## Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract) (2004)

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Venue: | J. COMPL |

Citations: | 17 - 10 self |

### BibTeX

@ARTICLE{Bürgisser04countingcomplexity,

author = {Peter Bürgisser and Felipe Cucker},

title = {Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)},

journal = {J. COMPL},

year = {2004},

volume = {22},

pages = {475--485}

}

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### Abstract

We define counting #P classes #P ¡ and in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤-complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥-complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACE-hardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.

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Citation Context ...mplete. This exhibited 1 All along this paper we use the words discrete, classical or Boolean to emphasize that we are refering to the theory of complexity over a finite alphabet as exposed in, e.g., =-=[2, 46]-=-.san unexpected difficulty for the computation of a function whose definition is only slightly different to that of the determinant, a well-known “easy” problem. This difficulty was highlighted by a r... |

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Citation Context ... complexes and their topology can be studied through the combinatorics of cell complexes. We briefly recall the definition of a finite cell complex (also called finite CW-complex), see, for instance, =-=[27]-=- for more details. We denote by D n the closed unit ball in ¢ n , and by S n−1 = ∂D n its boundary, the (n − 1)-dimensional unit sphere. An n-disk is a space homemorphic to D n . By an open n-cell we ... |

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Citation Context ...tional problems, whose associated functions count the number of solutions of some decisional problem. In classical complexity theory, counting classes were introduced by Valiant in his seminal papers =-=[55, 56]-=-. Valiant defined #P as the class of functions which count the number of accepting paths of NP-machines and proved that the computation of the permanent is #P-complete. This exhibited 1 All along this... |

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Citation Context ...nition. (Note that dimF Hk(X; F ) = 0 if k is greater than the dimension of X.) Consequently, the left-hand side of (2) is independent of the field F . For a general reference to homology we refer to =-=[27, 45]-=-. The following is proved using a relative version of PoincaréAlexander-Lefschetz duality, cf. [11, Thm. 8.3, p. 351].sLemma 6.2 Let Z be a compact n-dimensional real algebraic manifold and K ⊆ Z be a... |

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Citation Context ...utes its characteristic polynomial (a computation known to be in FPR, see [8, 16], and finally one uses Sturm’s algorithm to compute the number of real zeros in the interval (−∞,0) (again in FPR, see =-=[26]-=-). Comparing this number with k decides INDEX for (u,a,x,k,J). Given (u,a), let χ+(u,a) denote the number of (x,k,J) such that (u,a,x,k,J) ∈ INDEX and k + |J| is odd. Similarly, we define χ−(u,a) by r... |

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Citation Context ...tional problems, whose associated functions count the number of solutions of some decisional problem. In classical complexity theory, counting classes were introduced by Valiant in his seminal papers =-=[55, 56]-=-. Valiant defined #P as the class of functions which count the number of accepting paths of NP-machines and proved that the computation of the permanent is #P-complete. This exhibited 1 All along this... |

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Citation Context ... which can be algebraically computed according to [21]. Although Morse theory is not explicitly mentioned in [12, 52], the main idea behind these papers is an application of this theory as exposed in =-=[43]-=-. The single exponential time algorithm in [3] for computing the Euler characteristic uses Morse theory explicitly and in a crucial way. However, we note that the reduction in [3] from the case of an ... |

382 |
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Citation Context ...≤ i < r and dn−r = d r − deg(Σ). 9sProof. For a generic Lr we may write ϕ−1 (Lr ) = ZPn(g1, . . . , gr) \ Σ, where g1, . . . , gr form a generic linear combination of f0, . . . , fn. It is well known =-=[30]-=- that (g1, . . . , gr) is a regular sequence. Let C1, . . . , Cs be the irreducible components of Z := ZPn(g1, . . . , gr). Then all Cj � have codimension r and we have deg Z = s j=1 deg Cj = dr , see... |

375 |
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Citation Context ...s counting complexity classes, completeness, Euler characteristic, geometric degree, semialgebraic sets, Betti numbers 1. INTRODUCTION The theory of computation introduced by Blum, Shub, and Smale in =-=[9]-=- allows for computations over an arbitrary ring R. Emphasis was put, however, on the cases R = ¢ or R = £ . For these two cases, a major complexity result in [9] exhibited natural NP-complete problems... |

