## Randomization and Derandomization in Space-Bounded Computation (1996)

Venue: | In Proceedings of the 11th Annual IEEE Conference on Computational Complexity |

Citations: | 36 - 0 self |

### BibTeX

@INPROCEEDINGS{Saks96randomizationand,

author = {Michael Saks},

title = {Randomization and Derandomization in Space-Bounded Computation},

booktitle = {In Proceedings of the 11th Annual IEEE Conference on Computational Complexity},

year = {1996},

pages = {128--149},

publisher = {IEEE Computer Society}

}

### Years of Citing Articles

### OpenURL

### Abstract

This is a survey of space-bounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic space-bounded computations, and the application of such generators to obtain deterministic simulations.

### Citations

2343 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...ons given above will serve just as well. Also, there are the usual technicalities that s and t should be suitably well-behaved functions (see, e.g., the discussion of "proper" complexity fun=-=ctions in [38]-=-), and we always assume that the functions s and t satisfy whatever conditions are necessary. The usual definition of the space of a run is the number of distinct cells that are accessed on the work t... |

1871 | Randomized Algorithms
- MOTWANI, P
- 1995
(Show Context)
Citation Context ...uestions can be found within the related domains of finite state Markov chains and matrix computation. We will assume familiarity with the most elementary facts about finite Markov Chains (see, e.g., =-=[28]-=-). Associated to every finite Markov chain C on state space S is its transition probability matrix P = PC , where for i, j # S, P [i, j] is the probability, given that the chain is in state i at a par... |

317 |
Relationships between nondeterministic and deterministic tape complexities
- Savitch
- 1970
(Show Context)
Citation Context ...y, in DSPACE(s 6 ). This result was improved independently by Borodin, Cook and Pippenger, and Jung: Theorem 3.3 ([9, 20]) P rSPACE(s) # DSPACE(s 2 ). 9 This strengthens Savitch's fundamental theorem =-=[43]-=- that NSPACE(s) # DSPACE(s 2 ). The proofs of Theorem 3.3 and its antecedents were based on small space solutions to one of the associated matrix problems discussed in section 2.4. In particular, in [... |

288 | Hardness vs. randomness
- Nisan, Wigderson
- 1994
(Show Context)
Citation Context ...rm theoretical foundation, which, motivated largely by considerations from cryptography, began in the mid 1980's. A fundamental insight here, originating in [50] and expanded upon and systematized in =-=[36, 29], is the connection -=-between "hardness " and "randomness": the existence of problems whose instances can be generated e#ciently, and that are su#ciently hard to approximate within a particular computat... |

238 | Nondeterministic space is closed under complementation
- Immerman
- 1988
(Show Context)
Citation Context ...(s). Figure 4 suggests that the following two open questions have a#rmative answers. Question 3.1 Is NSPACE(s) = BPSPACE(s)? Question 3.2 Is BPHSPACE(s) # NSPACE(s)? In light of the celebrated result =-=[18, 48]-=- that the alternating space hierarchy collapses to NSPACE(s), the second question represents an analog to the result [46] that BPP is contained in the polynomial time hierarchy. However, a straightfor... |

222 |
Computational complexity of probabilistic Turing machines
- Gill
- 1977
(Show Context)
Citation Context ... for "non-random" problems such as primality testing ([47, 40]), probabilistic computation emerged as a major subfield of complexity theory during the late 1970's. Beginning with Gill's semi=-=nal paper [15]-=-, researchers defined models and built the foundations for a rigorous study of probabilistic computation and, in particular, of probabilistic time- and space-bounded complexity classes. Despite the co... |

188 | Pseudorandom generators for space-bounded computation
- Nisan
- 1992
(Show Context)
Citation Context ...on, it is enough to prove hardness results for a model of space-bounded computation in which the input is accessible only by a 1 one-way tape. This observation opened the way for a sequence of papers =-=[2, 7, 30, 19, 37, 6]-=-, presenting ingenious constructions of pseudorandom number generators that can be proved unconditionally to look random to space-bounded machines. These constructions provide the basis for some signi... |

170 |
Probabilistic Automata
- Rabin
- 1963
(Show Context)
Citation Context ...p problems and on complexity classes corresponding to space functions s(n) that are at least log n (thus omitting the notable body of work on very small space classes and probabilistic automata, e.g, =-=[39, 13, 17, 11, 24, 23]-=-). Within these restrictions, the aim is to be reasonably comprehensive, and apologies are o#ered in advance for the inevitable omissions. Section 2 presents definitions of the relevant models and com... |

