2007 European Summer Meeting of the Association for Symbolic Logic: Logic Colloquium '07

Abstract

s of invited and contributed talks given in person or by title by members of the Association follow. For the Program Committee Steffen Lempp Abstracts of invited joint ASL–LICS hour lecturess of invited joint ASL–LICS hour lectures MARTIN HYLAND, Combinatorics of proofs. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, DPMMS, CMS, Wilberforce Road, Cambridge CB3 0WB, UK. E-mail: m.hyland@dpmms.cam.ac.uk. Ideally interpretations of proofs should exhibit some essential combinatorial features in an interesting and appealing way. As a case study, one can consider the notion of innocent strategy which is the basis for a game semantical interpretation of proofs and programmes. Some combinatorial content of this notion is sketched in the joint LICS paper accompanying this talk, whose abstract reads as follows. We show how to construct the category of games and innocent strategies from a more primitive category of games. On that category we define a comonad and monad with the former distributing over the latter. Innocent strategies are the maps in the induced two-sided Kleisli category. Thus the problematic composition of innocent strategies reflects the use of the distributive law. The composition of simple strategies, and the combinatorics of pointers used to give the comonad and monad are themselves described in categorical terms. The notions of view and of legal play arise naturally in the explanation of the distributivity. The category-theoretic perspective provides a clear discipline for the necessary combinatorics. There are other instances of a kind of categorical combinatorics of proofs, but in this talk I shall restrict myself to the one instance. COLIN STIRLING, Higher-order matching, games and automata. School of Informatics, University of Edinburgh, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK. E-mail: cps@inf.ed.ac.uk. We describe a particular case wheremethods such asmodel-checking as used in verification are transferred to simply typed lambda calculus. Higher-order matching is the problem given t = u where t, u are terms of simply typed lambda-calculus and u is closed, is there a substitution S such that tS and u have the same normal formwith respect to beta eta-equality: can t be pattern matched to u? In the talk we consider the question: can we characterize the set of all solution terms to a matching problem? We provide an automata-theoretic account that is relative to resource: given a matching problem and a finite set of variables and constants, the (possibly infinite) set of terms that are built from those components and that solve the problem is regular. The characterization uses standard bottom-up tree automata. However, the technical proof uses a game-theoretic characterization of matching. LOGIC COLLOQUIUM ’07 125 Abstracts of invited joint ASL–LICS thirty-minute lecturess of invited joint ASL–LICS thirty-minute lectures CRISTIANO CALCAGNO, Can logic tame systems programs? Dept. of Computing, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK. E-mail: ccris@doc.ic.ac.uk. We report on our experience on designing and implementing tools for automatic reasoning about safety of systems programs using separation logic. We highlight some of the fundamental obstacles that need to be overcome, such as the complexity of data structures and scalability of the methods, on the path to realistic systems programs. MARTIN ESCARDÓ, Infinite sets that admit exhaustive search. School of Computer Science, University of Birmingham, Birmingham B15 2TT, UK. E-mail: m.escardo@cs.bham.ac.uk. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: (1) What kinds of infinite sets admit exhaustive search? (2) How do we systematically build such sets? (3) How fast can exhaustive search over infinite sets be performed? We give answers to them in the realm of Kleene–Kreisel higher-type computation: (1) involves the topological notion of compactness, (2) amounts to the usual closure properties of compact sets, including the Tychonoff theorem, (3) provides some fast algorithms and a conjecture. These two talks include my contributed LICS paper, but go beyond in two respects: a general introduction to the role of topology in computation is given, and a few new results are included, such as an Arzela–Ascoli type characterization of exhaustible sets. ROSALIE IEMHOFF, Skolemization in constructive theories. Department of Philosophy, University Utrecht, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands. E-mail: Rosalie.Iemhoff@phil.uu.nl. It has long been known that Skolemization is sound but not complete for intuitionistic logic. We will show that by slightly extending the expressive power of the logic one can define a translation that removes strong quantifiers from predicate formulas and that is related but not equal to Skolemization. Since the extended logic is constructive, the