This book constitutes the strictly refereed post-workshop proceedings of the 11th International Workshop on Computer Science Logic, CSL '97, held as the 1997 Annual Conference of the European Association on Computer Science Logic, EACSL, in Aarhus, Denmark, in August 1997. The volume presents 26 revised full papers selected after two rounds of refereeing from initially 92 submissions; also included are four invited papers. The book addresses all current aspects of computer science logics and its applications and thus presents (...) the state of the art in the area. (shrink)

These are the proceedings of the First International Conference on Compu- tional Logic (CL 2000) which was held at Imperial College in London from 24th to 28th July, 2000. The theme of the conference covered all aspects of the theory, implementation, and application of computational logic, where computational logic is to be understood broadly as the use of logic in computer science. The conference was collocated with the following events: { 6th International Conference on Rules and (...) Objects in Databases (DOOD 2000) { 10th International Workshop on Logic-based Program Synthesis and Tra- formation (LOPSTR 2000) { 10th International Conference on Inductive Logic Programming (ILP 2000). CL 2000 consisted of seven streams: { Program Development (LOPSTR 2000) { Logic Programming: Theory and Extensions { Constraints { Automated Deduction: Putting Theory into Practice { Knowledge Representation and Non-monotonic Reasoning { Database Systems (DOOD 2000) { Logic Programming: Implementations and Applications. The LOPSTR 2000 workshop constituted the program development stream and the DOOD 2000 conference constituted the database systems stream. Each stream had its own chair and program committee, which autonomously selected the papers in the area of the stream. Overall, 176 papers were submitted, of which 86 were selected to be presented at the conference and appear in these proceedings. The acceptance rate was uniform across the streams. In addition, LOPSTR 2000 accepted about 15 extended abstracts to be presented at the conference in the program development stream. (shrink)

~Et moi,.... si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point alle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non· The series is divergent; therefore we may be sense'. Eric T. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and (...) non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics...'; 'One service logic has rendered com­ puter science...'; 'One service category theory has rendered mathematics...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series. (shrink)

This is the 3rd edition. Although a number of new technological applications require classical deductive computation with non-classical logics, many key technologies still do well—or exclusively, for that matter—with classical logic. In this first volume, we elaborate on classical deductive computing with classical logic. The objective of the main text is to provide the reader with a thorough elaboration on both classical computing – a.k.a. formal languages and automata theory – and classical deduction with the classical first-order predicate (...) calculus with a view to computational implementations, namely in automated theorem proving and logic programming. The present third edition improves on the previous ones by providing an altogether more algorithmic approach: There is now a wholly new section on algorithms and there are in total fourteen clearly isolated algorithms designed in pseudo-code. Other improvements are, for instance, an emphasis on functions in Chapter 1 and more exercises with Turing machines. (shrink)

Reflecting Alan Robinson's fundamental contribution to computational logic, this book brings together seminal papers in inference, equality theories, and logic programming. It is an exceptional collection that ranges from surveys of major areas to new results in more specialized topics. Alan Robinson is currently the University Professor at Syracuse University. Jean-Louis Lassez is a Research Scientist at the IBM Thomas J. Watson Research Center. Gordon Plotkin is Professor of Computer Science at the University of Edinburgh. Contents: Inference. Subsumption, (...) A Sometimes Undervalued Procedure, Larry Wos, Ross Overbeek, and Ewing Lusk. The Markgraf Karl Refutation Procedure, Hans Jurgen Ohlbach and Jorg H. Siekmann. Modal Logic Should Say More than it Does, Melvin Fitting. Interactive Proof Presentation, W. W. Bledsoe. Intelligent Backtracking Revisited, Maurice Bruynooghe. A Science of Reasoning, Alan Bundy. Inductive Inference of Theories from Facts, Ehud Y. Shapiro. Equality. Solving Equations in Abstract Algebras: A Rule-based Survey of Unification, Jean-Pierre Jouannaud and Claude Kirchner. Disunification: A Survey, Hubert Comon. A Case Study of the Completion Procedure: Proving Ring Commutativity Problems, Deepak Kapur and Hantao Zhang. Computations in Regular Rewriting Systems I and II, Girard Huet and JeanJacques Levy. Unification and ML Type Reconstruction, Paris Kanellakis, Harry Mairson, and John Mitchell. Automatic Dimensional Analysis, Mitchell Wand. Logic Programming. Logic Programming Schemes and Their Implementations, Keith Clark. A Near-Horn Prolog for Compilation, Donald Loveland and David Reed. Unfold/Fold Transformations of Logic Programs, P. A. Gardner and J. C. Shepherdson. An Algebraic Representation of Logic Program Computations, Andrea Corradini and Ugo Montanari. Theory of Disjunctive Logic Programs, Jack Minker, Arcot Rajasekar, and Jorge Lobo. Bottom-Up Evaluation of Logic Programs, Jeffrey Naughton and Raghu Ramakrishnan. Absys, the First Logic Programming Language: A View of the Inevitability of Logic Programming, E. W. Elcock. (shrink)

This chapter contains sections titled: Introduction The ACL2 System A Modeling Problem Case Studies.

