Mathematical Logic: Proof Theory, Constructive Mathematics

Abstract

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexity. Mathematics Subject Classification (2010): 03Fxx. Introduction by the Organisers The workshop Mathematical Logic: Proof Theory, Constructive Mathematics was held November 16-22, 2014 and included two tutorials: (1) Thierry Coquand: Univalent Foundation and Constructive Mathematics (2 times 1 hour), (2) Ulrich Kohlenbach, Daniel Körnlein, Angeliki Koutsoukou-Argyraki, Laureņtiu Leu̧stean: Proof-Theoretic Methods in Nonlinear Analysis (2 times 50 min plus 2 times 30). Coquand’s tutorial gave a general introduction on the univalent foundation program of Voevodsky and discussed the construction of the cubical set model of type theory in a constructive metatheory. This model satisfies the computation rules for equality introduced by P. Martin-Löf as judgemental equality. The second tutorial developed the proof-theoretic framework for the unwinding of proofs in nonlinear analysis and outlined recent applications to: image recovery problems (Part I, Kohlenbach), fixed point theory of pseudocontractive mappings 2934 Oberwolfach Report 52/2014 (Part II, Körnlein), convex optimization (Part III, Leu̧stean) and abstract Cauchyproblems given by accretive operators (Part IV, Koutsoukou-Argyraki). In addition to these tutorials, 29 talks of mostly 25 minutes were given aiming: To promote the interaction of proof theory and computability theory with core areas of mathematics as well as computer science via the use of proof interpretations. J. Avigad’s talk studied the amount of algorithmic randomness needed in Weyl’s theorem on uniform distributions. H. Towsner showed how to arrive at Tao’s version of Szemerédi’s regularity lemma as the functional interpretation of a measure-theoretic Π3-statement. H. Schwichtenberg reported on a machine extracted program from the Nash-Williams minimal bad sequence argument for Higman’s lemma. V. Brattka introduced a concept of Las Vegas computable functions to calibrate the computational power of randomized computations on real numbers. A. Weiermann described a general formula for the computation of the maximal order types for well quasi orders arising in the combinatorics of finite multisets. P. Schuster showed how a reformulation of transfinite methods in algebra as admissible rules can be used to eliminate uses of such methods from proofs of sufficiently simple statements in abstract algebra. On the side of applications to concrete applications in computer science, M. Seisenberger reported on applications of logic to the verification of railway control systems and U. Berger developed a proposal to optimize programs extracted by proof-theoretic methods to be able to e.g. control their complexity, allow for partial data and to override data that are no longer used. To further develop foundational aspects of proof theory and constructive mathematics. S. Artemov talked on intuitionistic epistemic logic which is based on the BHK-semantics and treats intuitionistic knowledge as the result of a verification. F. Aschieri reported on a new proof-theoretic method to extract Herbrand disjunctions from classical first-order natural deduction proofs. B. Afshari’s talk also studied Herbrand’s theorem, this time in terms of certain tree grammars assigned to proofs of existential statements in first-order logic. The talk by G.E. Leigh addressed the issue of cut-elimination for first-order theories of truth. P. Oliva presented new results on a game-theoretic interpretation of Spector’s bar recursion, a more efficient novel variant of bar recursion and recent uses in the analysis of the Podelski-Rybalchenko termination theorem. F. Ferreira showed how a suitable functional interpretation can be used to give an ordinal analysis of Kripke-Platek set theory. B. van den Berg reported on new developments in the functional interpretation of systems of nonstandard analysis. T. Streicher talked on models of classical realizability (in the sense of J.-L. Krivine) arising from domain-theoretic models of λ-calculus with control. The talks by L.D. Beklemishev and J.J. Joosten addressed recent progress in the area of provability logic with applications to ordinal analysis. Also on the side of ordinal analysis was a talk by T. Strahm, who developed a so-called flexible type system in the spirit of S. Feferman whose strength is measured by the small Veblen ordinal. S. Berardi presented Mathematical Logic: Proof theory, Constructive Mathematics 2935 a new rule-learning based approach to the proof-theoretic analysis of second order arithmetic. A. Bauer talked about constructive homotopy theory and models of intensional type theory. I. Petrakis proposed a formalization of so-called Bishop spaces as a constructive foundation for point-function topology. A. Swan studied the existence property for intuitionistic set theories where this property has to be understood in terms of definability. M. Rathjen reported on his recent proof of a conjecture due to Feferman which states that the continuum hypothesis CH is not definite in the technical sense that a certain semi-intuitionistic set theory does not prove CH∨¬CH. To explore further the connections between logic and computational complexity. Talks in this area spanned the topics of propositional proof complexity, settheoretic computation, and complexity theoretic aspects of bounded arithmetic. P. Pudlák reported on work-in-progress and new conjectures for two propositional proof systems based on integer linear programming, the cutting planes proof system and the Lovász-Schrijver proof system. N. Thapen reported new results about size and width tradeoffs for propositional resolution refutations, including new lower bounds via the colored PLS (polynomial local search) principle. S. Buss presented a new framework of polynomial-time computation for set functions based on Cobham-style limited recursion using ∈-recursion. A. Beckmann described a proof-theoretic analysis for the polynomial-time computable set functions based on safe/normal ∈-recursion. L. Ko lodziejczyk discussed recent progress on complexity-theoretic aspects of the Paris-Wilkie problem on the relationship between bounded arithmetic, the (negation) of exponentiation, and collection. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Mathematical Logic: Proof theory, Constructive Mathematics 2937 Workshop: Mathematical Logic: Proof theory, Constructive Mathematics