Turing Machine Model Research Articles

Machine semantics

Using simple systems with a notion of discrete deterministic evolution over time, we study discrete causality via tools from theoretical computer science and logic. We consider the set of all representations (i.e. partial descriptions) of such systems, from algebraic, domain-theoretic, and categorical viewpoints. The order theory introduced, is based in the notion of comparing high-level and low-level descriptions of the same system. This is shown to give a Complete Partial Order where the down-closure of each element is a locale. This partial order has a very close connection to a categorical construction known as the ‘particle-style’ trace, via analogues of domain-theoretic equations. Thus, the trace may be thought of as the computation of suprema in this partial order. As a sample application, we show how to construct algebraic models of space-bounded Turing machines in these terms, and derive compositionality from the abstract properties of the trace.

A New Characterization of NP, P, and PSPACE with Accepting Hybrid Networks of Evolutionary Processors

We consider three complexity classes defined on Accepting Hybrid Networks of Evolutionary Processors (AHNEP) and compare them with the classical complexity classes defined on the standard computing model of Turing machine. By definition, AHNEPs are deterministic. We prove that the classical complexity class NP equals the family of languages decided by AHNEPs in polynomial time. A language is in P if and only if it is decided by an AHNEP in polynomial time and space. We also show that PSPACE equals the family of languages decided by AHNEPs in polynomial length.

Note on a universal quantum Turing machine

In this Letter, we construct a novel model of universal quantum Turing machine (QTM) by means of a property of Riemann zeta function, which is free from the specific time for an input data and efficiently simulates each step of a given QTM.

On the Complexity of Convex Hulls of Subsets of the Two-Dimensional Plane

We investigate the computational complexity of computing the convex hull of a two-dimensional set. We study this problem in the polynomial-time complexity theory of real functions based on the oracle Turing machine model. We show that the convex hull of a two-dimensional Jordan domain S is not necessarily recursively recognizable even if S is polynomial-time recognizable. On the other hand, if the boundary of a Jordan domain S is polynomial-time computable, then the convex hull of S must be NP-recognizable, and it is not necessarily polynomial-time recognizable if P≠NP. We also show that the area of the convex hull of a Jordan domain S with a polynomial-time computable boundary can be computed in polynomial time relative to an oracle function in #P. On the other hand, whether the area itself is a #P real number depends on the open question of whether NP=UP.

Consistency, Turing Computability and Gödel’s First Incompleteness Theorem

It is well understood and appreciated that Godel's Incompleteness Theorems apply to sufficiently strong, formal deductive systems. In particular, the theorems apply to systems which are adequate for conventional number theory. Less well known is that there exist algorithms which can be applied to such a system to generate a godel-sentence for that system. Although the generation of a sentence is not equivalent to proving its truth, the present paper argues that the existence of these algorithms, when conjoined with Godel's results and accepted theorems of recursion theory, does provide the basis for an apparent paradox. The difficulty arises when such an algorithm is embedded within a computer program of sufficient arithmetic power. The required computer program (an AI system) is described herein, and the paradox is derived. A solution to the paradox is proposed, which, it is argued, illuminates the truth status of axioms in formal models of programs and Turing machines.

On the complexity of the pancake problem

AbstractWe study the computational complexity of finding a line that bisects simultaneously two sets in the two‐dimensional plane, called the pancake problem, using the oracle Turing machine model of Ko. We also study the basic problem of bisecting a set at a given direction. Our main results are:(1) The complexity of bisecting a nice (thick) polynomial‐time approximable set at a given direction can be characterized by the counting class #P.(2) The complexity of bisecting simultaneously two linearly separable nice (thick) polynomial‐time approximable sets can be characterized by the counting class #P.(3) For either of these two problems, without the thickness condition and the linear separability condition (for the two‐set case), it is arbitrarily hard to compute the bisector, even if it is unique. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cooperative combinatorial optimization: Evolutionary computation case study

This paper presents a formalization of the notion of cooperation and competition of multiple systems that work toward a common optimization goal of the population using evolutionary computation techniques. It is proved that evolutionary algorithms are more expressive than conventional recursive algorithms, such as Turing machines. Three classes of evolutionary computations are introduced and studied: bounded finite, unbounded finite, and infinite computations. Universal evolutionary algorithms are constructed. Such properties of evolutionary algorithms as completeness, optimality, and search decidability are examined. A natural extension of evolutionary Turing machine (ETM) model is proposed to properly reflect phenomena of cooperation and competition in the whole population.