In this paper we solve the problem how to axiomatize a system of quantum computational gates known as the Poincaré irreversible quantum computational system. A Hilbert-style calculus is introduced obtaining a strong completeness theorem.

Alan Robinson This set of essays pays tribute to Bob Kowalski on his 60th birthday, an anniversary which gives his friends and colleagues an excuse to celebrate his career as an original thinker, a charismatic communicator, and a forceful intellectual leader. The logic programming community hereby and herein conveys its respect and thanks to him for his pivotal role in creating and fostering the conceptual paradigm which is its raison d’Œtre. The diversity of interests covered here reflects the variety (...) of Bob’s concerns. Read on. It is an intellectual feast. Before you begin, permit me to send him a brief personal, but public, message: Bob, how right you were, and how wrong I was. I should explain. When Bob arrived in Edinburgh in 1967 resolution was as yet fairly new, having taken several years to become at all widely known. Research groups to investigate various aspects of resolution sprang up at several institutions, the one organized by Bernard Meltzer at Edinburgh University being among the first. For the half-dozen years that Bob was a leading member of Bernard’s group, I was a frequent visitor to it, and I saw a lot of him. We had many discussions about logic, computation, and language. (shrink)

Computability logic (CL) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, "truth" means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves (...) its soundness and completeness with respect to the semantics of CL. Conservatively extending classical predicate calculus and offering considerable additional expressive and deductive power, CL12 presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories. To obtain a model example of such theories, this paper rebuilds the traditional, classical-logic-based Peano arithmetic into a computability-logic-based counterpart. Among the purposes of the present contribution is to provide a starting point for what, as the author wishes to hope, might become a new line of research with a potential of interesting findings—an exploration of the presumably quite unusual metatheory of CL-based arithmetic and other CL-based applied systems. (shrink)

This book constitutes the refereed proceedings of the 5th Kurt Gödel Colloquium on Computational Logic and Proof Theory, KGC '97, held in Vienna, Austria, in August 1997. The volume presents 20 revised full papers selected from 38 submitted papers. Also included are seven invited contributions by leading experts in the area. The book documents interdisciplinary work done in the area of computer science and mathematical logics by combining research on provability, analysis of proofs, proof search, and complexity.

This paper presents a soundness and completeness proof for propositional intuitionistic calculus with respect to the semantics of computability logic. The latter interprets formulas as interactive computational problems, formalized as games between a machine and its environment. Intuitionistic implication is understood as algorithmic reduction in the weakest possible — and hence most natural — sense, disjunction and conjunction as deterministic-choice combinations of problems , and “absurd” as a computational problem of universal strength.

This work is an attempt to lay foundations for a theory of interactive computation and bring logic and theory of computing closer together. It semantically introduces a logic of computability and sets a program for studying various aspects of that logic. The intuitive notion of computational problems is formalized as a certain new, procedural-rule-free sort of games between the machine and the environment, and computability is understood as existence of an interactive Turing machine that wins (...) the game against any possible environment. The formalism used as a specification language for computational problems, called the universal language, is a non-disjoint union of the formalisms of classical, intuitionistic and linear logics, with logical operators interpreted as certain—most basic and natural—operations on problems. Validity of a formula is understood as being “always computable”, and the set of all valid formulas is called the universal logic. The name “universal” is related to the potential of this logic to integrate, on the basis of one semantics, classical, intuitionistic and linear logics, with their seemingly unrelated or even antagonistic philosophies. In particular, the classical notion of truth turns out to be nothing but computability restricted to the formulas of the classical fragment of the universal language, which makes classical logic a natural syntactic fragment of the universal logic. The same appears to be the case for intuitionistic and linear logics . Unlike classical logic, these two do not have a good concept of truth, and the notion of computability restricted to the corresponding two fragments of the universal language, based on the intuitions that it formalizes, can well qualify as “intuitionistic truth” and “linear-logic truth”. The paper also provides illustrations of potential applications of the universal logic in knowledgebase, resourcebase and planning systems, as well as constructive applied theories. The author has tried to make this article easy to read. It is fully self-contained and can be understood without any specialized knowledge of any particular subfield of logic or computer science. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were (...) revised and updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ${\neg}$ , parallel conjunction ${\wedge}$ , parallel disjunction ${\vee}$ , branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (the present) Part I containing preliminaries and a soundness proof, and (the forthcoming) Part II containing a completeness proof.

