Strategic Issues, Problems and Challenges in Inductive Theorem Proving

Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP

Extending Context-Sensitivity in Term Rewriting

We propose a generalized version of context-sensitivity in term rewriting based on the notion of "forbidden patterns". The basic idea is that a rewrite step should be forbidden if the redex to be contracted has a certain shape and appears in a certain context. This shape and context is expressed through forbidden patterns. In particular we analyze the relationships among this novel approach and the commonly used notion of context-sensitivity in term rewriting, as well as the feasibility of rewriting with forbidden patterns from a computational point of view. The latter feasibility is characterized by demanding that restricting a rewrite relation yields an improved termination behaviour while still being powerful enough to compute meaningful results. Sufficient criteria for both kinds of properties in certain classes of rewrite systems with forbidden patterns are presented

Inductive Theorem Proving Using Refined Unfailing Completion Techniques

We present a brief overview on completion based inductive theorem proving techniques, point out the key concepts for the underlying "proof by consistency" - paradigm and isolate an abstract description of what is necessary for an algorithmic realization of such methods. In particular, we give several versions of proof orderings, which - under certain conditions - are well-suited for that purpose. Together with corresponding notions of (positive and negative) covering sets we get abstract "positive" and "negative" characterizations of inductive validity. As a consequence we can generalize known criteria for inductive validity, even for the cases where some of the conjectures may not be orientable or where the base system is terminating but not necessarily ground confluent. Furthermore we consider several refinements and optimizations of completion based inductive theorem proving techniques. In particular, sufficient criteria for being a covering set including restrictions of critical pairs (and the usage of non-equational inductive knowledge) are discussed. Moreover a couple of lemma generation methods are briefly summarized and classified. A new techniques of save generalization is particularly interesting, since it provides means for syntactic generalizations, i.e. simplifications, of conjectures without loosing semantic equivalence. Finally we present the main features and characteristics of UNICOM, an inductive theorem prover with refined unfailing completion techniques and built on top of TRSPEC, a term rewriting based system for investigating algebraic specifications

Termination of Rewriting with and Automated Synthesis of Forbidden Patterns

We introduce a modified version of the well-known dependency pair framework that is suitable for the termination analysis of rewriting under forbidden pattern restrictions. By attaching contexts to dependency pairs that represent the calling contexts of the corresponding recursive function calls, it is possible to incorporate the forbidden pattern restrictions in the (adapted) notion of dependency pair chains, thus yielding a sound and complete approach to termination analysis. Building upon this contextual dependency pair framework we introduce a dependency pair processor that simplifies problems by analyzing the contextual information of the dependency pairs. Moreover, we show how this processor can be used to synthesize forbidden patterns suitable for a given term rewriting system on-the-fly during the termination analysis.Comment: In Proceedings IWS 2010, arXiv:1012.533

On Semi-Completeness of Term Rewriting Systems

We investigate the question whether semi-completeness, i.e., weak termination plus confluence, of term rewriting systems is preserved under the normalization of right-hand sides. We give a simple counterexample showing that in general this transformation neither preserves weak termination nor (local) confluence. Moreover we present two conditions which are sufficient for the preservation of semi-completeness. In particular, we show that (almost) orthogonal systems enjoy this preservation property

Abstract involutions of algebraic groups and of Kac-Moody groups

Based on the second author's thesis in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established by Springer and by Helminck-Wang for algebraic groups and by Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by Devillers-Muhlherr in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised by Bennett-Gramlich-Hoffman-Shpectorov

Panelerhebungen der amtlichen Statistik als Datenquellen für die Sozialwissenschaften

Paneldaten haben gegenüber Querschnittsdaten zahlreiche Vorteile. Amtliche Daten sind zudem eine wichtige Quelle für die Sozial- und Wirtschaftswissenschaften. Viele amtliche Datenerhebungen sind als Panel konzipiert und durchgeführt oder können zu Panels zusammengefügt werden. Diese Arbeit gibt eine Übersicht über die Panelerhebungen oder zu Panels aufbereiteten Einzeldatensätze der deutschen amtlichen Statistik und beschreibt Erhebungsinhalte, Stichprobe sowie Zugangsmöglichkeiten. Panel data have numerous advantages to cross sectional data. Data from official statistical offices (and other public authorities) are a valuable data source for the social and economic sciences. Many of these data originally are panel data (or can be combined to form panel data). This article gives an overview over panel surveys and panel data conducted by German public authorities, describing topic and contents, sampling and access to these data.Paneldaten, amtliche Statistik, Datenzugang panel data, offcial statistics, data access

Panelerhebungen der amtlichen Statistik als Datenquellen für die Sozialwissenschaften

Panel data have numerous advantages to cross sectional data. Data from official statistical offices (and other public authorities) are a valuable data source for the social and economic sciences. Many of these data originally are panel data (or can be combined to form panel data). This article gives an overview over panel surveys and panel data conducted by German public authorities, describing topic and contents, sampling and access to these data.Paneldaten haben gegenüber Querschnittsdaten zahlreiche Vorteile. Amtliche Daten sind zudem eine wichtige Quelle für die Sozial- und Wirtschaftswissenschaften. Viele amtliche Datenerhebungen sind als Panel konzipiert und durchgeführt oder können zu Panels zusammengefügt werden. Diese Arbeit gibt eine Übersicht über die Panelerhebungen oder zu Panels aufbereiteten Einzeldatensätze der deutschen amtlichen Statistik und beschreibt Erhebungsinhalte, Stichprobe sowie Zugangsmöglichkeiten

Repräsentative Analyse der Lebenslagen einkommensstarker Haushalte

Die Studie von 2002 liefert eine Expertise für die Armuts- und Reichtumsberichterstattung der Bundesregierung auf Basis einer repräsentativen Erhebung einkommensstarker Haushalte. Die Datenbasis zur Analyse der Lebenslage, der Einkommen und der Vermögensbildung von Haushalten im oberen Einkommensbereich umfasst eine Zufallstichprobe von 1.224 Haushalten mit 2.671 Befragungspersonen, die über ein Haushaltsnettoeinkommen von 3.835 EUR und mehr verfügen, bzw. 505 Haushalte mit 1.130 Personen mit einem Haushaltsnettoeinkommen von 5.113 EUR und mehr. Die telefonische Befragung der einkommensstarken Haushalte wurde im Rahmen der SOEP-Befragung des Jahres 2002 durchgeführt. Die empirischen Ergebnisse zu den Lebenslagen gliedern sich in folgende Aspekte: (1) Struktur und Lebenslage von Hocheinkommensbeziehern, (2) subjektive Indikatoren, (3) Hocheinkommen und Gesundheit, (4) die individuelle Vermögensbilanz von Hocheinkommensbeziehern, (5) Spezifika einkommensstarker Haushalte, (6) Verteilung des Vermögens auf die Gruppen nach Hausnettoeinkommen sowie (7) Hocheinkommensbezieherhaushalte in der äquivalenzgewichteten Perspektive. (ICG2

Twilight Front

Describes the Murmansk Front of the Polar region