Space-bounded Turing Machines Research Articles

Simple programming languages and restricted classes of turing machines

In this paper we investigate the computational power of simple programming languages and provide characterizations (or partial characterizations) of the functions computable by such programs in terms of some space and/or time complexity classes of Turing machines. Call a function f( x 1,…, x 1) over the nonnegative integers linearly bounded if f( x 1,…, x 1)< O( x 1+…+ x 1). We show that any linearly bounded function f( x 1,…, x 1) computable by a Turing machine in O( n) space and O(2 λn ) time, where λ<1 and n=¦x 1+…+x 1¦ is the length of the binary representation of x 1+…+ x 1, can be computed by a program without nested loops using only the instruction set R=x←x−1, if x=0 then y←y+1, do x…lcubendrcub . Thus, functions like ¦x y ¦, ged{x,y}, x k , [log x] , etc. can be computed by R-programs. Any function computable by an R -program can be computed by a Turing machine in O( n) space and O(2 n ) time. It is open whether or not the time can be reduced to O(2 λ ) for some λ<1 (which may depend on the program). We also introduce simple programming languages which are complete characterizations of the functions computable by certain space bounded (linear bounded, polynomial bounded. etc.) classes of Turing machines. For example, we show that a function f( x 1,…, x 1 over the nonnegative integers is computable by a linear space bounded Turing machine if and only if it is computable by a program using only the constructs x←1, x←x+y, x←x−y, do x… end such that (i) there are no nested loops, and (ii) the occurence of a construct x← x+ y in a loop precludes the occurrence of a construct y← x+ z within the same loop. Extensions are also made for some time complexity classes of Turing machines.

Two-dimensional alternating turing machines with only universal states

Several properties of two-dimensional alternating Turing machines are investigated. The first part of this paper investigates the relationship between the classes of sets accepted by space-bounded and finitely leaf-size bounded three-way two-dimensional alternating Turing machines and the classes of sets which are finite intersections of sets accepted by space-bounded three-way two-dimensional nondeterministic Turing machines. The second part of this paper investigates the accepting power and closure properties (under Boolean operations) of two-dimensional alternating Turing machines with only universal states.

On time versus space II

Every t(n)-time bounded RAM (assuming the logarithmic cost measure) can be simulated by a t(n)/log t(n)-space bounded Turing machine and every t(n)-time bounded Turing machine with d-dimensional tapes by a t(n)5 d· log ∗t(n) / log t(n)- space bounded machine, where n is the length of the input. A class E of storage structures which generalizes multidimensional tapes is defined. Every t(n)-time bounded Turing machine whose storage structures are in E can be simulated by a t(n) loglog t(n)/log t(n)-space bounded Turing machine.

The covering problem for linear context-free grammars

Language equivalence, grammatical covering and structural equivalence are all notions of similarity defined on context-free grammars. We show that the problem of determining whether an arbitrary linear context-free grammar covers another is complete for the class of languages accepted by polynomially space bounded Turing machines. We then compare the complexity of this problem with the analogous problems for language equivalence and structural equivalence, not only for linear grammars, but also for regular grammars and unrestricted context-free grammars. As a step in obtaining the main result of this paper, we show that the equivalence problem for linear s-grammars is decidable in polynomial time.