Science involves descriptions of the world we live in. It also depends on nature exhibiting what we can best describe as a high aLgorithmic content. The theme running through this collection of papers is that of the interaction between descriptions, in the form of formal theories, and the algorithmic content of what is described, namely of the modeLs of those theories. This appears most explicitly here in a number of valuable, and substantial, contributions to what has until recently been known as 'recursive model theory' - an area in which researchers from the former Soviet Union (in particular Novosibirsk) have been pre-eminent. There are also articles concerned with the computability of aspects of familiar mathematical structures, and - a return to the sort of basic underlying questions considered by Alan Turing in the early days of the subject - an article giving a new perspective on computability in the real world. And, of course, there are also articles concerned with the classical theory of computability, including the first widely available survey of work on quasi-reducibility. The contributors, all internationally recognised experts in their fields, have been associated with the three-year INTAS-RFBR Research Project "Com putability and Models" (Project No. 972-139), and most have participated in one or more of the various international workshops (in Novosibirsk, Heidelberg and Almaty) and otherresearch activities of the network."

Science involves descriptions of the world we live in. It also depends on nature exhibiting what we can best describe as a high aLgorithmic content. The theme running through this collection of papers is that of the interaction between descriptions, in the form of formal theories, and the algorithmic content of what is described, namely of the modeLs of those theories. This appears most explicitly here in a number of valuable, and substantial, contributions to what has until recently been known as 'recursive model theory' - an area in which researchers from the former Soviet Union (in particular Novosibirsk) have been pre-eminent. There are also articles concerned with the computability of aspects of familiar mathematical structures, and - a return to the sort of basic underlying questions considered by Alan Turing in the early days of the subject - an article giving a new perspective on computability in the real world. And, of course, there are also articles concerned with the classical theory of computability, including the first widely available survey of work on quasi-reducibility. The contributors, all internationally recognised experts in their fields, have been associated with the three-year INTAS-RFBR Research Project "Com putability and Models" (Project No. 972-139), and most have participated in one or more of the various international workshops (in Novosibirsk, Heidelberg and Almaty) and otherresearch activities of the network.

This book constitutes the refereed proceedings of the 9th International Conference on Theory and Applications of Models of Computation, TAMC 2012, held in Beijing, China, in May 2012. The conference was combined with the Turing Lectures 2012, dedicated to celebrating Alan Turing s unique impact on mathematics, computing, computer science, informatics, morphogenesis, philosophy, and the wider scientific world. Eight Turing Lectures were given at the TAMC 2012. The 40 revised full papers presented together with invited talks were carefully reviewed and selected from 86 submissions. The papers address 4 special sessions at TAMC 2012 which were algorithms and information in networks, complexity and cryptography, models of computing and networking, programming and verification."

This book constitutes the refereed proceedings of the Turing Centenary Conference and the 8th Conference on Computability in Europe, CiE 2012, held in Cambridge, UK, in June 2012. The 53 revised papers presented together with 6 invited lectures were carefully reviewed and selected with an acceptance rate of under 29,8%. The CiE 2012 Turing Centenary Conference will be remembered as a historic event in the continuing development of the powerful explanatory role of computability across a wide spectrum of research areas. The papers presented at CiE 2012 represent the best of current research in the area, and forms a fitting tribute to the short but brilliant trajectory of Alan Mathison Turing. Both the conference series and the association promote the development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences such as physics and biology, and also including the promotion of related non-scientific fields such as philosophy and history of computing.

This book constitutes the refereed proceedings of the 6th International Conference on Theory and Applications of Models of Computation, TAMC 2009, held in Changsha, China in May 2009.

The 39 full papers presented together with 7 invited papers as well as 3 plenary talks were selected from 86 submissions. The papers address the three main themes of the conference which were Computability, Complexity, and Algorithms. The conference aimed to bring together researchers with interests in theoretical computer science, algorithmic mathematics, and applications to the physical sciences.

This book constitutes the refereed proceedings of the Third International Conference on Computability in Europe, CiE 2007, held in Sienna, Italy, in June 2007.

The 50 revised full papers presented together with 36 invited papers were carefully reviewed and selected from 167 submissions. Among them are papers corresponding to 12 plenary talks and papers of 8 special sessions entitled doing without turing machines: constructivism and formal topology, approaches to computational learning, real computation, computability and mathematical structure, complexity of algorithms and proofs, logic and new paradigms of computability, computational foundations of physics and biology, as well as a women in computability workshop.

This book constitutes the refereed proceedings of the Third International Conference on Theory and Applications of Models of Computation, TAMC 2006, held in Beijing, China, in May 2006. The 75 revised full papers presented together with 7 plenary talks were carefully reviewed and selected from 319 submissions. All major areas in computer science, mathematics (especially logic) and the physical sciences particularly with regard to computation and computability theory are addressed.

CiE 2005: New Computational Paradigms http: //www. illc. uva. nl/CiE/ ThecooperationComputabilityinEurope (CiE)isaninformalEuropeannetwork covering computability in theoretical computer science and mathematical logic, ranging from application of novel approaches to computation to set-theoretic analysesofin?nitarycomputingmodels. Thecooperationconsistsofelevenmain nodesandincludesover400researchers;itiscoordinatedfromLeeds(UK). More information about CiE can be found in Barry Cooper's introductory paper to this volume (p. 1) and at http: //www. amsta. leeds. ac. uk/pure/staff/cooper/cie. html CiE 2005 was a conference on the special topic "New Computational Pa- digms" and was held in Amsterdam in June 2005. It was initiated by and served as a focus point for the informal cooperation CiE. The topic of "New Com- tational Paradigms" covers connections between computation and physical s- tems (e. g., quantum computation, neural nets, molecular computation) but also higher mathematical models of computation (e. g., in?nitary computation or real computation). Computability theory is central to large areas of theoretical computer science and mathematical logic. Traditionally, the computational model of the Turing machine (or mathematically equivalent models) has been used to reason about computation or computability. For general computability inquiries (with - bounded resources), the choice of the model of computation hardly matters (this fact is encapsulated in the so-called "Church-Turing thesis"); this could change as soon as questions of e?ciency are investigated.

This book constitutes the refereed proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007.

The 67 revised full papers presented together with 2 plenary lectures were carefully reviewed and selected from over 500 submissions. All major areas in computer science, mathematics (especially logic) and the physical sciences particularly with regard to computation and computability theory are addressed. The papers - featuring this crossdisciplinary character - particularly focus on algorithms, complexity and computability theory, giving the conference a special flavor and distinction.