L.J. Stockmeyer, "The Polynomial - Time Hierarchy", Theoretical Computer Science 3 (1977), 1-22.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....into problems on nite automata. This will be the topic of our conversation today and you will see that reality is quite di erent, though. If you take stronger logics, like full second order, you can de ne famous complexity classes like L, NL, P , NP , PH, PSPACE, etc. For instance, Stockmeyer [40] proved that full second order corresponds to PH, the polynomial hierarchy. There is a lot to say, and this would deserve a column by itself, but I don t want to talk about this today. Q: All right. So you are only interested today in logics that de ne regular languages. I am trying to remember ....

L. Stockmeyer, 1977, The polynomial time hierarchy, Theoret. Comp. Sci. 3, 1-22.

....maximum flows in AND OR graphs with only one # vertex. 4 The problem of finding critical vertices in an AND OR graph In this section, we assume the familiarity with the complexity classes within the Polynomial time Hierarchy like # n and # n . For more details, it is referred to Stockmeyer [17]. Let us consider the following scenarios: A redundant computation system (or a problem solving process) with multiple inputs (e.g. the electrical power distribution systems, the air tra#c control system, etc. is modeled by an AND OR graph G with a capacity function c associated with it. And an ....

L. Stockmeyer. The polynomial time hierarchy. Theoret. Comput. Sci., 3:1--22, 1977.

....With fixed density, depth, modal fraction, and number of atomic propositions, the complexity of the resulting formula can be varied by adjusting the density L=N . We used d = 1; 2, C = 3 and p = 0:5 in our experiments. A reduction of K to QBF Both K and QBF have PSPACE complete decision problems [16, 31]. This implies that the two problems are polynomially reducible to each other. A natural reduction from QBF to K is described in [12] In the last few years extensive effort was carried out into the development of highly optimized QBF solvers [17, 5] One motivation for this effort is the hope of ....

L.J. Stockmeyer. The polynomial-time hierarchy. Theo. Comp. Sci., 3:1--22, 1977.

....proved NP complete, researchers turn to other ways of trying to solve the problem, usually using approximation algorithms to give an approximate solution or probabilistic methods to solve the problem in most cases. Another fundamental step was taken around 1970 by Meyer and Stockmeyer [MS72] Sto76] They de ned the polynomial hierarchy in analogy with the arithmetic hierarchy of Kleene. This hierarchy is de ned by iterating the notion of polynomial jump, in analogy with the Turing jump operator. This hierarchy has proven useful in classifying many hard combinatorial problems which do not ....

....structural complexity theory. 4.2 The Polynomial Hierarchy While numerous hard decision problems have been proved NP complete, a small number are outside NP and have escaped this classi cation. An extended classi cation, the polynomial time hierarchy (PH) was provided by Meyer and Stockmeyer [Sto76] They de ned the hierarchy, a collection of classes between P and PSPACE, in analogy with Kleene s arithmetic hierarchy. The polynomial time hierarchy (PH) consists of an in nite sequence of classes within PSPACE. The bottom (0 th ) level of the hierarchy is just the class P. The rst level ....

L. Stockmeyer. The polynomial-time hierarchy. Theor. Computer Science, 3:1-22, 1976.

....SAT and co CLAUSE HSUB to QSAT 2 . From a theoretical point of view, these polynomial translations are optimal because they reduce an NP complete problem to SAT and a problem complete for 2 (which is the complementary class to 2 ) to QSAT 2 which is itself a 2 complete problem [38] . In this work, equational problems constitute the crucial link between H subsumption problems and (quantified) satisfiability. Consider Figure 1 for the following discussion. First of all, subsumption coproblems are naturally expressible by means of equational problems (ep) Therefore, our ....

Stockmeyer, L.: 1977, `The Polynomial-Time Hierarchy'. Theoretical Computer Science 3, 1--22.

.... we call the first order query hierarchy) constitute a structural clasification of the firstorder queries (and hence also of the relational algebra queries) of Codd [4] We show the strictness of the first order query hierarchy and a correspondence with the polynomial time hierarchy of Stockmeyer [21]. The suggestions of Zloof [25] and of Aho and Ullman [2] concerning transitive closure transcend the first order query hierarchy, but are easily seen to be embedded in the full hierarchy, as do those of Kowalski [17] see also [8] The full hierarchy to level o) 2 is then shown to have natural ....

.... set S of queries, SLJ S cE o S =E o S =E o (SU S) We are now ready to define a hierarchy of sets of expressions, which induces a hierarchy of sets of queries, similar in structure to such known hierarchies as the arithmetical and analytical hierarchies [20] and the polynomial time hierarchy [21 ]. DEFINITION. Define the collection of sets of expressions of First Z, t o, as Zo = I quantifier free , Z i = Exist o H, Ii = Z i . We are interested, not so much in the sets of formulas Z, but rather in the corresponding sets of queries denoted by Z O, O: DEFINITION. The ....

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L. J. STOCKMEYER, The polynomial-time hierarchy, Theoret. Cornput. Sci. 3 (1977), 1-22.