345 |
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Citation Context ...sent problem instances of arbitrarily high dimension. For x ∈ ¢ n ¢ ⊂ ∞, we call n the size of x and we denote it by size(x). In this paper we will consider BSS-machines over ¢ as they are defined in =-=[8, 9]-=-. Roughly speaking, such a machine takes an input from ¢ ∞ , performs a number of arithmetic operations and comparisons following a finite list of instructions, and halts returning an element in ¢ ∞ (... |

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Citation Context ...how that HNZ C ∈ PSPACE, and an even more recent result of Koiran [42] shows that, assuming the generalized Riemann hypothesis, HNZ C is in the Arthur-Merlin class AM. (The class AM was introduced in =-=[1]-=- and should be interpreted as a randomized version of NP that is “close” to NP.) On the other hand, it is well-known (and rather trivial) that HNZ C is NP-hard. The complexity of problems like FeasZ R... |

297 |
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Citation Context ...d dpϕ: TpX → T ϕ(p)Y is an isomorphism (and hence ϕ(p) is smooth in Y ). We call a point q ∈ Y a regular value of ϕ if all p ∈ ϕ −1 (q) are regular points of ϕ. The following lemma adapts a result in =-=[31]-=- to our situation. Lemma A.2 Let ϕ: X → Y be a surjective morphism of irreducible complex projective varieties of the same dimension. Then all fibres of regular values of ϕ have the same finite cardin... |

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Citation Context ...in size(x). 2.2 Algebraic and semialgebraic sets We very briefly recall some definitions and facts from algebraic geometry, which will be needed later on. Standard textbooks on algebraic geometry are =-=[26, 44, 49]-=-. For information about real algebraic geometry we refer to [6, 10]. An algebraic set (or affine algebraic variety) Z is defined as the zero set Z = Z(f1, . . . , fr) := {x ∈ £ n | f1(x) = 0, . . . , ... |

227 |
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Citation Context ... numbers, rank of (sheaf) cohomology groups, Euler characteristic, etc. To our best knowledge, besides [15, 16], the only known non-trivial complexity lower bounds for some of these quantities are in =-=[1, 47]-=-. For other attempts to characterize the intrinsic complexity of problems of algebraic geometry, especially elimination, we refer to [40, 29]. 1. Counting classes The class #P is defined to be the cla... |

183 |
PP is as Hard as the Polynomial-Time Hierarchy
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Citation Context ...d difficulty for the computation of a function whose definition is only slightly different to that of the determinant, a well-known “easy” problem. This difficulty was highlighted by a result of Toda =-=[53]-=- proving that PH ⊆ P #P , i.e., that #P has at least the power of the polynomial hierarchy. In the continuous setting, i.e., over the reals, counting classes were first defined by Meer in [41]. Here a... |

173 | On the computational complexity and geometry of the first-order theory of the reals - Renegar - 1992 |

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Citation Context ...mplete. This exhibited 1 All along this paper we use the words discrete, classical or Boolean to emphasize that we are refering to the theory of complexity over a finite alphabet as exposed in, e.g., =-=[2, 46]-=-.san unexpected difficulty for the computation of a function whose definition is only slightly different to that of the determinant, a well-known “easy” problem. This difficulty was highlighted by a r... |

148 |
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Citation Context ...in size(x). 2.2 Algebraic and semialgebraic sets We very briefly recall some definitions and facts from algebraic geometry, which will be needed later on. Standard textbooks on algebraic geometry are =-=[26, 44, 49]-=-. For information about real algebraic geometry we refer to [6, 10]. An algebraic set (or affine algebraic variety) Z is defined as the zero set Z = Z(f1, . . . , fr) := {x ∈ £ n | f1(x) = 0, . . . , ... |

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Citation Context ... entangled with more than one century of attempts to prove a statement of Euler asserting that in a polyhedron, the number of vertices plus the number of faces minus the number of edges equals 2 (see =-=[39]-=- for a vivid account of this history). A precise definition of a generalization of this sum is today known with the name of Euler or Euler-Poincaré characteristic. The Euler characteristic of X, denot... |

