In recent years, classical computability has expanded beyond its original scope to address issues related to computability and complexity in algebra, analysis, and physics. The deep interconnection between "computation" and "proof" has originated much of the most significant work in constructive mathematics and mathematical logic of the last 70 years. Moreover, the increasingly compelling necessity to deal with computability in the real world (such as computing on continuous data, biological computing, and physical models) has brought focus to new paradigms of computation that are based on biological and physical models. These models address questions of efficiency in a radically new way and even threaten to move the so-called Turing barrier, i.e. the line between the decidable and the un-decidable.

This book examines new developments in the theory and practice of computation from a mathematical perspective, with topics ranging from classical computability to complexity, from biocomputing to quantum computing. The book opens with an introduction by Andrew Hodges, the Turing biographer, who analyzes the pioneering work that anticipated recent developments concerning computations allegedly new paradigms. The remaining material covers traditional topics in computability theory such as relative computability, theory of numberings, and domain theory, in addition to topics on the relationships between proof theory, computability, and complexity theory. New paradigms of computation arising from biology and quantum physics are also discussed, as well as the computability of the real numbers and its related issues.

This book is suitable for researchers and graduate students in mathematics, philosophy, and computer science with a special interest in logic and foundational issues. Most useful to graduate students are the survey papers on computable analysis and biological computing. Logicians and theoretical physicists will also benefit from this book.

New Computational Paradigms: Changing Conceptions of What is Computable

In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Holder-Lipschitz, uniformly continuous, bounded and weakly quasi-nonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

https://www.ams.org/tran/2008-360-05/S0002-9947-07-04429-7/S0002-9947-07-04429-7.pdf

General logical metatheorems for functional analysis

Priced timed (game) automata extend timed (game) automata with costs on both locations and transitions. In this paper we focus on reachability priced timed game automata and prove that the optimal cost for winning such a game is computable under conditions concerning the non-zenoness of cost. Under stronger conditions (strictness of constraints) we prove that in case an optimal strategy exists, we can compute a state-based winning optimal strategy.

https://hal.archives-ouvertes.fr/hal-00350483/document

Optimal strategies in priced timed game automata

http://cs.au.dk/~danvy/DSc/10_ager-danvy-midtgaard_ipl-2004.pdf

A functional correspondence between call-by-need evaluators and lazy abstract machines

This article demonstrates a method for composing a programming language by combining action-semantics modules. Each module is defined separately, and then a programming-language module is defined by combining existing modules. This method enables the language designer to gradually develop a language by defining, selecting and combining suitable modules. The resulting modular structure is substantially different from that previously employed in action-semantic descriptions.It also discusses how to resolve the conflicts that may arise when combining modules, and indicates some advantages that action semantics has over other approaches in this respect.

/pdf/composing-programming-languages-by-combining-action-wv3d9kff8h.pdf

Composing programming languages by combining action-semantics modules

This paper establishes explicit quantitative bounds on the computation of approximate fixed points of asymptotically (quasi-) nonexpansive mappings f by means of iterative processes. Here f is a selfmapping of a convex subset C of a uniformly convex normed space X. We consider general Krasnoselski-Mann iterations with and without error terms. As a consequence of our quantitative analysis we also get new qualitative results which show that the assumption on the existence of fixed points of f can be replaced by the existence of approximate fixed points only. We explain how the existence of effective uniform bounds in this context can be inferred already a-priorily by a logical metatheorem recently proved by the first author. Our bounds were in fact found with the help of the general logical machinery behind the proof of this metatheorem. The proofs we present here are, however, completely self-contained and do not require any tools from logic.

/pdf/bounds-on-iterations-of-asymptotically-quasi-nonexpansive-519xgt8bo6.pdf

Bounds on Iterations of Asymptotically Quasi-Nonexpansive Mappings

We present a new approach to computability of real numbers in which real functions have type-1 representations, which also includes the ability to reason about the complexity of real numbers and functions. We discuss how this allows efficient implementations of exact real numbers and also present a new real number system that is based on it.

/pdf/a-two-layer-approach-to-the-computability-and-complexity-of-2ufdzw9ve1.pdf

Branimir Lambov

Papers

Bounds on Iterations of Asymptotically Quasi-Nonexpansive Mappings

A Two-Layer Approach to the Computability and Complexity of Real Numbers