Proving Looping and Non-Looping Non-Termination by Finite Automata

A new technique is presented to prove non-termination of term rewriting. The basic idea is to find a non-empty regular language of terms that is closed under rewriting and does not contain normal forms. It is automated by representing the language by a tree automaton with a fixed number of states, and expressing the mentioned requirements in a SAT formula. Satisfiability of this formula implies non-termination. Our approach succeeds for many examples where all earlier techniques fail, for instance for the S-rule from combinatory logic

The Algebraic Intersection Type Unification Problem

The algebraic intersection type unification problem is an important component in proof search related to several natural decision problems in intersection type systems. It is unknown and remains open whether the algebraic intersection type unification problem is decidable. We give the first nontrivial lower bound for the problem by showing (our main result) that it is exponential time hard. Furthermore, we show that this holds even under rank 1 solutions (substitutions whose codomains are restricted to contain rank 1 types). In addition, we provide a fixed-parameter intractability result for intersection type matching (one-sided unification), which is known to be NP-complete. We place the algebraic intersection type unification problem in the context of unification theory. The equational theory of intersection types can be presented as an algebraic theory with an ACI (associative, commutative, and idempotent) operator (intersection type) combined with distributivity properties with respect to a second operator (function type). Although the problem is algebraically natural and interesting, it appears to occupy a hitherto unstudied place in the theory of unification, and our investigation of the problem suggests that new methods are required to understand the problem. Thus, for the lower bound proof, we were not able to reduce from known results in ACI-unification theory and use game-theoretic methods for two-player tiling games

Mixin Composition Synthesis based on Intersection Types

We present a method for synthesizing compositions of mixins using type inhabitation in intersection types. First, recursively defined classes and mixins, which are functions over classes, are expressed as terms in a lambda calculus with records. Intersection types with records and record-merge are used to assign meaningful types to these terms without resorting to recursive types. Second, typed terms are translated to a repository of typed combinators. We show a relation between record types with record-merge and intersection types with constructors. This relation is used to prove soundness and partial completeness of the translation with respect to mixin composition synthesis. Furthermore, we demonstrate how a translated repository and goal type can be used as input to an existing framework for composition synthesis in bounded combinatory logic via type inhabitation. The computed result is a class typed by the goal type and generated by a mixin composition applied to an existing class

On the enumeration of closures and environments with an application to random generation

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size $n$. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures

A survey of information-based complexity

AbstractWe survey some recent results in information-based complexity. We focus on the worst case setting and also indicate some average case results

Closed nominal rewriting and efficiently computable nominal algebra equality

We analyse the relationship between nominal algebra and nominal rewriting, giving a new and concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality. This subclass includes specifications of the lambda-calculus and first-order logic.Comment: In Proceedings LFMTP 2010, arXiv:1009.218