Nondeterministic Linear Time Research Articles

Time-space lower bounds for satisfiability

We establish the first polynomial time-space lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic random-access Turing machine can solve satisfiability in time n c and space n d , where d approaches 1 when c does. On conondeterministic instead of deterministic machines, we prove the same for any constant c less than √2.Our lower bounds apply to nondeterministic linear time and almost all natural NP-complete problems known. In fact, they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space n 1/c .Our proofs follow the paradigm of indirect diagonalization. We also use that paradigm to prove time-space lower bounds for languages higher up in the polynomial-time hierarchy.

The Expressive Power of Bijections over Weakly Arithmetized Structures

We investigate the expressive power of existential second-order formulas whose second-order quantifiers range over bijective unary functions. We show that as long as interpretations are taken over structures with a built-in linear order relation and an addition function, then quantifying over bijections is as expressive as quantifying over arbitrary unary functions. The originality of our result is that it remains true even if the first-order part of a formula contains exactly one variable (which is universally quantified). Our result immediately provides a new characterization of non-deterministic linear time on RAMs. It also permits us to derive a corollary on the Skolem normal form of first-order formulas over weakly arithmetized structures.

A theory for Log-Space and NLIN versus co-NLIN

AbstractThe use of Nepomnjaščiǐ's Theorem in the proofs of independence results for bounded arithmetic theories is investigated. Using this result and similar ideas, it is shown that at least one of S1 or TLS does not prove the Matiyasevich-Robinson-Davis-Putnam Theorem. It is also established that TLS does not prove a statement that roughly means nondeterministic linear time is equal to co-nondeterministic linear time. Here S1 is a conservative extension of the well-studied theory IΔ0 and TLS is a theory for LOGSPACE reasoning.

Linear Time and the Power of One First-Order Universal Quantifier

The aim of this paper is to give two new logical characterizations of NLIN (nondeterministic linear time) improving significantly a previous characterization of E. Grandjean (1994, SIAM J. Comput. 23 , 573–597; and F. Olive, 1998, Comput. Complexity 7 , 54–97). It is known that NLIN coincides with the class of problems definable by formulas of the prenex form ∃ f 1 …∃ f k ϕ where the f i are unary function symbols and ϕ is first-order, prenex, with only one universal quantifier. We show that the characterization remains true in the two following cases: (a) Unary functions are replaced by a single binary relation whose outdegree is bounded by some fixed constant h . (b) When only ordered structures are considered, unary functions are restricted to be bijective.

Linear Time and the Power of One First-Order Universal Quantifier

The aim of this paper is to give two new logical characterizations of NLIN (nondeterministic linear time) improving significantly a previous characterization of E. Grandjean (1994, SIAM J. Comput. 23 , 573–597; and F. Olive, 1998, Comput. Complexity 7 , 54–97). It is known that NLIN coincides with the class of problems definable by formulas of the prenex form ∃ f 1 …∃ f k ϕ where the f i are unary function symbols and ϕ is first-order, prenex, with only one universal quantifier. We show that the characterization remains true in the two following cases: (a) Unary functions are replaced by a single binary relation whose outdegree is bounded by some fixed constant h . (b) When only ordered structures are considered, unary functions are restricted to be bijective.

Monadic logical definability of nondeterministic linear time

We answer a question asked by J. F. Lynch by proving that existential monadic second-order logic with addition captures not only the class NTIME(n) but also the class NLIN (i.e., linear time on nondeterministic RAMs), so enlarging considerably the set of natural problems expressible in this logic, since most combinatorial NP-complete problems belong to NLIN. Moreover, our result still holds if the first-order part of the formulas is required to be \( \forall^{\ast} \exists ^{\ast} \), so improving the recent similar result by J. F. Lynch about NTIME(n).¶In addition, we explicitly state that a graph problem is recognizable in nondeterministic linear time O(n + e) (where n and e are the numbers of vertices and edges, respectively) if and only if it can be defined in existential second-order logic with unary functions and only one variable on the vertices-edges domain.

The Power of the Queue

Queues, stacks, and tapes are basic concepts that have direct applications in compiler design and the general design of algorithms. Whereas stacks (pushdown store or last-in–first-out storage) have been thoroughly investigated and are well understood, this is much less the case for queues (first-in-first-out storage). In this paper a comprehensive study comparing queues to stacks and tapes (off-line and with a one-way input tape) is presented. The techniques used rely on Kolmogorov complexity. In particular, one queue and one tape (or stack) are incomparable: (1) Simulating one stack (and hence one tape) by one queue requires $\Omega (n^{4/3} /\log n)$ time in both the deterministic and the nondeterministic cases. A corollary of this lower bound states that for this model of one-queue machines, nondeterministic linear time is not closed under complement. (2) Simulating one queue by one tape requires $\Omega (n^2 )$ time in the deterministic case and requires $\Omega (n^{4/3}/(\log n)^{2/3} )$ in the nonde...