129 | On the Betti numbers of real varieties
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Citation Context ...ds for the number of connected components of such basic semialgebraic sets (see e.g. [16, Thm. 11.1] or [8, Prop. 7, Chapt. 16]), which follow from the well-known Oleĭnik-Petrovski-Milnor-Thom bounds =-=[52, 56, 57, 67]-=-. □ We next locate the newly defined counting complexity classes within the landscape of known complexity classes. Theorem 3.7 We have FP #PR R element of R − N.) ⊆ FPARR. (To interpret this, represen... |

126 |
Some algebraic and geometric computations in pspace
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Citation Context ...the discretized versions of the above problems, where the input polynomials have integer coefficients, and study these problems in the Turing model. The following upper bound was first shown by Canny =-=[18]-=-. Theorem 8.12 The problem #CC 0 R is in FPSPACE. From a result by Reif [60, 61] on the PSPACE-hardness of a generalized movers problem in robotics, it follows easily that the problem #CC 0 R is in fa... |

124 | editors. Constraint Databases
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Citation Context ...ch that for all a ′ ∈ V the intersection Z ∩ Aa ′ ∩ U contains exactly one point. 6 This is closely related to the question of the expressive power of query languages for constraint spatial databases =-=[46]-=-. 7 Mumford considers projective varieties, but the following local considerations clearly hold in the affine setting as well. 22Proof. (a) ⇒ (b). Assume that ϕ1(x ′ ) = 0,... ,ϕn−d(x ′ ) = 0 are loc... |

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(Show Context)
Citation Context ...complexity lower bounds for some of these quantities are in [1, 47]. For other attempts to characterize the intrinsic complexity of problems of algebraic geometry, especially elimination, we refer to =-=[40, 29]-=-. 1. Counting classes The class #P is defined to be the class of functions f : {0, 1} ∞ → ¦ for which there exists a polynomial time Turing machine M and a polynomial p with the property that for all ... |

117 |
Completeness classes in algebra
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Citation Context ...n hardly studied. So far, the most systematic approach to study the complexity of certain functional problems within a framework of computations over the reals is Valiant’s theory of VNP-completeness =-=[13, 54, 57]-=-. However, the relationship of this theory to the more general BSS-setting is, as of today, poorly understood. A recent departure from the situation above is the work focusing on complexity classes re... |

89 |
Sur l’homologie des variétés algébriques réelles
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Citation Context ...ds for the number of connected components of such basic semialgebraic sets (see e.g. [16, Thm. 11.1] or [8, Prop. 7, Chapt. 16]), which follow from the well-known Oleĭnik-Petrovski-Milnor-Thom bounds =-=[52, 56, 57, 67]-=-. □ We next locate the newly defined counting complexity classes within the landscape of known complexity classes. Theorem 3.7 We have FP #PR R element of R − N.) ⊆ FPARR. (To interpret this, represen... |

87 | Singularities and Topology of Hypersurfaces. Universitext - Dimca - 1992 |

75 |
Algebraic complexity theory, volume 315 of Grundlehren Math
- Bürgisser, Clausen, et al.
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Citation Context ...variety occurs in many results in algebraic geometry, the most celebrated of them being Bézout’s Theorem. It also occurs in the algorithmics of algebraic geometry [22, 28] and in lower bounds results =-=[14, 51]-=-. The birth of algebraic topology is entangled with more than one century of attempts to prove a statement of Euler asserting that in a polyhedron, the number of vertices plus the number of faces minu... |

71 |
Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics
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Citation Context ...urns out that d0, . . . , dn−1 are the projective degrees of ϕ. The preceding discussion can also be made in the context of algebraic geometry, using divisors and Chow groups instead of homology, see =-=[16]-=-. 4 Generic quantifiers and generic reductions Our goal here is to develop a formal complexity framework for the analysis of general position arguments related to counting problems in (semi)algebraic ... |

