K. W. Wagner. Bounded query classes. SIAM J. Comput., 19(5):833--846, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Am Hubland, D 97074 Wurzburg, Germany Key words: Computational complexity, unambiguous computation, function classes 1 Introduction Unambiguous computation according to UP has become a classical notion in computational complexity theory. Unambiguity is also used in a theorem of Wagner [14]. A set L is in P iff there are a set A 2 NP and a polynomial p such that for all x and y with jyj p(jxj) if (x; y) 2 A then (x; y Gamma 1) 2 A, and x 2 L iff the maximal y with (x; y) 2 A is odd. Here, essentially for each x all positive y s must be neighbored, they form a cluster, where 0 ....

....(x; y) 2 A is odd. Here, essentially for each x all positive y s must be neighbored, they form a cluster, where 0 is always in the cluster if one exists. Unambiguity is reflected in the existence of at most one such cluster. This construction is crucially relevant for parallel access to NP (see [14]) The same idea occurs in the definition of the complexity theoretic operator F [13] F DeltaK is the class of all functions f for which there exist an A 2 K and polynomial p as above with f(x) supfyjjyj p(jxj) x; y) 2 Ag. This operator plays a key role as part of an operator pair to ....

K. W. Wagner. Bounded query classes. SIAM Journal on Computing, 19:833-- 846, 1990. 11

....NEXP) We meet this vein of generalization by writing BH2 (K) and feel free to use the term in our sense instead. Another good choice would be extended boolean hierarchy. This term however, has a very clear meaning in the literature as a boolean hierarchy intertwined with bounded query classes [42]. 1.3 Reducing the Set of Solutions of NP Problems The results from the study of the generalized bolean hierarchy have some consequences in further areas of complexity theory. For instance, using our results, we resolve in some sense an open question concerning certain possibilities to reduce ....

K. W. Wagner. Bounded query classes. SIAM Journal on Computing, 19:833-846, 1990.

....The class V4T uengt gw wL , is the conjunction of ; in particular, is the familiar class . The class , contains the decision problems which can be solved in polynomial time with parallel calls to a oracle, and is part of the Refined PH [50]. According to the current belief in complexity theory, Fig. 2 shows a strict hierarchy of inclusions. # # , # # Figure 2: Containment between Complexity Classes For classifying problems that compute an output value (e.g. the set of atoms that are ....

....by a classical formula ) function classes similar to the classes above have been introduced (cf. 45, 31] Among these are ( Q , which are the functional analogs of , a respectively. For further background on these complexity classes, we refer to [31, 32, 37, 45, 50]. In this paper, unless stated otherwise, completeness for a decision class is with respect to standard polynomial time transformations. Completeness for a function class is understood in terms of a natural generalization of polynomial time transformations: The problem O reduces to , if ....

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K. Wagner. Bounded query classes. SIAM Journal of Computing, 19(5):833--846, 1990.

....2 ]g. c) For all m 1, coDIFFm (C) fL L 2 DIFFm (C)g. d) BH(C) the boolean hierarchy over C, is m 1 DIFFm (C) The relationship between the levels of the boolean hierarchy over k , bounded access to k , and various symmetric di erence classes is as follows. Proposition 2.2 1. [25]) For each k 1 and each m 1, P DIFFm 1( coDIFFm 1( m 1 tt : 2. 16] For all k 1 and all m 1, DIFFm( 3. 26] For all k 1 and all m 1, P m tt = P DIFFm( k ) P . Regarding symmetric di erence classes, we point out an immediate, but in ....

K. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):833-846, 1990.

....possible with O(log m) 1 = O(log m) oracle calls; from this the claim follows immediately. This result suggests that the deduction problem under GCWA and under CCWA is not m complete for 3 , since it seems rather unlikely that a problem in P is m complete for 3 , cf. [16, 34]. 4 Conclusion Our main results are summarized in Table 1. Answering the question in [5] we have shown that the deduction problem with ECWA and with circumscription is 2 complete for propositional theories, even for theories in 3XCNF and a single literal. Moreover, we proved the same ....

