L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of accepting paths is the same. We can produce similar characterizations of other polynomial time bounded complexity classes. An interesting example is the class UP of languages accepted by unambiguous Turing machines (those which, if they accept, have a unique accepting path, introduced in [Val1976]) We can similarly de ne unambiguous relation set expressions as being those with at most one accepting branch and, on ordered structures, these unambiguous relation set expressions de ne exactly the UP properties. The ordering is required to simulate a Turing machine; while in the case of PP, ....

L.G. Valiant, Relative Complexity of Checking and Evaluating, Information Processing Letters, vol. 5, pp. 20-23, 1976.

....This is an immediate consequence from Corollary 3.9 and the Propositions 3.3.4 and 3.10. 2 We provide a picture of the mentioned classes sinclusion structure, that is established when we take the above results into account, at the end of section 4. 4 Relations to Other Classes In 1976 Valiant [Val76] introduced the class UP of languages that are decidable in unambiguous polynomial time. This means that UP consists of all languages that can be accepted in polynomial time by a nondeterministic machine satisfying the promise that each computation has at most one accepting path. Equivalently, L 2 ....

L. G. Valiant. Relative complexity of checking and evaluation. Information Processing Letters, 5:20--23, 1976.

.... was investigated by Kolaitis [Kol90] where it was shown that LFP IMP(FO) and IMP(FO) IMP(LFP) Moreover, in [Kol90] it was established that if C is a class of ordered structures, then IMP(FO) C) UP co UP on C; where UP is the class of unambiguous NP problems introduced by Valiant [Val76]. Thus, separating LFP from IMP(FO) on ordered structures is equivalent to separating PTIME from UP co UP. In view of the above, in [Kol90] the question was raised of whether LFP is properly contained in IMP(FO) on the class of all nite structures. This question is settled in the next ....

L. Valiant. Relative complexity of checking and evaluating. Information Processing Letters, 5:20-23, 1976.

....unsolved, in particular in the polynomial time case. In order to throw light upon that problem it is meaningful to analyze intermediate concepts like unambiguity or fewness. On the side of classes of sets these concepts have lead to various complexity classes located between P and NP such as UP [Val76] FewP [All86, AR88] or R [Gil77] VPP in Gill s notion) However, the same questions about function classes kept widely unnoticed, function classes between FP and #P [Val79a, Val79b] were not often considered (as an exception: HV95] Trying to close the gap a suitable machine concept will be ....

L. G. Valiant. Relative complexity of checking and evaluation. Information Processing Letters, 5:20--23, 1976.

.... . For a class K of sets, let FP K (P K , resp. be the class of functions (sets resp. that can be computed in polynomial time with an oracle from K. Next we review the de nitions of some complexity classes of sets, already existing in the literature, that we will use in this paper. [22] UP is the class of all sets L such that c L 2 #P. 3, 14] NP is the class of all sets L for which there exists a function f 2 #P such that x 2 L , f(x) 0 for all x 2 . 20, 5] PP is the class of all sets L for which there exist functions f 2 #P and g 2 FP such that x 2 L , f(x) ....

L. G. Valiant. Relative complexity of checking and evaluation. Information Processing Letters, 5:20-23, 1976.

....is the complexity of generating proofs that a string is a member of a given language. For NPcomplete sets, it is well known that the lexicographically smallest witness of membership can be generated in FP NP , the class of functions computable in polynomial time with access to an oracle in NP [Va76]. For example, if we consider the NP complete set SAT, the following function is in FP NP . f left ( 8 : the lexicographically smallest satisfying assignment of ; if 2 SAT; if 62 SAT; where is some special symbol to denote that a function is undefined at a point. ....

L. Valiant. The Relative Complexity of Checking and Evaluating. Information Processing Letters 5:20--23, 1976. 21

....spectrum a categorical spectrum If P=NP, so that (as noted above) E=NE, we know from Theorem 5.4 that the open problem is true, that is, every spectrum is a categorical spectrum. This makes the question all the more intriguing, as to whether such a hypothesis is really needed. Following Valiant [Val76], we say that an unambiguous Turing machine is a nondeterministic Turing machine that has at most one accepting computation for each input. Valiant defined the complexity class UP to consist of those languages accepted by an unambiguous Turing machine in polynomial time [Val76] Similarly, let us ....