The Complexity of Tensor Circuit Evaluation

We continue the study of tensor calculus over semirings in terms of complexity theory initiated by Damm et al. (2003). First, we look at tensor circuits, a natural generalization of tensor formulas; we show that the problem to determine whether the circuit output over a certain semiring is non-zero is complete for NE = NTime(2 O(n)) over the Boolean semiring, for $$\oplus$$ E over the field $$\mathbb{F}_{2}$$ , and for analogous classes over other semirings. Moreover, common-sense restrictions such as imposing bounds on circuit and/or tensor depth, are shown to elegantly capture the classes P, NTime $$(2^{O(log^{k} n)})$$ , NTimeSpace( $$2^{O(log^{k} n)}$$ , n O(1)), for k ? 1, PSPACE, and their counting counterparts. The proofs of these results use a model of algebraic Turing machines over a semiring together with a predicate-based approach on counting, which is similar to that of Toda (1991). This allows characterizations of the classes $$\oplus$$ P, NP, co-NP, co-DP, C=P, SPP, USP, and UP, and their exponential time counterparts, in a single framework. Finally, we show that a number of natural problems concerning tensor formulas and circuits, such as asking whether the output of a formula/circuit is a diagonal matrix, or the identity matrix, or a permutation matrix, capture the classes $$\prod^{p}_{2}$$ for formulas and $$\prod^{e}_{2}$$ for circuits over the Boolean semiring; other semirings are also discussed.

The $-calculus process algebra for problem solving: A paradigmatic shift in handling hard computational problems

The $-calculus is the extension of the π -calculus, built around the central notion of cost and allowing infinity in its operators. We propose the $-calculus as a more complete model for problem solving to provide a support to handle intractability and undecidability. It goes beyond the Turing Machine model. We define the semantics of the $-calculus using a novel optimization method (the k Ω -optimization), which approximates a nonexisting universal search algorithm and allows the simulation of many other search methods. In particular, the notion of total optimality has been utilized to provide an automatic way to deal with intractability of problem solving by optimizing together the quality of solutions and search costs. The sufficient conditions needed for completeness, optimality and total optimality of problem solving search are defined. A very flexible classification scheme of problem solving methods into easy, hard and solvable in the limit classes has been proposed. In particular, the third class deals with non-recursive solutions of undecidable problems. The approach is illustrated by solutions of some intractable and undecidable problems. We also briefly overview two possible implementations of the $-calculus.

On the strongly generic undecidability of the Halting Problem

It has been shown in [J.D. Hamkins, A. Miasnikov, The halting problem is decidable on a set of asymptotic probability one, Notre Dame J. Formal Logic 47(4) (2006) 515–524] that the classical Halting Problem for Turing machines with one-way tape is decidable on a “large” set of Turing machines (a so-called generic set). However, here we prove that the Halting Problem remains undecidable when restricted to an arbitrary “very large” set of Turing machines (a so-called strongly generic set). Our proof is independent of a Turing machine model.

On the Complexity of the Pancake Problem

We study the computational complexity of finding a line that bisects simultaneously two sets in the two-dimensional plane, called the pancake problem, using the oracle Turing machine model of Ko. W...

On the Complexity of the Pancake Problem

We study the computational complexity of finding a line that bisects simultaneously two sets in the two-dimensional plane, called the pancake problem, using the oracle Turing machine model of Ko. We also study the basic problem of bisecting a set at a given direction. Our main results are: (1) the complexity of bisecting a nice (thick) polynomial-time approximable set at a given direction can be characterized by the counting class #P; (2) the complexity of bisecting simultaneously two linearly separable nice (thick) polynomial-time approximable sets can be characterized by the counting class #P; and (3) for either of these two problems, without the thickness condition and the linear separability condition (for the two-set case), it is arbitrarily hard to compute the bisector (even if it is unique).

