRAPHS We chose to change some terminology of CGs. The reason is that some notions like generic, universal, existential markers have been in conflict in the CGs literature. We decided to use the logical symbol to denote both the generic (# as in [1] and the standard universal ....

....defining the lattice. Definite Horn Clauses which represent the graph h and the ontology on the graph. We reduce the question answering problem to the derivation of a conjunction of propositional literals corresponding to the graph g (propositional datalog) Hence the complexity is polynomial [17, 18, 19, 20]. Let C k 1 (a k 1 ) C n (a n ) 9x 1 . 9x k C 1 (x 1 ) C k (x k ) r(x 1 ,x 2 , x k ,a k 1 , a n ) be the translation of a single well formed part of a conceptual graph (up to a permutation of the terms) where r is of arity n. We suppose that the translation conforms to the ....

Jones, N. and Laaser, W. (1977) Complete problems in deterministic polynomial time. Theor. Comput. Sci., 3, 105-- 1177.

....the grammar logics in [FdCP88] it is worth observing that the regular grammar logics are too expressive to encode the generation of strings by regular grammars. Indeed, whether a string belongs to a context free language (de ned by a context free grammar) is a P complete problem (see e.g. [JL76, Corollary 11]) whereas the satis ability problem of any regular grammar logic shall be shown to be PSPACE hard. In a sense, introducing regular grammar logics to analyze regular languages is not very ecient. However, we claim that it is more interesting to take advantage of the wealth of knowledge about ....

....; One can show that this very has a satis ability problem in PSPACE. The condition for i 2 N , SF i (G) is nite implies that L(G) is nite but the converse is generally not true. It is decidable whether a context free grammar generates an in nite language and the problem is P complete [JL76, Gol81]. So it is tractable to check whether the languages of sentential forms generated by a context free grammar are all nite. Lemma 33 does not imply that the general satis ability problem for the class REG of regular grammars such that for i 2 N , SF i (G) is nite, is in PSPACE since e SF i (G) ....

N. Jones and W. Laaser. Complete problems for deterministic polynomial-time. Theoretical Computer Science, 3(1):105-117, 1976.

....the hierarchy of complexity classes. Among their results we list in Table 1 those which are most significant to our work. This is one more way of characterizing complexity classes in algebraic terms, which comes after the problem of evaluating formulas and circuits over the Boolean semiring (see [11, 7, 8]) and the computational models of programs over monoids (see [3, 4] and leaf languages (see [6] among others. Using tensor calculus in this context is especially appealing, if only because of the many applications matrix algebra finds in various areas, such as the specification of parallel ....

N. D. Jones and W. T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105--117, 1976.

....the rules of a monadic stringrewriting system are length reducing. This means that the general membership problem for mon McNL is also LOG(CFL) complete. This sharply contrasts the corresponding result for context free grammars, as the general membership problem for them is P complete in general [JL77]. Thus, in summary we have the following results for the class mon McNL. Theorem 36. The xed and the general membership problem for the class mon McNL are LOG(CFL) complete. Next we turn our attention to the con uent monadic McNaughton languages. Although we have con mon McNL DCFL, the ....

N.D. Jones and W.T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105-117, 1977.

....with nitely bounded modal depth in KD , T , KB , KDB , and B is decidable in PTIME. These PTIME results can further be categorized as PTIME complete, because the satis ability problem of sets of Horn clauses in the classical propositional logic is PTIME complete, as proved by Jones and Laaser [6]. In this paper, we show that the satis ability problem of sets of Horn modal clauses with modal depth bounded by k 2 in the modal logics K4 and KD4 is PSPACE complete, and in K is NP complete. We also show that the satis ability problem of modal formulae with modal depth bounded by 1 in K 4, ....

....depth bounded by 1 in K 4, KD4, and S4 belongs to the NP class. Proposition 4. The complexity of the satis ability problem of sets of Horn formulae with modal depth bounded by 1 in K, K4, KD4, and S4 is PTIMEcomplete. Proof. The lower bound PTIME hard follows from the result by Jones and Laaser [6] that the complexity of the satis ability problem of sets of Horn formulae in the classical propositional logic is PTIME complete. By the result of [9] every positive modal logic program with modal depth bounded by 1 has the least KD4 model and the least S4 model, which can be constructed in ....

N.D. Jones and T.W. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105-112, 1976.

