# On Pure Space vs Catalytic Space

18 Oct 2020-pp 439-451

TL;DR: In this article, the power of catalytic computation when the full memory for which the content needs to be restored to original content at the end of the computation is much more than exponential in the pure space (s(n), the empty memory which does not have any access/restoration constraints).

Abstract: This paper explores the power of catalytic computation when the catalytic space (c(n), the full memory for which the content needs to be restored to original content at the end of the computation) is much more than exponential in the pure space (s(n), the empty memory which does not have any access/restoration constraints). We study the following three regimes of the relation between s(n) and c(n) and explore the class \(\mathsf {CSPACE}(s(n),c(n))\) in each of them.

##### Citations

More filters

•

TL;DR: In this paper, the authors define the notion of catalytic-space computation, which is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished.

Abstract: We define the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. We show that the extra space can be used in a nontrivial way, to compute uniform TC1-circuits with just a logarithmic amount of clean space. The extra space thus works analogously to a catalyst in a chemical reaction. TC1-circuits can compute for example the determinant of a matrix, which is not known to be computable in logspace. In order to obtain our results we study an algebraic model of computation, a variant of straight-line programs. We employ register machines with input registers x1,..., xn and work registers r1,..., rm. The instructions available are of the form ri ← ri±u×v, with u, v registers (distinct from ri) or constants. We wish to compute a function f(x1,..., xn) through a sequence of such instructions. The working registers have some arbitrary initial value ri = τi, and they may be altered throughout the computation, but by the end all registers must be returned to their initial value τi, except for, say, r1 which must hold τ1 + f(x1,..., xn). We show that all of Valiant's class VP, and more, can be computed in this model. This significantly extends the framework and techniques of Ben-Or and Cleve [6]. Upper bounding the power of catalytic computation we show that catalytic logspace is contained in ZPP. We further construct an oracle world where catalytic logpace is equal to PSPACE, and show that under the exponential time hypothesis (ETH), SAT can not be computed in catalytic sub-linear space.

4 citations

##### References

More filters

•

01 Jan 1996TL;DR: This chapter surveys the theory of two-party communication complexity and presents results regarding the following models of computation: • Finite automata • Turing machines • Decision trees • Ordered binary decision diagrams • VLSI chips • Networks of threshold gates.

Abstract: In this chapter we survey the theory of two-party communication complexity. This field of theoretical computer science aims at studying the following, seemingly very simple, scenario: There are two players Alice who holds an n-bit string x and Bob who holds an n-bit string y. Their goal is to communicate in order to compute the value of some boolean function f(x, y), while exchanging a number of bits which is as small as possible. In the first part of this survey we present, mainly by giving examples, some of the results (and techniques) developed as part of this theory. We put an emphasis on proving lower bounds on the amount of communication that must be exchanged in the above scenario for certain functions f . In the second part of this survey we will exemplify the wide applicability of the results proved in the first part to other areas of computer science. While it is obvious that there are many applications of the results to problems in which communication is involved (e.g., in distributed systems), we concentrate on applications in which communication does not appear explicitly in the statement of the problems. In particular, we present results regarding the following models of computation: • Finite automata • Turing machines • Decision trees • Ordered binary decision diagrams (OBDDs) • VLSI chips • Networks of threshold gates We provide references to many other issues and applications of communication complexity which are not discussed in this survey.

2,004 citations

•

01 Jan 2006

TL;DR: This innovative text focuses primarily on computational complexity theory: the classification of computational problems in terms of their inherent complexity.

Abstract: This textbook is uniquely written with dual purpose. It cover cores material in the foundations of computing for graduate students in computer science and also provides an introduction to some more advanced topics for those intending further study in the area. This innovative text focuses primarily on computational complexity theory: the classification of computational problems in terms of their inherent complexity. The book contains an invaluable collection of lectures for first-year graduates on the theory of computation. Topics and features include more than 40 lectures for first year graduate students, and a dozen homework sets and exercises.

98 citations

••

TL;DR: Since L is accepted by a two-way deterministic pushdown automation, it is shown that one pushdown stack is more powerful than one counter for deterministic two way machines.

40 citations

••

31 May 2014TL;DR: Upper bounding the power of catalytic computation is shown, and it is shown that catalytic logspace is contained in ZPP, and that under the exponential time hypothesis (ETH), SAT can not be computed in catalytic sub-linear space.

Abstract: We define the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. We show that the extra space can be used in a nontrivial way, to compute uniform TC1-circuits with just a logarithmic amount of clean space. The extra space thus works analogously to a catalyst in a chemical reaction. TC1-circuits can compute for example the determinant of a matrix, which is not known to be computable in logspace. In order to obtain our results we study an algebraic model of computation, a variant of straight-line programs. We employ register machines with input registers x1,..., xn and work registers r1,..., rm. The instructions available are of the form ri ← ri±u×v, with u, v registers (distinct from ri) or constants. We wish to compute a function f(x1,..., xn) through a sequence of such instructions. The working registers have some arbitrary initial value ri = τi, and they may be altered throughout the computation, but by the end all registers must be returned to their initial value τi, except for, say, r1 which must hold τ1 + f(x1,..., xn). We show that all of Valiant's class VP, and more, can be computed in this model. This significantly extends the framework and techniques of Ben-Or and Cleve [6]. Upper bounding the power of catalytic computation we show that catalytic logspace is contained in ZPP. We further construct an oracle world where catalytic logpace is equal to PSPACE, and show that under the exponential time hypothesis (ETH), SAT can not be computed in catalytic sub-linear space.

40 citations