Nondeterministic Turing Machines Research Articles

Weighted automata and logics meet computational complexity

Complexity classes such as #P, ⊕P, GapP, OptP, NPMV, or the class of fuzzy languages realised by polynomial-time fuzzy nondeterministic Turing machines, can all be described in terms of a class NP[S] for a suitable semiring S, defined via weighted Turing machines over S similarly as NP is defined in the unweighted setting. Other complexity classes can be lifted to the quantitative world as well, the resulting classes relating to the original ones in the same way as weighted automata or logics relate to their unweighted counterparts. The article surveys these too-little-known connexions implicit in the existing literature and suggests a systematic approach to the study of weighted complexity classes. An extension of SAT to weighted propositional logic is proved to be complete for NP[S] when S is finitely generated. Moreover, a class FP[S] is introduced for each semiring S as a counterpart to P, and the relations between FP[S] and NP[S] are considered.

Designing of Reversible Functions in Classical and Quantum Domains

First sections of the paper contain some considerations relevant to the reversibility of quantum gates. The Solovay-Kitayev theorem shows that using proper set of quantum gates one can build a quantum version of the nondeterministic Turing machine. On the other hand the Gottesmann-Knill theorem shows the possibility to simulate the quantum machine consisting of only Clifford/Pauli group of gates. This paper presents also an original method of designing the reversible functions. This method is intended for the most popular gate set with three types of gates CNT (Control, NOT and Toffoli). The presented algorithm leads to cascade with minimal number CNT gates. This solution is called optimal reversible circuits. The paper is organized as follows. Section 5 recalls basic concepts of reversible logic. Section 6 contain short description of CNT set of the reversible gates. In Section 7 is presented form of result of designing as the cascade of gates. Section 8 describes the algorithm and section 9 simple example.

Guest column: A panorama of counting problems the decision version of which is in P 3

Since Valiant's seminal work, where the complexity class #P was defined, much research has been done on counting complexity, from various perspectives. A question that permeates many of these efforts concerns the approximability of counting problems, in particular which of them admit an efficient approximation scheme ( fpras ). A counting problem (a function from Σ * to N) that admits an fpras must necessarily have an easy way to decide whether the output value is nonzero. Having this in mind, we focus our attention on classes of counting problems, the decision version of which is in P (or in RP). We discuss structural characterizations for classes of such problems under various lenses: Cook and Karp reductions, path counting in non-deterministic Turing machines, approximability and approximation-preserving reductions, easy decision by randomization, descriptive complexity, and interval-size functions. We end up with a rich landscape inside #P, revealing a number of inclusions and separations among complexity classes of easy-to-decide counting problems.

A nondeterministic Turing machine variant to compute functions

Formally defining how outputs of nondeterministic computations may be combined, we propose a nondeterministic Turing machine variant to compute functions. First, we describe our inspiration, the association of a computation tree with the divide and conquer strategy. Next, we define the variant and prove its equivalence with the deterministic Turing machine model. Then, we give some examples and relate the variant to alternating, counting and metric Turing machines. After, we define the variant's time complexity classes and compare them to deterministic space, giving a new characterization of the ▪ and ▪ classes. Finally, we outline future work.

On Turing Machines Deciding According to the Shortest Computations

In this paper we propose and analyse from the computational complexity point of view several new variants of nondeterministic Turing machines. In the first such variant, a machine accepts a given input word if and only if one of its shortest possible computations on that word is accepting; on the other hand, the machine rejects the input word when all the shortest computations performed by the machine on that word are rejecting. We are able to show that the class of languages decided in polynomial time by such machines is PNP[log]. When we consider machines that decide a word according to the decision taken by the lexicographically first shortest computation, we obtain a new characterization of PNP. A series of other ways of deciding a language with respect to the shortest computations of a Turing machine are also discussed.