....side of rules so strategies such as unit resolution may be applied. Unit resolution is complete for classical propositional Horn formulae [WCR64] and algorithms have been developed that test for satisfiability of classical propositional Horn formulae based on unit resolution in polynomial time [JL77] Thus although we must repeatedly perform classical style resolution proofs on the right hand side of rules (because when g false is detected the rule is rewritten) the resolution steps to derive each g false on the right hand sides of rules will be more efficient as less intermediate steps ....

N. D. Jones and W. T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:107--117, 1977.

....LP The simulation of a DTM by a propositional logic program, as described in Section 3.1 is almost all we need in order to determine the complexity of propositional LP, i.e. the complexity of deciding whether P j= A holds for a given logic program P and ground atom A. Theorem 3. 2 (implicit in [80, 127, 76]) Propositional LP is P complete under logspace reductions. Proof. a) Membership. It obvious that the least fixpoint T 1 P of the operator TP , given program P , can be computed in polynomial time: the number of iterations (i.e. applications of TP ) is bounded by the number of rules plus one. ....

N. Jones and W. Laaser. Complete Problems in Deterministic Polynomial Time. Theoretical Computer Science, 3:105--117, 1977.

....the following way. Problem: Gen SubAlg Generators and Nongenerators 5 Instance: #A, X, a# in which A is a finite algebra of finite similarity type, X is a subset of A, and a is an element of A. Question: Is a # Sg A (X) In the literature, this problem is usually referred to as Gen. In [14] Jones and Laaser proved the following theorem. Theorem 2.1 (Jones and Laaser, 1977) Gen SubAlg is complete for P. Since the class P is closed under complements, we note that the complementary problem (Gen SubAlg) c is also complete for P. We now modify Gen SubAlg to ask about generating ....

N. D. Jones and W. T. Laaser, Complete problems for deterministic polynomial time, Theoret. Comput. Sci. 3 (1977), 105--117.

....algebra of finite similarity type, X # A and a # Sg A (X) 2. Gen SubSg is the subset of Gen SubAlg in which A is a finite semigroup, i.e. A = #A, #,where is an associative binary operation. In the literature, Gen SubAlg has usually been referred to simply as Gen. Jones and Laaser [7] proved that Gen SubAlg is complete for PTIME. On the other hand, Jones, Lien and Laaser [8] showed that GenSubSg is complete for NLOGSPACE. Now let #A, F,h# be an instance of Gen Clo and let A be the algebra #A, F#. Theorem 3.2 can be interpreted as asserting that Clo A n (F)isthe ....

N. D. Jones and W. T. Lasser, Complete problems for deterministic polynomial time, Theoretical Computer Science 3 (1977), 105--117.

....it is natural to define the problem Gen SubAlg = # #A, X, a# : X # A, a # A and a lies in the subuniverse of A generated by X # . It is easy to find a reduction of Gen Con to Gen SubAlg, see for example, 3, Theorem 5.5] Thus Gen Con can be no harder than Gen SubAlg. However, in [14] Jones and Laaser proved that Gen SubAlg is complete for P (the class of problems solvable in polynomial time) It is known that NL is contained in the class of problems solvable in polynomial time, and it is generally believed that the inclusion is proper. Thus Gen SubAlg is apparently strictly ....

N. D. Jones and W. T. Laaser, Complete problems for deterministic polynomial time, Theoret. Comput. Sci. 3 (1977), 105--117. 17

.... k ) provided that such a k exists) Finally, are there natural classes of non total top down tree transducers for which the exponential output size problem is at least solvable on polynomial space Acknowledgement I thank Joost Engelfriet, who told me where to find the completeness results in [11] and pointed out the related work in [1] as well as Helmut Seidl and an anonymous referee for their careful reading of the manuscript and helpful lists of suggested improvements. ....

Neil D. Jones and William T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105--117, 1977.

....A Boolean formula F in conjunctive normal form. Problem: Can the empty clause 2 be deduced from F by unit resolution A unit is a clause with only one term. For example, the unit resolvent of F = A B 1 Delta Delta Delta Bm and the unit G = A is B 1 Delta Delta Delta Bm . Reference: [JL76] Hint: Jones and Laaser provide a poly time algorithm for unit resolution [JL76] To show B follows from the assumption A 1 Delta Delta DeltaA m , negate B, add it to the set of clauses and derive the empty clause. Reduce CVP to UNIT as described below. A gate in the circuit v k v i v j is ....