....Following Valiant [Val76] we say that an unambiguous Turing machine is a nondeterministic Turing machine that has at most one accepting computation for each input. Valiant defined the complexity class UP to consist of those languages accepted by an unambiguous Turing machine in polynomial time [Val76]. Similarly, let us define the complexity class UE to consist of those languages accepted by an unambiguous Turing machine in exponential time. Thus, the complexity class UE lies between E and NE. Theorem 5.4 can be strengthened to say that every set of positive integers in UE is a categorical ....

L. Valiant. Relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....definition of C=P, it follows that co NP C=P. Also, as an intermediate step in the proof of Toda s theorem it is shown that PH BP. Phi P (the class obtained by applying the BP operator to PhiP) Tod91] In general, it holds that PH BP.Mod k P [TO92] The class UP introduced by Valiant [Val76] is another important complexity class. UP consists of those languages in NP accepted by nondeterministic polynomial time machines having at most one accepting path. Valiant defined UP for studying the relative complexity of checking and evaluating. This class later found applications in the ....

L. G. Valiant. Relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976. Bibliography 112

....that accepts SAT there is a function f 2 FP such that for each ff encoding an accepting path of N on input , f(ff) is a satisfying assignment of . 3. sat is a p optimal proof system for SAT. It is known that the assumption NP = P implies NPMV t c FP which in turn implies NP co NP = P (cf. [24]) Also, in [9] it has been shown that the converse of these implications is not true in suitable relativized worlds. The consequence NP co NP = P also shows that the assumption that sat is poptimal is presumably stronger than the assumption that SAT has a p optimal proof system. Namely the ....

Leslie G. Valiant. Relative complexity of checking and evaluating. Information Processing Letters, 5(1):20--23, 1976.

....bounded by some # 1 2 for inputs in the language. It follows immediately from the definitions that P#R#BPP#NP, and Lautemann [25] and Sipser [31] showed that BPP# # P 2 # # P 2 . 2 Unambiguous computation was introduced by Valiant as a moderate form of nondeterminism. Definition 2 ([33]) 1. A nondeterministic Turing machine is unambiguous if, for every input, the machine has at most one accepting computation (accepting path) 2. UP is the class of languages accepted by polynomial time unambiguous Turing machines. Unambiguous polynomial time falls between P and NP; P#UP#NP. ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

.... of those languages recognizable by non deterministic Turing Machines that satisfy a restriction: for every string in the language, the number of accepting computations (witness strings) is very small (exactly one for UP and polynomially bounded for FewP) The class UP was introduced by Valiant [16] in 1976 and was intensively studied. It is not known whether UP is strictly larger than P, although this is believed to be the case, for related results see [18] In contrast, Yannakakis [19] proved that in communication complexity this 4 restriction is as severe as can be, namely, that P cc ....

L. G. Valiant, Relative Complexity of Checking and Evaluating, Information Processing Letters 5, (1976), 20--23.

....types of counting classes. 4 Some Rice Style Theorems for Counting Problems In this section we will state and prove the theorems which show UP hardness of all nontrivial counting problems of the three types de ned in the previous chapter. Remember that the class UP, which was de ned rst in [28], is the promise class consisting of the languages L such that there is a polynomial time nondeterministic machine M such that on every input the machine M has at most one accepting path and an input x is in L if M running on input x has an accepting path. Such machines are called unambiguous. By ....

Leslie G. Valiant, The relative complexity of checking and evaluating, Information Processing Letters 5, 1976, 20-23.

....the size. This is contrary to the situation for probabilistic Turing machines and for randomized general BPs, where the error probability may be decreased below an arbitrary small constant while maintaining polynomial size by probability amplification. 8 UP versus NP for Read Once BPs. Valiant [52] has introduced the subclass UP of NP which contains the languages decidable by nondeterministic Turing machines with at most one accepting computation for each input. Obviously, P # UP # NP, but it is not known whether any of these inclusions are proper. The results of Valiant and Vazirani ....

