Leslie G. Valiant. Reducibility by algebraic projections. L'Enseignement Mathematique, 28:253--268, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....derived a switchinglemma using simple restrictions that limit the space of truth assignments to a subcube where certain variables are set to 0 or to 1. While this fails with 2n pigeons, a more general class of restrictions may suffice. Possible generalizations include the projections suggested in [22], which also allow identification of variables, or restrictions given by linear equations. Two important results ( 12] and [6] for bounded depth Frege systems already employ such generalized switching lemmas in cases where direct restrictions fail (although the latter use is implicit) ....

Leslie G. Valiant. Reducibility by algebraic projections. L'Enseignement Mathematique, XXVIII:253--268, 1982. 10

....Hilbert function HK and the related information we want to compute to some rank computations involving the matrices J( 1; 0) in a suitable eld. Now, we are going to prove that these rank computations can be performed in polynomial time in the input size. Unfortunately, as showed in [18], the arithmetic complexity of computing multiple partial derivatives is likely to be exponential in the order of derivation i. If the equations de ning system (1) are represented by a slp of length L, Theorem 4 shows that the computation of the matrix J( 1; 0) requires at least (5n) i L ....

Valiant, L. G. Reducibility by algebraic projections. L'enseig. Math. IIe Series 28, 3-4 (1982), 253-268.

....) 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. Reducibility by algebraic projections. Logic and Algorithms. L'ensignment Mathmatique, 30:365--380, 1982.

....the unique minimum reduced Grobner basis of size 257,576 on one processor of a 12 processor , while a conservative estimate is that the same computation given the 35 generators of Jac(p5) would take over 100 years to halt. 2 be a Valiant projection of the determinant polynomial d s 1 (see [Val79, Val82, vzG87]; the latter gives an elegant projection from d s 2 ) The best lower bound on s n such that the permanent polynomials p n are not Valiant projections of d sn , however, is s n p 2n [Cai90] It is possible that a good upper bound on the increase in some numerical invariant of polynomial ideals ....

L. Valiant. Reducibility by algebraic projections. L'Enseignement math'ematique, 28:253--268, 1982.

....RCS Revision: 1.46) A.4 Combinatorial Optimization and Flow Problems A.4.1 Linear Inequalities (LI) Given: An integer n Theta d matrix A and an integer n Theta 1 vector b. Problem: Is there a rational d Theta 1 vector x 0 such that Ax b (It is not required to find such an x. Reference: [Coo82, Val82a, Kha79] Hint: LI is in P by [Kha79] The following reduction of CVP to LI is due to [Coo82] 1. If input x i is true (false) it is represented by the equation x i = 1 (x i = 0) 2. A not gate with input u and output w, computing w :u is represented by the inequalities w = 1 Gamma u and 0 w 1. 3. ....

....to force z = 1. If the system has a solution then the output is true and otherwise the output is false. A.4.2 Linear Equalities (LE) Given: An integer n Theta d matrix A and an integer n Theta 1 vector b. Problem: Is there a rational d Theta 1 vector x 0 such that Ax = b Reference: [Coo82, Val82a, Kha79] Hint: LE is NC 1 reducible to LI since Ax b and GammaAx Gammab if and only if Ax = b. Thus LE is in P. For completeness an instance of LI can be reduced to LE as follows: for each inequality in LI there is a corresponding equality in LE with an additional slack variable that is used to ....

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L. G. Valiant. Reducibility by algebraic projections. L'Enseignement Math'ematique, XXVIII:253--268, 1982. Also in [L'E82, pages 365--380].

....there is a corresponding first order language L( built up from the symbols of and the logical relation symbols and constant symbols 2 : BIT; 0; m, using logical connectives: variables: x; y; z; and quantifiers: 8; 9. First Order Interpretations and Projections In [Val], Valiant defined the projection, an extremely low level many one reduction. Definition 3.1 A k ary projection from S to T is a sequence of maps fpng, n = 1; 2; such that for all n and for all binary strings s of length n, pn (s) is a binary string of length n k and, s 2 S , pn (s) 2 T ....

