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Medvedev degrees of 2dimensional subshifts of finite type. Ergodic Theory and Dynamical Systems
"... In this paper we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new counterexample in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial recursive functionals from X into Y and vice versa. The Medvedev deg ..."
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Cited by 24 (9 self)
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In this paper we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new counterexample in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial recursive functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursiontheoretic literature on the lattice of Medvedev degrees of nonempty Π 0 1 subsets of {0, 1} N. This lattice is known as Ps. We prove that Ps consists precisely of the Medvedev degrees of 2dimensional subshifts of finite type. We use this result to obtain an infinite collection of 2dimensional subshifts of finite type which are, in a certain sense, mutually incompatible. Definition 1. Let A be a finite set of symbols. The full 2dimensional shift on A is the dynamical system consisting of the natural action of Z2 on the compact set AZ2. A 2dimensional subshift is a nonempty closed set X ⊆ AZ2 which is invariant under the action of Z2. A 2dimensional subshift X is said to be of finite type if it is defined by a finite set of forbidden configurations. An interesting paper on 2dimensional subshifts of finite type is Mozes [22]. A standard reference for the 1dimensional case is the book of Lind/Marcus [20], which also includes an appendix [20, §13.10] on the 2dimensional case.
An Eunification algorithm for analyzing protocols that use modular exponentiation
, 2003
"... Modular multiplication and exponentiation are common operations in modern cryptography. Uni cation problems with respect to some equational theories that these operations satisfy are investigated. Two dierent but related equational theories are analyzed. A uni cation algorithm is given for one of ..."
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Cited by 22 (0 self)
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Modular multiplication and exponentiation are common operations in modern cryptography. Uni cation problems with respect to some equational theories that these operations satisfy are investigated. Two dierent but related equational theories are analyzed. A uni cation algorithm is given for one of the theories which relies on solving syzygies over multivariate integral polynomials with noncommuting indeterminates. For the other theory, in which the distributivity property of exponentiation over multiplication is assumed, the uni ability problem is shown to be undecidable by adapting a construction developed by one of the authors to reduce Hilbert's 10th problem to the solvability problem for linear equations over semirings. A new algorithm for computing strong Grobner bases of right ideals over the polynomial semiring Z<X 1 ; : : : ; Xn> is proposed; unlike earlier algorithms proposed by Baader as well as by Madlener and Reinert which work only for right admissible term orderings with the boundedness property, this algorithm works for any right admissible term ordering. The algorithms for some of these uni cation problems are expected to be integrated into Research supported in part by the NSF grant nos. CCR0098114 and CDA9503064, the ONR grant no. N000140110429, and a grant from the Computer Science Research Institute at Sandia National Labs.
Four Small Universal Turing Machines
, 2009
"... We present universal Turing machines with statesymbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bitag system and are the smallest known singletape universal Turing machines with 5, 4, 3 and 2symbols, respectively. Our 5symbol machin ..."
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Cited by 21 (7 self)
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We present universal Turing machines with statesymbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bitag system and are the smallest known singletape universal Turing machines with 5, 4, 3 and 2symbols, respectively. Our 5symbol machine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. Also, all of the universal machines we present here simulate Turing machines in polynomial time.
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 19 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Discrete Loops and Worst Case Performance
 Computer Languages
, 1994
"... In this paper socalled discrete loops are introduced which narrow the gap between general loops (e.g. while or repeatloops) and forloops. Alt hough discrete loops can be used for applications that would otherwise require general loops, discrete loops are known to complete in any case. Furthe ..."
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Cited by 18 (7 self)
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In this paper socalled discrete loops are introduced which narrow the gap between general loops (e.g. while or repeatloops) and forloops. Alt hough discrete loops can be used for applications that would otherwise require general loops, discrete loops are known to complete in any case. Furthermore it is possible to determine the number of iterations of a discrete loop, while this is trivial to do for forloops and extremely difficult for general loops. Thus discrete loops form an ideal framework for determining the worst case timing behavior of a program and they are especially useful in implementing realtime systems and proving such systems correct.
Improving Network System Security with Function Extraction Technology for Automated Calculation of Program Behavior
 In Proceedings of the 37th Annual Hawaii International Conference on System Sciences. IEEE
, 2004
"... Malicious attacks on systems are a threat to business, government, and defense. Many attacks exploit system behavior unknown to the developers who created it. In today’s state of art, software engineers have no practical means to determine how a sizable program will behave in all circumstances of us ..."
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Cited by 18 (10 self)
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Malicious attacks on systems are a threat to business, government, and defense. Many attacks exploit system behavior unknown to the developers who created it. In today’s state of art, software engineers have no practical means to determine how a sizable program will behave in all circumstances of use. This sobering reality lies at the heart of many problems in security and survivability. If full behavior is unknown, so too are embedded errors, vulnerabilities, and malicious code. This paper describes functiontheoretic foundations for automated calculation of full program behavior. These foundations treat program control structures as mathematical functions or relations. The function, or behavior, of control structures can be abstracted in a stepwise process into procedurefree expressions that specify their net functional effects. Problems of computability and complexities of language semantics appear to have engineering solutions. Automated behavior calculation will add rigor to security and survivability engineering. 1. Understanding Program Behavior Traditional engineering disciplines depend on rigorous methods to evaluate the expressions (equations, for example) that represent and manipulate their subject matter. Yet the discipline of software engineering has no practical means to fully evaluate the expressions it produces. In this case, the expressions are computer programs, and evaluation means understanding their full behavior, right or wrong, intended or malicious. Short of substantial time and effort, no software engineer can say for sure what a sizable program does in all circumstances of use. Yet modern society is dependent on the correct functioning of countless largescale systems composed of programs whose full behavior and security properties are
The ChurchTuring Thesis: Breaking the myth
 CiE 2005: New Computational Paradigms, volume 3526 of LNCS
, 2005
"... Abstract. According to the interactive view of computation, communication happens during the computation, not before or after it. This approach, distinct from concurrency theory and the theory of computation, represents a paradigm shift that changes our understanding of what is computation and how i ..."
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Abstract. According to the interactive view of computation, communication happens during the computation, not before or after it. This approach, distinct from concurrency theory and the theory of computation, represents a paradigm shift that changes our understanding of what is computation and how it is modeled. Interaction machines extend Turing machines with interaction to capture the behavior of concurrent systems, promising to bridge these two elds. This promise is hindered by the widespread belief, incorrectly known as the ChurchTuring thesis, that no model of computation more expressive than Turing machines can exist. Yet Turing's original thesis only refers to the computation of functions and explicitly excludes other computational paradigms such as interaction. In this paper, we identify and analyze the historical reasons for this widespread belief. Only by accepting that it is false can we begin to properly investigate formal models of interaction machines. We conclude the paper by presenting one such model, Persistent Turing Machines (PTMs). PTMs capture sequential interaction, which is a limited form of concurrency; they allow us to formulate the Sequential Interaction Thesis, going beyond the expressiveness of Turing machines and of the ChurchTuring thesis. 1
A reformulation of Hilbert’s tenth problem through Quantum Mechanics
, 2001
"... Inspired by Quantum Mechanics, we reformulate Hilbert’s tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shrödinger propagator with some appropriate kernel. Either way, Mathematics and Physi ..."
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Inspired by Quantum Mechanics, we reformulate Hilbert’s tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shrödinger propagator with some appropriate kernel. Either way, Mathematics and Physics could be combined for Hilbert’s tenth problem and for the notion of effective computability. 1