Showing papers on "Turing machine published in 1995"

Computation beyond the turing limit.

Abstract: Extensive efforts have been made to prove the Church-Turing thesis, which suggests that all realizable dynamical and physical systems cannot be more powerful than classical models of computation. A simply described but highly chaotic dynamical system called the analog shift map is presented here, which has computational power beyond the Turing limit (super-Turing); it computes exactly like neural networks and analog machines. This dynamical system is conjectured to describe natural physical phenomena.

Computing with First-Order Logic

Abstract: We study two important extensions of first-order logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff PTIME = PSPACE. These and other expressibility results are obtained using a powerful normal form for while which shows that each while computation over an unordered domain can be reduced to a while computation over an ordered domain via a fixpoint query. The fixpoint query computes an equivalence relation on tuples which is a congruence with respect to the rest of the computation. The same technique is used to show that equivalence of tuples and structures with respect to FO formulas with bounded number of variables is definable in fixpoint. Generalizing fixpoint and while, we consider more powerful languages which model arbitrary computation interacting with a database using a finite set of FO queries. Such computation is modeled by a relational machine called loosely coupled generic machine, GMloose. GMloose consists of a Turing machine augmented with a finite set of fixed-arity relations forming a relational store. The connection with while is emphasised by a result showing that GMloose is equivalent to while augmented with integer variables and arithmetic. The normal form for while extends to these languages. We study the expressive power of GMloose and its PTIME and PSPACE restrictions. We argue that complexity measures based on the size of the input may not be best suited for database computation. Instead, we suggest an alternative measure based on the discerning power of the machine, i.e., its ability to distinguish between tuples given its input. With this measure of the input, it is shown that fixpoint and while are the PTIME and PSPACE restrictions of GMloose.

Parallel molecular computation

Abstract: This paper describes techniques for massively parallel computation at the molecular scale, which we refer to as molecular parallelism. While this may at first appear to be purely science fiction, already Adleman [A 94] has employed molecular parallelism in the solution of the Hamiltonian path problem of n nodes using O(n2) lab steps on recombinant DNA of length O(n log n) base pairs. He successfully tested his techniques in a small lab experiment on DNA for a 7 node Hamiltonian path problem, Lipton [L 94] showed that finding the satisfying inputs to a Boolean circuit of size n can be done in O(n) lab steps using DNA of length O(n log n) base pairs. This recent work by Adleman and Lipton in molecular parallelism considered only the solution of NP search problems, and provided no way of quickly executing lengthy computations by purely molecular means; the number of lab steps depended linearly on the size of the simulated circuit. This paper describes techniques for quickly executing lengthy computations bg the use of molecular parallelism. We demonstrate that molecular computations can be done using short DNA strands by more or less conventional biotechnology engineering techniques within a small number of lab steps. We propose two abstract models of molecular computation. The first, the Parallel Associative Memory (PAM) Model, is a very high level model which includes a Parallel Associative Matching (PA-Match) operation, that appears to improve the power of molecular parallelism beyond the operations previously considered by Lipton [L 94]. We give some simulations of conventional sequential and parallel computational models by our PAM model. Each of the simulations use strings of length O(s) over an alphabet of size n (which correspond to DNA of length 0(s log n) baae pairs). Using O(t) PA-Match operations as well as 0(s logs) PAM operations that are not PA-Match, we can: (1) simulate a nondeterministic Turing Machine computation with space bound s < tand time bound 2°1sJ, (2) simulate a CREW PRAM with time bound D, with A4 memory cells, and processor bound P, where here s = *Surface address: Department of Computer Science, Duke University, Durham, NC 2770 S-0129. E-mail: reifOcs.duke. edu. Supported by NSF Grant NSF-IRI-91-006S1, Rome Labs Contracts F30602-94-C0037, ARPA/SISTO contracts NOO014-91-J-1985, and NOOO14-92-C01S2 under subcontract KI-92-01-0182. This paper, complete with figures, can be found in http://www.cs.duke. edu/ reif/HomePage.html. Permission to make d\gital/lmrd copies of al] or part of this nmtcrial without fee is granted prowded tlmt the copies are nnt made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication and its date oppefir, and notice is given that copyright is by permission of the Associating for Computing h40chinery, Inc. (ACM). To copy otherwise, to rcpubhsh ,to post on servers or to redistribute to lists, requires specific permission wld/or fee. SPAA’95 Santa Barbara CA USA@ 1995 ACM 0-89791-717-0/95/07.$3.50 O(log(PiM)) and t = D + s, (3) find the satisfying inputs to a Boolean circuit constructible in s space with n inputs, unbounded fan-out, and depth D, where here t = D + s, and (4) do List and Tree Contraction for inputs of size L constructible in s space (using here tubes each with O(B + L/ log L) aggregates and assuming a precomputed tube of B aggregates for the Contraction operation) where here t =

