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Claus-Peter Wirth

Publications and source records attributed to Claus-Peter Wirth.

18 recordsLinked to original sources
arXiv2024

A Simplified and Improved Free-Variable Framework for Hilbert's epsilon as an Operator of Indefinite Committed Choice

Free variables occur frequently in mathematics and computer science with ad hoc and altering semantics. We present the most recent version of our free-variable framework for two-valued logics with properly improved functionality, but only two kinds of free variables left (instead of three): implicitly universally and implicitly existentially quantified ones, now simply called "free atoms" and "free variables", respectively. The quantificational expressiveness and the problem-solving facilities of our framework exceed standard first-order and even higher-order modal logics, and directly support Fermat's descente infinie. With the improved version of our framework, we can now model also Henkin quantification, neither using quantifiers (binders) nor raising (Skolemization). We propose a new semantics for Hilbert's epsilon as a choice operator with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports reductive proof search optimally.

cs.AI↗
arXiv2021

A Most Interesting, but Revoked Draft for Hilbert and Bernays' "Grundlagen der Mathematik" that never found its way into any publication, and 2 CV of Gisbert Hasenjaeger

In 1934, in Bernays preface to the 1st edn. of the 1st vol. of Hilbert-Bernays "Grundlagen der Mathematik", a nearly completed draft of the the finally two-volume monograph is mentioned, which had to be revoked because of the completely changed situation in the area of proof theory after Herbrand and Goedels revolutionary results. Nothing at all seems to be known about this draft and its whereabouts. A third of a century later, Bernays preface to the 2nd edn. (1968) of the 1st vol. of Hilbert-Bernays mentions joint work of Hasenjaeger and Bernays on the second edition. Bernays states there that it became obvious that the integration of the many new results in the area of proof theory would have required a complete reorganization of the book, i.e. that the inclusion of the intermediately found new results in the area of proof theory turned out to be unobtainable by a revision, but would have required a complete reorganization of the entire textbook. We document that - even after the need for a complete reorganization had become obvious - this joint work went on to a considerable extent. Moreover, we document when Hasenjaeger stayed in Zurich to assist Bernays in the completion of the 2nd edn. In May 2017, we identified an incorrectly filed text in Bernays scientific legacy at the archive of the ETH Zurich as a candidate for the beginning of the revoked draft for the 1st edn. or of a revoked draft for the 2nd edn. In a partial presentation and careful investigation of this text we gather only some minor evidence that this text is the beginning of the nearly completed draft of the 1st edn., but ample evidence that this text is part of the work of Hasenjaeger and Bernays on the 2nd edn. We provide some evidence that this work has covered a complete reorganization of the entire 1st vol., including a completely new version of its last chapter on the iota.

math.HO↗
arXiv2016

The Explicit Definition of Quantifiers via Hilbert's epsilon is Confluent and Terminating

We investigate the elimination of quantifiers in first-order formulas via Hilbert's epsilon-operator (or -binder), following Bernays' explicit definitions of the existential and the universal quantifier symbol by means of epsilon-terms. This elimination has its first explicit occurrence in the proof of the first epsilon-theorem in Hilbert-Bernays in 1939. We think that there is a lacuna in this proof w.r.t. this elimination, related to the erroneous assumption that explicit definitions always terminate. Surprisingly, to the best of our knowledge, nobody ever proved confluence or termination for this elimination procedure. Even myths on non-confluence and the openness of the termination problem are circulating. We show confluence and termination of this elimination procedure by means of a direct, straightforward, and easily verifiable proof, based on a new theorem on how to obtain termination from weak normalization.

cs.LO↗
arXiv2015

The RatioLog Project: Rational Extensions of Logical Reasoning

Higher-level cognition includes logical reasoning and the ability of question answering with common sense. The RatioLog project addresses the problem of rational reasoning in deep question answering by methods from automated deduction and cognitive computing. In a first phase, we combine techniques from information retrieval and machine learning to find appropriate answer candidates from the huge amount of text in the German version of the free encyclopedia "Wikipedia". In a second phase, an automated theorem prover tries to verify the answer candidates on the basis of their logical representations. In a third phase - because the knowledge may be incomplete and inconsistent -, we consider extensions of logical reasoning to improve the results. In this context, we work toward the application of techniques from human reasoning: We employ defeasible reasoning to compare the answers w.r.t. specificity, deontic logic, normative reasoning, and model construction. Moreover, we use integrated case-based reasoning and machine learning techniques on the basis of the semantic structure of the questions and answer candidates to learn giving the right answers.