This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\neg}}$$\end{document}, parallel conjunction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\wedge}}$$\end{document}, parallel disjunction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\vee}}$$\end{document}, branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (...) (the previous) Part I containing preliminaries and a soundness proof, and (the present) Part II containing a completeness proof. (shrink)

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature | K | ≤ 1. Let Met (M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met (M) such as the diameter. They showed that for every Turing machine T e , e ∈ ω, there is a sequence (uniformly effective in e) of homology (...) n-spheres {P k e } k ∈ ω which are also hypersurfaces, such that P k e is diffeomorphic to the standard n-sphere S n (denoted P k e ≈ diff S n ) iff T e halts on input k, and in this case the connected sum N k e =M ♯ P k e ≈ diff M , so N k e ∈ Met(M), and N k e is associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time σ e (k) of T e on inputs y i } ∈ ω of c.e. sets so that for all i the settling time of the associated Turing machine for A i dominates that for A i + 1 , even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a "fractal" like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as "the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization." From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structural complexity of the geometry and topology, not merely undecidability results as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they use nontrivial information about c.e. sets, the Soare sequence {A i } i ∈ ω above, not merely G öodel's c.e. noncomputable set K of the 1930's; and (3) without using computability theory there is no known proof that local minima exist even for simple manifolds like the torus T 5 (see §). (shrink)

We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and "(general) recursiveness". We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. (...) After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of "computability" and "recursion." Specifically we recommend that: the term "recursive" should no longer carry the additional meaning of "computable" or "decidable;" functions defined using Turing machines, register machines, or their variants should be called "computable" rather than "recursive;" we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be "Computability Theory" or simply Computability rather than "Recursive Function Theory.". (shrink)

We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related. A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output (...) a finite sequence $\langle f_0, \ldots,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left$ for $i \le n$ form a covering of $2^ $. A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.Our study of yields exotic objects in computability theory, while leads to surprising results in Reverse Mathematics. We stress that and are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa. (shrink)

This paper shows that the basic logic induced by the parallel recurrence $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ of computability logic (i.e., the one in the signature $\{\neg,$\wedge$,\vee,\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}},\hspace {-2pt}\mbox {\raisebox {0.12cm}{\@setfontsize \small {7}{8}$\vee$}\hspace {-3.6pt}\raisebox {0.02cm}{\tiny $\mid$}\hspace {2pt}}\}$ ) is a proper superset of the basic logic induced by the branching recurrence $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ (i.e., the one in the signature (...) $\{\neg,$\wedge$,\vee,\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}},\mbox {\raisebox {0.12cm}{$\circ$}\hspace {-0.115cm}\raisebox {0.02cm}{\tiny $\mid$}\hspace {2pt}}\}$ ). The latter is known to be precisely captured by the cirquent calculus system CL15 , conjectured by Japaridze to remain sound—but not complete—with $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ instead of $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ . The present result is obtained by positively verifying that conjecture. A secondary result of the paper is showing that $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ is strictly weaker than $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ in the sense that, while $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}F$ logically implies $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}F$ , the reverse does not hold. (shrink)

Unlike the standard Quantum Computational Logic, where the carrier of information is conventionally assumed to be only the last qubit over a sequence of many qubits, here we propose an extended version of the QCL where the number and the position of the target qubits are arbitrary.

This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis’ work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert’s tenth (...) problem. The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis’ work. As a whole, this book shows how Davis’ scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic. (shrink)

This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and proven (...) to be extremely useful in computer science. The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students. (shrink)

The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n (...) ∈ ω with its set of predecessors, so n = {0, 1, 2, …, n − 1}. (shrink)

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in (...) question, we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question. (shrink)

We prove a number of elementary facts about computability in partial combinatory algebras. We disprove a suggestion made by Kreisel about using Friedberg numberings to construct extensional pca’s. We then discuss separability and elements without total extensions. We relate this to Ershov’s notion of precompleteness, and we show that precomplete numberings are not 1–1 in general.

