Richard M. Karp and Richard J. Lipton. Some Connections Between Nonuniform and Uniform Complexity Classes. In Proceedings of the 12th Annual ACM Symposium on Theory of Computing, pages 302--309, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... under reasonable complexity theoretic assumptions, RE #m hardness does not imply NP # m hardness (though, in fact, under other complexity theoretic assumption we will see that RE # m hardness does imply NP # m hardness) We first state a useful definition and result due to Karp and Lipton [KL80] Definition 4.4 [KL80] and each function f : N N, is defined as follows. #g) #L 1 # C) #x) g(1 x ) f( x ) x ## #x, g(1 x )# L 1 ) and each function class F is defined as follows. #f # F) L # C f ] Theorem 4.5 [KL80] If SAT P O(log n) ....

....assumptions, RE #m hardness does not imply NP # m hardness (though, in fact, under other complexity theoretic assumption we will see that RE # m hardness does imply NP # m hardness) We first state a useful definition and result due to Karp and Lipton [KL80] Definition 4. 4 [KL80] and each function f : N N, is defined as follows. #g) #L 1 # C) #x) g(1 x ) f( x ) x ## #x, g(1 x )# L 1 ) and each function class F is defined as follows. #f # F) L # C f ] Theorem 4.5 [KL80] If SAT P O(log n) then P = NP. We can now ....

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R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309. ACM Press, April 1980. An extended version has also appeared as: Turing machines that take advice, L'Enseignement Mathematique, 2nd series, 28, 1982, pages 191--209.

....asymmetry is not present in the relativized proof. In order to rectify this problem we propose a model of computation that is more stringent than the usual relativization computation. This turns out to be equivalent to a generalization of the notion of advice strings proposed by Karp and Lipton [KL80]. Intuitively, any relativized result can be regarded as a comparison between complexity classes under a certain nonuniform setting provided by an (infinite) advice, namely an oracle. Here we generalize the advice string formulation of Karp and Lipton by allowing random access to the advice ....

....status of a collapse of PSPACE to P under random access to advice is particularly interesting in view of a result of Kozen [Ko78] If PSPACE P, then there exists a proof of this fact by diagonalization. 2 Random Access to Advice Strings Recall the definition of C poly by Karp and Lipton [KL80]. We generalize this notion by allowing the underlying machines to have random access to an advice string. Let us fix any length function # from N to N. A function s : n ## 0, is called an advice function of size #(n) Given any advice function s of size #(n) we say a language L is in the ....

R. Karp and R. Lipton, Some connections between nonuniform and uniform complexity classes, in Proc. 12th ACM Symposium on Theory of Computing (STOC'80), ACM, 302--309, 1980. (An extended version appeared as: Turing machines that take advice, in L'Enseignement Mathematique (2nd series) 28, 191--209, 1982.)

....[Wat94] has taken a first step towards a resolution. He shows that one may relax the hypothesis to polynomially valued functions if one adds a certain constraint on the behavior of h. Below, we quickly summarize some of the other main results of Hemaspaandra et al. HNOS94] Definition 2.5 1. KL80] For any class of sets C, C=poly denotes the class of sets L for which there exist a set A 2 C and a polynomially length bounded function h : Sigma Sigma such that for every x, it holds that x 2 L if and only if hx; h(0 jxj )i 2 A. 2. BBS86] Let ExtendedLow 2 denote fL j NP L ....

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309, April 1980. An extended version has also appeared as: Turing machines that take advice, L'Enseignement Math'ematique, 2nd series 28, 1982, pages 191--209.

....if the polynomial hierarchy is strict, then each of these statements is false. Parts 1) 2) 3) and 5) of this corollary are immediate from Theorems 2.3, 3.1, 3.2, 3.3, and 3.4, and Propositions 2.5 and 2.6. Part 4) is an implication of part 3) using methods of Adleman [37] and Karp and Lipton [38]. Part 6) is proved from part 4) as follows. The decision version of circuit minimization has instances where is a circuit and , and the question is whether there is a circuit equivalent to with . This problem has a form: there exists a such that and, for all inputs , A standard argument ....