70 |
Definability and fast quantifier elimination in algebraically closed fields
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(Show Context)
Citation Context ... of £ n . The degree of an algebraic variety occurs in many results in algebraic geometry, the most celebrated of them being Bézout’s Theorem. It also occurs in the algorithmics of algebraic geometry =-=[22, 28]-=- and in lower bounds results [14, 51]. The birth of algebraic topology is entangled with more than one century of attempts to prove a statement of Euler asserting that in a polyhedron, the number of v... |

60 |
Completeness and Reduction in Algebraic Complexity Theory
- Bürgisser
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(Show Context)
Citation Context ...n hardly studied. So far, the most systematic approach to study the complexity of certain functional problems within a framework of computations over the reals is Valiant’s theory of VNP-completeness =-=[13, 54, 57]-=-. However, the relationship of this theory to the more general BSS-setting is, as of today, poorly understood. A recent departure from the situation above is the work focusing on complexity classes re... |

51 | Counting complexity
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(Show Context)
Citation Context ...The sum, the difference, and the product of two functions C ∞ → � Z is defined pointwise. The following definition is motivated by the class GapP, which is studied in classical complexity theory, see =-=[14]-=-. 16sDefinition 4.8 The class GapC consists of all functions γ : C ∞ → � Z of the form γ = ϕ − ψ for ϕ, ψ ∈ #P C. It is easy to see that the class Gap∗ C just consists of the difference of functions i... |

50 | On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets
- Basu
- 1999
(Show Context)
Citation Context ...tified formulae. Several algorithms to compute the Euler characteristic of a semialgebraic set reduce first to the case of a smooth hypersurface and then apply the fundamental theorem of Morse theory =-=[3, 12, 52]-=-. We proceed similarly. It should be noted, however, that our reduction to the smooth hypersurface case is different from those in the references above since the latter can not be carried out within t... |

50 | La détermination des points isolés et de la dimension d'une variété algebrique peut se faire en temps polynomial - Giusti, Heintz - 1991 |

48 |
On the topology of real algebraic surfaces
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(Show Context)
Citation Context ...ds for the number of connected components of such basic semialgebraic sets (see e.g. [16, Thm. 11.1] or [8, Prop. 7, Chapt. 16]), which follow from the well-known Oleĭnik-Petrovski-Milnor-Thom bounds =-=[52, 56, 57, 67]-=-. □ We next locate the newly defined counting complexity classes within the landscape of known complexity classes. Theorem 3.7 We have FP #PR R element of R − N.) ⊆ FPARR. (To interpret this, represen... |

39 | Hilbert’s Nullstellensatz is in the Polynomial Hierarchy,” DIMACS
- Koiran
- 1996
(Show Context)
Citation Context ...omplex solutions of a system of integer polynomial equations and #HN ¡ is the problem of counting the number ¡ of these solutions. It is known that HN ¡ ∈ PSPACE, and a more recent ¡ result of Koiran =-=[34]-=- shows that, assuming the generalized Riemann hypothesis, HN ¡ ∈ ¡ RP NP . On the other hand, it is well-known (and rather trivial) that HN ¡ is NP-hard. ¡ The complexity of problems like Feas ¡ ¡ ors... |

37 | Computing roadmaps of semi-algebraic sets on a variety
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(Show Context)
Citation Context ...ng Betti numbers. State-of-the-art algorithmics for computing the Euler characteristic or the number of connected components of a semialgebraic set suggests that the former is simpler than the latter =-=[3, 4]-=-. In a recently published book [5, page 547] it is explicitly observed that the Euler characteristic of real algebraic sets (which is the alternating sum of the Betti numbers) can be currently more ef... |

34 |
Solving Systems of Polynomial Inequalities in Subexponential Time
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(Show Context)
Citation Context ...technique we want to highlight is the application of Morse theory for the computation of the Euler characteristic. The use of Morse functions as an algorithmic tool in algebraic geometry goes back to =-=[23, 24]-=- where the “critical points method” was developed to decide quantified formulae. Several algorithms to compute the Euler characteristic of a semialgebraic set reduce first to the case of a smooth hype... |

33 |
D.Yu.: Complexity of deciding Tarski algebra
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- 1988
(Show Context)
Citation Context ...technique we want to highlight is the application of Morse theory for the computation of the Euler characteristic. The use of Morse functions as an algorithmic tool in algebraic geometry goes back to =-=[23, 24]-=- where the “critical points method” was developed to decide quantified formulae. Several algorithms to compute the Euler characteristic of a semialgebraic set reduce first to the case of a smooth hype... |