Undecidable boundedness problems for datalog programs

Abstract: A given Datalog program is bounded if its depth of recursion is independent of the input database. Deciding boundedness is a basic task for the analysis of database logic programs. The undecidability of Datalog boundedness was first demonstrated by Gaifman et al. [7]. We introduce new techniques for proving the undecidability of (various kinds of) boundedness, which allow us to considerably strengthen the results of Gaifman et al. [7]. In particular, (1) we use a new generic reduction technique to show that program boundedness is undecidable for arity 2 predicates, even with linear rules; (2) we use the mortality problem of Turing machines to show that uniform boundedness is undecidable for arity 3 predicates and for arity 1 predicates when ≠ is also allowed; (3) by encoding all possible transitions of a two-counter machine in a single rule, we show that program (resp., predicate) boundedness is undecidable for two linear rules (resp., one rule and a projection) and one initialization rule, where all predicates have small arities (6 or 7).

First Order Bounded Arithmetic and Small Boolean Circuit Complexity Classes

Abstract: A well known result of proof theory is the characterization of primitive recursive functions ƒ as those provably recursive in the first order theory of Peano arithmetic with the induction axiom restricted to Σ1 formulas. In this paper, we study a variety of weak theories of first order arithmetic, whose provably total functions (with graphs of a certain form) are exactly those computable within some resource bound on a particular computation model (boolean circuits, with possible parity or MOD 6 gates, or threshold circuits, or alternating Turing machines, or ordinary Turing machines). To establish these kinds of results for small complexity classes, we provide a recursion-theoretic characterization of the complexity class, prove how one can encode sequences in very weak theories, and use the witnessing technique of [7].

Quasi-Delay-Insensitive Circuits are Turing-Complete

Abstract: Quasi-delay-insensitive (QDI) circuits are those whose correct operation does not depend on the delays of operators or wires, except for certain wires that form isochronic forks In this paper we show that quasi-delay-insensitivity, stability and noninterference, and strong confluence are equivalent properties of a computation In particular, this shows that QDI computations are deterministic We show that the class of Turing-computable functions have QDI implementations by constructing a QDI Turing machine

On real Turing machines that toss coins

Abstract: In this paper we consider real counterparts of classical probabilistic complexity classes in the framework of real Turing machines as introduced by Blum, Shub, and Smale [2]. We give an extension of the well-known “BPP ~ P/poly” result from discrete complexity theory to a very general setting in the real number model. This result holds for real inputs, real outputs, and random elements drawn from an arbitrary probability distribution over lR~. Then we turn to the study of Boolean parts, that is, classes of languages of zero-one vectors accepted by real machines. In particular we show that the classes BPP, PP, PH, and PSPACE are not enlarged by allowing the use of real constants and arithmetic at unit cost provided we restrict branching to equality tests.

Towards a mathematical theory of machine discovery from facts

Abstract: This paper intends to give a theoretical foundation of machine discovery from facts. We point out that the essence of a computational logic of scientific discovery or a logic of machine discovery is the refutability of the entire spaces of hypotheses. We discuss this issue in the framework of inductive inference of length-bounded elementary formal systems (EFSs), which are a kind of logic programs over strings of characters and correspond to context-sensitive grammars in Chomsky hierarchy. First we present some characterization theorems on inductive inference machines that can refute hypothesis spaces. Then we show differences between our inductive inference and some other related inferences such as in the criteria of reliable identification, finite identification and identification in the limit. Finally we show that for any n , the class, i.e. hypothesis space, of length-bounded EFSs with at most n axioms is inferable in our sense, that is, the class is refutable by a consistently working inductive inference machine. This means that sufficiently large hypothesis spaces are identifiable and refutable.