....the symbol ; denote this set of words containing One Two by For those readers familiar with complexity theory, the argument follows. Using that 6 MBE D, it follows as in [37] that if MBE D2 P 6 P poly. Hemaspaandra et al. 39, Theorem 11] have observed that a result of Karp and Lipton [38] relativizes; in particular, if 6 P poly then 6 =6 for all k 3. As will become clear shortly, the symbol serves as a synchronization symbol ; for example, the DIFs we construct have the property that, in any block codebook of symbol words, the (unique) occurrence of appears at the same ....

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R. M. Karp and R. J. Lipton, "Some connections between nonuniform and uniform complexity classes," in Proc. 12st ACM Symp. Theory of Computing, 1980, pp. 302--309.

.... etant donn e IC pouvait toujours etre repr esent e par une formule equivalente dont la taille est polynomiale en jEj jICj alors on aurait NP P poly [5] ce qui est consid er e comme peu vraisemblable puisque cela entrainerait l effondrement de la hi erarchie polynomiale au deuxi eme niveau [9]. Pour en revenir a la complexit e de l interrogation, nous avons obtenu le r esultat tr es g en eral suivant pour les op erateurs a partir de distances : Proposition 1 Soit 4 un op erateur de fusion a partir de distances, soit un ensemble de croyances E et soient deux formules IC et ff ....

R.M. Karp and R.J. Lipton. Some connections between non-uniform and uniform complexity classes. In Proceedings of the Twelfth ACM Symposium on Theory of Computing (STOC'80), pages 302--309, 1980.

....every NP set has polynomial size circuits Kushilevitz and Mansour [KM91] have shown that depth O(logn) decision trees are exactly learnable with high probability using membership queries. then the polynomial hierarchy collapses to ZPP . This improves the previous result of Karp and Lipton [KL80] where Sigma 2 appears in place of ZPP . Let us comment on a particular technique used in this paper. For our results concerning learning with equivalence queries and an NP oracle, our approach builds on the investigation of the query complexity of learning by Kannan in [K93] Kannan shows ....

.... 22(1) rules out the possibility for monotone read twice DNF (since the CNF size might be exponentially large) 7 Applications to Structural Complexity Theory Watanabe [W94] has observed that a consequence of Theorem 7(b) in Section 3 is an improvement of the following result of Karp and Lipton [KL80]. Theorem 23 [KL80] If every NP set has polynomial size circuits then the polynomial hierarchy collapses to Sigma 2 . It is clear that ZPP is contained in Sigma 2 but the other direction of containment is not known (and would be surprising) Watanabe has observed the following, and we ....

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Richard M. Karp and Richard J. Lipton. Some Connections Between Nonuniform and Uniform Complexity Classes. In Proceedings of the 12th Annual ACM Symposium on Theory of Computing, pages 302--309, 1980.

....of size O(n ) for some fixed k (page 41) The proof of the lemma proceeds as follows: If SAT (the problem of deciding the satisfiability of boolean formulae) does not have polynomial circuits, then the lemma is true. In the case of SAT having polynomial circuits, by Karp and Lipton s theorem [11], the polynomial hierarchy collapses to 2 . This implies that # 3 can be simulated in # O(1) 2 , and since it was already established that # 3 contains functions of superpolynomial circuit complexity for any superpolynomial f , the lemma follows. After proving the above lemma, ....

R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. Proceedings of the 12th ACM Symposium on Theory of Computing, (1980), pp. 302--309.

....problem seems to well solved. Bruck and Naor [6] present a code (not well known, but nevertheless easily presented) for which they show that the existence of small size maximum likelihood decoding circuits would imply the collapse of the polynomial hierarchy (using a result of Karp and Lipton s [14]) However the codes presented by Bruck and Naor do not have large distance. It still remains open if the maximum likelihood decoding problem is hard for any constant distance code. The Reed Solomon codes would have formed a good 15 candidate to show hardness of this problem, except that it is ....

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. Proceedings of the 12th Annual ACM Symposium on Theory of of Computing, pp. 302-309, 1980.