Inductive Time-Space Lower Bounds for Sat and Related Problems

We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism and alternating computation, on both subpolynomial (n o(1)) space RAMs and sequential one-tape machines with random access to the input. We obtain improved lower bounds for Boolean satisfiability (SAT), as well as all NP-complete problems that have efficient reductions from SAT, and k-SAT, for constant k 2. For example, SAT cannot be solved by random access machines using time and subpolynomial space. 2. We show how indirect diagonalization leads to time-space lower bounds for computation with bounded nondeterminism. For both the random access and multitape Turing machine models, we prove that for all k 1, there is a constant c k > 1 such that linear time with n 1/k nondeterministic bits is not contained in deterministic $$n^{{c}_{k}}$$ time with subpolynomial space. This is used to prove that satisfiability of Boolean circuits with n inputs and n k size cannot be solved by deterministic multitape Turing machines running in $${n^{{k \cdot {c}}_{k}}}$$ time and subpolynomial space.

On the complexity of finding circumscribed rectangles and squares for a two-dimensional domain

We investigate the computational complexity of finding the minimum-area circumscribed rectangle of a given two-dimensional domain. We study this problem in the polynomial-time complexity theory of real functions based on the oracle Turing machine model. We show that a bounded domain S with a polynomial-time computable Jordan curve Γ as the boundary may not have a computable minimum-area circumscribed rectangle. We also show that the problem of finding the minimum area of a circumscribed rectangle of a polynomial-time computable Jordan curve Γ is equivalent to a discrete Σ 2 P -complete problem. The related problem of finding the circumscribed squares of a Jordan curve Γ is also studied. We show that for any polynomial-time computable Jordan curve Γ , there must exist at least one computable circumscribed square (not necessarily of the minimum area), but this square may have arbitrarily high complexity.

Toward a theory of evolutionary computation

We outline a theory of evolutionary computation using a formal model of evolutionary computation – the Evolutionary Turing Machine – which is introduced as the extension of the Turing Machine model. Evolutionary Turing Machines provide a better and a more complete model for evolutionary computing than conventional Turing Machines, algorithms, and Markov chains. The convergence and convergence rate are defined and investigated in terms of this new model. The sufficient conditions needed for the completeness and optimality of evolutionary search are investigated. In particular, the notion of the total optimality as an instance of the multiobjective optimization of the Universal Evolutionary Turing Machine is introduced. This provides an automatic way to deal with the intractability of evolutionary search by optimizing the quality of solutions and search costs simultaneously. Based on a new model a very flexible classification of optimization problem hardness for the evolutionary techniques is proposed. The expressiveness of evolutionary computation is investigated. We show that the problem of the best evolutionary algorithm is undecidable independently of whether the fitness function is time dependent or fixed. It is demonstrated that the evolutionary computation paradigm is more expressive than Turing Machines, and thus the conventional computer science based on them. We show that an Evolutionary Turing Machine is able to solve nonalgorithmically the halting problem of the Universal Turing Machine and, asymptotically, the best evolutionary algorithm problem. In other words, the best evolutionary algorithm does not exist, but it can be potentially indefinitely approximated using evolutionary techniques.

A New Iteration-Complexity Bound for the MTY Predictor-Corrector Algorithm

In this paper we present a new iteration-complexity bound for the Mizuno--Todd--Ye predictor-corrector (MTY P-C) primal-dual interior-point algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program min{cTx:Ax = b, x \ge 0} with decision variable $x\in \Re^n$, e show that the MTY P-C algorithm, started from a well-centered interior-feasible solution with duality gap $n \mu_0$, finds an interior-feasible solution with duality gap less than $n\eta$ in ${\cal O}(T(\mu_0/\eta)+n^{3.5}\log({{\bar\chi^*_A}}))$ iterations, where $T(t)\equiv\min\{n^2\log(\log t) ,\log t\}$ for all $t > 0$ and ${{\bar\chi^*_A}}$ is a scaling invariant condition number associated with the matrix A. More specifically, ${{\bar\chi^*_A}}$ is the infimum of all the conditions numbers ${\bar\chi}_{AD}$, where D varies over the set of positive diagonal matrices. Under the setting of the Turing machine model, our analysis yields an ${\cal O}(n^{3.5}L_A+ \min\{n^2\log L,L\})$ iteration-complexity bound for the MTY P-C algorithm to find a primal-dual optimal solution, where LA and L are the input sizes of the matrix A and the data (A,b,c), respectively. This contrasts well with the classical iteration-complexity bound for the MTY P-C algorithm, which depends linearly on L instead of log L.