L. G. Valiant. Relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....one to one function. For example, we can order the elements of the (multi)set, double each character of each element (except for , which we denote as 10) and separate each element with 01. Throughout this paper, we will use log x to mean log 2 x. A language L is in UP [Val76] if and only if there exists a nondeterministic Turing machine M that accepts L, runs in polynomial time, and has for all inputs at most one accepting path. A language L is in FewP [AR88] if and only if there exists a polynomial p and a nondeterministic Turing machine M that accepts L, ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20-23, 1976. 18

....M is a nondeterministic polynomial time Turing machineg: We review the de nitions of some complexity classes that we will discuss in this paper. De nition 2.2 1. 13,32] NP is the class of sets L for which there exists a function f 2 #P such that for every x 2 , x 2 L ( f(x) 0. 2. [53] UP is the class of sets L for which there exists a function f 2 #P such that for every x 2 , i) x 2 L ( f(x) 1, and (ii) x 62 L ( f(x) 0. 4 3. 45,55] C=P is the class of sets L for which there exist functions f 2 #P and g 2 PF such that for every x 2 , x 2 L ( f(x) g(x) ....

....v.d.s. is now removed. 8 A machine M is categorical relative to A if for every input x, it holds that M A (x) has at most one accepting path; thus, a set is in UP A if and only if it is accepted by some polynomial time nondeterministic Turing machine that is categorical relative to A (see [53]) 22 Subcase IIb [There exists some y 2 D that is not troublesome. Pick one such y and put 1#x#y into A . has been satis ed, though the v.d.s. remains active. End of Construction of A From the above construction, we have L(A) 62 UP A and U(A) 2 NP A . Hence the proof is ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20-23, 1976.

....USATQ (x) 8 : SAT(x) if the number of satisfying assignments of x is 0 or 1 Q(x) otherwise. De nition 2.4 a) Val79] A function f is in #P i there is some NPTM, N , such that for each x it holds that N(x) i.e. the computation of N on input x) has exactly f(x) accepting paths. 2 b) Val76] A set B is in UP i there is a #P function f satisfying (8x) f(x) 1] such that B = fx j f(x) 0g. c) AR88] A set B is in FewP i there is a #P function f satisfying (9 polynomial q) 8x) f(x) q(jxj) such that B = fx j f(x) 0g. d) CH90] A set B is in Few i there is a polynomial time ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20-23, 1976.

....functions are also possible. There are many restrictions of polynomial time NTM transducers that generate such classes [Sel91] one such restriction is F g , the subset of functions f in class F such that graph(f) fhx; yijf(x) 7 yg 2 P i.e. outputs can be checked in polynomial time [Sel91, Val76] Define FMPH = S k=1 F Sigma p k and FM g PH = S k=1 (F Sigma p k ) g . Classes F Sigma p 1 = FNP , F Sigma p 1 ) g = FNP g , and F Sigma p k are called NPMV, NPMV g , and NPMV Sigma p k Gamma1 in [FHOS92, Sel91] and NTM in FNP g which compute total functions are ....

....to NPMV g . ffl (NPMV ffi FP NP ) g is the set of all functions f 2 NPMV ffi FP NP such that graph(f) 2 P . The NPMV composition class notation is adapted from that in [FHOS92] Valiant noticed that all solution problems associated with NP decision problems are in NPMV g ( Sel91, p. 4] Val76] Class NPMV g ffi FP NP is useful because it corresponds to those solution problems whose associated decision and evaluation problems are in NP and OptP, respectively. The following class relations are known: Lemma 17 ( Sel91] Proposition 7) If f 2 FP NP [O(log n) and graph(f) 2 P then f ....

Valiant, L. G. The Relative Complexity of Checking and Evaluating. Information Processing Letters, 5, 20--23, 1976.