.... formula # into a polysize data structure # such that: # and # entail the same set of clauses, and clausal entailment on # can be decided in time polynomial in its size, unless NP # P poly [33, 5] This last assumption implies the collapse of the polynomial hierarchy at the second level [19], which is considered very unlikely. We use this classical result from knowledge compilation in some of our proofs of Proposition 3.1, which explains why some of its parts are conditioned on the polynomial hierarchy not collapsing. We have excluded the subsets BDD, s NNF, d NNF and f NNF from ....

R.M. Karp and R.J. Lipton. Some connections between non-uniform and uniform complexity classes. In Proc. of the 12 th ACM Symposium on Theory of Computing (STOC'80), pages 302--309, 1980. 29

....solvable. In this paper we give a negative answer to this question, under standard complexity assumptions. In particular, we show that if the closest vector problem with preprocessing can be solved in polynomial time, then NP is contained in P=poly and the polynomial hierarchy collapses (see [18]) Our result is analogous to similar results for the nearest codeword problem [19] and the subset sum problem [20] and is based on a new proof of the NP hardness of the closest vector problem. A related result is presented in [17] where it is proved that for recursive cube search (RCS) ....

Richard M. Karp and Richard J. Lipton, \Some connections between nonuniform and uniform complexity classes," in Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, Los Angeles, California, 28-30 Apr.

.... structure such that: and entail the same set of clauses, and clausal entailment on can be decided in time polynomial in its size, unless NP P poly [Selman and Kautz, 1996; Cadoli and Donini, 1997] This last assumption implies the collapse of the polynomial hierarchy at the second level [Karp and Lipton, 1980] , which is considered very unlikely. We use this classical result from knowledge compilation in some of our proofs of Proposition 3.1, which explains why some of its parts are conditioned on the polynomial hierarchy not collapsing. We have excluded the subsets BDD, s NNF, d NNF and f NNF from ....

R.M. Karp and R.J. Lipton. Some connections between non-uniform and uniform complexity classes. In Proc. of ACM Symposium on Theory of Computing (STOC'80), pages 302--309, 1980.

....of sets that are truth table reducible to NP. But in fact, P NP tt = P NP[log] Hem89] The polynomial hierarchy [MS72,Sto77] is defined as follows. Sigma p 1 = NP; Sigma p k 1 = NP Sigma p k (for k 2 f1; 2; 3; Delta Delta Deltag) and PH = k1 Sigma p k : P=poly [KL80] denotes the class of sets having small circuits. For a nondeterministic polynomial time Turing machine M , let acc M (x) rej M (x) denote the number of accepting (rejecting) paths of M on input x and let total M (x) denote 4 the total number of paths of M on input x. #P is the class of ....

....input x are accepting, if x 2 L(M ) and M 0 has no accepting paths, otherwise. This shows L(M) 2 R path . Corollary 2.5 NP BPP path . It follows that if BPP path is equal to BPP, then BPP path , and hence NP, has small circuits, which in turn, by the result of Karp, Lipton, and Sipser (see [KL80]) implies that the polynomial hierarchy collapses. So we cannot expect BPP path to have normalized machines. For different, contemporaneous work related to normalized versus non normalized computation, see Hertrampf et al. HLS 93] and Jenner, McKenzie, and Th erien [JMT94] We have seen now ....

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309, April

....proof uses universal hashing, approximate counting and witness sampling. We also discuss the problem of finding irrefutable proofs in ZPP NP . There is an interesting consequence of this result with respect to the well known KarpLipton Theorem concerning sparse sets (with contribution by Sipser) [KL80]. This theorem says, if NP is Cook reducible (# p T ) to sparse sets, or equivalently, if SAT has polynomial size circuits, then the Polynomial time Hierarchy collapses to its second level: PH = # p 2 # # p 2 . Many researchers have since tried to improve on this signature theorem To ....

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309. ACM Press, April

....problem seems to well solved. Bruck and Naor [6] present a code (not well known, but nevertheless easily presented) for which they show that the existence of small size maximum likelihood decoding circuits would imply the collapse of the polynomial hierarchy (using a result of Karp and Lipton s [13]) However the codes presented by Bruck and Naor do not have large distance. It still remains open if the maximum likelihood decoding problem is hard for any constant distance code. The Reed Solomon codes would have formed a good candidate to show hardness of this problem, except that it is not so ....