Some applications of logic to feasibility in higher types

While it is commonly accepted that computability on a Turing machine in polynomial time represents a correct formalization of the notion of a feasibly computable function, there is no similar agreement on how to extend this notion on functionals , that is, what functionals should be considered feasible. One possible paradigm was introduced by Mehlhorn, who extended Cobham's definition of feasible functions to type 2 functionals. Subsequently, this class of functionals (with inessential changes of the definition) was studied by Townsend who calls this class POLY , and by Kapron and Cook who call the same class basic feasible functionals . Kapron and Cook gave an oracle Turing machine model characterisation of this class. In this article, we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalise the corresponding properties of the class of feasible functions, thus giving further evidence that the notion of feasibility of functionals mentioned above is correctly chosen. We also improve the Kapron and Cook result on machine representation.Our proofs are based on essential applications of logic. We introduce a weak fragment of second order arithmetic with second order variables ranging over functions from N N which suitably characterises basic feasible functionals, and show that it is a useful tool for investigating the properties of basic feasible functionals. In particular, we provide an example how one can extract feasible programs from mathematical proofs that use nonfeasible functions.

Lorentz gas cellular automata on graphs

Lorentz Gas Cellular Automata (LGCA) are models of parallel many-tape Turing machines generated by the motion of localized (point) objects (particles, signals, wave trains, etc.) on a graph. In each vertex v of the graph we store a symbol. A read/write head of the Turing machine is represented as an object that hops from one vertex of the graph to another according to a rule (symbol) stored in the vertex. Thus, the symbols stored in the vertices represent scattering rules (or simply scatterers) of the lattice gas. It is assumed that initially the scatterers are randomly distributed among the vertices of the graph. The random environment formed by the scatterers may either be fixed or evolve due to collisions with the moving object. The collisions simulate storing of a new symbol in the vertex of the graph. We investigate these models with fixed environment in general types of graphs and the models with evolving environments on trees. Remarkably, it occurred that all models with fixed environments and many models with evolving environment behave like a depth-first search on the underlying graph. Observe that such behavior occurs for any initial random distribution of scatterers rather than for some “specially prepared” initial configurations of scatterers. We also give estimates of periods of orbits on finite graphs.

Transcending Turing Computability

It has been argued that neural networks and other forms of analog computation may transcend the limits of Turing-machine computation; proofs have been offered on both sides, subject to differing assumptions. In this article I argue that the important comparisons between the two models of computation are not so much mathematical as epistemological. The Turing-machine model makes assumptions about information representation and processing that are badly matched to the realities of natural computation (information representation and processing in or inspired by natural systems). This points to the need for new models of computation addressing issues orthogonal to those that have occupied the traditional theory of computation.

Some aspects of studying an optimization or decision problem in different computational models

In this paper we want to discuss some of the features coming up when analyzing a problem in different complexity theoretic frameworks. The focus will be on two problems. The first is related to mathematical optimization. We consider the quadratic programming problem of minimizing a quadratic polynomial on a polyhedron. We discuss how the complexity of this problem might change if we consider real data together with an algebraic model of computation (the Blum–Shub–Smale model) instead of rational inputs together with the Turing machine model. The results obtained will lead us to the second problem; it deals with the intrinsic structure of complexity classes in models over real- or algebraically closed fields. A classical theorem by Ladner for the Turing model is examined in these different frameworks. Both examples serve well for working out in how far different approaches to the same problem might shed light upon each other. In some cases this will lead to quite diverse results with respect to the different models. On the other hand, for some problems the more general approach can also give a unifying idea why results hold true in several frameworks. The paper is of tutorial character in that it collects some results into the above direction obtained previously.