Higher-Order Termination: from Kruskal to Computability

Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct. In the early days of logic, strong normalization was usually shown by assigning ordinals to expressions in such a way that eliminating a cut would yield an expression with a smaller ordinal. In the early days of verification, computer scientists used similar ideas, interpreting the arguments of a program call by a natural number, such as their size. Showing the size of the arguments to decrease for each recursive call gives a termination proof of the program, which is however rather weak since it can only yield quite small ordinals. In the sixties, Tait invented a new method for showing cut elimination of natural deduction, based on a predicate over the set of terms, such that the membership of an expression to the predicate implied the strong normalization property for that expression. The predicate being defined by induction on types, or even as a fixpoint, this method could yield much larger ordinals. Later generalized by Girard under the name of reducibility or computability candidates, it showed very effective in proving the strong normalization property of typed lambda-calculi..

CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

Termination is an important property of programs; notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting systems, where many methods and tools have been developed over the years to address this problem. Ensuring reliability of those tools is therefore an important issue. In this paper we present a library formalizing important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools

A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets

We give a sharpening of a recent result of Aschenbrenner and Pong about the maximal order type of the term ordering on the finite multisets over a wpo. Moreover we discuss an approach to compute maximal order types of well-partial orders which are related to tree embeddings

Abstract Canonical Inference

An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation, saturation, redundancy elimination, and rewrite-system reduction are connected to proof orderings. Fairness of deductive mechanisms is defined in terms of proof orderings, distinguishing between (ordinary) "fairness," which yields completeness, and "uniform fairness," which yields saturation.Comment: 28 pages, no figures, to appear in ACM Trans. on Computational Logi

Constructive Reasoning for Semantic Wikis

One of the main design goals of social software, such as wikis, is to support and facilitate interaction and collaboration. This dissertation explores challenges that arise from extending social software with advanced facilities such as reasoning and semantic annotations and presents tools in form of a conceptual model, structured tags, a rule language, and a set of novel forward chaining and reason maintenance methods for processing such rules that help to overcome the challenges. Wikis and semantic wikis were usually developed in an ad-hoc manner, without much thought about the underlying concepts. A conceptual model suitable for a semantic wiki that takes advanced features such as annotations and reasoning into account is proposed. Moreover, so called structured tags are proposed as a semi-formal knowledge representation step between informal and formal annotations. The focus of rule languages for the Semantic Web has been predominantly on expert users and on the interplay of rule languages and ontologies. KWRL, the KiWi Rule Language, is proposed as a rule language for a semantic wiki that is easily understandable for users as it is aware of the conceptual model of a wiki and as it is inconsistency-tolerant, and that can be efficiently evaluated as it builds upon Datalog concepts. The requirement for fast response times of interactive software translates in our work to bottom-up evaluation (materialization) of rules (views) ahead of time – that is when rules or data change, not when they are queried. Materialized views have to be updated when data or rules change. While incremental view maintenance was intensively studied in the past and literature on the subject is abundant, the existing methods have surprisingly many disadvantages – they do not provide all information desirable for explanation of derived information, they require evaluation of possibly substantially larger Datalog programs with negation, they recompute the whole extension of a predicate even if only a small part of it is affected by a change, they require adaptation for handling general rule changes. A particular contribution of this dissertation consists in a set of forward chaining and reason maintenance methods with a simple declarative description that are efficient and derive and maintain information necessary for reason maintenance and explanation. The reasoning methods and most of the reason maintenance methods are described in terms of a set of extended immediate consequence operators the properties of which are proven in the classical logical programming framework. In contrast to existing methods, the reason maintenance methods in this dissertation work by evaluating the original Datalog program – they do not introduce negation if it is not present in the input program – and only the affected part of a predicate’s extension is recomputed. Moreover, our methods directly handle changes in both data and rules; a rule change does not need to be handled as a special case. A framework of support graphs, a data structure inspired by justification graphs of classical reason maintenance, is proposed. Support graphs enable a unified description and a formal comparison of the various reasoning and reason maintenance methods and define a notion of a derivation such that the number of derivations of an atom is always finite even in the recursive Datalog case. A practical approach to implementing reasoning, reason maintenance, and explanation in the KiWi semantic platform is also investigated. It is shown how an implementation may benefit from using a graph database instead of or along with a relational database

A Transfinite Knuth-Bendix Order for Lambda-Free Higher-Order Terms

International audienceWe generalize the Knuth-Bendix order (KBO) to higher-order terms without λ-abstraction. The restriction of this new order to first-order terms coincides with the traditional KBO. The order has many useful properties, including transitivity, the subterm property, compatibility with contexts (monotonicity), stability under substitution, and well-foundedness. Transfinite weights and argument coefficients can also be supported. The order appears promising as the basis of a higher-order superposition calculus

REST: Integrating Term Rewriting with Program Verification (Extended Version)

We introduce REST, a novel term rewriting technique for theorem proving that uses online termination checking and can be integrated with existing program verifiers. REST enables flexible but terminating term rewriting for theorem proving by: (1) exploiting newly-introduced term orderings that are more permissive than standard rewrite simplification orderings; (2) dynamically and iteratively selecting orderings based on the path of rewrites taken so far; and (3) integrating external oracles that allow steps that cannot be justified with rewrite rules. Our REST approach is designed around an easily implementable core algorithm, parameterizable by choices of term orderings and their implementations; in this way our approach can be easily integrated into existing tools. We implemented REST as a Haskell library and incorporated it into Liquid Haskell's evaluation strategy, extending Liquid Haskell with rewriting rules. We evaluated our REST implementation by comparing it against both existing rewriting techniques and E-matching and by showing that it can be used to supplant manual lemma application in many existing Liquid Haskell proofs

Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma

Dickson's Lemma is a simple yet powerful tool widely used in termination proofs, especially when dealing with counters or related data structures. However, most computer scientists do not know how to derive complexity upper bounds from such termination proofs, and the existing literature is not very helpful in these matters. We propose a new analysis of the length of bad sequences over (N^k,\leq) and explain how one may derive complexity upper bounds from termination proofs. Our upper bounds improve earlier results and are essentially tight

Proof Theory at Work: Complexity Analysis of Term Rewrite Systems

This thesis is concerned with investigations into the "complexity of term rewriting systems". Moreover the majority of the presented work deals with the "automation" of such a complexity analysis. The aim of this introduction is to present the main ideas in an easily accessible fashion to make the result presented accessible to the general public. Necessarily some technical points are stated in an over-simplified way.Comment: Cumulative Habilitation Thesis, submitted to the University of Innsbruc