313 research outputs found
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
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 -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 -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
Nominal Unification from a Higher-Order Perspective
Nominal Logic is a version of first-order logic with equality, name-binding,
renaming via name-swapping and freshness of names. Contrarily to higher-order
logic, bindable names, called atoms, and instantiable variables are considered
as distinct entities. Moreover, atoms are capturable by instantiations,
breaking a fundamental principle of lambda-calculus. Despite these differences,
nominal unification can be seen from a higher-order perspective. From this
view, we show that nominal unification can be reduced to a particular fragment
of higher-order unification problems: Higher-Order Pattern Unification. This
reduction proves that nominal unification can be decided in quadratic
deterministic time, using the linear algorithm for Higher-Order Pattern
Unification. We also prove that the translation preserves most generality of
unifiers
Nominal Logic Programming
Nominal logic is an extension of first-order logic which provides a simple
foundation for formalizing and reasoning about abstract syntax modulo
consistent renaming of bound names (that is, alpha-equivalence). This article
investigates logic programming based on nominal logic. We describe some typical
nominal logic programs, and develop the model-theoretic, proof-theoretic, and
operational semantics of such programs. Besides being of interest for ensuring
the correct behavior of implementations, these results provide a rigorous
foundation for techniques for analysis and reasoning about nominal logic
programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as
of July 23, 200
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
Nominal disunification
Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2019.Propõe-se uma extensão para problemas de disunificação de primeira-ordem adicionando suporte a operadores de ligação de acordo com a abordagem nominal. Nesta abordagem, abstração é implementada usando átomos nominais ao invés de variáveis de ligação como na representação clássica de termos e renomeamento de átomos é implementado por permutações. Em lógica nominal problemas de unificação consistem de perguntas equacionais da forma s ≈α ? t (lê-se: s é α-equivalente a t?) consideradas sobre problemas de freshness da forma a# ? t (lê-se: a é fresco em t?) que restringem soluções proibindo ocorrências livres de átomos na instanciação de variáveis. Além dessas questões equacionais e freshness, problemas de disunificação nominal incluem restrições na forma de disequações s ̸≈α ? t (lê-se: s é αdiferente de t?) com soluções dadas por pares consistindo de uma substituição σ e um conjunto de restrições de freshness na forma a#X tal que sobre estas restrições a σ-instanciação de equações, disequações, e problemas de freshness são válidas. Mostra-se, reutilizando noções de unificação nominal, como decidir se dois termos nominais podem ser feitos diferentes módulo α-equivalência. Isso é feito extendendo resultados anteriores sobre disunificação de primeira ordem e definindo a noção de soluções com exceção na linguagem nominal. Uma discussão sobre a semântica de restrições em forma de disequações também é apresentada.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).An extension of first-order disunification problems is proposed by taking into account binding operators according to the nominal approach. In this approach, bindings are implemented through nominal atoms used instead of binding variables and renaming of atoms are implemented by atom permutations. In the nominal setting, unification problems consist of equational questions of the form s ≈α ? t (read: is s α-equivalent to t?) considered under freshness problems a# ? t (read: is a fresh for t?) that restrict solutions by forbidding free occurrences of atoms in the instantiations of variables. In addition to equational and freshness problems, nominal disunification problems also include nominal disunification constraints in the form of disequations s ̸≈α ? t (read: is s α-different to t?) and their solutions consist of pairs of a substitution σ and a finite set of freshness constraints in the form of a#X such that under these restrictions the σ-instantiation of the equations, disequations, and freshness problems holds. By re-using nominal unification techniques, it is shown how to decide whether two nominal terms can be made different modulo α-equivalence. This is done by extending previous results on first-order disunification and by defining the notion of solutions with exceptions in the nominal syntax. A discussion on the semantics of disunification constraints is also given
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
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