translation can be considered as an alternative to Skolemization for constructive settings. The result easily implies an analogue ofHerbrand’s theorem. Wewill apply themethod to various constructive theories and compare it to other Skolemization methods and related translations like the Dialectica Interpretation. ALEX SIMPSON, Non-well-founded proofs. LFCS, School of Informatics, University of Edinburgh, Edinburgh, UK. E-mail: Alex.Simpson@ed.ac.uk. I will discuss various situations, arising in computer science, mathematics and logic, in which one is naturally led to consider associated proof systems involving interesting forms of non-well-founded proof. Abstracts of invited tutorial talkss of invited tutorial talks STEVE JACKSON, Cardinal Arithmetic in L(R). Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas, 76203-1430, USA. E-mail: jackson@unt.edu. 126 LOGIC COLLOQUIUM ’07 In this series of talks we will survey the cardinal structure of the model L(R) assuming the axiom of determinacy. We describe the close relationship between the cardinal structure and partition properties of the oddprojective ordinals. Wewill present some recent simplifications to the presentation of this theory, as well as a result connecting the cardinal structure ofL(R) to that of the background universe V . We will attempt to make the talks as self contained as possible. BAKHADYR KHOUSSAINOV, Automatic structures. Computer Science Department, The University of Auckland, New Zealand. E-mail: bmk@cs.auckland.ac.nz. We study automatic structures. These are infinite structures that have automata presentations in a precise sense. By automata we mean any of the following: finite automata, tree automata, Büchi automata and Rabin automata. Automatic structures possess a number of interesting algorithmic, algebraic and modeltheoretic properties. For example, the first order theory of every automatic structure s decidable; automatic structures are closed under the first order interpretations; also, there are characterizations theorems for automatic well-founded partially ordered sets, Boolean algebras, trees, and finitely generated groups. Most of these theorems have algorithmic implications. For instance, the isomorphism problem for automatic Boolean algebras is decidable. The first lecture covers basic definitions and presents many examples. We explain the decidability theorem that describes extensions of the FO logic in which each automatic structure has a decidable theory. The second lecture surveys techniques for proving whether or not a given structure can be presented by automata. We also talk about logical characterizations of automatic structures. The last lecture concentrates on complexities of automatic structures in terms of well-known concepts of logic and model theory such as heights of well-founded relations, Scott ranks of structures, and Cantor–Bendixson ranks of trees. Most of the results are joint with Liu, Minnes, Nies, Nerode, Rubin, Semukhin, and Stephan. YA′ACOV PETERZIL, The infinitesimal subgroup of a definably compact group. Mathematics Department, University of Haifa, Haifa 31905, Israel. E-mail: kobi@math.haifa.ac.il. Consider the compact Linear groupG = SO(3,R). WhenG is viewed in any nonstandard real closed field, the setG of all matrices inG which are infinitesimally close to the identity forms a normal subgroup. Endow the quotient G/G with a “logic topology”, whose closed sets are those whose preimages in G are type-definable. It is easy to see that G/G, with this logic topology, is isomorphic to SO(3,R), with the Euclidean topology. Several years ago, A. Pillay conjectured that a similar phenomenon should be true for every “definably compact” group in an arbitrary o-minimal structure, even if the group itself was not defined over the real numbers. Roughly speaking, Pillay conjectured that every definably compact group G in a sufficiently saturated o-minimal structure has a canonical type-definable normal subgroup G such that the group G/G, when endowed with the logic topology as above, is isomorphic to a compact real Lie group. Moreover, the real dimension of this Lie group equals the o-minimal dimension of G . My goal in these talks is to show, with the help of examples, how the interaction between different notions, such as o-minimality, Lie groups, compactness, measure theory, and Shelah’s Independence property, yields a solution to the conjecture. For background on o-minimality, see van den Dries’s book [2] below. For Pillay’s conjecture, see [6]. For key-steps in the solution to the conjecture, see [1, 5, 3, 4]. [1] A. Berarducci, M. Otero, Y. Peterzil, and A. Pillay, A descending chain condition for groups in -minimal structures, Annals of Pure and Applied Logic, vol. 134 (2005), pp. 303–313. LOGIC COLLOQUIUM ’07 127 [2] L. v. d. Dries, Tame topology and o-minimal structures, Cambridge University Press, New York, 1998. [3]M. Edmundo and M. Otero, Definably compact abe