....and NSF grant CCR 8957604. string is a member of a given language. 1 For NP complete sets, it is well known that the lexicographically least, or leftmost, witness of membership can be generated in FP NP , the class of functions computable in polynomial time with access to an oracle in NP [Va76]. For example, if we consider the NP complete set SAT, the following function is in FP NP . f left ( 8 : the leftmost satisfying assignment of ; if 2 SAT; if 62 SAT; where is some special symbol to denote that f(x) is undefined. Krentel showed that every function in FP ....

L. Valiant. The Relative Complexity of Checking and Evaluating. Information Processing Letters, 5, pages 20--23, 1976.

....of one to one one way functions (see the excellent survey by Selman [Sel92] the history is much clearer. The analogous theorem for those is the following. Theorem 3. 2 [GS88,Ko85,Ber77] One to one one way functions exist if and only if P 6= UP, where UP is Valiant s unambiguous polynomial time [Val76]. This theorem was found independently by Grollmann and Selman [GS88] and Ko [Ko85] and Berman s thesis [Ber77] independently obtained essentially the same result (see [Sel92] To avoid possible confusion, we mention that though our De nition 2.1 (and this entire article) does not require ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20-23, 1976.

.... however, a large field of interesting classes that do not seem to be characterizable in terms of locally definable acceptance types or general predicates of computation trees, though they can intuitively be described by simple local conditions, among them the class unambiguous polynomial time UP [28]. Locally definable acceptance types were partially motivated by the wish to characterize as many complexity classes as possible in a uniform way. Demanding an outcome of 0 or 1 enables us to characterize more classes by local conditions, in particular unambiguous classes. Borchert uses the ....

....e.g. unambiguous existential and universal quantification and unambiguous hierarchies. 4.1 Characterizations of UP k and MODZ k P For a TM M we denote by Accept(M;w) the number of accepting paths of M on input w. The class UP was defined by Valiant via unambiguous polynomial time bounded TM s [28]. Formally, a language L is a member of UP, if there exists a polynomial time bounded NTM M such that Accept(M;w) 8 : 1 if w 2 L; 0 if w = 2 L: Unambiguity plays an important role in complexity theory (cf. 5,21,24] for some recent results) and, in particular, in cryptography where it is ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....types of counting classes. 4 Some Rice Style Theorems for Counting Problems In this section we will state and prove the theorems which show UP hardness of all nontrivial counting problems of the three types defined in the previous chapter. Remember that the class UP, which was defined first in [19], is the promise class consisting of the languages L such that there is a polynomial time nondeterministic machine M such that on every input the machine M has at most one accepting path and an input x is in L if M running on input x has an accepting path. Such machines are called unambiguous. By ....

L. G. Valiant. The relative complexity of checking and evaluating, Information Processing Letters 5, 1976, pp. 20--23

....problem. In particular, if the minimal order is not implicitly definable in least fixpoint logic LFP on R, then UP coUP 6= NP and, hence, PTIME 6= NP, where UP is the subclass of NP consisting of problems that are computable by unambiguous nondeterministic Turing machines in polynomial time [Val76]. On the other hand, if the minimal order turns out to be implicitly definable in least fixpoint logic LFP on R, then the graph automorphism problem is in the class UP and, hence, it is also in the class NP coNP. 2 Preliminaries This section contains the definitions of the basic notions and a ....

....Turing machine such that there is at most one accepting computation for each input string. A query Q is in the class UP if it is computable by an unambiguous polynomial time Turing machine. Furthermore, Q is in coUP if its complement :Q is in UP. These concepts were introduced by Valiant [Val76] and turned out to be of interest and use in cryptography (cf. GS88] It is clear from the definitions that the following inclusions hold: UP NP and PTIME UP coUP NP coNP: The relationship between implicit definability in first order logic and unambiguous computation was investigated ....

L. Valiant. Relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....between the notions of optimal proof systems and complete sets. In Section 4 we show that if p optimal proof systems exist, then the class UP (unambiguous NP) of problems in NP that can be accepted by nondeterministic polynomial time machines with at most one accepting path for every input [18], has complete problems under the logarithmic space many one reductions. The existence of complete problems for UP has been studied in [9] where the authors show the existence of a relativization under which this is not possible. Considering that p optimal proof systems exist, we also show the ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5, pp. 20--23, 1976.