L.G. Valiant, "Reducibility By Algebraic Projections," L'Enseignement math'ematique, 28, 3-4 (1982), 253-68.

....C that we have defined are closed under ae reductions. In other words, if K 1 ae K 2 then K 2 2 C ) K 1 2 C 2. ae reductions are closed under composition, i.e. K 1 ae K 2 ae K 3 ) K 1 ae K 3 Moreover, it should be clear that fop s and qfp s are indeed projections in the sense of Valiant [Val82], i.e. each bit of the output structure depends on at most one bit of the input structure. This notion can be made precise and formal, but since we don t use this fact further in the paper we leave it as an informal statement. 4 A Problem Complete for L We now describe the problem ORD, which ....

L. G. Valiant. Reducibility by algebraic projections. L'Enseignment Math`ematic, 28(3-4):253--268, 1982.

....P (Preliminary: RCS Revision: 1.46) A.4 Combinatorial Optimization and Flow Problems A.4.1 Linear Inequalities (LI) Given: An integer n 2 d matrix A and an integer n 2 1 vector b. Problem: Is there a rational d 2 1 vector x 0 such that Ax b (It is not required to find such an x. Reference: [Coo82, Val82a, Kha79] Hint: LI is in P by [Kha79] The following reduction of CVP to LI is due to [Coo82] 1. If input x i is true (false) it is represented by the equation x i = 1 (x i = 0) 2. A not gate with input u and output w, computing w :u is represented by the inequalities w = 1 0 u and 0 w 1. 3. An and ....

....inequalities required to force z = 1. If the system has a solution then the output is true and otherwise the output is false. A.4.2 Linear Equalities (LE) Given: An integer n 2 d matrix A and an integer n 2 1 vector b. Problem: Is there a rational d 2 1 vector x 0 such that Ax = b Reference: [Coo82, Val82a, Kha79] Hint: LE is NC 1 reducible to LI since Ax b and 0Ax 0b if and only if Ax = b. Thus LE is in P. For completeness an instance of LI can be reduced to LE as follows: for each inequality in LI there is a corresponding equality in LE with an additional slack variable that is used to make the ....

[Article contains additional citation context not shown here]

L. G. Valiant. Reducibility by algebraic projections. L'Enseignement Math'ematique, XXVIII:253--268, 1982. Also in [L'E82, pages 365--380].

....interpretations a standard concept from logic for translating one language into another [End, Imm87] We define a very weak version of first order interpretations, namely first order projections. These are first order interpretations that are at the same time projections in the sense of Valiant [Val]. We will see that the completeness results listed in Figure 2 all hold via first order projections. The value of this observation is that first order projections are sufficiently low level that they retain the full algebraic character of the problems. In [Imm87] a reduction called projection ....

....is expressible in FO, but not in FO(wo BIT) See also [Lin] for further discussion of BIT. In fact, the following are completely syntactic definitions for the uniform classes AC 0 and ThC 0 : Fact 3.1 ( BIS] FO = AC 0 and FOM = ThC 0 . First Order Interpretations and Projections In [Val], Valiant defined the projection, an extremely low level many one reduction. Projections are weak enough to preserve the algebraic structure of problems such as iterated multiplications. For this reason we find it particularly interesting that projections suffice for proving completeness ....

L.G. Valiant (1982), Reducibility By Algebraic Projections, L'Enseignement math'ematique, 28, 3-4, 253-68.

....and Biswas, is the most general theorem known for the 1 L reductions. Theorem 11 ( AB93] Let C be a complexity class that is closed under lin log reductions, e.g. P, NP, PSPACE. The sets complete for C under 1 L reductions are all p isomorphic. The first order projection (defined by Valiant in [Val82]) is another example of a very strong form of reduction. Allender, Balc azar and Immerman showed in [ABI93] that for the first order projections an isomorphism theorem holds Theorem 12 ( ABI93] Let C be a nice complexity class, e.g. P, NP, PSPACE. All sets complete for C under first order ....

