A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55(1--3):80--88, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....similar as the proof of Theorem 1.2 apart that we only have g(H) 3 and thus only g(G 1 ) g(H) Gamma 1. 4. Discussion There is couple of natural complexity theoretic questions related to unique colourability. The problem given a graph G, is G uniquely k colourable for example is NP hard [2], the problem given a graph G and a k colouring of G, is there another k colouring of G is NP complete [6] If there exists a polynomial time algorithm which finds a 3 colouring of a uniquely 3 colourable graph then NP=RP [24] For complexity classes related to uniqueness problems and a survey ....

Blass, A., Gurevich, Y., On the unique satisfiability problem, Inform. Control 55, 1982, 80-88.

....of the information presented in [2] concerning the sets KU v [s(n) n ] is erroneous. It is stated in [2] that, for all v and k,KU v [s(n) n ] is in UP, whereas in fact all that is known is that this set is in US, where US is the class corresponding to Unique Satisfiability , as studied in [8]. 15 1. NP P contains a subset of KU v [ks(n) n ] 2. 3. Proof: 1) # (3) is immediate (with no restriction on s) and ( 2) # (1) follows from Proposition 2 (requiring only that s(n) O(log n) 3) # (2) This is where the upward separation technique is ....

A. Blass and Y. Gurevich, On the unique satisfiability problem, Information and Control 55 (1982) 80-88.

....Let B be any NP complete set. We have shown that extra queries to B allow us to compute extra functions in polynomial time, provided that P 6= NP. However, it is not known whether extra queries to B allow us to solve extra decision problems in polynomial time. For example, Blass and Gurevich [BG82], Valiant and Vazirani [VV86] and Papadimitriou and Yannakakis [PY84] have considered the Others have defined NP hardness in terms of Turing reductions. The results to follow do not apply to that kind of NP hardness. class D = fL 1 Gamma L 2 j L 1 ; L 2 2 NPg; it is not known whether P 6= ....

Andreas Blass and Yuri Gurevich. On the unique satisfiability problem. Inf. & Control, 55:80--88, 1982.

....MOD q EXP, defined similarly with respect to counting modulo q, 2 the classes GapP = f f g j f; g 2 #P g, GapPSPACE, GapE, and GapEXP. We also use the class C=P of sets of the type f x j f(x) g(x) 0 g for some f; g 2 #P, the class USP of sets of type f x j f(x) 1 g for some f 2 #P (see [5]) and DP, the class of those languages which can be written as the difference of two NP languages, or equivalently as the intersection of an NP and a co NP language (see [13] The exponential time classes C=E, USE and DE are similarly defined. Finally, recall that a function h is polynomial time ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 82:80--88, 1982.

....all these problems is also open. Another well studied set is USAT, the set with formulas that have exactely one satisfying assignment. As an upper bound, USAT is in D P , the class of sets that are the difference of two NP sets. But it is not known to be complete for D P . Blass and Gurevich [BG82] showed that USAT is complete for D P if and only if it is hard for NP. Furthermore, they constructed an oracle such that USAT is not complete for D P . Note, however, that Valiant and Vazirani [VV86] showed that USAT is NP hard under randomized many one reductions. As a lower bound, USAT is ....

Blass, A., Gurevich, Y.: On the unique satisfiability problem. Information and Control 55 (1982) 80-88

....formulas, USAT, is contained in D P . This is easily seen from the definition by letting L 1 be SAT and L 2 be the set of formulas with two or more satisfying assignments. Then, USAT = L 1 Gamma L 2 . So, the natural question to ask is: Can USAT be complete for D P Blass and Gurevich [BG82] answered this question partially. They noticed that since SAT P m USAT and since the conjunction of uniquely satisfiable formulas is also uniquely satisfiable, USAT is P m complete for D P ( SATSAT P m USAT ( SAT P m USAT: So, the question of whether USAT can be P m complete ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55(1--3):80--88, 1982.

....may lack complete sets, even with respect to Turing reductions [22, 20] Unambiguous computation is related to and motivated by cryptography; Grollmann and Selman have shown that P#=UP if and only if one way functions exist [17] A related concept is that of unique acceptance. Blass and Gurevich [7] defined a class that they called UNIQUE SOLUTION or US, which models the sets accepted by nondeterministic Turing machines that by definition accept if and only if they have exactly one accepting path. Definition 3 ( 7] US= L there is a polynomial predicate P and an integer k such that for ....

