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.Comment: ii + 35 page

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 "Th\'eor\`eme 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], L\"owenheim'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 th\'eorie de la d\'emonstration --- submitted by Jacques Herbrand (1908-1931) in 1929 at the University of Paris. Herbrand's Fundamental Theorem is, together with G\"odel'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.Comment: ii + 16 page

Full First-Order Sequent and Tableau Calculi With Preservation of Solutions and the Liberalized delta-Rule but Without Skolemization

We present a combination of raising, explicit variable dependency representation, the liberalized delta-rule, and preservation of solutions for first-order deductive theorem proving. Our main motivation is to provide the foundation for our work on inductive theorem proving, where the preservation of solutions is indispensable.Comment: ii + 40 page

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".Comment: ii + 47 page

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.Comment: ii+34 page

Automation of Mathematical Induction as part of the History of Logic

We review the history of the automation of mathematical inductionComment: ii+107 page

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.Comment: ii + 187 page

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.Comment: ii + 36 pages, French abstract (R\'esum\'e) included in pape

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.Comment: ii + 48 page

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.Comment: ii + 21 page