L. Valiant. Reducibility by algebraic projections. L'Enseignement math'ematique, 28:3--4, 1982.

....being that of expanding symbolic determinants. One way to combat this problem is to t allow the sharing of common subexpressions, and almost all major systems have some facili ies to accommodate this. With the advent of probabilistic zero testing [21] 9] and [8] and e n hash coding [18] and [4], canonical representations for solving the zero identity problem wer ot required any longer. Moreover, should the expressions after being manipulated lead to a p sparse polynomial answer, that answer could be retrieved by Zippel s sparse polynomial inter olation procedure [27] or, where ....

....rm i n a n t p : d e t e rm i n a n t (ma t r i x ( 1 , x1 , x1 2 , x1 3 , x1 4 ] 1 , x2 , x2 2 , x2 3 , x2 4 ] 1 , x3 , x3 2 , x3 3 , x3 4 ] 1 , x4 , x4 2 , x4 3 , x4 4 ] T i me= 16 . 7 ms e c s . 1 , x5 , x5 2 , x5 3 , x5 4 ] 11 ] 2 3 4 1 x1 x1 x1 x1 ] [ 2 3 4 ] ] 1 x2 x2 x2 x2 ] d4 ) d e t e rm i n a n t ( 2 3 4 ] 1 x3 x3 x3 x3 ] 2 3 4 ] 1 x4 x4 x4 x4 ] 2 3 4 1 x5 x5 x5 x5 ] D ( c 5 ) c : s t r a i gh t c o e f f ( po l y t o s t r a i gh t ( p ) x1 ) i r e c t e s t i ma t e : 920 8 C I n t e r po l a t i on e s t i ....

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Valiant, L., "Reducibility by algebraic projections," L'Enseignement mathematique, vol. 28, pp. 253-268, 2 1982.

....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. Reducibility by algebraic projections. In Logic and Algorithmic (an International Symposium held in honour of Ernst Specker), pages 365-380. Monographie n o 30 de L'Enseignement Mathematique, 1982.

....language A f : f (x; i; b) bit i of f(x) equals b g belongs to DLOGTIME, except for the extra clause about M computing the length of f(x) which is met in all instances that we know. Also noteworthy is that (d) is equivalent to a uniform notion of projection reductions as defined by Valiant [Val82] (see also [SV85] A projection reduction is given by a family of mappings n : f 1; n 0 g f 0; 1; x 1 ; x 1 ; x n ; x n g. Intuitively, n (j) either sets bit j of f(x) to 0 or 1 depending only on n, or chooses some input bit x i that the output bit depends on, and ....

L. Valiant. Reducibility by algebraic projections. L'Enseignement math'ematique, 28:253--268, 1982.

....language A f : f (x; i; b) bit i of f(x) equals b g belongs to DLOGTIME, except for the extra clause about M computing the length of f(x) which is met in all instances that we know. Also noteworthy is that (d) is equivalent to a uniform notion of projection reductions as defined by Valiant [Val82] (see also [SV85] A projection reduction is given by a family of mappings n : f 1; n 0 g f 0; 1; x 1 ; x 1 ; x n ; x n g. Intuitively, n (j) either sets bit j of f(x) to 0 or 1 depending only on n, or chooses some input bit x i that the output bit depends on, and ....

L. Valiant. Reducibility by algebraic projections. L'Enseignement math'ematique, 28:253--268, 1982.

....on this claim, note that the class NP arises naturally in the study of logic and can be defined entirely in terms of logic, without any mention of computation [Fa] Thus it is natural to have a notion of NP completeness that is formulated entirely in terms of logic. On another front, Valiant [Val] noticed that reducibility can be formulated in algebra using the natural notion of a projection, again with no mention of computation. The sets that are complete under fops are complete in all of these different ways of formulating the notion of NP completeness. Since natural complete problems ....

.... there is a corresponding first order language L( built up from the symbols of and the numeric relation symbols and constant symbols 2 : BIT; 0; m, using logical connectives: variables: x; y; z; and quantifiers: 8; 9. First Order Interpretations and Projections In [Val], Valiant defined the projection, an extremely low level many one reduction. Definition 3.1 Let S; T f0; 1g . A k ary projection from S to T is a sequence of maps fp n g, n = 1; 2; that satisfy the following properties. First, for all n and for all binary strings s of length n, p n ....

