L.G. Valiant. Completeness classes in algebra. ACM Symp. on the Theory of Comput. 249-261 (1979).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on V is irreducible. The above descriptions of the isotropy subgroups of det n and perm n are irreducible representations, so these polynomials are stable under the action by SL n 2 (F ) 3 Application to Complexity Theory The jumping o point for the Mulmuley Sohoni method is Valiant s method [Val79] (see also [vzG87] of reducing any polynomial size family of arithmetical circuits to a polynomial size family of determinant computations. This extends to saying that functions believed to be intractable, such as the permanent polynomials, have polynomial size (arithmetical) circuits i they ....

L. Valiant. Completeness classes in algebra. Technical Report CSR-40-79, Dept. of Computer Science, University of Edinburgh, April 1979.

....approaches could be successfully applied. 1 Introduction The computation of the permanent of a matrix is a challenging task. The problem is computationally very hard, even for (0; 1) matrices. In fact, Valiant proved that computing the permanent of a (0; 1) matrix is #P complete (see [V179] and [V279]) The class #P contains those functions that can be computed in polynomial time by a counting (nondeterministic) Turing machine, and the #P complete problems represent the hardest problems within the class. The existence of a polynomial time algorithm for a #P complete problem would not only ....

L.G. Valiant. Completeness classes in algebra. ACM Symp. on the Theory of Comput. 249-261 (1979).

....which allows us to compute permanents of very sparse circulants of size up to 200. 1 Introduction The computation of the permanent of a matrix seems to be a very hard task, even for (0; 1) matrices. Valiant proved that computing the permanent of a (0; 1) matrix is #P complete (see [V179] and [V279]) The class #P contains those functions that can be computed in polynomial time by a counting (nondeterministic) Turing machine, and the #P complete problems represent the hardest problems within the class. More recently, several authors have found even stronger negative results [DLMV88, FL92] ....

L.G. Valiant. Completeness classes in algebra. ACM Symp. on the Theory of Comput. 249-261 (1979).

....dealt with by means of radically di erent approaches. 1 Introduction The computation of the permanent seems to be a very hard task, even for sparse (0; 1) matrices. A number of results show that it is extremely unlikely that there is a polynomial time algorithm for computing the permanent (see [18, 19], and also [3, 4] The best known algorithm is due to Ryser [16] and takes O(n 2 ) operations, where n is the matrix size. It is a striking fact that the computation of the permanent essentially keeps its general diculty when restricted to (0; 1) matrices with only three nonzero entries per ....

Valiant, L.G. (1979) Completeness classes in algebra. 10th ACM Symp. on the Theory of Comput. 249-261. 23

....between matrix multiplication and determinant computation [52, 53, 7] See also the link with matrix powering and the complexity class GapL following Toda, Vinay, Damm and Valiant as explained in [3] for example. We may also mention Valiant s theorem that the determinant is universal for formulas [54]. For integer matrices, computing the sign of the determinant is a priori an easier problem than computing its value. We will try to identify the di erences between these two problems even if it is not known whether the two complexities are asymptotically di erent in the worst case. Numerical ....

L.G. Valiant. Completeness classes in algebra. In Proc. 11th Annual ACM Symp. Theory Comput., pages 249261. ACM Press, 1979.

....3 (S n n ) n 2 ) 2 Although this model seems a bit obscure, it is interesting to note that similar models are already known. Our model can be viewed as SYM( 1 ; m ) the similar models DET( 1;1 ; m;m ) and PERM( 1;1 ; m;m ) have already been studied. In [22] Valiant shows that the formula size of a polynomial f is at least (up to a small constant) the minimal rank of a matrix A, whose entries are linear forms, such that det(A) f . In [22, 23, 24] Valiant studied p computable families of polynomials. In this model Valiant shows that the permanent is ....

....) the similar models DET( 1;1 ; m;m ) and PERM( 1;1 ; m;m ) have already been studied. In [22] Valiant shows that the formula size of a polynomial f is at least (up to a small constant) the minimal rank of a matrix A, whose entries are linear forms, such that det(A) f . In [22, 23, 24] Valiant studied p computable families of polynomials. In this model Valiant shows that the permanent is p complete (for p definable polynomials under p projections) i.e every p definable polynomial can be represented as the permanent of a matrix with linear functions as entries, such that the ....

L. G. Valiant. Completeness classes in algebra. In STOC, pages 249--261, 1979.

....Step 2 (Subsection 3.6) which reduces the last component by a constant factor while keeping the other components smaller than 2 b . Such an extension would provide, at least in principle, an extremely general solution to robustness in geometric computation since, by a result of Valiant [Val79], any algebraic expression of size e can be constructively written as an (e 2) Theta (e 2) determinant whose entries are either variables or constants. Acknowledgments Jean Pierre Merlet is acknowledged for supplying to us his interactive drawing preparation system J p draw . 21 ....