Verification of Composed Web Service Using Synthesized Nondeterministic Turing Model (SNTMM) With Multiple Tapes and Stacks

To verify the composed web services, a general view of what traits of a service need to be identified is still lacking. The existing verification model did not address any mechanism for getting alternative services if we failed to reach the desired service and partially concentrated on the reachability problem for a deterministic and non-deterministic system in sequential. This paper proposes a synthesised non-deterministic turing machine model (SNTMM) by combining the multistacked non-deterministic Turing machine (MSNTM) model and multitaped non-deterministic Turing machine (MTNTM) model to verify the composed web services for both deterministic and non-deterministic systems in parallel. The deceased transition and departed service marking algorithm have been proposed to address each participated service's reachability in composing service for all possible input in parallel. This article shows an example to demonstrate the meticulousness of the model. The experimental results show that the performance of the proposed model is measured efficiently.

Clocked population protocols

We define an extension to the standard population protocol that provides each agent with a clock signal that indicates when the agent has waited long enough for the protocol to have converged. We represent “long enough” as an infinite time interval, and treat the protocol as operating over transfinite time. Over finite time intervals, the protocol behaves as in the standard model. At nonzero limit ordinals, corresponding to clock ticks, the protocol switches to a limit of previous configurations supplemented by a clock signal appearing as an extra component in some of the agents' states. Fairness generalizes to transfinite executions straightforwardly. We show that a clocked population protocol running in less than ωk time for any fixed k≥2 is equivalent in power to a nondeterministic Turing machine with space complexity logarithmic in the population size, and can be simulated using only finitely many clock ticks.

A Short Essay towards if P not equal NP

We find a computational algorithmic task and prove that it is solvable in polynomial time by a non-deterministic Turing machine and cannot be solved in polynomial time by any deterministic Turing machine. The point is that our task does not look as very canonical one and if it may be classified as computational problem in standard terms

Axiomatizing Rectangular Grids with no Extra Non-unary Relations

We construct a first-order formula φ such that all finite models of φ are non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A ⊆ ℕ is a spectrum of a formula which has only planar models if numbers n ∈ A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t(n) and space s(n), where t(n)s(n) ≤ n and t(n); s(n) = Ω(log(n)).

Complexity-theoretic aspects of expanding cellular automata

The expanding cellular automata (XCA) variant of cellular automata is investigated and characterized from a complexity-theoretical standpoint. An XCA is a one-dimensional cellular automaton which can dynamically create new cells between existing ones. The respective polynomial-time complexity class is shown to coincide with {le _{tt}^p}(textsf {NP}), that is, the class of decision problems polynomial-time truth-table reducible to problems in textsf {NP}. An alternative characterization based on a variant of non-deterministic Turing machines is also given. In addition, corollaries on select XCA variants are proven: XCAs with multiple accept and reject states are shown to be polynomial-time equivalent to the original XCA model. Finally, XCAs with alternative acceptance conditions are considered and classified in terms of {le _{tt}^p}(textsf {NP}) and the Turing machine polynomial-time class textsf {P}.

Non-Closure Properties of Multi-Inkdot Nondeterministic Turing Machines with Sublogarithmic Space

Non-Closure Properties of Multi-Inkdot Nondeterministic Turing Machines with Sublogarithmic Space

A structured view on weighted counting with relations to counting, quantum computation and applications

Weighted counting problems are a natural generalization of counting problems where a weight is associated with every computational path of polynomial-time non-deterministic Turing machines. The goal is to compute the sum of weights of all paths (instead of number of accepting paths). Useful properties and plenty of applications make them interesting. The definition captures even undecidable problems, but obtaining an exponentially small additive approximation is just as hard as solving conventional counting. We present a structured view by defining classes that depend on the functions that assign weights to paths and by showing their relationships and how they generalize counting problems. Weighted counting is flexible and allows us to cast a number of famous results of computational complexity, including quantum computation, probabilistic graphical models and stochastic combinatorial optimization. Using the weighted counting terminology, we are able to simplify and to answer some open questions.