31 |
Precise sequential and parallel complexity bounds for quantifier elimination over algebraically closed fields
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- 1990
(Show Context)
Citation Context ... of £ n . The degree of an algebraic variety occurs in many results in algebraic geometry, the most celebrated of them being Bézout’s Theorem. It also occurs in the algorithmics of algebraic geometry =-=[22, 28]-=- and in lower bounds results [14, 51]. The birth of algebraic topology is entangled with more than one century of attempts to prove a statement of Euler asserting that in a polyhedron, the number of v... |

29 |
Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten
- Strassen
- 1973
(Show Context)
Citation Context ...variety occurs in many results in algebraic geometry, the most celebrated of them being Bézout’s Theorem. It also occurs in the algorithmics of algebraic geometry [22, 28] and in lower bounds results =-=[14, 51]-=-. The birth of algebraic topology is entangled with more than one century of attempts to prove a statement of Euler asserting that in a polyhedron, the number of vertices plus the number of faces minu... |

29 | A weak version of the Blum, Shub & Smale model
- Koiran
- 1997
(Show Context)
Citation Context ...y classes Determining Boolean parts amounts to characterize, in terms of classical complexity classes, the power of resource bounded machines over R or C when their inputs are restricted to be binary =-=[6, 9, 10, 11, 21, 24]-=-. The formal definition is the following. Definition 6.1 Let C be a complexity class over C. (i) If C is a class of counting functions C ∞ → N ∪ {∞}, then the class of functions {0, 1} ∞ → N obtained ... |

28 |
Homology theory for locally compact spaces
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- 1960
(Show Context)
Citation Context ...i numbers are not unrelated. One has χ(X) = ∑ k∈N (−1)kbk(X). Just as with the Euler characteristic, a version of the Betti numbers satisfying an additivity property was introduced by Borel and Moore =-=[11]-=- for locally closed spaces X. These Borel-Moore Betti numbers bBM k (X) are invariant under homeomorphisms and are related to the modified Euler characteristic as follows: for locally 4closed spaces ... |

27 |
Computing over the reals with addition and order
- Koiran
- 1994
(Show Context)
Citation Context ...rete complexity classes, which are obtained from real or complex complexity classes by the operation of taking the Boolean part. A natural restriction for real or complex machines (considered e.g. in =-=[20, 33, 36]-=-) is the requirement that no constants other than 0 and 1 appear in the machine program. Complexity classes arising by considering such constant-free machines are denoted by a superscript 0 as in P 0 ... |

26 |
An algebraic formula for the degree of a C” map germ
- EISENBUD, LEVINE
- 1977
(Show Context)
Citation Context ...In the papers [12, 52], the Euler characteristic of a real algebraic variety is expressed by the index of an associated gradient vector field at zero, which can be algebraically computed according to =-=[21]-=-. Although Morse theory is not explicitly mentioned in [12, 52], the main idea behind these papers is an application of this theory as exposed in [43]. The single exponential time algorithm in [3] for... |

26 |
Algebraic Decision Trees and Euler Characteristics
- Yao
- 1992
(Show Context)
Citation Context ...ogy) plays a key role in the Riemann-Roch Theorem for non-singular projective varieties [32]. The Euler characteristic has also played a role in complexity lower bounds results. For this purpose, Yao =-=[58]-=- introduced a minor variation of the Euler characteristic. This modified Euler characteristic has a desirable additivity property and coincides with the usual Euler characteristic in many cases, e.g.,... |

24 | Randomized and deterministic algorithms for the dimension of algebraic varieties
- Koiran
- 1997
(Show Context)
Citation Context ...etry allows one to eliminate generic quantifiers in parameterized formulae. The basic idea behind this method appeared already in [31] and was used also in [7], but the method itself was developed in =-=[35, 37, 38]-=- to prove that the problem of computing the dimension of a semialgebraic (or complex algebraic) set is complete in NPs(resp. NP¡ ). We extend this method and use this in the completeness proofs of bot... |