/pdf/on-a-theory-of-computation-and-complexity-over-the-real-2s93av83k1.pdf

On a theory of computation and complexity over the real numbers: $NP$- completeness, recursive functions and universal machines

An $(n,s)$ Davenport--Schinzel sequence, for positive integers $n$ and $s$, is a sequence composed of $n$ symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation $a \cdots b \cdots a \cdots b \cdots$ of length $s+2$ between two distinct symbols $a$ and $b$. The close relationship between Davenport--Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive, because a wide variety of geometric problems can be formulated in terms of lower envelopes. A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems. This paper gives a comprehensive survey on the theory of Davenport--Schinzel sequences and their geometric applications.

http://www.math.tau.ac.il/~michas/dssurvey.pdf

Davenport-Schinzel sequences and their geometric applications

In this paper a microscopic quantum mechanical model of computers as represented by Turing machines is constructed. It is shown that for each numberN and Turing machineQ there exists a HamiltonianH
N
Q and a class of appropriate initial states such that if c is such an initial state, thenψ
Q
N
(t)=exp(−1H

 N
 
Q
t)ψ
Q
N
(0) correctly describes at timest
3,t
6,⋯,t
3N model states that correspond to the completion of the first, second, ⋯, Nth computation step ofQ. The model parameters can be adjusted so that for an arbitrary time intervalΔ aroundt
3,t
6,⋯,t
3N, the “machine” part ofψ
Q
N
(t) is stationary.

The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines

Model-checking is a method of verifying concurrent systems in which a state-transition graph model of the system behavior is compared with a temporal logic formula. This paper extends model-checking for the branching-time logic CTL to the analysis of real-time systems, whose correctness depends on the magnitudes of the timing delays. For specifications, we extend the syntax of CTL to allow quantitative temporal operators such as ?? <5, meaning "possibly within 5 time units." The formulas of the resulting logic, Timed CTL (TCTL), are interpreted over continuous computation trees, trees in which paths are maps from the set of nonnegative reals to system states. To model finite-state systems we introduce timed graphs-state-transition graphs annotated with timing constraints. As our main result, we develop an algorithm for model-checking, for determining the truth of a TCTL-formula with respect to a timed graph. We argue that choosing a dense domain instead of a discrete domain to model time does not significantly blow up the complexity of the model-checking problem. On the negative side, we show that the denseness of the underlying time domain makes the validity problem for TCTL ?11-hard. The question of deciding whether there exists a timed graph satisfying a TCTL-formula is also undecidable.

Model-Checking in Dense Real-Time

This article argues that the agent-based computational model permits a distinctive approach to social science for which the term “generative” is suitable. In defending this terminology, features distinguishing the approach from both “inductive” and “deductive” science are given. Then, the following specific contributions to social science are discussed: The agent-based computational model is a new tool for empirical research. It offers a natural environment for the study of connectionist phenomena in social science. Agent-based modeling provides a powerful way to address certain enduring—and especially interdisciplinary—questions. It allows one to subject certain core theories—such as neoclassical microeconomics—to important types of stress (e.g., the effect of evolving preferences). It permits one to study how rules of individual behavior give rise—or “map up”—to macroscopic regularities and organizations. In turn, one can employ laboratory behavioral research findings to select among competing agent-based (“bottom up”) models. The agent-based approach may well have the important effect of decoupling individual rationality from macroscopic equilibrium and of separating decision science from social science more generally. Agent-based modeling offers powerful new forms of hybrid theoretical-computational work; these are particularly relevant to the study of non-equilibrium systems. The agentbased approach invites the interpretation of society as a distributed computational device, and in turn the interpretation of social dynamics as a type of computation. This interpretation raises important foundational issues in social science—some related to intractability, and some to undecidability proper. Finally, since “emergence” figures prominently in this literature, I take up the connection between agent-based modeling and classical emergentism, criticizing the latter and arguing that the two are incompatible. ! 1999 John Wiley & Sons, Inc.

/pdf/agent-based-computational-models-and-generative-social-1m1ncpv05f.pdf

Agent-based computational models and generative social science

Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular. A second group of topics has to do with generalizations of recursion theory. The third topics group mentioned is subrecursive computability and subrecursive hierarchies

Jr. Hartley Rogers

Papers

Theory of Recursive Functions and Effective Computability