cs.AI↗
arXiv2015

Herbrand's Fundamental Theorem - an encyclopedia article

Herbrand's Fundamental Theorem provides a constructive characterization of derivability in first-order predicate logic by means of sentential logic. Sometimes it is simply called "Herbrand's Theorem", but the longer name is preferable as there are other important "Herbrand theorems" and Herbrand himself called it "Théorème fondamental". It was ranked by Bernays [1957] as follows: "In its proof-theoretic form, Herbrand's Theorem can be seen as the central theorem of predicate logic. It expresses the relation of predicate logic to propositional logic in a concise and felicitous form." And by Heijenoort [1967]: "Let me say simply, in conclusion, that Begriffsschrift [Frege, 1879], Löwenheim's paper [1915], and Chapter 5 of Herbrand's thesis [1930] are the three cornerstones of modern logic." Herbrand's Fundamental Theorem occurs in Chapter 5 of his PhD thesis [1930] --- entitled Recherches sur la théorie de la démonstration --- submitted by Jacques Herbrand (1908-1931) in 1929 at the University of Paris. Herbrand's Fundamental Theorem is, together with Gödel's incompleteness theorems and Gentzen's Hauptsatz, one of the most influential theorems of modern logic. Because of its complexity, Herbrand's Fundamental Theorem is typically fouled up in textbooks beyond all recognition. As we are convinced that there is still much more to learn for the future from this theorem than many logicians know, we will focus on the true message and its practical impact. This requires a certain amount of streamlining of Herbrand's work, which will be compensated by some remarks on the actual historical facts.

math.LO↗
arXiv2014

Herbrand's Fundamental Theorem: The Historical Facts and their Streamlining

Using Heijenoort's unpublished generalized rules of quantification, we discuss the proof of Herbrand's Fundamental Theorem in the form of Heijenoort's correction of Herbrand's "False Lemma" and present a didactic example. Although we are mainly concerned with the inner structure of Herbrand's Fundamental Theorem and the questions of its quality and its depth, we also discuss the outer questions of its historical context and why Bernays called it "the central theorem of predicate logic" and considered the form of its expression to be "concise and felicitous".

math.LO↗
arXiv2014

Lectures on Jacques Herbrand as a Logician

We give some lectures on the work on formal logic of Jacques Herbrand, and sketch his life and his influence on automated theorem proving. The intended audience ranges from students interested in logic over historians to logicians. Besides the well-known correction of Herbrand's False Lemma by Goedel and Dreben, we also present the hardly known unpublished correction of Heijenoort and its consequences on Herbrand's Modus Ponens Elimination. Besides Herbrand's Fundamental Theorem and its relation to the Loewenheim-Skolem-Theorem, we carefully investigate Herbrand's notion of intuitionism in connection with his notion of falsehood in an infinite domain. We sketch Herbrand's two proofs of the consistency of arithmetic and his notion of a recursive function, and last but not least, present the correct original text of his unification algorithm with a new translation.

cs.LO↗
arXiv2013

David Poole's Specificity Revised

In the middle of the 1980s, David Poole introduced a semantical, model-theoretic notion of specificity to the artificial-intelligence community. Since then it has found further applications in non-monotonic reasoning, in particular in defeasible reasoning. Poole tried to approximate the intuitive human concept of specificity, which seems to be essential for reasoning in everyday life with its partial and inconsistent information. His notion, however, turns out to be intricate and problematic, which --- as we show --- can be overcome to some extent by a closer approximation of the intuitive human concept of specificity. Besides the intuitive advantages of our novel specificity ordering over Poole's specificity relation in the classical examples of the literature, we also report some hard mathematical facts: Contrary to what was claimed before, we show that Poole's relation is not transitive. The present means to decide our novel specificity relation, however, show only a slight improvement over the known ones for Poole's relation, and further work is needed in this aspect.

cs.AI↗
arXiv2012

Hilbert's epsilon as an Operator of Indefinite Committed Choice

Paul Bernays and David Hilbert carefully avoided overspecification of Hilbert's epsilon-operator and axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the epsilon-operator underspecified. In the meanwhile, there have been several suggestions for semantics of the epsilon as a choice operator. After reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports proof search optimally and is natural in the sense that it does not only mirror some cases of referential interpretation of indefinite articles in natural language, but may also contribute to philosophy of language. Finally, we ask the question whether our epsilon within our free-variable framework can serve as a paradigm useful in the specification and computation of semantics of discourses in natural language.

cs.AI↗
arXiv2010

A Self-Contained and Easily Accessible Discussion of the Method of Descente Infinie and Fermat's Only Explicitly Known Proof by Descente Infinie

We present the only proof of Pierre Fermat by descente infinie that is known to exist today. As the text of its Latin original requires active mathematical interpretation, it is more a proof sketch than a proper mathematical proof. We discuss descente infinie from the mathematical, logical, historical, linguistic, and refined logic-historical points of view. We provide the required preliminaries from number theory and develop a self-contained proof in a modern form, which nevertheless is intended to follow Fermat's ideas closely. We then annotate an English translation of Fermat's original proof with terms from the modern proof. Including all important facts, we present a concise and self-contained discussion of Fermat's proof sketch, which is easily accessible to laymen in number theory as well as to laymen in the history of mathematics, and which provides new clarification of the Method of Descente Infinie to the experts in these fields. Last but not least, this paper fills a gap regarding the easy accessibility of the subject.

cs.AI↗
arXiv2010

Progress in Computer-Assisted Inductive Theorem Proving by Human-Orientedness and Descente Infinie?