.... how much k completeness is in a 13 CWR rule lower bound upper bound CWA(T ) j= F NP hard GCWA(T ) j= F EGCWA(T ) j= F CCWA(T ; P ; Q;Z) j= F ECWA(T ; P ; Q;Z) j= F CIRC(T ; Q;P ; Z) j= F Table 1: Complexity Results for Propositional Closed World Deduction problem, cf. [19, 34]. This may be measured by the number of necessary calls to a k oracle [19, 17, 33] Closure computation with O(n) oracle calls is straightforward for CWA, GCWA, and CCWA. For CWA, it is not dicult to show that closure computation is (under suitable polynomial transformability) equivalent to the ....

K. Wagner. Bounded query classes. SIAM J. Comp., 19(5):833-846, 1990.

.... P coincide (and they further coincide with P #O(1) where a fixed number of parallel rounds is allowed) BH91] Wagner showed that many di#erent and natural ways of restricting # 2 all lead to the same P class (e.g. P coincide with L ) for which he introduced the name # 2 [Wag90]. Further variants were introduced by Castro and Seara, who proved that, for all k N, P coincide with P n) where a succession of O(log ) parallel querying rounds are allowed) CS96] 2.2 Branching time logics and NP hard path modalities We assume familiarity with temporal logic ....

K. W. Wagner. Bounded query classes. SIAM J. Computing, 19(5):833--846, 1990.

....to compute one value of this function. With bounded query complexity we look at the set of functions that can be calculated if we put an upper bound on the number of queries that we allow the computer to ask the oracle. This notion has been extensively studied both in the resource bounded setting [2, 4, 5, 13, 12, 11, 17, 60, 75, 104] and in the recursive setting[15, 16] This notion and its variants has lead to a series of techniques and tools that are used throughout complexity theory. In this chapter we combine some of the bounded query notions with quantum computation. The main goal is to further as was done by Fortnow ....

....the extra power that a polynomial time, exact quantum computer yields compared to classical deterministic computation when querying a set in the class NP. In the case of deterministic computation the following equality between adaptive and non adaptive queries to NP is well known. Fact 6 [13, 29, 104] 1. For all k 0 we have P . 2. For any polynomial q(n) 1 the equality P NP[O(log(q(n) holds. Proof: Both items are proved in a similar way which has two parts. The first part shows that computing a function in P can be reduced to computing the parity of 2 1 other ....

Klaus W. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):833--846, October 1990.

....the tools to the case of function problems. The main contributions of this paper can be thus summarized as follows: Firstly, we determine the complexity of data disjunctions. We obtain natural and simple logical inference problems complete for the class 2 of the re ned polynomial hierarchy [46], and, in their computational variants, complete problems for the function classes FP k and FL log [log] and their exponential analogs. Secondly, we provide upgrading techniques for determining the complexity of function computations. They generalize available tools for decision problems ....

....employed in the paper; we assume however some familiarity with basic notions of complexity theory such as oracle computations, NP, PSpace, L etc. The class 2 contains the languages which are polynomial time truth table reducible to sets in NP. It has a wide range of di erent characterizations [46,22]. In particular, the following classes coincide with 2 : k[k] polynomial time computation with k rounds of parallel queries to an NP oracle [28,9] log : polynomial time computation where the number of queries to an NP oracle is at most logarithmic in the input size [26] L log : ....

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K. Wagner. Bounded Query Classes. SIAM Journal of Computing, 19(5):833{ 846, 1990. 34

....of complexity theory, such as the classes P, NP, L (logspace) oracle computations, and the polynomial hierarchy (PH) whose classes are Sigma k ; Pi k , and Delta k , k 1; consult [20, 46] for background. Intertwined with the levels Sigma k , Pi k of PH are the classes Theta k [63] ( Sigma k k 1 Sigma k 1 ) which are now considered intermediate constituents of PH. The k 1 classes have natural complete problems in terms of optimization variants of Sigma k complete problems. The class P is the class of problems decidable by polynomial time Turing machines ....