....[17] A related concept is that of unique acceptance. Blass and Gurevich [7] defined a class that they called UNIQUE SOLUTION or US, which models the sets accepted by nondeterministic Turing machines that by definition accept if and only if they have exactly one accepting path. Definition 3 ([7]) US= L there is a polynomial predicate P and an integer k such that for all x: x # L if and only if there is exactly one element in the set y P (x, y) # y # x k . Though UP machines can never have more than one accepting path, US machines can this simply causes them to reject. ....

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A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55:80--88, 1982.

....of Horn clauses of propositional logic can be determined in linear time, whereas satisfiability for sets of arbitrary clauses of propositional logic is NP hard. Unique satisfiability on Horn sets is one problem that can be solved in quadratic time [7] even though the general problem is co NP hard [5]. Using ideas based on those of [7] we can improve the latter result, finding a nearly linear time algorithm for testing unique satisfiability for sets of propositional Horn clauses. This work is based on research supported in part by the National Security Agency, Grant No. MDA904 91 H 0016. y ....

A. Blass and Y. Gurevich, "On the unique satisfiability problem," Information and Control 55 (1982), 80--88.

....a formula F = F (x 1 ; x n ) is in BA if and only if for some pair i; j 2 f1; ng; i 6= j we have that (F [i] F [j] is in BI. It follows that BA is many one reducible to BI. Corollary 4.10 BA p m BI. We have already seen that BI is coNP hard. Unique Satisfiability (USAT) [BG82] is the set of all Boolean formulas that have exactly one satisfying assignment. The function F (x 1 ; x n ) 7 (F z) V n i=1 x i z) reduces unsatisfiable formulas to uniquely satisfiable ones. Therefore USAT is coNP hard. Since USAT can be written as the difference of two NP sets, ....

A. Blass, Y. Gurevich. On the unique satisfiability problem. Information and Control 55, 80-88, 1982.

....last years. These complexity classes are defined in terms of nondeterministic machines for which the accepting mechanism is a predicate on the number of accepting paths or solutions of the machine. Examples of these classes are NP, PP (probabilistic polynomial time) 10] US (unique solutions) [8], C =P (exact counting) 29] PhiP (parity P) 16] and MOD k P (modulo classes) 6] we include later definitions of the classes treated in this survey) All these classes have complete problems, and are included in PSPACE. Although the classes are interesting in themselves, an important further ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control 55 (1982), 80--88.

....classes 1NSPACE(logn) and 1NP consist of languages defined by machines that accept their inputs if there is exactly one accepting path. Thus, the existence of two or more accepting computations is not forbidden, but simply leads to rejection. In the polynomial time case we have Co NP 1NP [3]. In the logspace case inductive counting [13, 25] shows 1NSPACE(logn) NSPACE(logn) A more restrictive form of unambiguity is Strong Unambiguity. A nondeterministic machine is said to be strongly unambiguous, if for every pair of configurations there exits at most one computational path ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Inform. and Control, 55:80--88, 1982.

....Let B be any NP complete set. We have shown that extra queries to B allow us to compute extra functions in polynomial time, provided that P 6= NP. However, it is not known whether extra queries to B allow us to solve extra decision problems in polynomial time. For example, Blass and Gurevich [BG82], Valiant and Vazirani [VV86] and Papadimitriou and Yannakakis [PY84] have considered the 1 Others have defined NP hardness in terms of Turing reductions. The results to follow do not apply to that kind of NP hardness. CHAPTER 5. POLYNOMIAL TIME BOUNDED REDUCTIONS 117 class D P = fL 1 ....

Andreas Blass and Yuri Gurevich. On the unique satisfiability problem. Inf. & Control, 55:80--88, 1982.

....an 9, and (3) has at most i Gamma 1 alternations of quantifiers, returns jf b 2 f0; 1g n : OE( b) 1gj: iii. C = PSPACE and f = #QBF, the function that, given a quantified Boolean formula OE(x 1 ; x n ) returns jf b 2 f0; 1g n : OE( b) 1gj: iv. Let C = US, defined by [18]) which is the class of all sets of the form fx j there is exactly one path such that N(x) accepts g where N is a nondeterministic polynomial time Turing machine. Let f be the function that, when given a Boolean formula OE(x 1 ; x n 1 ; y 1 ; y n 2 ) returns jf b 2 f0; 1g n ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55:80--88, 1982.

