Irreversible computable functions

International audienceThe strong relationship between topology and computations has played a central role in the development of several branches of theoretical computer science: foundations of functional programming, computational geometry, computability theory, computable analysis. Often it happens that a given function is not computable simply because it is not continuous. In many cases, the function can moreover be proved to be non-computable in the stronger sense that it does not preserve computability: it maps a computable input to a non-computable output. To date, there is no connection between topology and this kind of non-computability, apart from Pour-El and Richards ''First Main Theorem'', applicable to linear operators on Banach spaces only. In the present paper, we establish such a connection. We identify the discontinuity notion, for the inverse of a computable function, that implies non-preservation of computability. Our result is applicable to a wide range of functions, it unifies many existing ad hoc constructions explaining at the same time what makes these constructions possible in particular contexts, sheds light on the relationship between topology and computability and most importantly allows us to solve open problems. In particular it enables us to answer the following open question in the negative: if the sum of two shift-invariant ergodic measures is computable, must these measures be computable as well? We also investigate how generic a point with computable image can be. To this end we introduce a notion of genericity of a point w.r.t. a function, which enables us to unify several finite injury constructions from computability theory

Negative Interactions in Irreversible Self-Assembly

This paper explores the use of negative (i.e., repulsive) interaction the abstract Tile Assembly Model defined by Winfree. Winfree postulated negative interactions to be physically plausible in his Ph.D. thesis, and Reif, Sahu, and Yin explored their power in the context of reversible attachment operations. We explore the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Omega(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate a Turing machine with space bound s and time bound t, while ensuring that no intermediate assembly grows larger than O(s), rather than O(s * t) as required by the standard Turing machine simulation with tiles

Estimation of a dynamic discrete choice model of irreversible investment

In this paper we propose and estimate a dynamic structural model of fixed capital investment at the firm level. Our dataset consists of an unbalanced panel of Spanish manufacturing firms. Two important features are present in this dataset. There are periods in which firms decide not to invest and periods of large investment episodes. These empirical evidence of infrequent and lumpy investment provides evidence in favour of irreversibilities and nonconvex capital adjustment costs. We consider a dynamic discrete choice model of irreversible investment with a general specification of adjustment costs including convex and nonconvex components. We use a two stage estimation procedure. In a first stage, we obtain GMM estimates of technological parameters. In the second stage, we obtain partial maximum likelihood estimates for the adjustment cost parameters. The estimation strategy builds on the representation of conditional value functions as a computable function of conditional choice probabilities. It is in the line of structural estimation techniques which avoid the solution of the dynamic programming problem

Possible Roles of Neural Electron Spin Networks in Memory and Consciousness

Spin is the origin of quantum effects in both Bohm and Hestenes quantum formulism and a fundamental quantum process associated with the structure of space-time. Thus, we have recently theorized that spin is the mind-pixel and developed a qualitative model of consciousness based on nuclear spins inside neural membranes and proteins. In this paper, we explore the possibility of unpaired electron spins being the mind-pixels. Besides free O2 and NO, the main sources of unpaired electron spins in neural membranes and proteins are transition metal ions and O2 and NO bound/absorbed to large molecules, free radicals produced through biochemical reactions and excited molecular triplet states induced by fluctuating internal magnetic fields. We show that unpaired electron spin networks inside neural membranes and proteins are modulated by action potentials through exchange and dipolar coupling tensors and spin-orbital coupling and g-factor tensors and perturbed by microscopically strong and fluctuating internal magnetic fields produced largely by diffusing O2. We argue that these spin networks could be involved in brain functions since said modulation inputs information carried by the neural spike trains into them, said perturbation activates various dynamics within them and the combination of the two likely produce stochastic resonance thus synchronizing said dynamics to the neural firings. Although quantum coherence is desirable, it is not required for these spin networks to serve as the microscopic components for the classical neural networks. On the quantum aspect, we speculate that human brain works as follows with unpaired electron spins being the mind-pixels: Through action potential modulated electron spin interactions and fluctuating internal magnetic field driven activations, the neural electron spin networks inside neural membranes and proteins form various entangled quantum states some of which survive decoherence through quantum Zeno effects or in decoherence-free subspaces and then collapse contextually via irreversible and non-computable means producing consciousness and, in turn, the collective spin dynamics associated with said collapses have effects through spin chemistry on classical neural activities thus influencing the neural networks of the brain. Thus, according to this alternative model, the unpaired electron spin networks are the “mind-screen,” the neural membranes and proteins are the mind-screen and memory matrices, and diffusing O2 and NO are pixel-activating agents. Together, they form the neural substrates of consciousness

A Local Deterministic Model of Quantum Spin Measurement

The conventional view, that Einstein was wrong to believe that quantum physics is local and deterministic, is challenged. A parametrised model, Q, for the state vector evolution of spin 1/2 particles during measurement is developed. Q draws on recent work on so-called riddled basins in dynamical systems theory, and is local, deterministic, nonlinear and time asymmetric. Moreover the evolution of the state vector to one of two chaotic attractors (taken to represent observed spin states) is effectively uncomputable. Motivation for this model arises from Penrose's speculations about the nature and role of quantum gravity. Although the evolution of Q's state vector is uncomputable, the probability that the system will evolve to one of the two attractors is computable. These probabilities correspond quantitatively to the statistics of spin 1/2 particles. In an ensemble sense the evolution of the state vector towards an attractor can be described by a diffusive random walk. Bell's theorem and a version of the Bell-Kochen_specker quantum entanglement paradox are discussed. It is shown that proving an inconsistency with locality demands the existence of definite truth values to certain counterfactual propositions. In Q these deterministic propositions are physically uncomputable and no non-algorithmic solution is either known or suspected. Adapting the mathematical formalist approach, the non-existence of definite truth values to such counterfactual propositions is posited. No inconsistency with experiment is found. Hence Q is not necessarily constrained by Bell's inequality.Comment: This paper has been accepted for publication in the Proceedings of the Royal Society of London (Proc.Roy.Soc.A) I will mail the paper's figures on request (write to [email protected]