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. STOC, 1980.

....empty. There are some important results related to this question. Theorem 2.6. 1) Ma82] If P # #= NP, then NP Pm (SPARSE) is not empty. 2) Wa88, Wa91] If R # #= NP, then NP P 1 tt (SPARSE) is not empty. 3) OW90] If P # #= NP, then NP P btt (SPARSE) is not empty. 4) [KL80] If # P 2 # #= PH, then NP PT(SPARSE) is not empty. 5) Wa87a, BK88] DEXT P btt (SPARSE) is not empty. 3. Structure of Complete Sets In the last section we identified the hardest problems in a given class C to be the sets that are C complete (i.e. # P m complete for ....

R. Karp and R. Lipton, Some connections between nonuniform and uniform complexity classes, in "Proc. 12th ACM Sympos. Theory of Computing", ACM (1980), 302-309.

....of Theorem 1.1 to classical structural complexity is the following corollary. Corollary 1.2 [LFKN92, BFL91] For K 2 fPP; PSPACE;EXPg, if K P=poly then K = MA. The above result improves previously known collapses to Sigma p 2 under the same assumption, obtained by Meyer, Karp, and Lipton [KL80]. This is particularly interesting since it is known that the collapse of PH to Sigma p 2 under the assumption NP P=poly [KL80] cannot be improved to Delta p 2 with relativizable proof methods [Hel86] and the LFKN protocol is known to be not relativizable [FS88, FRS88] The crux of the ....

....if K P=poly then K = MA. The above result improves previously known collapses to Sigma p 2 under the same assumption, obtained by Meyer, Karp, and Lipton [KL80] This is particularly interesting since it is known that the collapse of PH to Sigma p 2 under the assumption NP P=poly [KL80] cannot be improved to Delta p 2 with relativizable proof methods [Hel86] and the LFKN protocol is known to be not relativizable [FS88, FRS88] The crux of the proof of Corollary 1.2 is as follows: if K is contained in P=poly, then by Theorem 1.1, every language in K has an MIP protocol in ....

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R.M. Karp and R.J. Lipton. Some connections between nonuniform and uniform complexity classes. In Proc. 12th ACM Symposium Theory of Computer Science, 302-309, 1980.

....NP, E, EXP etc. can be found in standard books [7, 22] A relativized complexity class C with oracle A is denoted by either C A or C(A) Likewise, we denote an oracle Turing machine M with oracle A by M A or M(A) For a class C of sets and a class F of functions from 1 to , let C=F [15] be the class of sets A such that there is a set B 2 C and a function h 2 F such that for all x 2 , x 2 A , hx; h(1 jxj )i 2 B: The function h is called an advice function for A. We recall de nitions of AM and MA. A language L is in AM if there exist a polynomial p and a set B 2 P such ....

....single valued partial functions in NPMV. 3. NPMV t is the class of total functions in NPMV. 4. NPSV t is the class of total single valued functions in NPMV. The classes NPMV=poly, NPSV=poly, NPMV t =poly, and NPSV t =poly are the standard nonuniform analogs of the above classes de ned as usual [15]: for F 2 fNPMV;NPSV;NPMV t NPSV t g, a multivalued partial function f is in F=poly if there is a function g 2 F , a polynomial p, and an advice function h : 1 7 with jh(1 n )j = p(n) for all n, such that for all x 2 , set f(x) set g(hx; h(1 jxj )i) Before we connect ....

R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. In Proc. 12th ACM Symposium on Theory of Computing, pages 302-309. ACM Press, 1980.

.... a result of Kautz and Selman [25] that the existence of a data structure whose size is polynomial with respect to the size of the knowledge base, and from which one can correctly answer all queries in polynomial time, is equivalent to NP # P poly (which implies that # p 2 =PH, cf. [24], i.e. that the polynomial hierarchy collapses at the second level) However, the problem is compilable to P if we constrain # to be a single literal: the compilation can be done by caching all answers which are at most O( K ) in a boolean array. In Section 4.1 we show another restriction on ....

