The experimental logic of Moore and Mealy type automata is investigated. key words: automaton logic; partition logic; comparison to quantum logic; intrinsic measurements 1

We investigate the orthoalgebras of certain non-Boolean models which have a classical realization. Our particular concern will be the partition logics arising from the investigation of the empirical propositional structure of Moore and Mealy type automata. 1

The classical halting probability to quantum computations. introduced by Chaitin is generalized Chaitin's [1,2,3] is a magic number. It is a measure for arbitrary programs to take a nite number of execution steps and then halt. It contains the solution for all halting problems, and hence to questions codable into halting problems, such asFermat's theorem. It contains the solution for the question of whether or not a particular exponential Diophantine equation has in nitely many ora nite number of solutions. And, since is provable \algorithmically incompressible," it is Martin-Lof/Chaitin/Solovay random. Therefore, is both: a mathematicians \fair coin, " and a formalist's nightmare. Here, is generalized to quantum computations. Consider a (not necessarily universal) quantum computer C and its ith program pi, which, at time t 2 Z, can be described by a quantum state [4, 5,6,7,

This paper provides a brief introduction to automatic differentiation and relates it to the tangent linear model and adjoint approaches commonly used in meteorology. After a brief review of the forward and reverse mode of automatic differentiation, the ADIFOR automatic differentiation tool is introduced, and initial results of a sensitivity-enhanced version of the MM5 PSU/NCAR mesoscale weather model are presented. We also present a novel approach to the computation of gradients that uses a reverse mode approach at the time loop level and a forward mode approach at every time step. The resulting "pseudoadjoint" shares the characteristic of an adjoint code that the ratio of gradient to function evaluation does not depend on the number of independent variables. In contrast to a true adjoint approach, however, the nonlinearity of the model plays no role in the complexity of the derivative code.

Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid three-dimensional objects, (iii) in the theory of effective computability (Church-Turhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.

Algorithmic (KCS) complexity results can be interpreted as indicating some limits to software estimation. While these limits are abstract they nevertheless contradict enthusiastic claims occasionally made by commercial software estimation advocates. Specifically, if it is accepted that algorithmic complexity is an appropriate definition of the complexity of a programming project, then claims of purely objective estimation of project complexity, development time, and programmer productivity are necessarily incorrect.

Quantum coherence allows the computation of an arbitrary number of distinct computational paths in parallel. Based on quantum parallelism it has been conjectured that exponential or even larger speedups of computations are possible. Here it is shown that, although in principle correct, any speedup is accompanied by an associated attenuation of detection rates. Thus, on the average, no effective speedup is obtained relative to classical (nondeterministic) devices. 1 Recent findings in quantum complexity theory suggest an exponential speedup of discrete logarithms and factoring [1] and the travelling salesman problem [2] with respect to classical complexity. (The best classical estimate for the computing time for factoring is e cn 1/3 , where n is the number of bits in the number to be factored and c is a constant; the travelling salesman problem is NP-complete). At the heart of these types of speedups is quantum parallelism. Roughly stated, quantum parallelism assures that a ...

constructive re-interpretation of physical undecidability

this paper (see the discussion in [51, 58, 70, 126, 127, 120, 121, 130]); in what follows we shall superficially review this topic in connection with the related question: can computers think?

This paper is an attempt to unify classical automata theory and dynamical systems theory. We present a notion of generalized dynamical systems, allowing us to compare properties of both types of systems. Then we etablish a hierachy of dynamical systems, including Turing machines, cellular automata and classical dynamical systems. We finish with some conclusions and motivations for future work. 1 Introduction The theory of finite automata has always been an important field of computer science. Automata, languages, grammars, have been extensively studied [1, 2]. Automata recognize languages and grammars produce languages. Among others, the most studied are finite automata, pushdown automata, Turing machines. The Chomsky hierarchy for languages and grammars is also very well established: regular, context-free, context-sensitive, general grammars generate languages with the same names. The theory of dynamical systems, chaos, attractors [3, 4], is an important field of mathematics and phys...