Alan turing and the central limit theorem

Abstract: Because the English mathematician Alan Mathison Turing (1912–1954) is remembered today primarily for his work in mathematical logic (Turing machines and the “Entscheidungsproblem”), machine computation, and artificial intelligence (the “Turing test”), his name is not usually thought of in connection with either probability or statistics. One of the basic tools in both of these subjects is the use of the normal or Gaussian distribution as an approximation, one basic result being the Lindeberg-Feller central limit theorem taught in first-year graduate courses in mathematical probability. No-one associates Turing with the central limit theorem, but in 1934 Turing, while still an undergraduate, rediscovered a version of Lindeberg's 1922 theorem and much of the Feller-Levy converse to it (then unpublished). This paper discusses Turing's connection with the central limit theorem and its surprising aftermath: his use of statistical methods during World War II to break key German military codes. 1 Introduction Turing went up to Cambridge as an undergraduate in the Fall Term of 1931, having gained a scholarship to King's College. (Ironically, King's was his second choice; he had failed to gain a scholarship to Trinity.) Two years later, during the course of his studies, Turing attended a series of lectures on the Methodology of Science, given in the autumn of 1933 by the distinguished astrophysicist Sir Arthur Stanley Eddington. One topic Eddington discussed was the tendency of experimental measurements subject to errors of observation to often have an approximately normal or Gaussian distribution.

Automata That Take Advice

Abstract: Karp and Lipton introduced advice-taking Turing machines to capture nonuniform complexity classes. We study this concept for automata-like models and compare it to other nonuniform models studied in connection with formal languages in the literature. Based on this we obtain complete separations of the classes of the Chomsky hierarchy relative to advices.

Symbolic Artificial Intelligence and Numeric Artificial Neural Networks: Towards A Resolution of the Dichotomy

Abstract: The attempt to understand intelligence entails building theories and models of brains and minds, both natural as well as artificial. From the earliest writings of India and Greece, this has been a central problem in philosophy. The advent of the digital computer in the 1950’s made this a central concern of computer scientists as well (Turing, 1950). The parallel development of the theory of computation (by John von Neumann, Alan Turing, EmilPost, Alonzo Church, Charles Kleene, Markov and others) provided a new set of tools with which to approach this problem — through analysis, design, and evaluation of computers and programs that exhibit aspects of intelligent behavior — such as the ability to recognize and classify patterns; to reason from premises to logical conclusions; and to learn from experience.

An $\cal NC$ Algorithm for Evaluating Monotone Planar Circuits

Abstract: Goldschlager first established that a special case of the monotone planar circuit problem can be solved by a Turing machine in $O(\log^2 n)$ space Subsequently, Dymond and Cook refined the argument and proved that the same class can be evaluated in $O(\log^2 n)$ time with a polynomial number of processors In this paper, we prove that the general monotone planar circuit value problem can be evaluated in $O(\log^4 n)$ time with a polynomial number of processors, settling an open problem posed by Goldschlager and Parberry

Using autoreducibility to separate complexity classes

Abstract: A language is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate exponential space from doubly exponential space by showing that all Turing complete sets for exponential space are autoreducible but there exists some Turing complete set for doubly exponential space that is not. We immediately also get a separation of logarithmic space from polynomial space. Although we already know how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's Program (E. Pos, 1944) to complexity theory. We feel such techniques may prove unknown separations in the future. In particular if we could settle the question as to whether all complete sets for doubly exponential time were autoreducible we would separate polynomial time from either logarithmic space or polynomial space. We also show several other theorems about autoreducibility.

Computable Quantifiers and Logics over Finite Structures

Abstract: We explore the notion of computable quantifiers over finite structures and use them to give a unified treatment of the theory of computable queries in databases and logics capturing complexity classes. We use this framework also to discuss generalized Ehrenfeucht—Fraisse games and their applications to complexity theory.