....variants of Sigma k complete problems. The class P is the class of problems decidable by polynomial time Turing machines whose oracle queries are first collected, and then all queries are forwarded to the oracle together. Thus, the queries are independent of each other. Proposition 2. 1 [63] Theta k 1 = P = L . 3 Logic Programs with Generalized Quantifiers Let 0 be a signature for describing the program input, and let be an extension of 0 by new relational symbols. A logic program with GQs (GQLP) on is a finite collection P of rules where A is a atom whose predicate ....

K. Wagner. Bounded Query Classes. SIAM Journal of Computing, 19(5):833--846, 1990.

....Preliminaries on expressive power We assume some familiarity with the basic concepts of complexity theory, such as P, NP, L (logspace) oracle computations, and the polynomial hierarchy (PH) Consult [18, 36] for an introduction. Intertwined with the levels , of PH are the classes [51] ( k k 1 ) which are now considered constituting intermediate classes of PH. It turns out that the classes have natural complete problems which are typically obtained as optimization variants of k complete problems. Since oracles can be seen as a logical concept of ....

....will be most useful for our investigations. The class P is the class of problems decidable by polynomial time Turing machines whose oracle queries are rst collected, and then all queries are forwarded to the oracle together. Thus, the queries are independent of each other. Proposition 6. 1 [51] = L k = P k [O(log n) Here, the term O(log n) in the brackets indicates that the number of queries posed to the oracle is bound by O(log n) where n is the size of the input. A Turing machine which computes an output rather than only telling yes or no is known as a transducer. ....

K. Wagner. Bounded Query Classes. SIAM J. Comp., 19(5):833-846, 1990.

....problems in a suitable way. Thus, the main contributions of this paper can be summarized as follows. Firstly, we determine the complexity of data disjunctions. We obtain natural and simple logical inference problems complete for the class Theta of the refined polynomial hierarchy [45], and, in their computational variants, complete problems for the function classes FP k and FL log [log] and their exponential analogs. Secondly, we provide upgrading techniques for determining the complexity of function computations. They generalize available tools for decisional problems and ....

....in the paper; we assume however some familiarity with basic notions of complexity theory such as oracle computations, NP, PSpace, L etc. The class Theta 2 contains the languages which are polynomial time truth table reducible to sets in NP. It has a wide range of different characterizations [45, 21]. In particular, the following classes coincide with 2 : k[k] polynomial time computation with k rounds of parallel queries to an NP oracle [27] polynomial time computation where the number of queries to an NP oracle is at most logarithmic in the input size [25] ffl L : ....

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K. Wagner. Bounded query classes. SIAM Journal of Computing, 19(5):833--846, 1990. 24

....M j= is easy if the values of all i in M are known; by (i) they can be determined in parallel with calls to k oracles. Thus, deciding M j= is possible in k 1 k, i.e. in polynomial time with one round of parallel oracle calls. Since, as well known, k 1 k = O(log n) see [50] for k = 1, which easily generalizes) this proves the membership part for (ii) INFSYS RR 1843 02 10 PSPACE hardness of deciding M j= for general and hardness for (i) can be shown by a reduction from evaluating suitable prenex QBFs. We exploit that nested abnormality theories (NATs) ....

....given m instances (M 1 ; 1 ) M n ; m ) of the model checking problem for circumscriptive atoms on disjoint alphabets At 1 ; Atm , respectively, whether the number of yes instances among them is even. The [O(log n) completeness of this problem is an instance of Wagner s [50] general result for all complete problems. Moreover, we may assume that m is even and use the assertion (cf. 50] that (M i ; i ) is a yes instance only if (M i 1 ; i 1 ) is a yes instance, for all i 2 f1; m 1g. Then, we can define = e 1 0 2i m ( 2i 2i 1 ) ....

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K. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):833--846, 1990.

..... The class P , k 1, contains the decision problems which can be solved in polynomial time with parallel calls to a k oracle; FP k is the analog for function computations. Note that P =P and FP k =FP . For further background on the complexity classes, see e.g. [23, 24, 30, 41]. 3 OVERVIEW OF RESULTS In this section, we give an overview on the complexity results that we derive, and discuss possible implications. 2 3 4 co NP NP Figure 2: Containment between Complexity Classes 3.1 PROBLEM STATEMENTS In our analysis, we focus on the following ....