....and Min Delta TSP. Successively, we apply the notion of APX completeness to the study of the relative complexity of evaluating an ffl approximate value and computing an ffl approximate solution for any ffl. The relative complexity of checking and evaluating a function was first considered in [14]. It is well known, for example, that checking whether an array is already sorted is simpler than sorting it. Valiant proved that, indeed, the two questions are equivalent if and only if P= NP. In [13] and, successively, in [5] the relative complexity of evaluating the optimum cost and ....

Valiant L.: Relative complexity of checking and evaluating. Information Processing Letters 5 (1976) 20--23

.... formalized by machines with an appropriate (so called chain respecting ) acceptance type [Wec85, GW86] Finally, a rich spectrum of complexity classes is based on counting the accepting paths of NPMs [Val79, Hem87, GW86, GW87, Wag86, Tor88, Tod91, FFK94] Unambiguous polynomial time [Val76], denoted by UP, is defined via NPMs that on no input have more than one accepting computation path. FewP [All86] is the class of sets that are accepted by NPMs that on no input have more than polynomially many accepting computations. Clearly, P UP FewP NP. Classes such as UP and FewP are ....

....thesis. In particular, we study to what extent, if any, results for the thoroughly investigated non promise class NP carry over to the promise classes UP and FewP. The study of UP is crucial in both cryptography and structural complexity theory. There has been a long line of research regarding UP [Val76, Rac82, GS88, HH88, HH91, Wat88, Wat91]. To pinpoint some of the most important results about UP, we mention the following. Grollmann and Selman [GS88] have shown that one way functions exist if and only if P 6= UP. Informally speaking, a one way function is one that is easy to compute but hard to invert. It is not known whether UP ....

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....and our definition are compatible as long as our computability admits subtraction. We also note that we let censusL map strings 1 n (as opposed to numbers n in binary notation) to j L =n j to emphasize that the input to the transducer computing censusL is given in unary. accepting path. UP [Val76] (respectively, UE) is the class of all languages accepted by some unambiguous Turing machine running in polynomial time (respectively, in time 2 cn for some constant c) For any nondeterministic Turing machine M and any input x 2 Sigma , let acc M (x) denote the number of accepting paths ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....between deterministic and nondeterministic complexity classes. One approach to solve this question in the case of polynomial time was to consider machines with a limited number of accepting computations. This led to the definition of the classes UP (unambiguous polynomial time) and FewP [Val76, All86] defined by NP machines with a number of accepting computations bounded by one and a polynomial, respectively. Unambiguous complexity classes play a crucial role in cryptography and for the existence of one way functions [GS88] A more general approach to find concepts between determinism ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....becomes obvious if one considers the computational power comprised in these classes. In Section 3 we prove that all standard problems of linear algebra over the finite ring Z kZ are complete for MOD k L. The notion of unambiguity is of great importance for nondeterministic computations. Valiant [20] used the term unambiguous computation for the definition of UP, the class of languages recognizable by polynomial time nondeterministic machines which never accept on more than one path. Today this type of machines is usually called unique. The power of space bounded unique computations was ....

Leslie G. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....to some set in PhiP (which, due to P PhiP = PhiP PhiP = PhiP [PZ83] is equivalent to SAT 2 PhiP, or NP PhiP, respectively) Valiant and Vazirani raised and settled the 1 The witnessing GapP function promises never to take on values other than 0 or 1. SPP is the gap analog of UP [Val76], so the same applies to the definition of UP, where the promise is with respect to the witnessing #P function. 94 reduction question in the context of randomized reductions by showing that each NP set is polynomial time randomized many one reducible to a set in PhiP [VV86] 2 and in a ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters 5, 1976, 20--23.

.... in these characterizations (and to our knowledge in all other similar results given up to now) the obtained class of functions is closed under substitution (simply since substitution is one of the defining operators) Therefore, it was of course not possible to characterize the classes #P [22] and Gap P [8] since these are most likely not closed under substitution. Thus, no similar characterization of #P was known up to now. Using the method of arithmetization of boolean formulae well known from Shamir s famous result [18, 1] we show that the functions from #P are exactly those ....

