## On span programs (1993)

Venue: | In Proc. of the 8th IEEE Structure in Complexity Theory |

Citations: | 122 - 6 self |

### BibTeX

@INPROCEEDINGS{Karchmer93onspan,

author = {M. Karchmer and A. Wigderson},

title = {On span programs},

booktitle = {In Proc. of the 8th IEEE Structure in Complexity Theory},

year = {1993},

pages = {102--111},

publisher = {IEEE Computer Society Press}

}

### OpenURL

### Abstract

We introduce a linear algebraic model of computation, the Span Program, and prove several upper and lower bounds on it. These results yield the following applications in complexity and cryptography: • SL ⊆ ⊕L (a weak Logspace analogue of N P ⊆ ⊕P). • The first super-linear size lower bounds on branching programs that count. • A broader class of functions which posses information-theoretic secret sharing schemes. The proof of the main connection, between span programs and counting branching programs, uses a variant of Razborov’s general approximation method. 1

### Citations

2419 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...lary 2 ⊕BP (MAJn) = Ω(n log log log ∗ n). Corollary 3 ⊕BP (T k n ) = O(n) iff either k = O(1) or n − k = O(1). 7 Monotone Span Programs Monotone analogues of Boolean complexity classes are studied in =-=[8]-=-. We will adopt their notation of mC to denote the monotone analogue of the class C. Recall that a branching program is monotone if we use only positive literals to label its edges. For the algebraic ... |

1926 | How to share a secret
- Shamir
- 1979
(Show Context)
Citation Context ...quivalent: Majority, Threshold-2 and all the rest require size exactly Θ(n log n). Third, we show that this model captures in a natural way information theoretic secret sharing in the sense of Shamir =-=[20]-=-. It enables us to extend the result of Rudich [19], and enlarge the class of functions for which such efficient secret sharing schemes exists. In the other direction, existing schemes can provide upp... |

288 | Algebraic methods in the theory of lower bounds for Boolean circuit complexity
- Smolensky
- 1987
(Show Context)
Citation Context ...ount is an old and fruitful theme in complexity theory. One such direction was to add mod m gates to unbounded fan-in circuits. This resulted in the exponential lower bounds of Razborov and Smolensky =-=[15, 21]-=- in the case when m is a prime, and the frustrating question of the power of ACC, when m is composite. Another direction was to let nondeterministic polynomial time Turing machines count the number of... |

215 |
NP is as easy as detecting unique solutions
- Valiant, Vazirani
- 1986
(Show Context)
Citation Context ...osite. Another direction was to let nondeterministic polynomial time Turing machines count the number of accepting paths. For counting mod 2, this defines ([12, 6]) the class ⊕P. Valiant and Vazirani =-=[23]-=- were first to show the power of this model by giving a probabilistic Turing reduction of N P to ⊕P. Toda [22] used this technique to prove a much stronger result, namely that the whole polynomial tim... |

175 |
The complexity of finite functions
- Boppana, Sipser
- 1990
(Show Context)
Citation Context ...d decides whether all n numbers are distinct. Corollary 1 ⊕BP (EDn) = Ω(n 3/2 / log n). Proof: The element distinctness function EDn is a canonical example of a function having many subfunctions (see =-=[4]-=-). The partition of variables is the natural one, a part for each integer (2 log n bits). The number k of parts is n/(2 log n), and for every part ci(EDn) = 2 Θ(n) . 3 The basic model: Span Programs W... |

75 |
Two remarks on the power of counting
- Papadimitriou, Zachos
- 1983
(Show Context)
Citation Context ...question of the power of ACC, when m is composite. Another direction was to let nondeterministic polynomial time Turing machines count the number of accepting paths. For counting mod 2, this defines (=-=[12, 6]-=-) the class ⊕P. Valiant and Vazirani [23] were first to show the power of this model by giving a probabilistic Turing reduction of N P to ⊕P. Toda [22] used this technique to prove a much stronger res... |

43 | Structure and importance of logspace-MOD-classes
- Buntrock, Damm, et al.
- 1992
(Show Context)
Citation Context ...e are interested in nondeterministic logspace machines that count the number of accepting paths (mod m). The analogues modmL of the polynomial counting classes modmP were defined and first studied in =-=[5]-=-. In [5] it was shown that most natural problems in linear algebra over GF (p), such as determinant, rank and solving linear systems, are logspace complete for the class modpL. Still, very little is k... |