....A i =fA i;1 ; A i;m i g and B i = fB i;1 ; B i;n i g, whether the number of valid formulas among 1 ; k is even. W.l.o.g. A 1 [ B 1 ; A k [ B k are pairwise disjoint, 1 is valid, and for each j2f2; kg, the validity of j implies the validity of j 1 [41]. We create such that X =x is an partial explanation of relative to (C; P ) iff the number of valid formulas among 1 ; k is even. Roughly, the main idea behind this construction is as follows. For each i , we construct an instance of weak cause, that is, M i = U i ; V i ; F i ....

K. Wagner. Bounded query classes. SIAM Journal of Computing, 19(5):833--846, 1990.

....queries are made in parallel, we use the notation P tt . We will use P tt when the machines are allowed polynomial many queries. The connections between the Boolean Hierarchy and the bounded queries to SAT is rich and varied. We ask the reader to consult the literature for a full accounting [WW85, KSW87, Wag88, Kad88, CGH 89, ABG90, Wag90, Bei91, Cha92, HN93, BCO93, CGL97, CK96, HHH99, BF99, Cha01]. For this paper, we make use of the following facts about the Boolean Hierarchy and bounded queries to SAT. SAT[k 1] tt BH k coBH k BH k [ coBH k P tt [KSW87; Bei91] tt [Bei91] BH k = coBH k = BH = BH k [CGH 88; CGH 89] BH k = coBH k = SAT 2 NP=poly [Kad88] 3 Proof ....

K. Wagner. Bounded query classes. SIAM Journal on Computing, 19:833-846, 1990.

....and is intuitively easier than a problem complete for . 2 [O(log n) contains the problems in 2 that can be solved with O(log n) many oracle calls, where n is the size of the problem input. This class, also named 2 , is very robust and has many different equivalent characterizations [90]. In particular, it coincides with L , logspace computability with an NP oracle, and with P , that is, polynomial time computability with an NP oracle where all oracle calls must be first prepared and then issued in parallel. Qualitatively speaking, membership in [O(log n) means that ....

....contained in D i . But this contradicts the assumption that D is the z partition of D. Hence, it holds D i = D i . That is, we get D i = D for all i = 0; k and thus also k = l. 2 Proof of Theorem 5. 6 (continued) a) It remains to show P from the following P complete problem [90]. Given the propositional formulas in CNF 1 ; m , we are asked whether the number of tautologies among 1 ; m is even. Without loss of generality, we can assume that 1 ; m are defined on pairwise disjoint sets of variables, that 1 and 2 are tautologies, that m ....

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K. Wagner. Bounded query classes. SIAM Journal of Computing, 19(5):833--846, 1990. 62

....mind changes in order to re derive a theorem of [29] obtaining as a corollary that (2 k Gamma1) tt = PF k T . Based on this, Buss and Hay were able to show that P bf [12] thus disproving a conjecture of Kobler, Schoning, and Wagner. Wagner has also independently disproved his conjecture [28]. The mind change technique also yields a new proof of Hemachandra s surprising result that P = NP . In Section 5. we examine a tradeoff between serial queries and parallel queries to general oracles (this is related to p terseness) We define k cheatable sets and prove an important ....

....Schoning, and Wagner had conjectured that these two classes were unequal, because it seemed that a proof would hinge on finding a polynomial time algorithm to convert Boolean circuits into equivalent Boolean formulas. We note that Wagner also disproved his conjecture independently of Buss and Hay [28]. The conjecture can also be disproved as a corollary to Hemachandra s result [16] that P O(logn) T . We say that a hierarchy is proper if all levels are distinct. We say that a hierarchy collapses at the kth level if the first k levels are distinct, but all levels after the kth are equal to ....

K. W. Wagner. Bounded query classes. SICOMP, 19(5):833--846, Oct. 1990. 28

....length bounded and graph( f ) belongs to coNP. Observe that the classes NPMV and coNPMV satisfy the nice symmetry that graphs of functions in the former class are in NP and those in the latter class are in coNP. Just as the definition of the Boolean hierarchy over NP leads to the class ) see [Wag90]) we now introduce the class NPMV(n ) It can be shown that a function h belongs to NPMV(k) if and only if there is a 2 ary function f 2 NPMV such that set h(x) set f (x;k) Gamma (set f (x;k Gamma 1) Gamma(set f (x; k Gamma 2) Gamma( Delta Delta Delta Gamma set f (x; 1) ....