Non-Erasing Turing Machines: A New Frontier Between a Decidable Halting Problem and Universality

Abstract: We define a new criterion which allows to separate cases when all non erasing Turing machines on {0, 1} have a decidable halting problem from cases where a universal non erasing machine can be constructed. It is the case of the number of left instructions in the machine program. In this paper we give the main ideas of the proof for both parts of the frontier result. We prove that there is a universal non-erasing Turing machine whose program has precisely 3 left instructions and that the halting problem is decidable for any non-erasing Turing machine on alphabet {0, 1}, the program of which contains at most 2 left instructions. For this latter result, we have a uniform decision algorithm.

On Gödel’s Theorems on Lengths of Proofs II: Lower Bounds for Recognizing k Symbol Provability

Abstract: This paper discusses a claim made by Godel in a letter to von Neumann which is closely related to the P versus NP problem. Godel’s claim is that k-symbol provability in first-order logic can not be decided in o(k) time on a deterministic Turing machine. We prove Godel’s claim and also prove a conjecture of S. Cook’s that this problem can not be decided in o(k/ log k) time by a nondeterministic Turing machine. In addition, we prove that the k-symbol provability problem is NP-complete, even for provability in propositional logic.

Molecular automata utilizing single- or double-strand oligonucleotides

Abstract: Single- or double-strand oligonucleotides are used to create a molecular automata. The preferred embodiment is a DNA Turing machine and a method of performing a transition in such a DNA Turing machine.

Effective procedures and computable functions

Abstract: Horsten and Roelants have raised a number of important questions about my analysis of effective procedures and my evaluation of the Church-Turing thesis. They suggest that, on my account, effective procedures cannot enter the mathematical world because they have a built-in component of causality, and, hence, that my arguments against the Church-Turing thesis miss the mark. Unfortunately, however, their reasoning is based upon a number of misunderstandings. Effective mundane procedures do not, on my view, provide an analysis of ourgeneral concept of an effective procedure; mundane procedures and Turing machine procedures are different kinds of procedure. Moreover, the same sequence ofparticular physical action can realize both a mundane procedure and a Turing machine procedure; it is sequences of particular physical actions, not mundane procedures, which “enter the world of mathematics.” I conclude by discussing whether genuinely continuous physical processes can “enter” the world of real numbers and compute real-valued functions. I argue that the same kind of correspondence assumptions that are made between non-numerical structures and the natural numbers, in the case of Turing machines and personal computers, can be made in the case of genuinely continuous, physical processes and the real numbers.

An analysis of Bennett's pebble game

Abstract: Bennett's pebble game was introduced to obtain better time/space tradeoffs in the simulation of standard Turing machines by reversible ones. So far only upper bounds for the tradeoff based on the pebble game have been published. Here we give a recursion for the time optimal solution of the pebble game given a space bound. We analyze the recursion to obtain an explicit asymptotic expression for the best time-space product.

Strong Optimal Lower Bounds for Turing Machines that Accept Nonregular Languages

Abstract: In this paper, simultaneous lower bounds on space and input head reversals for deterministic, nondeterministic and alternating Turing machines accepting nonregular languages are studied.

A Semantic Logic for CAST Related to Zuse, Deutsch and McCulloch and Pitts Computing Principles

Abstract: The goal of CAST research and development is to provide modelling tools for formal systems design in the field of information and systems engineering. This paper deals with such modelling tools for formal systems related to Zuse, Deutsch and McCulloch and Pitts computing principles. The semantic logic of such systems can be exhibited in replacing the differential equations by digital cellular automata. K. Zuse proposed such a method for representing physical systems by a computing space. I show that the digital wave equation exhibits waves by digital particles with interference effects. The logical table of the wave equation shows the conservation of the parity related to exclusive OR. The Fractal Machine proposed by the author deals with a cellular automata based on incursion, an inclusive recursion, with exclusive OR. In this machine, the superimposition of states is related to the Deutsch quantum computer. Finally, it is shown that the exclusive OR can be modelled by a fractal non-linear equation and a new method to design digital equations is proposed to create McCulloch and Pitts formal neurons.

Deux machines de Turing universelles à au plus deux instructions gauches

Abstract: On construit une machine de Turing universelle sur l'alphabet {0, 1} dont le programme contient exactement deux instructions de mouvement gauche et une machine de Turing universelle sur l'alphabet {0, 1, 2} dont le programme contient une unique instruction de mouvement gauche