41 |
On the construction of parallel computers from various bases of boolean functions
- Goldschlager, Parberry
- 1986
(Show Context)
Citation Context ...question of the power of ACC, when m is composite. Another direction was to let nondeterministic polynomial time Turing machines count the number of accepting paths. For counting mod 2, this defines (=-=[12, 6]-=-) the class ⊕P. Valiant and Vazirani [23] were first to show the power of this model by giving a probabilistic Turing reduction of N P to ⊕P. Toda [22] used this technique to prove a much stronger res... |

35 |
On the method of approximations
- Razborov
- 1989
(Show Context)
Citation Context ...amir’s construction. Finally we describe the evolution of the idea to use span programs for lower bounds. It was inspired by the papers [18] and [9], both of which have as a common ancestor the paper =-=[16]-=-. In [16], Razborov introduced his generalized approximation method. He showed how to assign to every Boolean function f a set cover problem (∆M (f), SM (f)). Here ∆M (f) is the universe to be covered... |

33 |
Applications of matrix methods to the theory of lower bounds in computational complexity, Combinatorica 10
- Razborov
- 1990
(Show Context)
Citation Context ...n terms of the affine dimension of a graph associated with f. This algebraic measure has been proposed as a source of lower bounds for formulae and Boolean branching programs, and has been studied in =-=[14, 17]-=-. Fix a field K and fix a partition of the variables of f into two sets A, B. Thus every sequence σ ∈ {0, 1} n decomposes in a natural way into σ A ∈ {0, 1} A and σ B ∈ {0, 1} B such that σ = σ A ◦ σ ... |

26 |
bounds on the size of bounded-depth networks over a complete basis with logical addition
- Lower
- 1987
(Show Context)
Citation Context ...ount is an old and fruitful theme in complexity theory. One such direction was to add mod m gates to unbounded fan-in circuits. This resulted in the exponential lower bounds of Razborov and Smolensky =-=[15, 21]-=- in the case when m is a prime, and the frustrating question of the power of ACC, when m is composite. Another direction was to let nondeterministic polynomial time Turing machines count the number of... |

17 |
Lower bounds to the complexity of symmetric boolean functions
- Babai, Pudlák, et al.
- 1990
(Show Context)
Citation Context ...ndeterministic branching programs was proved by Razborov [18], and indeed we use much of his machinery. In contrast, for the weaker deterministic branching programs, the best lower bound for Majority =-=[3]-=- is Ω(n log n/ log log n). The route to both types of results goes through the same device – the span program. The span program (over any field K) is a linear algebraic model that computes a function ... |

15 |
Random walks, universal sequences and the complexity of maze problems
- Aleliunas, Karp, et al.
- 1979
(Show Context)
Citation Context ... p (SL, symmetric logspace, is the class of all problems that are reducible in logspace to undirected stconnectivity). Previously, it was only known that SL ⊆ L/poly which follows from the results of =-=[1]-=-. We also prove the first nontrivial lower bounds on branching programs that count mod 2. The most interesting (though not the largest) is the slightly superlinear (Ω(n log log log ∗ n)) lower bound o... |

14 |
On the computational power of
- Toda
- 1989
(Show Context)
Citation Context ...ting paths. For counting mod 2, this defines ([12, 6]) the class ⊕P. Valiant and Vazirani [23] were first to show the power of this model by giving a probabilistic Turing reduction of N P to ⊕P. Toda =-=[22]-=- used this technique to prove a much stronger result, namely that the whole polynomial time hierarchy is probabilisticly Turing reducible to ⊕P. Moreover, the ∗ Partially supported by NSF grant CCR-92... |

12 |
private communication
- Rudich
- 1993
(Show Context)
Citation Context ... the pieces {si : i ∈ T } determine s, while if f(T ) = 0 these pieces give no information whatsoever about s. The size of such scheme is � i∈[n] di. Such a scheme was first described to us by Rudich =-=[19]-=-. Theorem 13 For every prime p, every monotone function has a secret sharing scheme (over GF (p)) of size mSPp(f) Proof: Fix a prime p, set K = GF (p) and let ˆM(M, ρ) be a monotone span program for a... |