....summation and modified subtraction. As an immediate consequence of the just given characterization, we see that #P is closed under modified subtraction if and only if the hierarchy of counting functions collapses to # P, which in turn is equivalent to a collapse of the counting hierarchy to UP [22]. An analogous consequence concerning division of #P functions is given. To prove such a result for an operation, say ffi, it is because of the above described characterization only necessary to show how modified subtraction can be simulated using the operation ffi and other operations which ....

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L. G. Valiant, Relative complexity of checking and evaluation, Information Processing Letters 5 (1976) 20--23.

....A is in # Delta C 6. A 2 X Delta C if the characteristic function of A can be written as the difference of two # Delta C functions. It is well known that the above operators capture, as special cases, the standard complexity classes NP, PP [Sim75] C P [Sim75,Wag86] Phi P [PZ83,GP86] UP [Val76] and SPP [OH93,FFK94] respectively, via 9 Delta P = NP, C Delta P = PP, C Delta P = C P, Phi Delta P = Phi P, U Delta P = UP, and X Delta P = SPP. A careful look at the definitions and a little thought (or a look at [Vol94a] reveal that for classes C and C 0 that fulfill some ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....nonuniform computation. Unambiguous computation has been the focus of much attention over the past three decades. Unambiguous context free languages form one of the most important subclasses of the class of context free languages. The complexity class UP was first defined and studied by Valiant [Val76] a necessary precondition for the existence of one way functions is for P to be properly contained in UP Supported in part by the DFG Project La 618 3 1 KOMET. y Supported in part by NSF grant CCR 9509603. This work was performed while this author was a visiting scholar at the ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....if and only if the difference between the number of accepting and non accepting computations of M is 1. From this fact follows immediately that the problem is in the classes PhiP and C= P. Observe also that the accepting behaviour of machine M is similar to one for a language in UP (unambiguos NP [40]) where a machine accepting the language has either 0 or 1 accepting paths. The difference is that in the case of UP we count the number of accepting paths (a #P function) and in our case we count the difference between the number of accepting and rejecting paths (a Gap P function) Graph ....

L.G. Valiant, The relative complexity of checking and evaluating, Information Processing Letters 5 (1976), 20--23.

....the most meaningful restrictions of nondeterminism to study. For example: ffl Unambiguous context free languages form one of the most important subclasses of the class of context free languages. ffl The complexity class UP (unambiguous polynomial time) was first defined and studied by Valiant [Val76] a necessary precondition for the existence of one way functions is for P to be properly contained in UP [GS88] Although UP is one of the most intensely studied subclasses of NP, it is neither known nor widely believed that UP contains any sets that are hard for NP under any interesting notion ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

.... however, a large field of interesting classes that do not seem to be characterizable in terms of locally definable acceptance types or general predicates of computation trees, though they can intuitively be described by simple local conditions, among them the class unambiguous polynomial time UP [33]. Locally definable acceptance types were partially motivated by the wish to characterize as many complexity classes as possible in a uniform way. Demanding an outcome of 0 or 1 enables us to characterize more classes by local conditions, in particular unambiguous classes. Borchert uses the ....

....locally definable. Note that for both these classes it does not seem to be possible to get a characterization in terms of (F )P . For a TM M we denote by Accept(M; w) the number of accepting paths of M on input w. The class UP was defined by Valiant via unambiguous polynomial time bounded TM s [33]. Formally, a language L is a member of UP , if there exists a polynomial time bounded NTM M such that Accept(M; w) 1 if w 2 L; 0 if w = 2 L: Unambiguity plays an important role in complexity theory (cf. 8, 25, 28] for some recent results) and, in particular, in cryptography where it is ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....would mean that every partial multivalued function in NPMV can be computed efficiently by some deterministic polynomial time transducer. The following proposition is known: Proposition 1 1. Sel90] NPMV c PF if and only if P = NP. 2. SXB83] NPSV PF if and only if P = NP. Following Valiant [Val76] given a class of partial functions F , let F g denote the class of all functions f 2 F such that graph(f) 2 P. 1 Valiant noticed that ordinary search problems associated with NP decision problems are functions in NPMV g : That is, let R(x; y) be an arbitrary relation in P, and let p be a ....

