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.

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 =

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

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.

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.

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.

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.

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.

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.

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.

Abstract: Lucas and Penrose have contended that, by displaying how any characterisation of arithmetical proof programmable into a machine allows of diagonalisation, generating a humanly recognisable proof which eludes that characterisation, Gddel's incompleteness theorem rules out any purely mechanical model of the human intellect. The main criticisms of this argument have been that the proof generated by diagonalisation (i) will not be humanly recognisable unless humans can grasp the specification of the object-system (Benacerraf); and (ii) counts as a proof only on the (iinproven) hypothesis that the object system is consistent (Putnam). The present paper argues that criticism (ii) may be met head-on by an intuitionistic proponent of the anti-mechanist argument; and that criticism (i) is simply mistaken. However the paper concludes by questioning the sufficiency of the situation for an interesting anti-mechanist conclusion. u Thanks to Bob Hale, Stewart Shapiro and Neil Tennant for criticisms of an earlier draft, and to the participants at the Conference on the philosophy of Michael Dummett held at Mussomeli, Sicily, in September 1992 at which a version of the principal argument was presented. Downloaded from https://academic.oup.com/philmat/article-abstract/3/1/86/1508324 by University of St Andrews user on 21 November 2017

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.

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.

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

Abstract: We characterize various complexity classes as the images in set$\sp2,$ set$\sp{V},$ and set$\sp3$ of categories initial in various complexity doctrines. (A doctrine consists of the models of a theory of theories.) We so characterize the linear time, P space, linear space, P time, and Kalmar elementary functions as well as the linear time hierarchy relations. (Our machine model is multi-tape Turing machines with constant number of tapes.) These doctrines extend, using comprehensions, the first order doctrines GM and JB. We show, using dependent product diagrams, how to so extend the higher order doctrine LCC. However, using Church numerals, we show that the resulting LCC comprehensions do not provide enough control over higher order types to characterize complexity classes. We also show how to use sketches and orthogonality for almost equational specification.

Abstract: Interaction machines, defined by extending Turing machines with input actions (read statements), are shown to be more expressive than computable functions, providing a counterexample to the hypothesis of Church and Turing that the intuitive notion of computation corresponds to formal computability by Turing machines. The negative result that interaction cannot be modeled by algorithms leads to positive principles of interactive modeling by interface constraints that support partial descriptions of interactive systems whose complete behavior is inherently unspecifiable. The unspecifiability of complete behavior for interactive systems is a computational analog of Godel incompleteness for the integers. Fortunately the complete behavior of interaction machines is not needed to harness their behavior for practical purposes. Interface descriptions are the primary mechanism used by software designers and application programmers for partially describing systems for the purpose of designing, controlling, predicting, and understanding them. Interface descriptions are an example of "harness constraints" that constrain interactive systems so their behavior can be harnessed for useful purposes. We examine both system constraints like transaction correctness and interface constraints for software design and applications. Sections 1, 2, and 3 provide a conceptual framework and theoretical foundation for interactive models, while sections 4, 5, and 6 apply the framework to distributed systems, software engineering, and artificial intelligence. Section 1 explores the design space of interaction machines, section 2 considers the relation between imperative, declarative, and interactive paradigms of computing, section 3 examines the limitations of logic and formal semantics. In section 4 we examine process models, transaction correctness, time, programming languages, and operating systems from the point of view of interaction. In section 5, we examine life-cycle models, object-based design, use-case models, design patterns, interoperability, and coordination. In section 6, we examine knowledge representation, intelligent agents, planning for dynamical systems, nonmonotonic logic, and "can machines think?". The interaction paradigm provides new ways of approaching each of these application areas, demonstrating the value of interac