11 |
Complexity of Contact Circuits Realizing a Function of Logical Algebra,” Dokl
- Krichevskii
- 1963
(Show Context)
Citation Context ...Θ(n log n). This theorem follows from the following two theorems. Theorem 11 mSP2(T 2 n) ≥ n log n. Proof: The proof is an algebraic variation on the n log n lower bound on the formula size for T 2 n =-=[10]-=-. Let ˆ M(M, ρ) be a monotone span program for T 2 n. Let t be the number of columns in M, and R the set of odd vectors in GF (2) t . Clearly |R| = 2t−1 . Let di = dim(X1 i ). For a subspace V of GF (... |

10 | Characterizing non-deterministic circuit size
- Karchmer, Wigderson
- 1993
(Show Context)
Citation Context ... O(n log n) upper bound for Majority was inspired by Shamir’s construction. Finally we describe the evolution of the idea to use span programs for lower bounds. It was inspired by the papers [18] and =-=[9]-=-, both of which have as a common ancestor the paper [16]. In [16], Razborov introduced his generalized approximation method. He showed how to assign to every Boolean function f a set cover problem (∆M... |

7 |
Private communication
- Pudlák
- 2003
(Show Context)
Citation Context ...There are no lower bounds known for algebraic branching programs. Nečiporuk [11] presented a method which yields lower bounds of the form Ω((n/ log n) 2 ) for deterministic branching programs. Pudlák =-=[13]-=- observed that the method yields lower bounds of the form Ω(n 3/2 / log n) for nondeterministic branching programs. Here we observe that Pudlák’s idea carries over to the algebraic model: Fix a partit... |

6 |
A combinatorial approach to complexity
- Pudlák, Rödl
- 1992
(Show Context)
Citation Context ...n terms of the affine dimension of a graph associated with f. This algebraic measure has been proposed as a source of lower bounds for formulae and Boolean branching programs, and has been studied in =-=[14, 17]-=-. Fix a field K and fix a partition of the variables of f into two sets A, B. Thus every sequence σ ∈ {0, 1} n decomposes in a natural way into σ A ∈ {0, 1} A and σ B ∈ {0, 1} B such that σ = σ A ◦ σ ... |

5 |
bounds on the size of switching-and-rectifier networks for symmetric Boolean functions (in Russian). Mathematical otes of the Academy of Sciences of the SSR
- Lower
- 1990
(Show Context)
Citation Context ...y function. In fact, the proof characterizes those threshold functions that admit linear size programs. The same lower bound for Majority on nondeterministic branching programs was proved by Razborov =-=[18]-=-, and indeed we use much of his machinery. In contrast, for the weaker deterministic branching programs, the best lower bound for Majority [3] is Ω(n log n/ log log n). The route to both types of resu... |

4 |
A note on the power of threshold circuts
- Allender
- 1989
(Show Context)
Citation Context ...ially supported by NSF grant CCR-9212184 and DARPA contract N00014-92-J-1799. same result can be obtained via the techniques used for the constant-depth circuits mentioned above, as shown by Allender =-=[2]-=-. Here we are interested in nondeterministic logspace machines that count the number of accepting paths (mod m). The analogues modmL of the polynomial counting classes modmP were defined and first stu... |

2 |
On a Boolean function. Doklady of the Academy of
- NECIPORUK
(Show Context)
Citation Context ...he monotone analogues of N L, SL and L defined by allowing only positive literals to label edges of the branching programs. There are no lower bounds known for algebraic branching programs. Nečiporuk =-=[11]-=- presented a method which yields lower bounds of the form Ω((n/ log n) 2 ) for deterministic branching programs. Pudlák [13] observed that the method yields lower bounds of the form Ω(n 3/2 / log n) f... |

1 |
Theory Wiley-Interscience
- Ramsey
- 1980
(Show Context)
Citation Context ...ine the coloring ψ : [n] s−d ↦→ [q] s−d as follows: if C = {i1, i2, · · · , is−d} with i1 < · · · < is−d, then ψ(C) = (χi1 (C), χi2 (C), · · · , χis−d (C)). It follows from Ramsey’s Theorem (see e.g. =-=[7]-=-) that there exists a subset A ∈ [n] s and a vector v ∈ [q] s−d such that every subset C ⊆ A, |C| = s − d satisfies ψ(C) = v. Now assume without loss of generality that A = [s]. To specify the subsets... |