L. Valiant. Relative complexity of checking and evaluating. Information Processing Letters, 5(1):20--23, May 1976.

.... P NP jj , or equivalently, using R notation, R p dtt (DP) P NP jj ; we claim that this is implicit and immediate from the fact that a certain set known as PARITY SAT that Buss and Hay [BH91] proved P NP jj complete is clearly in R p dtt (DP) Unambiguous Polynomial Time (UP) UP [Val76] is the class of those NP sets that are accepted via some NP machine that, on each input, has at most one accepting path. Clearly, P UP NP. Like R and C = P, UP is not known to be closed under Turing reductions (or even under complementation) Unlike C = P, however, UP (though clearly closed ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

....: Sigma N fi fi (9k 2 N) f 2 # k Delta C]g. 2. HV95] For each class C, # few Delta C = ff : Sigma N fi fi (9 polynomial s) f 2 # s Delta C]g. Definition 2.3 1. Val79] #P = f : Sigma N (9 NPTM N) 8x 2 Sigma ) N(x) has exactly f(x) accepting paths] 2. Val76] UP = fL fi fi (9f 2 # 1 Delta P) 8x 2 Sigma ) x 2 L ( f(x) 0]g. 3. Bei89] see also [Wat88] For each k 2 N Gamma f0g, UPk = fL fi fi (9f 2 # k Delta P) 8x 2 Sigma ) x 2 L ( f(x) 0]g. 4. HZ93] UP O(1) fL fi fi (9f 2 # const Delta P) 8x 2 Sigma ) x 2 L ( ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

.... be efficiently (i.e. in polynomial time) computed by sequentially checking function values, or in other words by adaptively testing yes no hypotheses about what values the function takes on on various inputs (e.g. questions of the form is the value of f on input a equal to b , which Valiant [Val76] has dubbed checking the value of a function) It has been Email: lane cs.rochester.edu. Supported in part by grants NSF CCR 9322513 and NSF INT9513368 DAAD 315 PRO fo ab. Work done in part while visiting Friedrich Schiller Universitat Jena. y Email: fhempel,vogelg informatik.uni jena.de. ....

....DAAD 315 PRO fo ab. Work done in part while visiting Le Moyne College. known for decades that even among polynomially length bounded functions whose graph is exponential time computable not all functions are polynomial time Turing reducible to their own graphs ( Tra75, p. 129] see also [Dek70,Val76]) This explains why, though mathematicians consider graphs of functions to be a central tool, computer scientists tend to represent functions as languages via different means, such as looking at left cuts of functions or encoding separately each bit of a function s output on each input. ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20--23, 1976.

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5, pp. 20--23, 1976. 13

....polynomial p such that the following hold for each x # # : a) g(x) f( x ) and (b) # y y = p( x ) #x, y# L = g(x) 3. HR00] For each class let # const (#k) g # k C] 4. HV95] For each class let # few (# polynomial q) g # q C] 5. Val76] UP = # 1 g(x) 0] 6. AR88] FewP = g(x) 0] 7. CH90] Few = P P[1] i.e. the class of languages accepted by P machines that on each input are allowed at most one query to a function from # few P. 8. OH93,FFK94] SPP is the class of all languages such that ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20--23, 1976.

....between the notions of optimal proof systems and complete sets. In Section 4 we show that if p optimal proof systems exist, then the class UP (unambiguous NP) of problems in NP that can be accepted by nondeterministic polynomial time machines with at most one accepting path for every input [17], has complete problems under the logarithmic space many one reductions. The existence of complete problems for UP has been studied in [9] where the authors show the existence of a relativization under which this is not possible. Considering that p optimal proof systems exist, we also show the ....

L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5, pp. 20--23, 1976.

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20-23, 1976.

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20-23, 1976.

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20-23, 1976.

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20--23, 1976.

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L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20--23, 1976.

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L. Valiant. Relative complexity of checking and evaluating. Information Processing Letters, 5(1):20-23, 1976.

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