IA Scholar Query: Helmut Schwichtenberg
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 18 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Limits of real numbers in the binary signed digit representation
https://scholar.archive.org/work/dzomtnxzazct5f6axl3yytkqde
We extract verified algorithms for exact real number computation from constructive proofs. To this end we use a coinductive representation of reals as streams of binary signed digits. The main objective of this paper is the formalisation of a constructive proof that real numbers are closed with respect to limits. All the proofs of the main theorem and the first application are implemented in the Minlog proof system and the extracted terms are further translated into Haskell. We compare two approaches. The first approach is a direct proof. In the second approach we make use of the representation of reals by a Cauchy-sequence of rationals. Utilizing translations between the two represenation and using the completeness of the Cauchy-reals, the proof is very short. In both cases we use Minlog's program extraction mechanism to automatically extract a formally verified program that transforms a converging sequence of reals, i.e.~a sequence of streams of binary signed digits together with a modulus of convergence, into the binary signed digit representation of its limit. The correctness of the extracted terms follows directly from the soundness theorem of program extraction. As a first application we use the extracted algorithms together with Heron's method to construct an algorithm that computes square roots with respect to the binary signed digit representation. In a second application we use the convergence theorem to show that the signed digit representation of real numbers is closed under multiplication.Franziskus Wiesnet, Nils Köppwork_dzomtnxzazct5f6axl3yytkqdeThu, 18 Aug 2022 00:00:00 GMTImpredicativity and Trees with Gap Condition: A Second Course on Ordinal Analysis
https://scholar.archive.org/work/42kc5ozo7vbcrcpgebbli7efta
These lecture notes introduce central notions of impredicative ordinal analysis, such as the Bachmann-Howard ordinal and the method of collapsing, which transforms uncountable proof trees into countable ones. Specifically, we analyze parameter-free Π^1_1-comprehension and show that it cannot prove the extended Kruskal theorem due to Harvey Friedman (not even for two labels). In terms of prerequisites, we build on a previous lecture on the ordinal analysis of Peano arithmetic. The present material is intended for 12 lectures and 6 exercise sessions of 90 minutes each.Anton Freundwork_42kc5ozo7vbcrcpgebbli7eftaWed, 10 Aug 2022 00:00:00 GMTType-Theoretic Approaches to Ordinals
https://scholar.archive.org/work/da7muzgsbfhlrnicudxll3pnau
In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with deciding extensional equality. Using homotopy type theory as the foundational setting, we develop an abstract framework for ordinal theory and establish a collection of desirable properties and constructions. We then study and compare three concrete implementations of ordinals in homotopy type theory: first, a notation system based on Cantor normal forms (binary trees); second, a refined version of Brouwer trees (infinitely-branching trees); and third, extensional well-founded orders. Each of our three formulations has the central properties expected of ordinals, such as being equipped with an extensional and well-founded ordering as well as allowing basic arithmetic operations, but they differ with respect to what they make possible in addition. For example, for finite collections of ordinals, Cantor normal forms have decidable properties, but suprema of infinite collections cannot be computed. In contrast, extensional well-founded orders work well with infinite collections, but almost all properties are undecidable. Brouwer trees take the sweet spot in the middle by combining a restricted form of decidability with the ability to work with infinite increasing sequences. Our three approaches are connected by canonical order-preserving functions from the "more decidable" to the "less decidable" notions. We have formalised the results on Cantor normal forms and Brouwer trees in cubical Agda, while extensional well-founded orders have been studied and formalised thoroughly by Escardo and his collaborators. Finally, we compare the computational efficiency of our implementations with the results reported by Berger.Nicolai Kraus and Fredrik Nordvall Forsberg and Chuangjie Xuwork_da7muzgsbfhlrnicudxll3pnauSun, 07 Aug 2022 00:00:00 GMTUnprovability in Mathematics: A First Course on Ordinal Analysis
https://scholar.archive.org/work/o3ltz2k2tveojp66slwxiau4uq
These are the lecture notes of an introductory course on ordinal analysis. Our selection of topics is guided by the aim to give a complete and direct proof of a mathematical independence result: Kruskal's theorem for binary trees is unprovable in conservative extensions of Peano arithmetic (note that much stronger results of this type are due to Harvey Friedman). Concerning prerequisites, we assume a solid introduction to mathematical logic but no specialized knowledge of proof theory. The material in these notes is intended for 12 lectures and 6 exercise sessions of 90 minutes each.Anton Freundwork_o3ltz2k2tveojp66slwxiau4uqThu, 21 Apr 2022 00:00:00 GMTAnother Combination of Classical and Intuitionistic Conditionals
https://scholar.archive.org/work/bc7yvuu4tjaetlu4jckul4sqwi
On the one hand, classical logic is an extremely successful theory, even if not being perfect. On the other hand, intuitionistic logic is, without a doubt, one of the most important non-classical logics. But, how can proponents of one logic view the other logic? In this paper, we focus on one of the directions, namely how classicists can view intuitionistic logic. To this end, we introduce an expansion of positive intuitionistic logic, both semantically and proof-theoretically, and establish soundness and strong completeness. Moreover, we discuss the interesting status of disjunction, and the possibility of combining classical logic and minimal logic. We also compare our system with the system of Caleiro and Ramos.Satoru Niki, Hitoshi Omoriwork_bc7yvuu4tjaetlu4jckul4sqwiThu, 14 Apr 2022 00:00:00 GMTGeometric Logic, Constructivisation, and Automated Theorem Proving (Dagstuhl Seminar 21472)
https://scholar.archive.org/work/gnnydk6kx5adlmegxfiwolipge
At least from a practical and contemporary angle, the time-honoured question about the extent of intuitionistic mathematics rather is to which extent any given proof is effective, which proofs of which theorems can be rendered effective, and whether and how numerical information such as bounds and algorithms can be extracted from proofs. All this is ideally done by manipulating proofs mechanically or by adequate metatheorems, which includes proof translations, automated theorem proving, program extraction from proofs, proof analysis and proof mining. The question should thus be put as: What is the computational content of proofs? Guided by this central question, the present Dagstuhl seminar puts a special focus on coherent and geometric theories and their generalisations. These are not only widespread in mathematics and non-classical logics such as temporal and modal logics, but also a priori amenable for constructivisation, e.g., by Barr's Theorem, and last but not least particularly suited as a basis for automated theorem proving. Specific topics include categorical semantics for geometric theories, complexity issues of and algorithms for geometrisation of theories including speed-up questions, the use of geometric theories in constructive mathematics including finding algorithms, proof-theoretic presentation of sheaf models and higher toposes, and coherent logic for automatically readable proofs.Thierry Coquand, Hajime Ishihara, Sara Negri, Peter M. Schusterwork_gnnydk6kx5adlmegxfiwolipgeMon, 11 Apr 2022 00:00:00 GMTDagstuhl Reports, Volume 11, Issue 10, October 2021, Complete Issue
https://scholar.archive.org/work/3w5nqw2gangnrkuqgfzp32cw4u
Dagstuhl Reports, Volume 11, Issue 10, October 2021, Complete Issuework_3w5nqw2gangnrkuqgfzp32cw4uMon, 11 Apr 2022 00:00:00 GMT