....there exist problems in UP for which there is no constructive proof. It is easy to verify (see Figure 3) that P UP NP. co UP is the class of decision problems whose complement is computable in UP, and UP co UP, the class of decision problems solvable in both UP and co UP. Blass and Gurevich [BG82] introduced the class US whose resemblance to UP is confusing. The class US consists of sets L that can be represented in the form L = fx j 9 y R(x; y)g where R is a polynomially bounded relation computable in polynomial time. US consists then of all unique version of problems in NP (Unique ....

....time. Moreover, the computation on I is described by R(I; G) where G encodes the guesses. For each I, G is unique since MP is unambiguous. 2 Moreover Unique Satisfiability (Example 2.6) which is co NP hard is complete for the class US. It follows that co NP US and that NP US implies US = DP [BG82]. Similarly, if US NP then NP = co NP = DP = P NP . Example 2.6 An example of a decision problem in US is Unique Satisfiability. ae INPUT : A propositional sentence OUTPUT : true if admits a unique model Pspace is the class of decision problems solvable by a deterministic Turing ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55:80--88, 1982.

....hierarchy (SDH) and the Boolean hierarchy (CH) remain equal to the Boolean closure (BC) even in the absence of the assumption of closure under union. That is, for any class K containing Sigma and ; and closed under intersection (e.g. UP, US, and FewP, first defined respectively in [Val76] BG82] and [All86] SDH(K) CH(K) BC(K) However, for the remaining two hierarchies, we show that not all classes containing Sigma and ; and closed under intersection robustly display equality. In particular, the Hausdorff hierarchy over UP and the nested difference hierarchy over UP both ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55:80--88, 1982.

....classes 1NSPACE(logn) and 1NP consists of languages defined by machines that accept their inputs if there is exactly one accepting path. Thus, the existence of two or more accepting computations is not forbidden, but simply leads to rejection. In the polynomial time case we have Co NP 1NP [2]. In the logspace case inductive counting [11,22] shows 1NSPACE(logn) NSPACE(logn) A more restrictive form of unambiguity is Strong Unambiguity. A nondeterministic machine is said to be strongly unambiguous, if for every pair of configurations there exits at most one computational path ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Inform. and Control, 55:80--88, 1982.

....next theorem. The complexity classes FewP and R mentioned in the result are well known and we refer to the standard literature for definitions. US is the class of languages computed by a polynomial time nondeterministic Turing machine that accepts an input if it produces exactly one accepting path [6]. Theorem 3.4 If the promise problem (1SAT; SAT) has a solution in P then FewP=P, NP=R and coNP=US. The first equality is due to [39] and the second one is due to [41] To our knowledge the third equality is new (and its proof is left as an easy exercise to the reader) The following corollary ....

A. Blass and Y. Gurevich, On the unique satisfiability problem, Information and Control 55 (1982) pp. 80--88.

....also follows from Property 8(iii) and Property 6(i) by the multinomial theorem. 3. Counting Classes Many well known complexity classes can be defined in terms of nondeterministic polynomialtime machines by appropriate interpretation of the results of all possible computation paths. NP, PP, US [6], and members of the polynomial time hierarchy [16, 23] are examples of such complexity classes. Counting classes consist of languages in which acceptance is determined by the number of accepting computations. Definition 10. For a relation R(x; ff; we define CP R(x;ff; to be the class of ....

....considered machines that are defined to accept when exactly t(x) paths accept; others have considered machines that accept when at least t(x) paths accept. Still others take t(x) to be a fraction of all paths, rather than an absolute number. Classes that arise from such considerations include US [6] and PP [9, 21] The following definition introduces notation for some of these counting classes. Definition 11. ffl CP =f(x) CP ff=f(x) ffl CP =f(x; CP ff=f(x; ffl CP f(x) CP fff(x) ffl CP f(x; CP fff(x; ffl CP= S f2PF CP =f(x) ffl CP = S f2PF CP f(x) ffl ....

A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55:80--88, 1982.

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A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55(1--3):80--88, 1982.

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A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55(1--3):80--88, 1982.

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