L. Valiant. Completeness classes in algebra. In Proc. 11th Annu. ACM Sympos. Theory Comput., pages 249261, 1979. 23

....between matrix multiplication and determinant computation [52,53,7] See also the link with matrix powering and the complexity class GapL following Toda, Vinay, Damm and Valiant as explained in [3] for example. We may also mention Valiant s theorem that the determinant is universal for formulas [54]. For integer matrices, computing the sign of the determinant is a priori an easier problem than computing its value. We will try to identify the di#erences between these two problems even if it is not known whether the two complexities are asymptotically di#erent in the worst case. Numerical ....

L.G. Valiant. Completeness classes in algebra. In Proc. 11th Annual ACM Symp. Theory Comput., pages 249--261. ACM Press, 1979.

....pathsK Input: A directed acyclic graph G with n vertices, and two vertices 1 i; j n: Output: wpath(G; i; j) Theorem 3.2. weighted pathsK is complete in m DETK : Proof. First we show that weighted pathsK is in m DETK by reducing it to determinantK . The idea of the proof goes back to Valiant[21]. Given a directed acyclic graph G with two distinguished vertices i and j first we construct a directed acyclic graph H as follows: We replace every edge e = u; v) by two consecutive edges (u; u e ) and (u e ; v) where u e is a new vertex. The weight of (u; u e ) is the weight of (u; v) and the ....

L.G. Valiant. Completeness classes in algebra. In Proc. Eleventh Ann. ACM Symp. Theor. Comput., 1979, 249--261.

....the permanent cannot be computed with a polynomial number of arithmetic operations. This hypothesis is supported by Valiant s famous result [23] stating that the problem to evaluate the permanent of a matrix with entries in f0; 1g is #Pcomplete, as well as his analogous VNP completeness result [22] in a framework of algebraic computations. Both permanents and determinants are special cases of immanants, introduced by Littlewood [16] To de ne these polynomial matrix functions, we have to rely on some basic facts about the characters of the symmetric groups, which can be found for instance ....

....by indeterminates and constants. A p family (f n ) is called a p projection of a family (g m ) i there is a p bounded function t: N N such that fn is a projection of g t(n) for all n. Finally, a p de nable family (g m ) is called VNP complete i any (f n ) 2 VNP is a p projection of (g m ) In [22] Valiant proved that the p families PER of permanents and HC of Hamilton cycles polynomials are VNP complete (over elds k of characteristic di erent from two, which is a general assumption in this paper) Thus PER is not p computable i Valiant s hypothesis VP 6= VNP is true. One can prove that ....

L. Valiant, Completeness classes in algebra, in Proc. 11th ACM STOC, 1979, pp. 249-261.

....REPRESENTATIONS 3 In a subsequent paper [4] we will complement this upper bound by intractability results. Strictly speaking, we will prove the completeness of certain families of immanants corresponding to hook or rectangular diagrams within the framework of Valiant s algebraic P NP theory [20, 21, 5]. This means that these families of immanants cannot be evaluated by a polynomial number of arithmetic operations, unless this is possible for the family of permanents. 2. Preliminaries on representations of GLm . We collect rst some facts about the representations of the complex general linear ....

....k m=2. For permanents, our Theorem 7. 2 yields the bound O(m 1:5 4 m log m) which is not too far away from the best known upper bound O(m2 m ) due to Ryser [19] In a subsequent paper [4] we will complement this upper bound by completeness results within Valiant s algebraic P NP theory [20, 21]. For a comprehensive account of this theory see [5] In fact, we will prove the completeness of certain families of immanants corresponding to hook or rectangular diagrams. We close by showing that the characters of GLm can be evaluated very rapidly. Let 2 3 denote the exponent of matrix ....

L. Valiant, Completeness classes in algebra, in Proc. 11th ACM STOC, 1979, pp. 249-261.

....circuits computing an explicit function, the determinant function, over an arbitrary finite field. In this paper, we interpret the arithmetic circuits in the polynomial algebra over the given field. The determinant function is especially interesting because of its algebraic universality property ([V79]) over arbitrary fields. We refer a general reader to [L84] and [H77] for all the needed notions used in our proof. We denote by F = F q a finite field with q elements. We shall study fields for q 3 (for q = 2, the boolean case, the lower bound could be derived from [R87] and [V79] We study ....

....property ( V79] over arbitrary fields. We refer a general reader to [L84] and [H77] for all the needed notions used in our proof. We denote by F = F q a finite field with q elements. We shall study fields for q 3 (for q = 2, the boolean case, the lower bound could be derived from [R87] and [V79]) We study the representation of Det = Det n = P oe ( Gamma1) sgn(oe) X 1;oe(1) Delta Delta Delta X n;oe(n) in the polynomial algebra F [X 1;1 ; X n;n ] in the form of a depth 3 arithmetic circuit, or equivalently, an expansion: Det = X 1 N Y m L ;m (1) where each L ;m = P ....