....so on. A concept of reduction for this hierarchy is defined, along with the definition of complete problems for each class of the hierarchy. The second hierarchy is the non uniform version of the first one. In this sense, it generalizes both the first one, and the classical non uniform hierarchy [24, 40]. A suitable concept of reduction is defined for this hierarchy. Some prototypical AI problems are complete for some classes of the non uniform hierarchy, but they do not seem to be complete for any class of the uniform hierarchy. Thus, the nonuniform hierarchy is better suited for assessing the ....

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R. M. Karp and R. J. Lipton. Some connections between non-uniform and uniform complexity classes. In Proceedings of the Twelfth ACM Symposium on Theory of Computing (STOC'80), pages 302--309, 1980.

....of compNP hardness. For example, the problem of diagnosis introduced in the last section is not compNP hard. However, we can prove its incompilability (its non membership in compP) using the non uniform classes and reductions. For languages of strings, the non uniform classes were introduced by Karp and Lipton [1980]. The extension for languages of pairs is given in [Cadoli et al. 1996] where the nonuniform compilability classes are also compared with Karp and Lipton s classes. Definition 7 A language of pairs S belongs to ##C (non uniform compC) i# there exists a polysize function f and a language of ....

R. M. Karp and R. J. Lipton. Some connections between non-uniform and uniform complexity classes. In Proceedings of the Twelfth ACM Symposium on Theory of Computing (STOC80) , pages 302--309, 1980.

....to view oracles as hardware upgrades and thus need a rather more restricted version of the general notion that only provides its upgrades in a general and largely input independent way. A good candidate are the advice functions. Turing machines with advice were rst considered by Karp and Lipton [19] in their fundamental study of non uniform complexity theory (cf. 5] 36] De nition 1. An advice function is a function f : N . An advice function f is called S(n) bounded if for all n, the length of f(n) is bounded by S(n) An advice Turing machine with input of size n will be ....

....on every input and accepts K. By a similar argument it follows that all recursively enumerable languages and their complements can be recognized by Turing machines with linear advice. It is easily seen that classes like P=poly and most other advice classes contain nonrecursive languages as well [19]. Many problems remain about the relative power of advice. In fact, several classical open problems in uniform complexity theory have their equivalents in the non uniform world. For example, Karp and Lipton [19] proved, among other results: Theorem 6. The following equivalences hold: i) NP ....

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R.M. Karp, R.J. Lipton. Some connections between nonuniform and uniform complexity classes, in: Proc. 12th Annual ACM Symposium on the Theory of Computing (STOC'80), 1980, pp. 302-309.

....iff for all problems A # ##C we have that A # nucomp S. Moreover, S is ##C complete if S is in ##C and is ##C hard. It is important to point out that the hierarchy formed by the compilability classes is proper if and only if the polynomial hierarchy is proper (Cadoli et al. 1996; Karp Lipton 1980; Yap 1983) a fact widely conjectured to be true. Informally, we may say that ##NP hard problems are not compilable to P . Indeed, if such compilation were possible, then it would be possible to define f as the function that takes the fixed part of the problem and gives the result of ....

Karp, R. M., and Lipton, R. J. 1980. Some connections between non-uniform and uniform complexity classes. In Proc. of STOC'80, 302--309.

....of size O(n k ) for some xed k (page 41) The proof of the lemma proceeds as follows: If SAT (the problem of deciding the satis ability of boolean formulae) does not have polynomial circuits, then the lemma is true. In the case of SAT having polynomial circuits, by Karp and Lipton s theorem [11], the polynomial hierarchy collapses to p 2 p 2 . This implies that f 3 can be simulated in f O(1) 2 , and since it was already established 1 The setting of the desired level of approximation to inverse polynomial is somewhat arbitrary; it avoids a third parameter besides time ....

R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. Proceedings of the 12th ACM Symposium on Theory of Computing, (1980), pp. 302-309.

....in part by an Alexander von Humboldt research fellowship. 1 1 Introduction Sparse sets play a central role in structural complexity theory. The question of the existence of sparse hard sets for various complexity classes under different sorts of reducibilities is well studied (see for example [KL80, Mah82, OW91, AHH 93, AKM92] Besides, much work has also been done on other issues concerning the complexity of sets reducible to sparse sets (see for example [Kad87, AH92, LS91, AHH 93] A central motivation for most earlier work (and also for this paper) can be seen as seeking ....