In this short position paper we briefly review the development history of automated inductive theorem proving and computer-assisted mathematical induction. We think that the current low expectations on progress in this field result from a faulty narrow-scope historical projection. Our main motivation is to explain--on an abstract but hopefully sufficiently descriptive level--why we believe that future progress in the field is to result from human-orientedness and descente infinie.

cs.AI↗
arXiv2009

Syntactic Confluence Criteria for Positive/Negative-Conditional Term Rewriting Systems

We study the combination of the following already known ideas for showing confluence of unconditional or conditional term rewriting systems into practically more useful confluence criteria for conditional systems: Our syntactical separation into constructor and non-constructor symbols, Huet's introduction and Toyama's generalization of parallel closedness for non-noetherian unconditional systems, the use of shallow confluence for proving confluence of noetherian and non-noetherian conditional systems, the idea that certain kinds of limited confluence can be assumed for checking the fulfilledness or infeasibility of the conditions of conditional critical pairs, and the idea that (when termination is given) only prime superpositions have to be considered and certain normalization restrictions can be applied for the substitutions fulfilling the conditions of conditional critical pairs. Besides combining and improving already known methods, we present the following new ideas and results: We strengthen the criterion for overlay joinable noetherian systems, and, by using the expressiveness of our syntactical separation into constructor and non-constructor symbols, we are able to present criteria for level confluence that are not criteria for shallow confluence actually and also able to weaken the severe requirement of normality (stiffened with left-linearity) in the criteria for shallow confluence of noetherian and non-noetherian conditional systems to the easily satisfied requirement of quasi-normality. Finally, the whole paper may also give a practically useful overview of the syntactical means for showing confluence of conditional term rewriting systems.

cs.AI↗
arXiv2009

An Algebraic Dexter-Based Hypertext Reference Model

We present the first formal algebraic specification of a hypertext reference model. It is based on the well-known Dexter Hypertext Reference Model and includes modifications with respect to the development of hypertext since the WWW came up. Our hypertext model was developed as a product model with the aim to automatically support the design process and is extended to a model of hypertext-systems in order to be able to describe the state transitions in this process. While the specification should be easy to read for non-experts in algebraic specification, it guarantees a unique understanding and enables a close connection to logic-based development and verification.

cs.AI↗
arXiv2009

lim+, delta+, and Non-Permutability of beta-Steps

Using a human-oriented formal example proof of the (lim+) theorem, i.e. that the sum of limits is the limit of the sum, which is of value for reference on its own, we exhibit a non-permutability of beta-steps and delta+-steps (according to Smullyan's classification), which is not visible with non-liberalized delta-rules and not serious with further liberalized delta-rules, such as the delta++-rule. Besides a careful presentation of the search for a proof of (lim+) with several pedagogical intentions, the main subject is to explain why the order of beta-steps plays such a practically important role in some calculi.

cs.AI↗
arXiv2009

Writing Positive/Negative-Conditional Equations Conveniently

We present a convenient notation for positive/negative-conditional equations. The idea is to merge rules specifying the same function by using case-, if-, match-, and let-expressions. Based on the presented macro-rule-construct, positive/negative-conditional equational specifications can be written on a higher level. A rewrite system translates the macro-rule-constructs into positive/negative-conditional equations.

cs.AI↗
arXiv2009

ASF+ --- eine ASF-aehnliche Spezifikationssprache

Maintaining the main aspects of the algebraic specification language ASF as presented in [Bergstra&al.89] we have extend ASF with the following concepts: While once exported names in ASF must stay visible up to the top the module hierarchy, ASF+ permits a more sophisticated hiding of signature names. The erroneous merging of distinct structures that occurs when importing different actualizations of the same parameterized module in ASF is avoided in ASF+ by a more adequate form of parameter binding. The new ``Namensraum''-concept of ASF+ permits the specifier on the one hand directly to identify the origin of hidden names and on the other to decide whether an imported module is only to be accessed or whether an important property of it is to be modified. In the first case he can access one single globally provided version; in the second he has to import a copy of the module. Finally ASF+ permits semantic conditions on parameters and the specification of tasks for a theorem prover.

cs.AI↗