....section from ESO to full second order logic over strings. Our focus of attention will be the SO prefix classes which are regular vs. those which are not. In particular, we are interested to know how adding second order variables affects regular fragments. Generalizing Fagin s theorem, Stockmeyer [16] has shown that full SO captures the polynomial hierarchy (PH) Second order variables turn out to be quite powerful. In fact, already two first order variables, a single binary predicate variable, and further monadic predicate variables are sufficient to express languages that are complete for ....

L. J. Stockmeyer. The polynomial-time hierarchy. Theoretical Comp. Sc., 3:1--22, 1977.

....(which is a row vector) denote the transpose of x. Given an alphabet (i.e. a finite set of symbols) E, we write E to denote the set of all finite length strings (including the empty 2 ii2P denotes the set of all languages whose complements are in the second level of the polynomial time hierarchy [15]. string ) using symbols from E. We write E to denote E . A Petri net (PN, for short) is a triple (P, T, 99) where P is a finite set of places, T is a finite set of transitions, and 99 is a flow function 99 ( P x T) U (T x P) N. In this paper, k and m will be reserved for IPI (the ....

L. Stockmeyer, The polynomiaLtime hierarchy, Theoret. Comput. Sci. 3 (1977) 1-22.

....is a Sigma ql 2 definition of L. Parts (b) and (c) follow from (a) by standard means. The case k = 1 of (d) is Schnorr s theorem about SAT . The cases k 1 follow by induction on k, inserting Schnorr s construction into the corresponding parts of the proofs for polynomial time reductions in [Sto77, Wra77] Part (e) follows because the language QBF = k B k of quantified Boolean formulas belongs to deterministic quasilinear (in fact, linear) space. 7 2.1 Remarks on SAT , QBF , reductions, and relativization We first note that it is unknown whether QBF is complete for quasilinear space ....

L. Stockmeyer. The polynomial time hierarchy. Theor. Comp. Sci., 3:1--22, 1977.

....of normed context free processes is decidable in polynomial time (Theorem 2. 10) For comparison, the best complexity bound previously known was due to Huynh and Tian [12] who showed that this decision problem is in Sigma P 2 , the second level of the polynomial hierarchy of Meyer and Stockmeyer [19]. The existence of a polynomial time algorithm is surprising, given that mere decidability was once in question. Our algorithm draws on a number of existing ideas and techniques: the notions of Caucal base (or self bisimulation ) 3] and decomposing function [12] and a unique prime decomposition ....

L.J. Stockmeyer, The polynomial time hierarchy, Journal of Theoretical Computer Science 3(1) (1977), pp. 1--22. 18

....of all considered semantics. Thus, adding (inequality) to DATALOG (giving DATALOG , suffices to gain full expressive power. This result is proved exploiting a strengthened form of Fagin s Theorem generalized to the Polynomial Hierarchy ( 1 p 5 NP, 2 p 5 NP NP , Fagin 1974; Stockmeyer 1977], which exists for finite structures only and classes ( k p where k 2 [Eiter et al. 1996] The same expressive power ( 2 p ) is gained by adding negation ( in case of both the SM and the PM semantics. For these semantics, symmetric results (P 2 p expressiveness) holds for certainty ....

....definable in . As common with query languages, we focus in our analysis of the expressive power of disjunctive Datalog on generic queries. It is well known that second order definability of properties on finite structures is closely connected to computability within the Polynomial Hierarchy (PH) [Stockmeyer 1977]. Recall that the classes of PH are defined as follows: 0 p 5P 0 p 5D 0 p 5P, and for k 0, D k11 p 5 P ( k p , k11 p 5 NP ( k p , and P k11 p 5 co ( k11 p . For a first order vocabulary s, denote by ( k 1 (s) the second order sentences of the form ( S 1 ) S 2 ) ....

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STOCKMEYER, L. J. 1977. The polynomial-time hierarchy. Theor. Comput. Sci. 3, 1--22.

....in its prenex quanti cation. Those known results are: 1) deciding the validity problem for a closed quanti ed Boolean formula is PSPACE complete; 2) deciding the validity problem for a closed quanti ed Boolean formula of alternation depth k whose outermost quanti er is 9 is P k complete (Stockmeyer 1976), where P k denotes the k th level of the polynomial time hierarchy. In particular, P 0 = P and P 1 = NP. We will use the following formula as a running example of a valid closed QBF formula: 8v 1 :9v 2 :9v 3 : v 1 v 2 v 3 ) v 1 v 2 v 3 ) v 3 4.1 The Full Calculus and ....

Stockmeyer, L. J. (1976). The polynomial-time hierarchy. Theoretical Computer Science 3 (1), 1-22.

....within space 2 O(l) in this case. As the last particular case let us consider a signature Sigma containing 2 constant symbols only. In this case the bounded theory of finite trees is equivalent to the first order theory of pure equality in a 2 element structure, known to be PSPACE complete [15, 14]. This finishes the proof of our Main Theorem. 9 Conclusion We introduced the bounded theory of trees and proved that it can be decided within elementary space (hence time) as contrasted to the usual theory of finite trees, which is of non elementary decision complexity. We thus demonstrated ....