L. Valiant, Completeness Classes in Algebra, Proc. 11th ACM STOC (1979), pp. 259--261.

....and Bell Labs. DIMACS is an NSF Science and Technology Center, funded under contract STC 91 19999; and also receives support from the New Jersey Commission on Science and Technology. ABSTRACT Continuing a line of investigation that has studied the function classes #P [Val79b] #SAC 1 [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94] and #NC 1 [CMTV96] we study the class of functions #AC 0 . One way to define #AC 0 is as the class of functions computed by constant depth polynomial size arithmetic circuits of unbounded fan in addition and multiplication gates. In contrast to the preceding ....

L. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on Theory of Computing (STOC), pages 249--261, 1979.

....the other hand showing that two apparently different classes are in fact the same is a stronger result in the monotone world than in general, this justifies the checking of simulations in section 2.4. For nonuniform reductions we use polynomially bounded Boolean projections, from Skyum and Valiant [44, 39]. Given two Boolean functions f and g on n and p Boolean inputs respectively, we say f is a projection of g if there is some way to substitute the inputs of f (along with the constants 0 and 1) as positive or negative inputs to g, so that g then computes f . More precisely, a projection is a map ....

L. Valiant. Completeness classes in algebra. In Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing, pages 249--261, 1979.

....Mahajan (personal communication) pointed out that this actually requires some proof. More information is available in [A97] 7 #NC 1 #NC 1 coincides with the class of functions that have arithmetic formulae of polynomial size. As such, it has been studied as a complexity class at least since [V79a]. Evaluating arithmetic formulae over N (Z) is complete for #NC 1 (GapNC 1 , respectively) BCGR92] Probably the most important and fascinating open question regarding #NC 1 is the question of whether or not it is identical to the class of functions having Boolean NC 1 circuits. We have ....

L. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on Theory of Computing (STOC), pages 249--261, 1979.

....algebraic settings this goal has not been achieved yet. For instance, it is not known whether the resultant of two sparse univariate polynomials can be computed by straight line programs of polynomial length (see [11] for a motivation) the problem VP = VNP in Valiant s model of computation [12, 13] is still open; and the same is true of the P = NP problem in the most interesting versions of the Blum Shub Smale model. It is not always clear whether these algebraic questions are easier than the well known open questions from discrete complexity theory. Indeed, it was shown in [3] that ....

L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on Theory of Computing, pages 249-261, 1979.

.... in [Va92, Theorem 2] Also, this class of functions is precisely the class computed by polynomial size skew arithmetic circuits over the integers [To92] For additional related results, see [AO94] The question of the relationship between #L and #LOGCFL is thus exactly the question asked in [Va79], concerning the relationship between the determinant and circuits of polynomial size and degree. It is worth mentioning that Immerman and Landau [IL95] have conjectured that TC 1 is exactly the class of sets reducible to #SAC 1 ; in fact they make the stronger conjecture that computing the ....

L. Valiant, Completeness classes in algebra, Proc. 11th STOC (1979) 249--261.

....so must E of course) The results of Table 2 fall in three groups according to the proof technique used. The random polynomial time upper bounds use a result due to Schwartz [17] The undecidability result for Z uses a combination of Valiant s result that the determinant is universal [20] and Matiyasevich s proof that Hilbert s Tenth Problem is unsolvable [14] All the remaining problems of the result table (those that are not marked either RP or undecidable) are equivalent (under polynomial time transformations) to deciding the existential first order theory over the field S. The ....

....is bounded by 1=2 r Gamma1 . 6 Universality of the determinant In this section, we prove a result that underlies all our lower bounds for the singularity and minrank problems: that any multivariate polynomial is the determinant of a fairly small matrix. The result was first proven by Valiant [20], but since we need a slightly modified construction and the result is fundamental to our lower bound proofs, we make this paper self contained and give the details of the construction. To state the result, we need a few definitions. Let an arithmetic formula F be a wellformed formula using ....

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L. G. Valiant. Completeness classes in algebra. In Proc. Eleventh Ann. ACM Symp. Theor. Comput., pp. 249--261, 1979.

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Leslie G. Valiant. Completeness classes in algebra. In Proc. 11th Int. ACM Symp. on Theory of Computing (STOC), pages 249--261, 1979.

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L.G. Valiant. Completeness classes in algebra. ACM Symp. on the Theory of Comput. 249-261 (1979).

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L. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on Theory of Computing (STOC), pages 249--261, 1979.

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L.G. Valiant. Completeness classes in algebra. ACM Symp. on the Theory of Comput. 249-261 (1979).

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L. G. Valiant, `Completeness classes in algebra', Proc 11th ACM Symp. on the Theory of Comput., 1979, 249--261. 11

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L. Valiant. Completeness classes in algebra. In 11th ACM Symposium on Theory of Computing, pages 249-261, 1979. 13

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Valiant, L.G., Completeness classes in algebra. Conf. Rec. of 11 th Ann. ACM Stoc. (1979) pp. 249--261. ACM, N.Y.

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