Leslie Valiant, "Reducibility By Algebraic Projections," L'Enseignement math'ematique, 28, 3-4 (1982), 253-68.

....and verified that it is quite inefficient. The ensuing search for efficien lgorithms to factor polynomials is a fine example in the discipline of the design and analysis r t of algorithms as well as complexity theory and exhibits many of the techniques developed fo hese subjects. In 1967 Berlekamp [2] found an algorithm to factor univariate polynomials over . B moderately sized finite fields in time proportional to the cube of the input degrees erlekamp s algorithm is the first evidence that polynomial factorization is not as complex a l problem as is integer factoring. However, his algorithm ....

Valiant, L., "Reducibility by algebraic projections," L'Enseignement mathematique, vol. 28, pp. 253-268, 6 1982.

....A. Similarly we distinguish the between DETn and det for the determinant of a matrix A. 1. 1 Algebraic complexity of HC We shall work in Valiant s non uniform model of algebraic computation and assume the reader is familiar with the corresponding complexity classes VP(k) and VNP(k) as defined in [Val82] and [BCS97, chapter 21] Briefly, VP(k) consists of the non uniform class of families of functions computable by straight line algebraic programs over k algebras where k is a field and the computations use only the k algebra operations. VNP(k) is its non deterministic analogue. However, we shall ....

L. Valiant. Reducibility by algebraic projections. In Logic and Arithmetic: An International Symposium held in honour of Ernst Specker, volume 30 of L'enseignement Math'ematique, pages 365--380. Universit 'e de Gen`eve, 1982.

.... usual RAM programs can be converted into RAMs of polynomially related binary p asymptotic complexity which generate the straight line programs corresponding to these com utations (4) The GCD problem for dense multivariate polynomials was first made feasible by work of G Collins [8] and Brown [5]. Moses and Yun [34] showed how to apply the Hensel lemma to CD computations. Zippel [47] invented an important technique to preserve sparsity of the t Z multivariate GCD during Brown s interpolation scheme, though it should be noted tha ippel s approach is not random polynomial time. The reason ....

.... in len(P ) el size(P ) d , log( e # 1 # ) e a Notice that the parameters n and r are dominated by len(P ) We do not know how th pproach of repeated GCD computations GCD( f , f ) GCD(GCD( f , f ) f ) x 1 2 1 2 3 r o 1 r that of extracting cont (f ) 1 r r , cf. Brown [5]) can lead to a polynomial time solution. We first restrict ourselves to r = 2, that is the GCD problem for two polynomials. We t will later show that the GCD problem for many polynomials can be probabilistically reduced o that for two. In order to avoid the content computation we will work with ....

Valiant, L., "Reducibility by algebraic projections," L'Enseignement mathematique, vol. 28, pp. 253-268, 4 1982.

....reductions. Suppose that I is a many one reduction from S to T , i.e. for all A in STRUC[oe] A 2 S , I(A) 2 T Then we say that I is a k ary first order reduction of S to T . Furthermore, if the i s are quantifier free and do not include BIT then I is a quantifier free reduction. 2 Valiant [V] defined a very low level, non uniform reduction called a projection. A projection is a many one reduction f : f0; 1g f0; 1g such that each bit of f(w) depends on at most one bit of w. It can be thought of as a reduction computed by a circuit of depth zero, depending on the details of ....

L.G. Valiant, "Reducibility by algebraic projections," L'Enseignement math'ematique, 28 (1982), 253-268.

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Leslie G. Valiant. Reducibility by algebraic projections. L'Enseignement Mathematique, 28:253--268, 1982.

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L. Valiant. Reducibility by algebraic projections. L'Enseignement math'ematique, 28:3--4, 1982.

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L.G. Valiant, "Reducibility by algebraic projections, " L'Enseignement math'ematique, 28 (1982), 253-268.

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L.G. Valiant, "Reducibility By Algebraic Projections," L'Enseignement math'ematique, T. XXVIII, 3-4, 1982, (253-268).

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