....A is low for Delta p 2 . An immediate corollary of the above result is: ffl If an NP complete set disjunctively reduces to a sparse set then PH = Delta p 2 . The above result is interesting since the previous best known collapse of PH under the above assumption was to Sigma p 2 [KL80] Also it is known [Sal93] that in some relativized world the well known left set technique cannot be used to get a collapse to P under the above assumption. The paper is organized as follows. In Section 2 we describe the notation and give basic definitions. In Section 3 we establish simplicity ....

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R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. Proceedings of the 12th ACM Symposium on Theory of Computing, 302-309, April 1980.

....complexity theory. In research from the early 1980s to the present, a variety of results has been obtained showing that this is impossible under plausible assumptions (see, e.g. the survey [18] A typical model for nonuniform computations are circuit families. In the notation of Karp and Lipton [22], sets decidable by polynomial size circuits are precisely the sets in P poly; i.e. they are decidable in polynomial time with the help of a polynomial length bounded advice function [32] Karp and Lipton (together with Sipser) 22] proved that no NP complete set has polynomial size circuits (in ....

....are circuit families. In the notation of Karp and Lipton [22] sets decidable by polynomial size circuits are precisely the sets in P poly; i.e. they are decidable in polynomial time with the help of a polynomial length bounded advice function [32] Karp and Lipton (together with Sipser) [22] proved that no NP complete set has polynomial size circuits (in symbols NP ## P poly) unless the polynomial time hierarchy collapses to its second level. The proof given in [22] exploits a certain kind of self reducibility of the well known NP complete problem SAT. More generally, it is shown ....

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R. M. Karp and R. J. Lipton, Some connections between nonuniform and uniform complexity classes, in Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302--309.

....does not have small circuits. By small circuits are meant circuits of size O(n k ) for some xed k (page 41) The proof of the lemma proceeds as follows: If SAT does not have polynomial circuits, then the lemma is true. In the case of SAT having polynomial circuits, by Karp and Lipton s theorem [12], the polynomial hierarchy collapses to p 2 p 2 . This implies that t(n) 3 can be simulated in t(n) O(1) 2 , and since it was already established that t(n) 3 t(n) 3 contains functions of superpolynomial circuit complexity for any superpolynomial t(n) the lemma ....

R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. Proceedings of the 12th ACM Symposium on Theory of Computing, (1980), pp. 302-309.

....NP, E, EXP etc. can be found in standard books [7, 22] A relativized complexity class C with oracle A is denoted by either C A or C(A) Likewise, we denote an oracle Turing machine M with oracle A by M A or M(A) For a class C of sets and a class F of functions from 1 to Sigma , let C=F [13] be the class of sets A such that there is a set B 2 C and a function h 2 F such that for all x 2 Sigma , x 2 A , hx; h(1 jxj )i 2 B: The function h is called an advice function for A. We recall definitions of AM and MA. A language L is in AM if there exist a polynomial p and a set B 2 P ....

....single valued partial functions in NPMV. 3. NPMV t is the class of total functions in NPMV. 4. NPSV t is the class of total single valued functions in NPMV. The classes NPMV poly, NPSV poly, NPMV t poly, and NPSV t poly are the standard nonuniform analogs of the above classes defined as usual [13]: for F 2 fNPMV;NPSV;NPMV t NPSV t g, a multivalued partial function f is in F=poly if there is a function g 2 F , a polynomial p, and an advice function h : 1 7 Sigma with jh(1 n )j p(n) for all n, such that for all x 2 Sigma , set f(x) set g(hx; h(1 jxj )i) 9 Before we connect ....

R. M. Karp and R. J. Lipton, Some connections between nonuniform and uniform complexity classes, in Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, 1980, pp. 302--309.

....then C poly is the class of languages defined by Turing machines with the same time bounds as C, augmented by polynomial advice. Any class C poly is also known as non uniform C, where non uniformity is due to the presence of the advice. Non uniform and uniform complexity classes are related: Karp and Lipton (1980) proved that if NP # P poly then # p 2 = # p 2 = PH, i.e. the polynomial hierarchy collapses at the second level, while Yap (1983) generalized their results, in particular by showing that if NP # coNP poly then # p 3 = # p 3 = PH, i.e. the polynomial hierarchy collapses at the third ....

