Space hierarchy theorem revised

AbstractWe show that, for an arbitrary function h(n) and each recursive function ℓ(n), that are separated by a nondeterministically fully space constructible g(n), such that h(n)∈Ω(g(n)) but ℓ(n)∉Ω(g(n)), there exists a unary language L in NSPACE(h(n)) that is not contained in NSPACE(ℓ(n)). The same holds for the deterministic case.The main contribution to the well-known Space Hierarchy Theorem is that (i) the language L separating the two space classes is unary (tally), (ii) the hierarchy is independent of whether h(n) or ℓ(n) are in Ω(logn) or in o(logn), (iii) the functions h(n) or ℓ(n) themselves need not be space constructible nor monotone increasing, (iv) the hierarchy is established both for strong and weak space complexity classes. This allows us to present unary languages in such complexity classes as, for example, NSPACE(loglogn·log∗n)⧹NSPACE(loglogn), using a plain diagonalization

The RAM equivalent of P vs. RP

One of the fundamental open questions in computational complexity is whether the class of problems solvable by use of stochasticity under the Random Polynomial time (RP) model is larger than the class of those solvable in deterministic polynomial time (P). However, this question is only open for Turing Machines, not for Random Access Machines (RAMs). Simon (1981) was able to show that for a sufficiently equipped Random Access Machine, the ability to switch states nondeterministically does not entail any computational advantage. However, in the same paper, Simon describes a different (and arguably more natural) scenario for stochasticity under the RAM model. According to Simon's proposal, instead of receiving a new random bit at each execution step, the RAM program is able to execute the pseudofunction $\textit{RAND}(y)$, which returns a uniformly distributed random integer in the range $[0,y)$. Whether the ability to allot a random integer in this fashion is more powerful than the ability to allot a random bit remained an open question for the last 30 years. In this paper, we close Simon's open problem, by fully characterising the class of languages recognisable in polynomial time by each of the RAMs regarding which the question was posed. We show that for some of these, stochasticity entails no advantage, but, more interestingly, we show that for others it does.Comment: 23 page

A Sharp Separation of Sublogarithmic Space Complexity Classes

We present very sharp separation results for Turing machine sublogarithmic space complexity classes which are of the form: For any, arbitrarily slow growing, recursive nondecreasing and unbounded function s there is a k in N and an unary language L such that L in SPACE(s(n)+k) setminus SPACE(s(n-1)). For a binary L the supposition łims = infty is sufficient. The witness languages differ from each language from the lower classes on infinitely many words. We use so called demon (Turing) machines where the tape limit is given automatically without any construction. The results hold for deterministic and nondeterministic demon machines and also for alternating demon machines with a constant number of alternations, and with unlimited number of alternations. The sharpness of the results is ensured by using a very sensitive measure of space complexity of Turing computations which is defined as the amount of the tape required by the simulation (of the computation in question) on a fixed universal machine. As a proof tool we use a succint diagonalization method

Passively mobile communicating machines that use restricted space

We propose a new theoretical model for passively mobile Wireless Sensor Networks, called PM, standing for Passively mobile Machines. The main modification w.r.t. the Population Protocol model [Angluin et al. 2006] is that the agents now, instead of being automata, are Turing Machines. We provide general definitions for unbounded memories, but we are mainly interested in computations upper-bounded by plausible space limitations. However, we prove that our results hold for more general cases. We focus on complete interaction graphs and define the complexity classes PM-SPACE(f(n)) parametrically, consisting of all predicates that are stably computable by some PM protocol that uses O(f(n)) memory in each agent. We provide a protocol that generates unique identifiers from scratch only by using O(log n) memory, and use it to provide an exact characterization of the classes PMSPACE(f(n)) when f(n) = Ω(log n): they are precisely the classes of all symmetric predicates in NSPACE(nf(n)). As a consequence, we obtain a space hierarchy of the PM model when the memory bounds are Ω(log n). Finally, we establish that the minimal space requirement for the computation of non-semilinear predicates is O(log log n). © 2011 ACM.FOM

On the computational complexity of the languages of general symbolic dynamical systems and beta-shifts

AbstractWe consider the computational complexity of languages of symbolic dynamical systems. In particular, we study complexity hierarchies and membership of the non-uniform class P/poly. We prove: 1.For every time-constructible, non-decreasing function t(n)=ω(n), there is a symbolic dynamical system with language decidable in deterministic time O(n2t(n)), but not in deterministic time o(t(n)).2.For every space-constructible, non-decreasing function s(n)=ω(n), there is a symbolic dynamical system with language decidable in deterministic space O(s(n)), but not in deterministic space o(s(n)).3.There are symbolic dynamical systems having hard and complete languages under ≤mlogs- and ≤mp-reduction for every complexity class above LOGSPACE in the backbone hierarchy (hence, P-complete, NP-complete, coNP-complete, PSPACE-complete, and EXPTIME-complete sets).4.There are decidable languages of symbolic dynamical systems in P/poly for every alphabet of size |Σ|≥1.5.There are decidable languages of symbolic dynamical systems not in P/poly iff the alphabet size is >1.For the particular class of symbolic dynamical systems known as β-shifts, we prove that: 1.For all real numbers β>1, the language of the β-shift is in P/poly.2.If there exists a real number β>1 such that the language of the β-shift is NP-hard under ≤Tp-reduction, then the polynomial hierarchy collapses to the second level. As NP-hardness under ≤mp-reduction implies hardness under ≤Tp-reduction, this result implies that it is unlikely that a proof of existence of an NP-hard language of a β-shift will be forthcoming.3.For every time-constructible, non-decreasing function t(n)≥n, there is a real number 1<β<2 such that the language of the β-shift is decidable in time O(n2t(logn+1)), but not in any proper time bound g(n) satisfying g(4n)=o(t(n)/16n).4.For every space-constructible, non-decreasing function s(n)=ω(n2), there is a real number 1<β<2 such that the language of the β-shift is decidable in space O(s(n)), but not in space g(n) where g is any function satisfying g(n2)=o(s(n)).5.There exists a real number 1<β<2 such that the language of the β-shift is recursive, but not context-sensitive