....2 be deduced from F by unit resolution A unit is a clause with only one term. For example, the unit resolvent of F = A B 1 Delta Delta Delta Bm and the unit G = A is B 1 Delta Delta Delta Bm . Reference: JL76] Hint: Jones and Laaser provide a poly time algorithm for unit resolution [JL76]. To show B follows from the assumption A 1 Delta Delta DeltaA m , negate B, add it to the set of clauses and derive the empty clause. Reduce CVP to UNIT as described below. A gate in the circuit v k v i v j is represented by the clauses of v k , v i v j , that is, v k v i ) v k v j ) ....

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N. D. Jones and W. T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3(2):105--117, 1976.

....assignments to almost satisfiable 2 SAT formula As far as we know, no one considered this question before. It does not seem to have a simple answer and we consider it to be an intriguing open problem. The satisfiability of HORN SAT formulae can be checked in the following way (see [DG84] [JL76], YD83] As long as there are unit clauses, i.e. clauses of size 1, assign the variables appearing in these clauses the required values. If a variable and its negation both appear, at some stage, as unit clauses, the formula is unsatisfiable. If a clause contains a literal that was already ....

N.D. Jones and W.T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105--117, 1976.

.... C # C) is also know to be solvable e#ciently (see e.g. 4,13,16] and is used successfully in many of the fastest exact SAT solvers (see e.g. 9] The satisfiability problem for Horn CNFs (i.e. for which C # V # 1 for all C # C) is well known to be solvable in linear time (see e.g. [15,23,24]) Horn subformulae were used systematically in a SAT solver by [19] There are a few other extensions of the above special classes, for which SAT is known to be solvable in polynomial time, see e.g. 5,6,11,18,25,27] None of these classes however, as far as the author knows, were put in a ....

N. D. Jones and W.T. Laaser, Complete problems for deterministic polynomial time, Theoretical Computer Science 3 (1977) pp. 105--117.

....in i must be extendible. As O(log n) space bounded ATMs capture exactly the complexity class PTIME, the theorem follows. Theorem 2 The fair controllability problem is PTIME hard. Proof: The lower bound is shown by reducing from a known PTIMEcomplete problem, namely, the path system problem [2]. Due to space limitations, the details are omitted here. ....

Jones, N. and Laaser, W., Complete problems for deterministic polynomial time, Theoret. Comput. Sci., 3: 105-117, 1977.

....bound implies lower bounds for the communication complexity of many monotone Karchmer Wigderson s games, and hence gives lower bounds for the monotone depth of many functions. The separation of the monotone NC hierarchy is then obtained by proving a lower bound for a variant, called GEN (see [JoLa77]) of the monotone P complete problem Path Systems (see [Co74] As mentioned above, our argument is general enough to prove lower bounds for many other functions. In particular, we get a new proof for Karchmer Wigderson s Omega Gammamer 2 n) lower bound for st connectivity, on a graph with n ....

....capturing the difficulty of the class monotone P. Let us describe a variant of the very first P complete function known [Co74] which Cook called: Path Systems. In this paper we call this function GEN (for GENeration ) in analogy with Jones and Laaser s non monotone version of the function [JoLa77] (see also [BaMc91] The GEN function: The input for GEN is a string of l 3 bits (t ijk ) 1i;j;kl . For 1 k l, we say that 1 generates k if k = 1, or for some i and j such that t ijk = 1, 1 generates i and 1 generates j (where 1 generates i and 1 generates j are defined recursively ....

N.D. Jones and W.T. Laaser, Complete problems for deterministic polynomial time, Theoretical Computer Science 3 (1977), pp. 105--117.

....hXi generated by X CGM is a special case of the more general problem where the multiplication table need not be that of a group. With an arbitrary table (for a groupoid, which need not even be associative) the membership problem was shown to be complete for polynomial time by Jones and Laaser [19]. If the table obeys the associative law (a semigroup) Jones et al. showed the membership problem to be NLcomplete [20] This completeness result holds even if the semigroups are group free, making groups the natural domain to explore further. Barrington and McKenzie first investigated the ....

N.D. Jones and W.T. Laaser, Complete problems for deterministic polynomial time, Theoretical Computer Science 3 (1977), pp. 105-117.

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N. Jones and W. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105--117, 1977.

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N. D. Jones and W. T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science 3, pages 105--117, 1977.

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N.D. Jones and W.T. Laaser. Complete problems for deterministic polynomial time. Theoretical Computer Science, 3:105-117, 1976.

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N.D. Jones and W.T. Laaser, "Complete problems for deterministic polynomial time," Theoretical Computer Science 3 (1976) 105--117.

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N. D. Jones and T. Laaser. Complete Problems for Deterministic Polynomial Time. Theoretical Computer Science, 3:105--117, 1977.

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