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

From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions

Permissive-Nominal Logic (PNL) extends first-order predicate logic with term-formers that can bind names in their arguments. It takes a semantics in (permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are just term-formers satisfying axioms, and their denotation is functions on nominal atoms-abstraction. Then we have higher-order logic (HOL) and its models in ordinary (i.e. Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full or partial function spaces. This raises the following question: how are these two models of binding connected? What translation is possible between PNL and HOL, and between nominal sets and functions? We exhibit a translation of PNL into HOL, and from models of PNL to certain models of HOL. It is natural, but also partial: we translate a restricted subsystem of full PNL to HOL. The extra part which does not translate is the symmetry properties of nominal sets with respect to permutations. To use a little nominal jargon: we can translate names and binding, but not their nominal equivariance properties. This seems reasonable since HOL---and ordinary sets---are not equivariant. Thus viewed through this translation, PNL and HOL and their models do different things, but they enjoy non-trivial and rich subsystems which are isomorphic

Nominal Unification of Higher Order Expressions with Recursive Let

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems.Comment: Pre-proceedings paper presented at the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh, Scotland UK, 6-8 September 2016 (arXiv:1608.02534

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

A Variant of Higher-Order Anti-Unification

We present a rule-based Huet's style anti-unification algorithm for simply-typed lambda-terms in η-long -normal form, which computes a least general higher-order pattern generalization. For a pair of arbitrary terms of the same type, such a generalization always exists and is unique modulo α-equivalence and variable renaming. The algorithm computes it in cubic time within linear space. It has been implemented and the code is freely available. © Alexander Baumgartner, Temur Kutsia, Jordi Levy, and Mateu Villaret; licensed under Creative Commons License CC-BY 24th International Conference on Rewriting Techniques and Applications (RTA'13).This research has been partially supported by the projects HeLo (TIN2012-33042) and TASSAT (TIN2010-20967-C04-01), by the Austrian Science Fund (FWF) with the project SToUT (P 24087-N18) and by the Generalitat de Catalunya with the grant AGAUR 2009-SGR-1434.Peer Reviewe

On the Limits of Second-Order Unification

Second-Order Unification is a problem that naturally arises when applying automated deduction techniques with variables denoting predicates. The problem is undecidable, but a considerable effort has been made in order to find decidable fragments, and understand the deep reasons of its complexity. Two variants of the problem, Bounded Second-Order Unification and Linear Second-Order Unification ¿where the use of bound variables in the instantiations is restricted¿, have been extensively studied in the last two decades. In this paper we summarize some decidability/undecidability/complexity results, trying to focus on those that could be more interesting for a wider audience, and involving less technical details.Peer Reviewe

Unifying Nominal Unification

Nominal unification is proven to be quadratic in time and space. It was so by two different approaches, both inspired by the Paterson-Wegman linear unification algorithm, but dramatically different in the way nominal and first-order constraints are dealt with. To handle nominal constraints, Levy and Villaret introduced the notion of replacing while Calves and Fernandez use permutations and sets of atoms. To deal with structural constraints, the former use multi-equations in a way similar to the Martelli-Montanari algorithm while the later mimic Paterson-Wegman. In this paper we abstract over these two approaches and genralize them into the notion of modality, highlighting the general ideas behind nominal unification. We show that replacings and environments are in fact isomorphic. This isomorphism is of prime importance to prove intricate properties on both sides and a step further to the real complexity of nominal unification

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

From nominal to higher-order rewriting and back again

We present a translation function from nominal rewriting systems (NRSs) to combinatory reduction systems (CRSs), transforming closed nominal rules and ground nominal terms to CRSs rules and terms, respectively, while preserving the rewriting relation. We also provide a reduction-preserving translation in the other direction, from CRSs to NRSs, improving over a previously defined translation. These tools, together with existing translations between CRSs and other higher-order rewriting formalisms, open up the path for a transfer of results between higher-order and nominal rewriting. In particular, techniques and properties of the rewriting relation, such as termination, can be exported from one formalism to the other.Comment: 41 pages, journa