....coNPMV and NPMV . 4.4 Theorem. For all k 2, we have Furthermore, all of the inclusions are strict unless the polynomial time hierarchy collapses. Proof. It remains to show the last inclusion. Let f 2 NPMV(n ) Then the graph of f is in NP(n ) which is known to be equal to P [Wag90]. Obviously, f can be computed by an NPMV algorithm with access to a P oracle: simply guess an output of f and, querying its graph, check that the guess is correct. Thus, NPMV(n . 2 Under the likely assumption that NP 6= coNP, we see, by Theorem 4.2, that the class NPMV is not included ....

K. Wagner. Bounded query classes. SIAM Journal on Computing, 19:833--846, 1990. 20

....Supported in part by NSF Research Grant CCR 88 23053. Dept. of Computer Science, University of Electro Communications, Chofu si, Tokyo 182, Japan. This work was done while the author was at Dept. of Information Science, Tokyo Institute of Technology. 1. Introduction Numerous researchers [3, 5, 8, 9, 10, 11, 16, 17, 24, 25, 26, 27] have studied the Boolean hierarchy over NP. This hierarchy intertwines the query hierarchies over NP, and is identical to the Haussdorf and the difference hierarchies over NP. Similar relations hold among hierarchies over many classes other than NP [7] A central question is whether these ....

K. W. Wagner. Bounded query classes. SIAM J. Comput., 19(5):833--846, Oct. 1990.

....of a formula making nonadaptive queries to SAT. Buhrman and Thierauf [BT96] give a relativized world where this fails deterministically. Theorem 1.4 gives the first relativized counterexample to Hypothesis 1.2. In fact Theorem 1.4 shows a stronger result. Buss and Hay [BH91] and Wagner [Wag90] show that languages computable with a polynomial number of nonadaptive queries to SAT are equivalent to those computable with 4 O(log n) adaptive queries to SAT. For functions the equivalence would imply we can distinguish unique solutions (see [BFT97] Theorem 1.4 gives a relativized world ....

K. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):833--846, 1990. 12

....P NP[k] tt of sets that are bounded truth table reducible to some NP set, where the generator produces at most k queries, for some k 0. These bounded query classes for NP are central topics of investigations in computational complexity theory [ABG90, Bei91, BH88, CGH 88, Ka88a, Ka88b, KT94, W90, Wec85] Especially the (extended version of the) paper by Amir, Beigel, and Gasarch [ABG90] gives a very broad overview on this topic and also provides an extensive list of references. Let us point out the following, rather obvious, property of truth table reductions: the evaluator, by getting ....

K. Wagner. Bounded query classes. SIAM J. on Computing 19(5), pages 833-846, 1990.

.... (x i ) x k ) is P NP k complete. is FNP OptP[O(log n) hard. Proof. This can be shown by a reduction from problem MAX 3SAT ODD, which asks whether max jfx i 2 X j (x i ) truegj, where ranges over the satisfying assignments for a given 3SAT instance over X , is odd (see e.g. [10]) Using further variables y i , whether jfx i 2 X j (x i ) truegj is odd can be expressed as y n , where y 1 x 1 and y i : y i 1 x i ) written in clausal form. Then, apply the reduction as in the proof of Lemma 4.4, and let x i be as there and x j be y n . Construct for the problem in ....

K. Wagner. Bounded Query Classes. SIAM Journal of Computing, 19(5):833--846, 1990.

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K. W. Wagner. Bounded query classes. SIAM Journal on Computing, 19:833-846, 1990.

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K. W. Wagner. Bounded query classes. SIAM J. Comput., 19(5):833--846, 1990.

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K. Wagner. Bounded query classes. SIAM Journal on Computing, 19:833-846, 1990.

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K. Wagner. Bounded query classes. SIAM Journal of Computing, 19(5):833--846, 1990.

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