Abstract: This talk will describe a voyage of discovery the discovery of Algorithmic Probability. But before I describe the voyage a few words about motivation. Motivation in science is roughly of two kinds: In one, the motivation is discovery itself the joy of "going where no one has gone before" the joy of creating new universes and exploring them. Another kind of discovery is motivated by a previously defined larger goal, and there may be many subsidiary discoveries on the path to this goal. In my own case both kinds of motivation were very strong. I first experienced the pure joy of mathematical discovery when I was very y o u n g learning algebra. I didn' t really see why one of the axioms was "obvious", so I tried reversing it and exploring the resultant system. For a few hours I was in this wonderful never-never land that I myself had created! The joy of exploring it only lasted until it became clear that my new axiom wouldn't work, but the motivation of the joy of discovery continued for the rest of my life. The motivation for the discovery of Algorithmic Probabili ty was somewhat different. It was the result of "goat motivated discovery" like the discovery of the double helix in biology, but with fewer people involved and relatively little political skullduggery. The goal I set grew out of my early interest in science and mathematics. I found that while the discoveries of the past were interesting to me, I was even more interested in how people discovered things. Was there a general technique to solve all mathematical problems? Was there a general method by which scientists could discover all scientific truths? The problems seemed closely related to induction and so around 1941 I first defined a general induction problem. It had two aspects:

Abstract: We can associate with each consistent formula F of first-order logic a computing device as its representation. This computing device is one which will calculate the Skolem functions of F (for a denumerable domain). When two such devices are operating in parallel, the resulting architecture does not necessarily represent any ordinary first-order formula, but it will represent a formula in independence-friendly (IF) logic, which hence can be considered as a true logic of parallel processing. In order to preserve representability by a digital automaton (Turing machine), a nonstandard (constructivistic) interpretation of the logic in question has to be adopted. It is obtained by restricting the Skolem functions available to verify a formula F to recursive ones, as in the Godel’s Dialectica interpretation.

Abstract: In this paper we extend the “Oracle Turing Machine” model to compute functionals of all finite types. Using this model we define analogs of class C 1 (type 2 basic feasible functionals), of [11], in all finite types. We get two apparently different classes which we call D and E when C 1 is generalized to finite types. These classes correspond to two different ways of querying higher type inputs. Both these classes are shown to satisfy the necessary conditions proposed by Cook [5], which any class of feasible functionals must satisfy. Class E, as expected, turns out to be the BFF, thus providing a more natural computational characterization of higher type basic feasible functionals. Class D is the same as BFF for type 2, but appears to be larger than BFF for types 3 and above; however, showing separation between these two classes is open. The question, “Is class D larger than class E?” is equivalent to the question “Does computing the indices of subprograms used to query higher type inputs add to the computational power compared to the case when only a fixed finite number of parameterizable functionals are used for querying all higher type inputs? ”.

Abstract: Alan Turing proposed an interactive test to replace the question "Can machines think?" This test has become known as the Turing Test and its validity for determining intelligence or thinking is still in question.Struggling with the validity of long proofs, program correctness, computational complexity and cryptography, theoreticians developed interactive proof systems. By formalizing the Turing Test as an interactive proof system and by employing results from complexity theory, this paper investigates the power and limitations of the Turing Test. In particular, assuming the notion of completeness for standard complexity classes carries over faithfully to human cognition, then we can say: if human intelligence subsumes machine intelligence, and human intelligence is not simulatable by any bounded machine, then the Turing Test can distinguish humans and machines to within arbitrarily high probability.This paper makes no claim about the Turing Test's sufficiency to distinguish humans and machines. Rather, through its formalization this paper gives several ramifications involving the acceptance or rejection of the Turing Test as sufficient for making any such distinction.

Abstract: On construit une machine de Turing universelle sur l'alphabet {0, 1} qui ne peut jamais transformer un 1 du ruban en 0 et telle que son programme contienne exactement 3 instructions gauches et 218 etats

Abstract: Recently, related to the open problem of whether deterministic and nondeterministic space (especially lower-level) complexity classes are separated, the inkdot Turing machine was introduced. An inkdot machine is a conventional Turing machine capable of dropping an inkdot on a given input tape for a landmark, but not to pick it up nor further erase it. In this paper, we introduce a finite state version of the inkdot machine as a weak recognizer of the properties of digital pictures, rather than a Turing machine supplied with a one-dimensional working tape. We first investigate the sufficient spaces of three-way Turing machines to simulate two-dimensional inkdot finite automaton, as preliminary results. Next, we investigate the basic properties of two-dimensional inkdot automaton, i.e. the hierarchy based on the number of inkdots and the relationship of two-dimensional inkdot automata to other conventional two-dimensional automata. Finally, we investigate the recognizability of connected pictures of two-dimensional inkdot finite machines.