Which function is computed by a Turing machine will depend on how the symbols it manipulates are interpreted. Further, by invoking bizarre systems of notation it is easy to define Turing machines that compute textbook examples of uncomputable functions, such as the solution to the decision problem for first-order logic. Thus, the distinction between computable and uncomputable functions depends on the system of notation used. This raises the question: which systems of notation are the relevant ones for determining whether (...) a function is computable? These are the acceptable notations. I argue for a use criterion of acceptability: the acceptable notations for a domain of objects D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document} are the notations that we can use for the usual D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document}-activities. Acceptable notations for natural numbers are ones that we can count with. Acceptable notations for logical formulas are the ones that we can use in inference and logical analysis. And so on. This criterion makes computability a relative notion—whether a function is computable depends on which notations are acceptable, which is relative to our interests and abilities. I argue that this is not a problem, however, since there is independent reason for taking the extension of computable function to be relative to contingent facts. Similarly, the fact that the use criterion makes a mathematical distinction depend on practical concerns is also not a problem because it dovetails with similar phenomena in other areas of computability theory, namely the roles of notation in computation over real numbers and of Extended Church’s Thesis in complexity theory. (shrink)

criticism defend the idea that an individual reader's understanding of a text can be a factor in determining the meaning of what is written in that text, and hence must play a part in determining the very identity conditions of works of literary art. We examine some accounts that have been given of the type of readerly ‘competence’ that a reader must have in order for her responses to a text to play this sort of constitutive role. We argue that (...) the analogy drawn by Stanley Fish and Jonathan Culler between literary and linguistic competence is philosophically flawed and explanatorily unfruitful, and that a better way of understanding the notion of literary competence can be constructed by appeal to some limitation results in formal logic and computability theory. (shrink)

Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computable function can make van der Waerden's intuitive notion of an explicit field precise. This (...) led to the notion of a computable structure. In 1960, Rabin proved that every computable field has a computable algebraic closure. However, not every classical result “effectivizes”. Unlike Vaught's theorem that no complete theory has exactly two nonisomorphic countable models, Millar's and Kudaibergenov's result establishes that there is a complete decidable theory that has exactly two nonisomorphic countable models with computable elementary diagrams. In the 1970's, Metakides and Nerode [58], [59] and Remmel [71], [72], [73] used more advanced methods of computability theory to investigate algorithmic properties of fields, vector spaces, and other mathematical structures. (shrink)

The paper discusses tools for teaching logic used in Logic & Proofs, a web-based introduction to modern logic that has been taken by more than 1,300 students since the fall of 2003. The tools include a wide array of interactive learning environments or cognitive mini-tutors; most important among them is the Carnegie Proof Lab. The Proof Lab is a sophisticated interface for constructing natural deduction proofs and is central, as strategically guided discovery of proofs is the distinctive (...) focus of the course. My discussion makes explicit the broader intellectual context, but also the pursuit of pedagogical goals and their experimental examination. The intellectual context includes i) the theoretical background for the proof search algorithm AProS and its use for a dynamic Proof Tutor, and ii) the programmatic expansion of the course to Computational Logic. 1. (shrink)

The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than non-ergodic ones. In a previous paper we studied the extent to which non-ergodic measures inherit the computability properties of ergodic ones, and introduced the notion of an effectively decomposable measure. We asked the (...) following question: if the ergodic decomposition of a stationary measure is finite, is this decomposition effective? In this paper we answer the question in the negative. (shrink)

Several results about the game of cops and robbers on infinite graphs are analyzed from the perspective of computability theory. Computable robber-win graphs are constructed with the property that no computable robber strategy is a winning strategy, and such that for an arbitrary computable ordinal \, any winning strategy has complexity at least \}\). Symmetrically, computable cop-win graphs are constructed with the property that no computable cop strategy is a winning strategy. Locally finite infinite trees and graphs are explored. (...) The Turing computability of a binary relation used to classify cop-win graphs is studied, and the computational difficulty of determining the winner for locally finite computable graphs is discussed. (shrink)

Topologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their results depended on the existence of certain sequences of c.e. sets, constructed at their request by Csima and Soare, whose settling times had the necessary dominating properties. (...) Although these computability results had been announced earlier, their proofs have been deferred until this paper. Computably enumerable sets have long been used to prove undecidability of mathematical problems such as the word problem for groups and Hilbert's Tenth Problem. However, this example by Nabutovsky and Weinberger is perhaps the first example of the use of c.e. sets to demonstrate specific mathematical or geometric complexity of a mathematical structure such as the depth and distribution of local minima. (shrink)

In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous models," (...) which generalize the older results and which include positive results on when a certain homogeneous model of T has an isomorphic copy of a given Turing degree. We then survey Lange's "A characterization of the 0-basis homogeneous bounding degrees" for negative results about when does not have such copies, generalizing negative results by Goncharov, Peretyat'kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare, "Bounding homogeneous models," about degrees d that are homogeneous bounding and explain their relation to the PA degrees. (shrink)

This volume contains several invited papers as well as a selection of the other contributions. The conference was the first meeting of the Soviet logicians interested in com- puter science with their Western counterparts. The papers report new results and techniques in applications of deductive systems, deductive program synthesis and analysis, computer experiments in logic related fields, theorem proving and logic programming. It provides access to intensive work on computer logic both in the USSR and in Western (...) countries. (shrink)