....We now have the right complexity class to completely characterize the problem pli. In fact pli is ##coNP complete (Cadoli et al. 1996b, Theorem 7) Furthermore, the hierarchy formed by the compilability classes is proper if and only if the polynomial hierarchy is proper (Cadoli et al. 1996b; Karp Lipton, 1980; Yap, 1983) a fact widely conjectured to be true. Informally, we may say that # #NP hard problems are not compilable to P , as from the above considerations we know that if there exists a preprocessing of their fixed part that makes them on line solvable in polynomial time, then the ....

Karp, R. M., & Lipton, R. J. (1980). Some connections between non-uniform and uniform complexity classes. In Proceedings of the Twelfth ACM Symposium on Theory of Computing (STOC'80), pp. 302--309.

....we can classify sets into some categories by using di erent reducibilities to sets of small density [6, 15, 16] Especially, a set having a census function bounded above by some polynomial is called sparse. Relative to this notion, the following questions have been considered by many researchers [5, 10, 13, 14, 22, 24, 35, 38, 40, 41]. For a class K and a reducibility r , is there any sparse set to which every set in K is r reducible , Suppose that every set in K is r reducible to some sparse set. Will then any unexpected inclusions follow As a matter of fact, after Berman and Hartmanis conjectured that all P ....

R. M. Karp and R. J. Lipton, Some connections between nonuniform and uniform complexity classes, Proceedings of the 12th Annual Symposium on Theory of Computing (ACM, 1980) 302-309.

.... small information content can be crisply formalized: R p T (SPARSE) is precisely the class more commonly referred to as P poly of sets having polynomial sized (nonuniform) circuits (Meyer, see [7] R p T (SPARSE) has been intensely studied, both in terms of the question NP R p T (SPARSE) [19,16, 10,18], and in terms of the robustness of R p T (SPARSE) R p T (SPARSE) is indeed quite robust; in addition to its characterization in terms of small circuits, R p T (SPARSE) is easily noted equivalent to R p T (TALLY) R p tt (TALLY) and R p tt (SPARSE) see [6] Nonetheless, Book and Ko ....

R. Karp and R. Lipton, Some connections between nonuniform and uniform complexity classes, in Proceedings of the 12th ACM Symposium on Theory of Computing, April 1980, pp. 302-309.

....active research area. For the case of Turing reductions, it is known that the existence of sparse Turing complete sets for NP would collapse the polynomial hierarchy to P NP[log] Kad89] and the existence of sparse Turing hard sets for NP would collapse the polynomial hierarchy to Sigma p 2 [KL80] both of these results are known to be essentially optimal with respect to relativizable proof techniques [Kad89,Hel86] As just noted, for the cases of many one and Turing reductions, the consequences of sparse NP complete sets are well understood. However, with respect to the reductions whose ....

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309, April 1980.

.... a result of Kautz and Selman in [26] that the existence of a data structure whose size is polynomial with respect to the size of the knowledge base, and from which one can correctly answer all queries in polynomial time, is equivalent to NP P poly (which implies that Sigma p 2 =PH, cf. [25], i.e. that the polynomial hierarchy collapses at the second level) However, the problem is compilable to P if we constrain fl to be a single literal: the compilation can be done by caching all answers which are at most O(jKj) in a boolean array. In Section 4.1 we show another restriction of ....

....so on. A concept of reduction for this hierarchy is defined, along with the definition of complete problems for each class of the hierarchy. The second hierarchy is the non uniform version of the first one. In this sense, it generalizes both the first one, and the classical non uniform hierarchy [25, 37]. A suitable concept of reduction is defined for this hierarchy. Some prototypical AI problems are complete for some classes of the non uniform hierarchy, but they do not seem to be complete for any class of the uniform hierarchy. Thus, the non uniform hierarchy is better suited for assessing the ....

