ABSTRACT. Automated termination analysis is a central topic in the research of term rewriting. In this talk, I will first review the theoretical foundation of termination of term rewriting, leading to the recently established tuple interpretation method. Then I will present an Isabelle/HOL formalization of the theory. Although the formalization is based on the existing library IsaFoR (Isabelle Formalization of Rewriting), the present work takes another approach of representing relations (predicates rather than sets) so that the notation is more human friendly. Then I will present a unified implementation of the termination analysis techniques via SMT encoding, leading to the termination prover NaTT. Many tools have been developed for termination analysis and have been competing annually in termCOMP (Termination Competition) for two decades. At the end of the talk, I will share my experience in organizing termCOMP in the last five years.

ABSTRACT. We propose a generalization of Newman's lemma which gives a criterion of confluence for a wide class of not-necessarily-terminating abstract rewriting systems. We show that ordinary Newman's lemma for terminating systems can be considered as a corollary of this criterion. We describe a formalization of the proposed generalized Newman's lemma in Isabelle proof assistant using HOL logic.

ABSTRACT. We introduce a new form of restricted term rewrite system, the graph-embedded term rewrite system. These systems, and thus the name, are inspired by the graph minor relation and are more flexible extensions of the well know homeomorphic embedded property of term rewrite systems. As a motivating application area, we consider the symbolic analysis of security protocols, and more precisely the two knowledge problems defined by the deduction problem and the static equivalence problem. In this field restricted term rewrite systems, such as subterm-convergent, have proven useful since the knowledge problems are decidable for such systems. However, many of the same decision procedures still work for examples of systems which are ``beyond subterm-convergent''. However, the applicability of the corresponding decision procedures to these examples must often be proven on an individual basis. This is due to the problem that they don't fit into an existing syntactic definition for which the procedures are known to work. Here we show that many of these systems belong to a particular subclass of graph embedded convergent systems, called contracting convergent systems. On the one hand, we show that the knowledge problems are decidable for the subclass of contracting convergent systems. On the other hand, we show that the knowledge problems are undecidable for the class of graph embedded systems.

ABSTRACT. This paper is concerned with automatically proving properties about the input-output relation of functional programs operating over algebraic data types. Recent results show how to approximate the image of a functional program using a regular tree language. Though expressive, those techniques cannot prove properties relating the input and the output of a function, e.g., proving that the output of a function reversing a list has the same length as the input list. In this paper, we built upon those results and define a procedure to compute or over-approximate such a relation. Instead of representing the image of a function by a regular set of terms, we represent (an approximation of) the input-output relation by a regular set of tuples of terms. Regular languages of tuples of terms are recognized using a tree automaton recognizing convolutions of terms, where a convolution transforms a tuple of terms into a term built on tuples of symbols. Both the program and the properties are transformed into predicates and Constrained Horn clauses (CHCs). Then, using an Implicational Counter Example procedure (ICE), we infer a model of the clauses, associating to each predicate a regular relation. In this ICE procedure, checking if a given model satisfies the clauses is undecidable in general. We overcome undecidability by proposing an incomplete but sound inference procedure for such relational regular properties. Though the procedure is incomplete, its implementation performs well on more than 50 examples. It efficiently proves non-trivial relational properties or finds counter-examples. Additionally, on examples where using a non-relational model suffices to prove the property, our solving technique is flexible enough to find such a model.

ABSTRACT. We present a new encoding of the Battle of Hercules and Hydra as a rewrite system with AC symbols. Unlike earlier term rewriting encodings, it faithfully models any strategy of Hercules to beat Hydra. To prove the termination of our encoding, we employ type introduction in connection with a new termination criterion for AC rewriting.

ABSTRACT. Higher-order rewriting is a framework in which higher-order programs can be described by transformation rules on expressions. A computation occurs by transforming an expression into another using such rules. This step-by-step computation model induced by rewriting naturally gives rise to a notion of complexity as the number of steps needed to reduce expressions to a normal form, i.e., an expression that cannot be reduced further. The study of complexity analysis focuses on the development of automatable techniques to provide bounds to this number. In this paper, we consider a form of higher-order rewriting with a call-by-value evaluation strategy, so as to model call-by-value programs. We provide cost-size denotational semantics: a class of algebraic interpretations to map terms to tuples which bound both the reduction cost and the size of normal forms.

ABSTRACT. Higher-order unification (HOU) concerns unification of (extensions of) λ-calculus and can be seen as an instance of equational unification (E-unification) modulo βη-equivalence of λ-terms. We study equational unification of terms in languages with arbitrary variable binding constructions modulo arbitrary second-order equational theories. Abstract syntax with general variable binding and parametrised metavariables allows us to work with arbitrary binders without committing to λ-calculus or use inconvenient and error-prone term encodings, leading to a more flexible framework. In this paper, we introduce E-unification for second-order abstract syntax and describe a unification procedure for such problems, merging ideas from both full HOU and general E-unification. We prove that the procedure is sound and complete.

ABSTRACT. The essence of compiling with continuations is that conversion to continuation-passing style (CPS) is equivalent to a source language transformation converting to administrative normal form (ANF). Taking as source language Moggi's computational lambda-calculus (lambdac), we define an alternative to the CPS-translation with target in the sequent calculus LJQ, named value-filling style (VFS) translation, and making use of the ability of the sequent calculus to represent contexts formally. The VFS-translation requires no type translation: indeed, double negations are introduced only when encoding the VFS target language in the CPS target language. This optional encoding, when composed with the VFS-translation reconstructs the original CPS-translation. Going back to direct style, the "essence" of the VFS-translation is that it reveals a new sublanguage of ANF, the value-enclosed style (VES), next to another one, the continuation-enclosing style (CES): such an alternative is due to a dilemma in the syntax of lambdac, concerning how to expand the application constructor. In the typed scenario, VES and CES correspond to an alternative between two proof systems for call-by-value, LJQ and natural deduction with generalized applications, confirming proof theory as a foundation for intermediate representations.