147 |
A complexity theoretic approach to randomness
- Sipser
- 1983
(Show Context)
Citation Context ...Question 3.2 Is BPHSPACE(s) # NSPACE(s)? In light of the celebrated result [18, 48] that the alternating space hierarchy collapses to NSPACE(s), the second question represents an analog to the result =-=[46]-=- that BPP is contained in the polynomial time hierarchy. However, a straightforward attempt to adapt the simulation for BPP to question 3.2 fails for at least two reasons: the simulation seems to requ... |

137 |
A Fast Monte-Carlo Test for Primality
- Solovay, Strassen
- 1977
(Show Context)
Citation Context ...ion of such generators to obtain deterministic simulations. 1 Introduction Inspired in part by the then-surprising use of randomness in algorithms for "non-random" problems such as primality=-= testing ([47, 40]-=-), probabilistic computation emerged as a major subfield of complexity theory during the late 1970's. Beginning with Gill's seminal paper [15], researchers defined models and built the foundations for... |

111 | Multi-party protocols
- Chandra, Furst, et al.
- 1983
(Show Context)
Citation Context ... # S) . The proof that their generator is a PRG for space bounded computation is based on a connection between small space computation and a model of multiparty communication complexity introduced in =-=[10]-=-. Let f(x 1 , x 2 , . . . , x k ) be a boolean function as above. In the multi-party communication model, there are k parties, and the i th party knows every input except x i . They communicate by mea... |

111 |
The method of forced enumeration for nondeterministic automata
- SzelepcsĂ©nyi
- 1988
(Show Context)
Citation Context ...(s). Figure 4 suggests that the following two open questions have a#rmative answers. Question 3.1 Is NSPACE(s) = BPSPACE(s)? Question 3.2 Is BPHSPACE(s) # NSPACE(s)? In light of the celebrated result =-=[18, 48]-=- that the alternating space hierarchy collapses to NSPACE(s), the second question represents an analog to the result [46] that BPP is contained in the polynomial time hierarchy. However, a straightfor... |

98 |
Deterministic simulation in logspace
- Ajtai, Komlos, et al.
- 1987
(Show Context)
Citation Context ...on, it is enough to prove hardness results for a model of space-bounded computation in which the input is accessible only by a 1 one-way tape. This observation opened the way for a sequence of papers =-=[2, 7, 30, 19, 37, 6]-=-, presenting ingenious constructions of pseudorandom number generators that can be proved unconditionally to look random to space-bounded machines. These constructions provide the basis for some signi... |

97 |
Two theorems on random polynomial time
- Adleman
- 1978
(Show Context)
Citation Context ...v. The two functions above map matrices to matrices, and by specifying a particular entry [i, j] of the output, we may view them as mapping matrices to real numbers (and in our case the range will be =-=[0, 1]-=-). For any real valued function f on some domain D, we define the threshold language L f associated to f to be the set of inputs x # D such that f(x) > 1/2. We will consider three versions of the memb... |

91 |
Probabilistic algorithms
- Rabin
- 1976
(Show Context)
Citation Context ...ion of such generators to obtain deterministic simulations. 1 Introduction Inspired in part by the then-surprising use of randomness in algorithms for "non-random" problems such as primality=-= testing ([47, 40]-=-), probabilistic computation emerged as a major subfield of complexity theory during the late 1970's. Beginning with Gill's seminal paper [15], researchers defined models and built the foundations for... |

81 |
Explicit constructions of linear-sized superconcentrators
- Gabber, Galil
- 1981
(Show Context)
Citation Context ...e in the following sense: there is an algorithm that given an S-bit index to a vertex v and an integer i between 1 and d, runs in space #(S) and returns the index of the i th neighbor of v (see, e.g. =-=[14, 27]-=-). Given a d-regular expander H and arbitrary positive integers J # K we define a partial mapping sending S + (log d)(K - 1) + K bits to JS bits as follows. Fix a (possibly redundant) encoding of the ... |

72 |
Explicit expanders and the Ramanujan conjectures
- Lubotzky, Phillips, et al.
- 1986
(Show Context)
Citation Context ...e in the following sense: there is an algorithm that given an S-bit index to a vertex v and an integer i between 1 and d, runs in space #(S) and returns the index of the i th neighbor of v (see, e.g. =-=[14, 27]-=-). Given a d-regular expander H and arbitrary positive integers J # K we define a partial mapping sending S + (log d)(K - 1) + K bits to JS bits as follows. Fix a (possibly redundant) encoding of the ... |

61 |
Parallel computation for well-endowed rings and space-bounded probabilistic machines
- Borodin, Cook, et al.
- 1983
(Show Context)
Citation Context ...ed on connections between these classes and matrix computations. These connections led to two of the main results: that (unbounded error) probabilistic space s is contained in deterministic space s 2 =-=[9]-=-, and that the power of unbounded error probabilistic space s is not diminished by imposing a time bound of 2 O(s) [22]. Much of the more recent progress in the area grew out of e#orts to place the th... |

54 | Two applications of inductive counting for complementation problems - Borodin, Cook, et al. - 1989 |

53 |
Multiparty protocols and logspace-hard pseudorandom sequences, 21st STOC
- Babai, Nisan, et al.
- 1989
(Show Context)
Citation Context ...on, it is enough to prove hardness results for a model of space-bounded computation in which the input is accessible only by a 1 one-way tape. This observation opened the way for a sequence of papers =-=[2, 7, 30, 19, 37, 6]-=-, presenting ingenious constructions of pseudorandom number generators that can be proved unconditionally to look random to space-bounded machines. These constructions provide the basis for some signi... |

52 | Undirected connectivity in O(log 1:5 n) space
- NISAN, SZEMEREDI, et al.
- 1992
(Show Context)
Citation Context ...l restriction on the number of random bits was proved by Saks and Zhou: Theorem 3.6 ([42]) BPH SPACE(s) # DSPACE(s 3/2 ). This result generalized the previous result of Nisan, Szemeredi and Wigderson =-=[34]-=- that USTCON could be solved in DSPACE((logn) 3/2 ). The central ingredient in the deterministic simulations that achieve Theorems 3.4, 3.5 and 3.6 are pseudorandom generators for space-bounded comput... |

45 |
Space-bounded hierarchies and probabilistic computation
- Ruzzo, Simon, et al.
- 1984
(Show Context)
Citation Context ... error) then there is a machine M # running in space s that computes the same language as L and runs in expected time 2 2 O(s) (and hence halts almost surely). 4 A simple construction of M # given in =-=[41]-=- is: choose constants c and d appropriately and perform the following loop: while the computation has not halted, do (i) run M for d s steps, and if M does not halt, (ii) toss (c + d) s coins and if a... |

41 |
More deterministic simulation in logspace
- Nisan, Zuckerman
- 1993
(Show Context)
Citation Context |

40 |
C.H.: Symmetric space-bounded computation
- Lewis, Papadimitriou
- 1982
(Show Context)
Citation Context ...cated in the figure hold. In addition to these classes and the standard classes DSPACE(s) and NSPACE(s), we will also refer to the class SSPACE(s), of languages computed by a symmetric Turing machine =-=[25]-=- running in space s. For our purposes it su#ces to know that any language in SSPACE(s) can be reduced in DSPACE(s) to an undirected (s, t)-connectivity (USTCON ) problem for a graph on 2 O(s) vertices... |

34 |
Probabilistic two-way machines
- Freivalds
- 1981
(Show Context)
Citation Context ...p problems and on complexity classes corresponding to space functions s(n) that are at least log n (thus omitting the notable body of work on very small space classes and probabilistic automata, e.g, =-=[39, 13, 17, 11, 24, 23]-=-). Within these restrictions, the aim is to be reasonably comprehensive, and apologies are o#ered in advance for the inevitable omissions. Section 2 presents definitions of the relevant models and com... |

31 | Approximation of general independent distributions
- EVEN, GOLDEICH, et al.
- 1992
(Show Context)
Citation Context ...ive. Note that this latter condition says that the set H which is the range of G(w) intersects or hits all (m, d)- rectangles of volume at least #. Versions of this problem were discussed in [30] and =-=[12]-=-. The problem of constructing such generators can be viewed as a special case of the problem of constructing small space pseudorandom generators as follows. Suppose we have a space O(S) computation wh... |

27 | Symmetric logspace is closed under complement
- Nisan, Ta-Shma
- 1995
(Show Context)
Citation Context ...) and P r HSPACE(s). For NSPACE(s) (and hence for RSPACE(s)), closure under complementation is, of course, from [18, 48]. Finally for SSPACE(s), the result follows from a very clever direct reduction =-=[35]-=- from the problemsco - USTCON to USTCON . This implies an earlier result of [8], which was proved using inductive counting arguments in the spirit of [18, 48], that SSPACE(s) # co - RHSPACE(s), i.e., ... |