L. J. Stockmeyer. The polynomial-time hierarchy. Theor. Comput. Sci., 3:1--22, 1977.

.... and plays the role of prototypical problem for such a class, while the problem of evaluating a kQBF 9 is Sigma p k complete, whereas the problem of evaluating a kQBF 8 is Pi p k complete, Sigma p k and Pi p k being the classes at the k th level of the Polynomial Hierarchy (Stockmeyer 1976). We remind that NP Sigma p 2 Delta Delta Delta Sigma p k Delta Delta Delta PSPACE and that most researchers in computational complexity believe these containments to be strict. There is an important structural difference in the nature of the solutions for SAT and QBF ....

Stockmeyer, L. J. 1976. The polynomial-time hierarchy.

....we use homomorphism techniques from database theory and logspace reductions from computational complexity. We use the standard complexity classes PTIME (polynomial time) and NP= Sigma p 1 , coNP= Pi p 1 , Sigma p 2 , Pi p 2 (from the first two levels of the polynomial time hierarchy [S], GJ] Let us explain our results using Figure 2. For the containment problem we have 49 cases. These cases depend on a choice among 7 kinds of representation for each dimension of this figure. The vertical dimension is the set of worlds tested for containment in the set of worlds of the ....

....and I 0 is represented by a single table. 5) 9 positive existential q 0 such that CONT(q 0 , is Pi p 2 complete even if I is represented by a single e table and I 0 is represented by a single table. Proof: 1) Reduction of 89 3CNF: We first state the Pi p 2 complete problem used ([S]) input: two disjoint sets X and Y of variables, and a conjunction H of or clauses over X[Y such that for each c in H , #(c) 3. question: does there exist for each truth assignment of X , a truth assignment of Y which makes H true Let H = fc k j k 2 [1: p]g be a conjunction of or clauses ....

Stockmeyer, L., "The Polynomial Time Hierarchy", Theoretical Computer Science, vol. 3, no. 1, pp. 1--22, 1976.

....Although each of these three steps can be considered separately, in this paper we consider the total problem: given find a small encoder (or decoder or encoder decoder) circuit for . We give several results classifying these problems in terms of the polynomial time hierarchy (see Stockmeyer [12]) The classes of this hierarchy extend the class NP. Intuitively, problems in NP can be defined using existential quantification over a large (exponentially sized) set of short (polynomially sized) objects. For example, a Boolean formula is satisfiable iff there exists a satisfying truth ....

....multiset of integers and a number : 1) whether the set contains an interval of size at least this is complete; and 2) whether the set contains an element of multiplicity at least this is NP complete. The succinct representations used in these two results are either integer expressions [12], 23] or the general hierarchic input language [24] More recently, Mundhenk, Goldsmith, Lusena, and Allender [25] have shown that certain problems concerning Markov decision processes are NP complete. They note that natural NP complete problems are very rare, and they suggest further study of ....

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L. J. Stockmeyer, "The polynomial-time hierarchy," Theoret. Comp. Sci., vol. 3, pp. 1--22, 1977.

.... of these three steps can be considered separately, in this paper we consider the total problem: given (G; p; q) find a small encoder (or decoder or encoder decoder) circuit for (G; p; q) We give several results classifying these problems in terms of the polynomial time hierarchy (see Stockmeyer [12]) The classes of this hierarchy extend the class NP. Intuitively, problems in NP can be defined using existential quantification over a large (exponentially sized) set of short (polynomially sized) objects. For example, a Boolean formula is satisfiable iff there exists a satisfying truth ....

....and a number K: 1) whether the set contains an interval of size at least K this is Sigma p 3 complete; and (2) whether the set contains an element of multiplicity at least K this is NP #P complete. The succinct representations used in these two results are either integer expressions [12, 23] or the general hierarchic input language [24] 4 More recently, Mundhenk, Goldsmith, Lusena, and Allender [25] have shown that certain problems concerning Markov decision processes are NP #P complete. They note that natural NP #P complete problems are very rare, and they suggest further ....

[Article contains additional citation context not shown here]

L. J. Stockmeyer, "The polynomial-time hierarchy," Theoret. Comp. Sci., vol. 3, pp. 1--22, 1977.

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L.J. Stockmeyer, "The Polynomial - Time Hierarchy", Theoretical Computer Science 3 (1977), 1-22.

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L. J. Stockmeyer. The Polynomial-Time Hierarchy. TCS, 3:1--22, 1977.

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Stockmeyer, L. J. (1976). The polynomial-time hierarchy. Theoretical Computer Science 3 (1), 1-22.

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Stockmeyer, L. J. (1976). The Polynomial-time Hierarchy. Theoretical Computer Science, 3(1):1--22.

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Stockmeyer, L.J. 1976. The polynomial time hierarchy. Theor. Comp. Sci., 3(1):1--22.

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L. J. Stockmeyer, The polynomial-time hierarchy, Theoretical Comput. Sci., 3 (1976), pp. 1--22. 54

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Stockmeyer, L.J., "The polynomial-time hierarchy," Theoretical C.S. 3(1977): 1-22.

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L. Stockmeyer, 1977, The polynomial time hierarchy, Theoret. Comp. Sci. 3, 1--22.

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