Networks of Uniform Splicing Processors: Computational Power and Simulation

We investigated the computational power of a new variant of network of splicing processors, which simplifies the general model such that filters remain associated with nodes but the input and output filters of every node coincide. This variant, called network of uniform splicing processors, might be implemented more easily. Although the communication in the new variant seems less powerful, the new variant is sufficiently powerful to be computationally complete. Thus, nondeterministic Turing machines were simulated by networks of uniform splicing processors whose size depends linearly on the alphabet of the Turing machine. Furthermore, the simulation was time efficient. We argue that the network size can be decreased to a constant, namely six nodes. We further show that networks with only two nodes are able to simulate 2-tag systems. After these theoretical results, we discuss a possible software implementation of this model by proposing a conceptual architecture and describe all its components.

Membership Problem for Two-Dimensional General Row Jumping Finite Automata

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

A restricted second-order logic for non-deterministic poly-logarithmic time

AbstractWe introduce a restricted second-order logic $\textrm{SO}^{\textit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin’s style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\textrm{SO}^{\textit{plog}}$ correspond exactly to the class of decision problems that can be computed by a non-deterministic Turing machine with random access to the input in time $O((\log n)^k)$ for some $k \ge 0$, i.e. to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin’s theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\textrm{SO}^{\textit{plog}}$ is too weak as to define a total order of the domain. Nevertheless, $\textrm{SO}^{\textit{plog}}$ provides natural levels of expressibility within poly-logarithmic space in a way which is closely related to how second-order logic provides natural levels of expressibility within polynomial space. Indeed, we show an exact correspondence between the quantifier prefix classes of $\textrm{SO}^{\textit{plog}}$ and the levels of the non-deterministic poly-logarithmic time hierarchy, analogous to the correspondence between the quantifier prefix classes of second-order logic and the polynomial-time hierarchy. Our work closely relates to the constant depth quasipolynomial size AND/OR circuits and corresponding restricted second-order logic defined by David A. Mix Barrington in 1992. We explore this relationship in detail.

The Gridlock-Proof Functionality in Real Time Gross Settlement Systems

The last two decades of the XX century marked the starting point for the central banks across the globe to move their payment and settlement systems into Real Time Gross Settlement (RTGS) mode. Despite the fact that RTGS systems can effectively eliminate the credit exposure between the paying bank and the receiving bank at the interbank level by means of fast final and irrevocable money transfer, there is another serious problem associated with these systems. RTGS systems turned out to be liquidity-demanding arrangements, as opposed to deferred net settlement systems. Thus, the efficiency of liquidity management arrangements is the precondition of smooth RTGS operation, especially in tough times when liquidity is a systemic shortage. If liquidity management is inefficient, the RTGS system may stop operating properly by terminating in the grid-lock state brining chaos to the national economy. In this research we suggest the practical approach to solve the problem of seeking the maximization of aggregate value of payment instructions under liquidity shortage, including the most severe scenarios. The classification of the RTGS queue statuses is suggested and discussed. Some complementary results are articulated, including: (a) the statement that the formal mathematical optimization problem lies in the NP class of the computational complexity (the class of problems solved in polynomial time by nondeterministic Turing machine); (b) the equivalence between MaxFlow-MinCost problem (from the network flow theory) and the dual linear problem of the linear program relaxation of the initial optimization problem; (c) the illustration that no optimization strategy, other than the suggested one, can deliver substantially better optimization results. Despite enormous efforts, there were no previous research results reasonably claiming the near optimality of liquidity optimization strategy in RTGS systems under severe liquidity shortages. The results of this research may help the central banks and other RTGS system operators to ensure the protection of their payment systems from future liquidity crises and bring the resilience of respective national economies to the next level of sustainability.

Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity

Compositions of tree-walking tree transducers form a hierarchy with respect to the number of transducers in the composition. As main technical result it is proved that any such composition can be realized as a linear-bounded composition, which means that the sizes of the intermediate results can be chosen to be at most linear in the size of the output tree. This has consequences for the expressiveness and complexity of the translations in the hierarchy. First, if the computed translation is a function of linear size increase, i.e., the size of the output tree is at most linear in the size of the input tree, then it can be realized by just one, deterministic, tree-walking tree transducer. For compositions of deterministic transducers it is decidable whether or not the translation is of linear size increase. Second, every composition of deterministic transducers can be computed in deterministic linear time on a RAM and in deterministic linear space on a Turing machine, measured in the sum of the sizes of the input and output tree. Similarly, every composition of nondeterministic transducers can be computed in simultaneous polynomial time and linear space on a nondeterministic Turing machine. Their output tree languages are deterministic context-sensitive, i.e., can be recognized in deterministic linear space on a Turing machine. The membership problem for compositions of nondeterministic translations is nondeterministic polynomial time and deterministic linear space. All the above results also hold for compositions of macro tree transducers. The membership problem for the composition of a nondeterministic and a deterministic tree-walking tree translation (for a nondeterministic IO macro tree translation) is log-space reducible to a context-free language, whereas the membership problem for the composition of a deterministic and a nondeterministic tree-walking tree translation (for a nondeterministic OI macro tree translation) is possibly NP-complete.

On counting functions and slenderness of languages

We study counting-regular languages — these are languages L for which there is a regular language L′ such that the number of strings of length n in L and L′ are the same for all n. We show that the languages accepted by unambiguous nondeterministic Turing machines with a one-way read-only input tape and a reversal-bounded worktape are counting-regular. Many one-way acceptors are a special case of this model, such as reversal-bounded deterministic pushdown automata, reversal-bounded deterministic queue automata, and many others, and therefore all languages accepted by these models are counting-regular. This result is the best possible in the sense that the claim does not hold for either 2-ambiguous NPDA's, unambiguous NPDA's with no reversal-bound, and other models.We also study closure properties of counting-regular languages, and we study decidability problems in regards to counting-regularity. For example, it is shown that the counting-regularity of even some restricted subclasses of NPDA's is undecidable. Lastly, k-slender languages — where there are at most k words of any length — are also studied. Amongst other results, it is shown that it is decidable whether a language in any semilinear full trio is k-slender.

What can be verified locally?

In the framework of distributed network computing, it is known that not all Turing-decidable predicates on labeled networks can be decided locally whenever the computing entities are Turing machines (TM). This holds even if nodes are running non-deterministic Turing machines (NTM). In contrast, we show that every Turing-decidable predicate on labeled networks can be decided locally if nodes are running alternating Turing machines (ATM). More specifically, we show that, for every such predicate, there is a local algorithm for ATMs, with at most two alternations, that decides whether the actual labeled network satisfies that predicate. To this aim, we define a hierarchy of classes of decision tasks, where the lowest level contains tasks solvable with TMs, the first level those solvable with NTMs, and the level k>1 contains those tasks solvable with ATMs with k−1 alternations. We characterize the entire hierarchy, and show that it collapses in the second level.

Small networks of polarized splicing processors are universal

In this paper, we consider the computational power of a new variant of networks of splicing processors in which each processor as well as the data navigating throughout the network are now considered to be polarized. While the polarization of every processor is predefined (negative, neutral, positive), the polarization of data is dynamically computed by means of a valuation mapping. Consequently, the protocol of communication is naturally defined by means of this polarization. We show that networks of polarized splicing processors (NPSP) of size 2 are computationally complete, which immediately settles the question of designing computationally complete NPSPs of minimal size. With two more nodes we can simulate every nondeterministic Turing machine without increasing the time complexity. Particularly, we prove that NPSP of size 4 can accept all languages in NP in polynomial time. Furthermore, another computational model that is universal, namely the 2-tag system, can be simulated by NPSP of size 3 preserving the time complexity. All these results can be obtained with NPSPs with valuations in the set \(\{-1,0,1\}\) as well. We finally show that Turing machines can simulate a variant of NPSPs and discuss the time complexity of this simulation.