24 |
A time-complexity gap for two-way probabilistic finite state automata
- Dwork, Stockmeyer
- 1990
(Show Context)
Citation Context ...p problems and on complexity classes corresponding to space functions s(n) that are at least log n (thus omitting the notable body of work on very small space classes and probabilistic automata, e.g, =-=[39, 13, 17, 11, 24, 23]-=-). Within these restrictions, the aim is to be reasonably comprehensive, and apologies are o#ered in advance for the inevitable omissions. Section 2 presents definitions of the relevant models and com... |

23 | The complexity of graph connectivity
- Wigderson
- 1992
(Show Context)
Citation Context ...his way from Nisan's generator. A thorough overview of the complexity theoretic aspects of the graph connectivity problem, including universal traversal sequences, can be found in Wigderson 's survey =-=[49]-=-. 5.3 Hitting sets and generators for combinatorial rectangles Another special case of the problem of constructing a small space pseudorandom generator can be formulated as follows. Let m = 2 q , d be... |

22 |
On probabilistic time and space
- Jung
- 1985
(Show Context)
Citation Context ...t (unbounded error) probabilistic space s is contained in deterministic space s 2 [9], and that the power of unbounded error probabilistic space s is not diminished by imposing a time bound of 2 O(s) =-=[22]-=-. Much of the more recent progress in the area grew out of e#orts to place the theory of pseudorandom generation on a firm theoretical foundation, which, motivated largely by considerations from crypt... |

22 | Efficient construction of a small hitting set for combinatorial rectangles in high dimension
- Linial, Luby, et al.
- 1997
(Show Context)
Citation Context ...ping. We briefly describe the AKS construction of a pseudorandom sampler, which introduced an important technique known as "expander walks". We present a simplified version of their that was=-= given in [26]-=- for another purpose. For # # (0, 1) and positive integers N , and d > 0, an undirected graph H is an (N, d, #)- expander if H has N vertices, maximum degree d and for any subset A of vertices, the fr... |

21 |
Extracting randomness: How and why
- Nisan
- 1996
(Show Context)
Citation Context ...with the naive deterministic simulation, yields Theorem 3.5. At the center of their construction is a combinatorial construction called an extractor. A thorough survey of extractors is given by Nisan =-=[33]; we conte-=-nt ourselves with a short discussion. Extractors first arose in the following context. Suppose we have a random source that outputs bits that are "faulty"; a k-bit string produced by the sou... |

18 | On read-once vs. multiple access to randomness in logspace
- Nisan
- 1993
(Show Context)
Citation Context ...ape. An alternative model is to have the random bits written on a two-way readable 18 tape, so that each bit can be accessed arbitrarily often. Such a model was proposed in [7] and studied further in =-=[32]-=-. Following [32], for each probabilistic complexity class XSPACE(s), we denote the corresponding class with multi-access random bits by X # SPACE(s). For non-halting probabilistic computation, multiac... |

16 |
Random walks, universal sequences and the complexity of maze problems
- ALELIUNAS, KARP, et al.
- 1979
(Show Context)
Citation Context ... we highlight various questions that point in the direction of figure 4. 3.1 Solving USTCON in RHL The most well known use of probabilism in spacebounded computation is the result of Aleliunas et al. =-=[3]-=- showing how USTCON can be decided in RHL. By our earlier remark that SSPACE(s) can be reduced to a USTCON problem on a graph of size 2 s , this implies that SSPACE(s) # RHSPACE(s). Given the n vertex... |

15 |
A lower bound for probabilistic algorithms for finite state machines
- Greenberg, Weiss
- 1986
(Show Context)
Citation Context |

14 |
Space-bounded probabilistic Turing machine complexity classes are closed under complement
- SIMON
- 1981
(Show Context)
Citation Context .... If it doesn't, then we can not reliably do repeated trials, since there is a nontrivial chance that the computation stalls during one trial. Thus, we need the following result of Simon: Theorem 2.1 =-=[45]-=- If M is a PTM running in space s with unbounded error (resp., bounded error, one-sided error) then there is a machine M # running in space s that computes the same language as L and runs in expected ... |

13 |
On probabilistic tape complexity and fast circuits for matrix inversion problems
- Jung
- 1984
(Show Context)
Citation Context ...is just the entry P H [START,ACC] of the hitting probability matrix. As explained earlier, this computation boils down to the matrix computation Q # = lim ##1 (I - #Q) -1 for some substochastic Q. In =-=[21]-=-, it is shown that if we choose # # 1 - 2 -Cs for some C > 0, then Q # [i, j] > T if and only if (I - #Q) -1 [i, j] > T . Thus we can reduce the problem Q # [i, j] > T to the problems(I -R) -1 [i, j] ... |

13 |
Minimal nontrivial space complexity of probabilistic one-way Turing machines
- Kaneps, Freivalds
- 1990
(Show Context)
Citation Context |

13 |
Using Hard Problems to Create Pseudorandom Generators
- Nisan
- 1991
(Show Context)
Citation Context ...rm theoretical foundation, which, motivated largely by considerations from cryptography, began in the mid 1980's. A fundamental insight here, originating in [50] and expanded upon and systematized in =-=[36, 29], is the connection -=-between "hardness " and "randomness": the existence of problems whose instances can be generated e#ciently, and that are su#ciently hard to approximate within a particular computat... |

13 |
RSPACE(S) ` DSPACE(S3=2
- Saks, Zhou
- 1995
(Show Context)
Citation Context ... randomized log space, polynomial time computation can be simulated by a deterministic polylog space, polynomial time computation [31], and also by an O((log n) 3/2 ) space deterministic computation (=-=[42]-=-), and that if the randomized computation uses only polylog many random bits then it has a log space deterministic simulation [37]. In this survey, we focus on language membership problems and on comp... |

11 |
Relationships between probabilistic and deterministic tape complexity
- JUNG
- 1981
(Show Context)
Citation Context ...n be simulated deterministically with only a polynomial increase in space, specifically, in DSPACE(s 6 ). This result was improved independently by Borodin, Cook and Pippenger, and Jung: Theorem 3.3 (=-=[9, 20]-=-) P rSPACE(s) # DSPACE(s 2 ). 9 This strengthens Savitch's fundamental theorem [43] that NSPACE(s) # DSPACE(s 2 ). The proofs of Theorem 3.3 and its antecedents were based on small space solutions to ... |

9 |
On the derandomization of space-bounded computations
- Armoni
- 1998
(Show Context)
Citation Context |

8 |
Pseudorandomness for network algorithms
- Impagliazza, Nisan, et al.
- 1994
(Show Context)
Citation Context |

5 |
Regularity of one-letter languages acceptable by 2-way finite probabilistic automata
- Kaneps
- 1991
(Show Context)
Citation Context |

4 |
On tape-bounded probabilistic Turing machine transducers (extended abstract
- SIMON, GILL, et al.
- 1978
(Show Context)
Citation Context ...lations The first deterministic simulation results for bounded-space probabilistic computation were for the most powerful class P rSPACE(s). Gill [15] showed that P rSPACE(s) # DSPACE(2 O(s) ). Simon =-=[44]-=- was the first to show that P rSPACE(s) can be simulated deterministically with only a polynomial increase in space, specifically, in DSPACE(s 6 ). This result was improved independently by Borodin, C... |

3 |
Relationships among pl, #l, and the determininant
- Allender, Ogihara
- 1994
(Show Context)
Citation Context ...CE(s). In contrast, for unbounded error computation, Jung showed that the two classes coincide: Theorem 3.1 ([22]) P rSPACE(s) = P r HSPACE(s). The proof of the non-trivial inclusion as simplified in =-=[4]-=-, follows from (1) the fact listed in figure 2 that P rSPACE(s) can be reduced to the threshold problem for the entry of some matrix inverse, (2) the fact that the threshold problem for a matrix inver... |

3 | Deterministic simulation of tape-bounded probabilistic Turing machine transducers, Theoretical Computer Science 12 - Gill, Hunt, et al. - 1980 |

1 |
Pseudorandom generators for combinatorial rectangles
- Armoni, Saks, et al.
- 1996
(Show Context)
Citation Context ...)-rectangles is open; the best known constructions are (1) a generator with seed length log m + O(log 2 d + log d log(1/#)) which can be constructed using the INW generator, and (2) a construction in =-=[5]-=-, which gives such a generator whose seed length is O(logm + log d + log 2 (1/#)). 5.4 Amplification and computing with weak random sources Some of the most interesting recent work in timeboundedsprob... |