Alternating on-line turing machines with only universal states and small space bounds

AbstractLet L[AONTM(L(n))] be the class of sets accepted by L(n) space bounded alternating on-line Turing machines, and L[UONTM(L(n))] be the class of sets accepted by L(n) space bounded alternating on-line Turing machines with only universal states. This note first shows that, for any L(n) such that L(n) ⩾ log log n and limn → ∞[L(n)/log n] = 0, (i) L[UONTM(L(n))] ⊋ L[AONTM(L(n))], (ii) L[UONTM(L(n))] is not closed under complementation, and (iii) L[UONTM(L(n))] is properly contained in the class of sets accepted by L(n) space bounded alternating Turing machines with only universal states. We then show that there exists an infinite hierarchy among L[UONTM(L(n))]'s with log log n ⩽ L(n) ⩽ log n

Lower Bounds for Alternating Online State Complexity

The notion of Online State Complexity, introduced by Karp in 1967, quantifies the amount of states required to solve a given problem using an online algorithm, which is represented by a deterministic machine scanning the input from left to right in one pass. In this paper, we extend the setting to alternating machines as introduced by Chandra, Kozen and Stockmeyer in 1976: such machines run independent passes scanning the input from left to right and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem , stating that the polynomial hierarchy of alternating online state complexity is infinite, and the second is a linear lower bound on the alternating online state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model

On space-bounded synchronized alternating Turing machines

AbstractWe continue the study of the computational power of synchronized alternating Turing machines (SATM) introduced in (Hromkovič 1986, Slobodová 1987, 1988a, b) to allow communication via synchronization among processes of alternating Turing machines. We are interested in comparing the four main classes of space-bounded synchronized alternating Turing machines obtained by adding or removing off-line capability and nondeterminism (1SUTM(S(n)), SUTM(S(n)), 1SATM(S(n)), and SATM(S(n)) against one another and against other variants of alternating Turing machines. Denoting the class of languages accepted by machines in C by L(C), we show as our main results that L(1SUTM(S(n))) ⊂ L(SUTM(S(n))) ⊂ L(1SATM(S(n)))= L(SATM(S(n))) for all space-bounded functions S(n)ϵo(n), and L(1SUTM(S(n)))= L(SUTM(S(n))) ⊂ L(1SATM(S(n)))=L(SATM(S(n))) for S(n)) ⩾ n. Furthermore, we show that for log log(n) ⩽ S(n)ϵo(log(n)), L(1SUTM(S(n))) is incomparable to L[1] ATM(S(n))). L(UTM(S(n))), L(1MUTM(S(n))), and L(MUTM(S(n))), where MATMs are alternating Turing machines with modified acceptance proposed in (Inoue 1989); in contrast, we show that these relationships become proper inclusions when log(n) ⩽ S(n)ϵo(n).For deterministic synchronized alternating finite automata with at most k processes (1DSA(k)FA and DSA(k)FA) we establish a tight hierarchy on the number of processes for the one-way case, namely, L(1DSA(n)FA) ⊂ L(1DSA(n+1)FA) for all n > 0, and show that L(1DFA(2)) − ∪k=1∞L(DSA(k)FA) ≠ ∅, where DFA(k) denotes deterministic k-head finite automata. Finally we investigate closure properties under Boolean operations for some of these classes of languages

Two-Way Automata Making Choices Only at the Endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2DFAs (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2DFAs. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2NFAs and 2DFAs can be obtained only for unrestricted 2NFAs using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2DFAs would imply LNL. In the same way, the alternating version of these machines is related to L =? NL =? P, the classical computational complexity problems.Comment: 23 page

A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences

Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity, and that lossless compression algorithms fall short at characterizing patterns other than statistical ones not different to entropy estimations, here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure $m$ calculated from the output distribution of small Turing machines. We introduce and justify finite approximations $m_k$ that have been used in some applications as an alternative to lossless compression algorithms for approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of the relevant properties of both $m$ and $m_k$ and compare them to Levin's Universal Distribution. We provide error estimations of $m_k$ with respect to $m$. Finally, we present an application to integer sequences from the Online Encyclopedia of Integer Sequences which suggests that our AP-based measures may characterize non-statistical patterns, and we report interesting correlations with textual, function and program description lengths of the said sequences.Comment: As accepted by the journal Complexity (Wiley/Hindawi

How Much Lookahead is Needed to Win Infinite Games?

Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. For $\omega$-regular winning conditions it is known that such games can be solved in doubly-exponential time and that doubly-exponential lookahead is sufficient. We improve upon both results by giving an exponential time algorithm and an exponential upper bound on the necessary lookahead. This is complemented by showing EXPTIME-hardness of the solution problem and tight exponential lower bounds on the lookahead. Both lower bounds already hold for safety conditions. Furthermore, solving delay games with reachability conditions is shown to be PSPACE-complete. This is a corrected version of the paper https://arxiv.org/abs/1412.3701v4 published originally on August 26, 2016

Time and Space Bounds for Reversible Simulation

We prove a general upper bound on the tradeoff between time and space that suffices for the reversible simulation of irreversible computation. Previously, only simulations using exponential time or quadratic space were known. The tradeoff shows for the first time that we can simultaneously achieve subexponential time and subquadratic space. The boundary values are the exponential time with hardly any extra space required by the Lange-McKenzie-Tapp method and the ($\log 3$)th power time with square space required by the Bennett method. We also give the first general lower bound on the extra storage space required by general reversible simulation. This lower bound is optimal in that it is achieved by some reversible simulations.Comment: 11 pages LaTeX, Proc ICALP 2001, Lecture Notes in Computer Science, Vol xxx Springer-Verlag, Berlin, 200

Complexity Hierarchies Beyond Elementary

We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a non-elementary complexity, which occur naturally in logic, combinatorics, formal languages, verification, etc., with complexities ranging from simple towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep updating the catalogue of problems from Section 6 in future revision

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature