Mathematische Logik

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Hilbert's Tenth Problem in Coq (Extended Version)

We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem -- in our case by a Minsky machine -- is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer. Additionally, we prove the reverse direction and show that every Diophantine relation is recognisable by $\mu$-recursive functions and give a certified compiler from $\mu$-recursive functions to Minsky machines.Comment: submitted to LMC

On Measure Quantifiers in First-Order Arithmetic

International audienceWe study the logic obtained by endowing the language of first-order arithmetic with second-order measure quantifiers. This new kind of quantification allows us to express that the argument formula is true in a certain portion of all possible interpretations of the quantified variable. We show that first-order arithmetic with measure quantifiers is capable of formalizing simple results from probability theory and, most importantly, of representing every recursive random function. Moreover, we introduce a realizability interpretation of this logic in which programs have access to an oracle from the Cantor space

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

The reverse mathematics of elementary recursive nonstandard analysis: a robust contribution to the foundations of mathematics

Reverse Mathematics (RM) is a program in the Foundations of Mathematics founded by Harvey Friedman in the Seventies ([17, 18]). The aim of RM is to determine the minimal axioms required to prove a certain theorem of ‘ordinary’ mathematics. In many cases one observes that these minimal axioms are also equivalent to this theorem. This phenomenon is called the ‘Main Theme’ of RM and theorem 1.2 is a good example thereof. In practice, most theorems of everyday mathematics are equivalent to one of the four systems WKL0, ACA0, ATR0 and Π1-CA0 or provable in the base theory RCA0. An excellent introduction to RM is Stephen Simpson’s monograph [46]. Nonstandard Analysis has always played an important role in RM. ([32,52,53]). One of the open problems in the literature is the RM of theories of first-order strength I∆0 + exp ([46, p. 406]). In Chapter I, we formulate a solution to this problem in theorem 1.3. This theorem shows that many of the equivalences from theorem 1.2 remain correct when we replace equality by infinitesimal proximity ‘≈’ from Nonstandard Analysis. The base theory now is ERNA, a nonstandard extension of I∆0 + exp. The principle that corresponds to ‘Weak K ̈onig’s lemma’ is the Universal Transfer Principle (see axiom schema 1.57). In particular, one can say that the RM of ERNA+Π1-TRANS is a ‘copy up to infinitesimals’ of the RM of WKL0. This implies that RM is ‘robust’ in the sense this term is used in Statistics and Computer Science ([25,35]). Furthermore, we obtain applications of our results in Theoretical Physics in the form of the ‘Isomorphism Theorem’ (see theorem 1.106). This philosophical excursion is the first application of RM outside of Mathematics and implies that ‘whether reality is continuous or discrete is undecidable because of the way mathematics is used in Physics’ (see paragraph 3.2.4, p. 53). We briefly explore a connection with the program ‘Constructive Reverse Mathematics’ ([30,31]) and in the rest of Chapter I, we consider the RM of ACA0 and related systems. In particular, we prove theorem 1.161, which is a first step towards a ‘copy up to infinitesimals’ of the RM of ACA0. However, one major aesthetic problem with these results is the introduction of extra quantifiers in many of the theorems listed in theorem 1.3 (see e.g. theorem 1.94). To overcome this hurdle, we explore Relative Nonstandard Analysis in Chapters II and III. This new framework involves many degrees of infinity instead of the classical ‘binary’ picture where only two degrees ‘finite’ and ‘infinite’ are available. We extend ERNA to a theory of Relative Nonstandard Analysis called ERNAA and show how this theory and its extensions allow for a completely quantifier- free development of analysis. We also study the metamathematics of ERNAA, motivated by RM. Several ERNA-theorems would not have been discovered without considering ERNAA first

Arithmetic and Modularity in Declarative Languages for Knowledge Representation

The past decade has witnessed the development of many important declarative languages for knowledge representation and reasoning such as answer set programming (ASP) languages and languages that extend first-order logic. Also, since these languages depend on background solvers, the recent advancements in the efficiency of solvers has positively affected the usability of such languages. This thesis studies extensions of knowledge representation (KR) languages with arithmetical operators and methods to combine different KR languages. With respect to arithmetic in declarative KR languages, we show that existing KR languages suffer from a huge disparity between their expressiveness and their computational power. Therefore, we develop an ideal KR language that captures the complexity class NP for arithmetical search problems and guarantees universality and efficiency for solving such problems. Moreover, we introduce a framework to language-independently combine modules from different KR languages. We study complexity and expressiveness of our framework and develop algorithms to solve modular systems. We define two semantics for modular systems based on (1) a model-theoretical view and (2) an operational view on modular systems. We prove that our two semantics coincide and also develop mechanisms to approximate answers to modular systems using the operational view. We augment our algorithm these approximation mechanisms to speed up the process of solving modular system. We further generalize our modular framework with supported model semantics that disallows self-justifying models. We show that supported model semantics generalizes our two previous model-theoretical and operational semantics. We compare and contrast the expressiveness of our framework under supported model semantics with another framework for interlinking knowledge bases, i.e., multi-context systems, and prove that supported model semantics generalizes and unifies different semantics of multi-context systems. Motivated by the wide expressiveness of supported models, we also define a new supported equilibrium semantics for multi-context systems and show that supported equilibrium semantics generalizes previous semantics for multi-context systems. Furthermore, we also define supported semantics for propositional programs and show that supported model semnatics generalizes the acclaimed stable model semantics and extends the two celebrated properties of rationality and minimality of intended models beyond the scope of logic programs

Refinement of Classical Proofs for Program Extraction

The A-Translation enables us to unravel the computational information in classical proofs, by first transforming them into constructive ones, however at the cost of introducing redundancies in the extracted code. This is due to the fact that all negations inserted during translation are replaced by the computationally relevant form of the goal. In this thesis we are concerned with eliminating such redundancies, in order to obtain better extracted programs. For this, we propose two methods: a controlled and minimal insertion of negations, such that a refinement of the A-Translation can be used and an algorithmic decoration of the proofs, in order to mark the computationally irrelevant components. By restricting the logic to be minimal, the Double Negation Translation is no longer necessary. On this fragment of minimal logic we apply the refined A-Translation, as proposed in (Berget et al., 2002). This method identifies further selected classes of formulas for which the negations do not need to be substituted by computationally relevant formulas. However, the refinement imposes restrictions which considerably narrow the applicability domain of the A-Translation. We address this issue by proposing a controlled insertion of double negations, with the benefit that some intuitionistically valid \Pi^0_2-formulas become provable in minimal logic and that certain formulas are transformed to match the requirements of the refined A-Translation. We present the outcome of applying the refined A-translation to a series of examples. Their purpose is two folded. On one hand, they serve as case studies for the role played by negations, by shedding a light on the restrictions imposed by the translation method. On the other hand, the extracted programs are characterized by a specific behaviour: they adhere to the continuation passing style and the recursion is in general in tail form. The second improvement concerns the detection of the computationally irrelevant subformulas, such that no terms are extracted from them. In order to achieve this, we assign decorations to the implication and universal quantifier. The algorithm that we propose is shown to be optimal, correct and terminating and is applied on the examples of factorial and list reversal.Die A-Übersetzung ermöglicht es, die rechnerische Information aus klassischen Beweisen einzuholen. Dennoch hat sie den Nachteil, dass die Programme, die man aus auf diese Weise transformierten Beweisen extrahiert, viele redundante Teile enthalten. Das liegt daran, dass die A-Übersetzung viele doppelte Negationen hinzufügt und alle diese Negationen durch die rechnerisch relevante Form der Ziel-Formel substituiert werden. In dieser Doktorarbeit werden Methoden dargestellt, um Teile der redundante Information in den extrahierten Programen zu entfernen. Einerseits wird das Einfügen der Negationen minimal gehalten und anderseits werden die nicht rechnerischen Teile als solche indentifiziert und ausgezeichnet. Wir bemerken zuerst, dass in der Minimallogik das Einfügen der doppelten Negationen nicht mehr nötig ist. Darüber hinaus, um das Ersetzen aller Negationen zu vermeiden, identifizieren (Berger et al., 2002) diejenigen, wo die Substitution nicht nötig ist. Diese verfeinerte A-Übersetzung hat aber den Nachteil, dass sie den Anwendungsbereich begrenzt. Um das zu beseitigen, wird in dieser Dissertation eine verfeinerte Doppel-Negation angewandt, die bestimmte Formeln so umsetzt, dass die verfeinerte A-Übersetzung darauf anwendbar ist. Als Zugabe kann diese Methode auch benutzt werden, um konstruktive Beweise mancher \Pi^0_2-Formeln in der Minimallogik durchzuführen. Dieses Verfahren wird durch Anwendung der verfeinerten A-Übersetzung auf eine Reihe von bedeutenden Fallstudien illustriert. Es werden das Lemma von Dickson, das unendliche Schubfachprinzip und das Erdös-Szekeres Theorem betrachtet. Dabei wird es festgestellt, dass ein Zusammenhang zu der Endrekursion und dem Rechnen mit Fortsezungen besteht. Ferner, um möglichst viel der überflüssigen Information zu entfernen, wird ein Dekorationsalgorithmus vorgelegt. Dadurch werden die rechnerisch irrelevanten Komponenten identifiziert und entsprechend annotiert, so dass sie während der Extraktion nicht berücksichtigt werden. Es wird gezeigt, dass das vorgeschlagene Dekorationsverfahren, das auf Beweisebene eingesetzt wird, optimal, korrekt und terminierend ist

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science