[Article contains additional citation context not shown here]

R. M. Karp and R. J. Lipton. Some connections between non-uniform and uniform complexity classes. In Proceedings of the Twelfth ACM Symposium on Theory of Computing (STOC-80), pages 302--309, 1980.

.... coNP complete set, then P = NP [For79] Mahaney settled the sparseness conjecture by proving that if any NP complete set many one reduces to a sparse set then P = NP [Mah82] From an entirely different angle of research, the possible existence of sparse Turing hard sets for NP was studied in [KL80] This question is equivalent to NP complete problems having nonuniform polynomial size circuits. Karp, Lipton, and Sipser proved that if NP has sparse Turinghard sets then the polynomial hierarchy collapses to Sigma p 2 [KL80] Discovering consequences of the existence of sparse complete sets ....

....possible existence of sparse Turing hard sets for NP was studied in [KL80] This question is equivalent to NP complete problems having nonuniform polynomial size circuits. Karp, Lipton, and Sipser proved that if NP has sparse Turinghard sets then the polynomial hierarchy collapses to Sigma p 2 [KL80] Discovering consequences of the existence of sparse complete sets for different kinds of truth table reducibilities has remained an active research area. The next important advance was made recently by Ogiwara and Watanabe [OW91] when they proved, using a new left set technique, that if NP ....

[Article contains additional citation context not shown here]

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309, April 1980. 22

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Richard M. Karp and Richard J. Lipton. Some Connections Between Nonuniform and Uniform Complexity Classes. In Proceedings of the 12th Annual ACM Symposium on Theory of Computing, pages 302--309, 1980.

No context found.

R. Karp and R. Lipton, Some connections between nonuniform and uniform complexity classes, in Proc. 12th ACM Symposium on Theory of Computing (STOC'80), ACM, 302--309, 1980. (An extended version appeared as: Turing machines that take advice, in L'Enseignement Mathematique (2nd series) 28, 191--209, 1982.)

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R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proc. 12th ACM Symp. Theory of Computer Science, pages 302--309, 1980.

No context found.

R.M. Karp and R.J. Lipton, Some connections between nonuniform and uniform complexity classes, Proceedings of the 12th ACM Symposium on the Theory of Computing, 1980, pp. 302--309.

No context found.

R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. In Twelfth Annual ACM Symposium on Theory of Computing (STOC '80), pages 302--309, New York, Apr. 1979. ACM.

No context found.

R.M. Karp and R.J. Lipton. Some connections between nonuniform and uniform complexity classes. In 12th STOC, pages 302-309, 1980.

No context found.

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pp. 302--309, 1980.

No context found.

R. M. Karp and R. J. Lipton. Some connections between non-uniform and uniform complexity classes. In Proceedings of the Twelfth ACM Symposium on Theory of Computing (STOC'80), pages 302--309, 1980.

No context found.

R.M. Karp and R.J. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309, 1980.

No context found.

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proc. 12th ACM Symposium on Theory of Computing, pages 302--309, 1980.

No context found.

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302-309. ACM Press, April 1980. An extended version has also appeared as: Turing machines that take advice, L'Enseignement Mathematique, 2nd series, 28, 1982, pages 191-209.

No context found.

R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings 12th ACM Symposium on Theory of Computing, pages 302-309, 1980.

No context found.

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309. ACM Press, April 1980. An extended version has also appeared as: Turing machines that take advice, L'Enseignement Mathematique, 2nd series, 28, 1982, pages 191--209.

No context found.

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proc. 12th ACM Symposium on Theory of Computing, pages 302--309, 1980.

No context found.

R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th ACM Symposium on Theory of Computing, pages 302--309, April 1980. An extended version has also appeared as: Turing machines that take advice, L'Enseignement Math'ematique, 2nd series 28, 1982, pages 191--209.

No context found.

R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. In Proc. 12th ACM Symposium on Theory of Computing, 302-309. ACM Press, 1980.

No context found.

R. M. Karp and R. J. Lipton, Some connections between nonuniform and uniform complexity classes, Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, (1980), 302-309.

No context found.

R. Karp and R. Lipton, Some Connections Between Nonuniform and Uniform Complexity Classes, Proceedings of Symposium on Theory of Computing, ACM, 1980, pp. 302-309

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