Nominal Narrowing

Nominal unification is a generalisation of first-order unification that takes alpha-equivalence into account. In this paper, we study nominal unification in the context of equational theories. We introduce nominal narrowing and design a general nominal E-unification procedure, which is sound and complete for a wide class of equational theories. We give examples of application

Fixed-Point Constraints for Nominal Equational Unification

We propose a new axiomatisation of the alpha-equivalence relation for nominal terms, based on a primitive notion of fixed-point constraint. We show that the standard freshness relation between atoms and terms can be derived from the more primitive notion of permutation fixed-point, and use this result to prove the correctness of the new alpha-equivalence axiomatisation. This gives rise to a new notion of nominal unification, where solutions for unification problems are pairs of a fixed-point context and a substitution. Although it may seem less natural than the standard notion of nominal unifier based on freshness constraints, the notion of unifier based on fixed-point constraints behaves better when equational theories are considered: for example, nominal unification remains finitary in the presence of commutativity, whereas it becomes infinitary when unifiers are expressed using freshness contexts

O problema da dedução do intruso para teorias AC-convergentes localmente estáveis

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013.Apresenta-se um algoritmo para decidir o problema da dedução do intruso (PDI) para a classe de teorias localmente estáveis normais, que incluem operadores associativos e comutativos (AC). A decidibilidade é baseada na análise de reduções de reescrita aplicadas na cabeça de termos que são construídos a partir de contextos normais e o conhecimento inicial de um intruso. Este algoritmo se baseia em um algoritmo eficiente para resolver um caso restrito de casamento módulo AC de ordem superior, obtido pela combinação de um algoritmo para Casamento AC com Ocorrências Distintas, e um algoritmo padrão para resolver sistemas de equações Diofantinas lineares. O algoritmo roda em tempo polinomial no tamanho de um conjunto saturado construído a partir do conhecimento inicial do intruso para a subclasse de teorias para a qual operadores AC possuem inversos. Os resultados são aplicados para teoria AC pura e a teoria de grupos Abelianos de ordem n dada. Uma tradução entre dedução natural e o cálculo de sequentes permite usar a mesma abordagem para decidir o problema da dedução elementar para teorias localmente estáveis com inversos. Como uma aplicação, a teoria de assinaturas cegas pode ser modelada e então, deriva-se um algoritmo para decidir o PDI neste contexto, estendendo resultados de decidibilidade prévios. ______________________________________________________________________________ ABSTRACTWe present an algorithm to decide the intruder deduction problem (IDP) for the class of normal locally stable theories, which include associative and commutative (AC) opera- tors. The decidability is based on the analysis of rewriting reductions applied in the head of terms built from normal contexts and the initial knowledge of the intruder. It relies on a new and efficient algorithm to solve a restricted case of higher-order AC-matching, obtained by combining the Distinct Occurrences of AC-matching algorithm and a stan- dard algorithm to solve systems of linear Diophantine equations. Our algorithm runs in polynomial time on the size of a saturation set built from the initial knowledge of the intruder for the subclass of theories for which AC operators have inverses. We apply the results to the Pure AC equational theory and Abelian Groups with a given order n. A translation between natural deduction and sequent calculus allows us to use the same approach to decide the elementary deduction problem for locally stable theories with inverses. As an application, we model the theory of blind signatures and derive an algorithm to decide IDP in this context, extending previous decidability results

Non-Deterministic Functions as Non-Deterministic Processes

We study encodings of the ?-calculus into the ?-calculus in the unexplored case of calculi with non-determinism and failures. On the sequential side, we consider ?^?_?, a new non-deterministic calculus in which intersection types control resources (terms); on the concurrent side, we consider ??, a ?-calculus in which non-determinism and failure rest upon a Curry-Howard correspondence between linear logic and session types. We present a typed encoding of ?^?_? into ?? and establish its correctness. Our encoding precisely explains the interplay of non-deterministic and fail-prone evaluation in ?^?_? via typed processes in ??. In particular, it shows how failures in sequential evaluation (absence/excess of resources) can be neatly codified as interaction protocols

Nominal C-Unification

Nominal unification is an extension of first-order unification that takes into account the \alpha-equivalence relation generated by binding operators, following the nominal approach. We propose a sound and complete procedure for nominal unification with commutative operators, or nominal C-unification for short, which has been formalised in Coq. The procedure transforms nominal C-unification problems into simpler (finite families) of fixpoint problems, whose solutions can be generated by algebraic techniques on combinatorics of permutations.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854

On Nominal Syntax and Permutation Fixed Points

We propose a new axiomatisation of the alpha-equivalence relation for nominal terms, based on a primitive notion of fixed-point constraint. We show that the standard freshness relation between atoms and terms can be derived from the more primitive notion of permutation fixed-point, and use this result to prove the correctness of the new $\alpha$-equivalence axiomatisation. This gives rise to a new notion of nominal unification, where solutions for unification problems are pairs of a fixed-point context and a substitution. Although it may seem less natural than the standard notion of nominal unifier based on freshness constraints, the notion of unifier based on fixed-point constraints behaves better when equational theories are considered: for example, nominal unification remains finitary in the presence of commutativity, whereas it becomes infinitary when unifiers are expressed using freshness contexts. We provide a definition of $\alpha$-equivalence modulo equational theories that take into account A, C and AC theories. Based on this notion of equivalence, we show that C-unification is finitary and we provide a sound and complete C-unification algorithm, as a first step towards the development of nominal unification modulo AC and other equational theories with permutative properties

Non-Deterministic Functions as Non-Deterministic Processes (Extended Version)

We study encodings of the lambda-calculus into the pi-calculus in the unexplored case of calculi with non-determinism and failures. On the sequential side, we consider lambdafail, a new non-deterministic calculus in which intersection types control resources (terms); on the concurrent side, we consider spi, a pi-calculus in which non-determinism and failure rest upon a Curry-Howard correspondence between linear logic and session types. We present a typed encoding of lambdafail into spi and establish its correctness. Our encoding precisely explains the interplay of non-deterministic and fail-prone evaluation in lambdafail via typed processes in spi. In particular, it shows how failures in sequential evaluation (absence/excess of resources) can be neatly codified as interaction protocols

Non-Deterministic Functions as Non-Deterministic Processes

We study encodings of the λ-calculus into the π-calculus in the unexplored case of calculi with non-determinism and failures. On the sequential side, we consider λ^↯_⊕, a new non-deterministic calculus in which intersection types control resources (terms); on the concurrent side, we consider sπ, a π-calculus in which non-determinism and failure rest upon a Curry-Howard correspondence between linear logic and session types. We present a typed encoding of λ^↯_⊕ into sπ and establish its correctness. Our encoding precisely explains the interplay of non-deterministic and fail-prone evaluation in λ^↯_⊕ via typed processes in sπ. In particular, it shows how failures in sequential evaluation (absence/excess of resources) can be neatly codified as interactio

Termination in Concurrency, Revisited

Termination is a central property in sequential programming models: a term is terminating if all its reduction sequences are finite. Termination is also important in concurrency in general, and for message-passing programs in particular. A variety of type systems that enforce termination by typing have been developed. In this paper, we rigorously compare several type systems for $\pi$-calculus processes from the unifying perspective of termination. Adopting session types as reference framework, we consider two different type systems: one follows Deng and Sangiorgi's weight-based approach; the other is Caires and Pfenning's Curry-Howard correspondence between linear logic and session types. Our technical results precisely connect these very different type systems, and shed light on the classes of client/server interactions they admit as correct