# Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques

@article{Dowek2010PermissiveNT, title={Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques}, author={Gilles Dowek and Murdoch James Gabbay and Dominic P. Mulligan}, journal={Log. J. IGPL}, year={2010}, volume={18}, pages={769-822} }

Nominal terms extend first-order terms with binding. They lack some properties of first- and higher-order terms: Terms must be reasoned about in a context of ‘freshness assumptions’; it is not always possible to ‘choose a fresh variable symbol’ for a nominal term; it is not always possible to ‘α-convert a bound variable symbol’ or to ‘quotient by α-equivalence’; the notion of unifier is not based just on substitution. Permissive nominal terms closely resemble nominal terms but they recover…

## 33 Citations

### Nominal Terms and Nominal Logics: From Foundations to Meta-mathematics

- Computer Science
- 2014

This chapter draws together material from several papers to deliver a coherent account of a journey from the foundations of a mathematics with names, via logical systems based on those foundations, to concrete applications in axiomatising systems with binding.

### Permissive-nominal logic: First-order logic over nominal terms and sets

- PhilosophyTOCL
- 2012

Permissive-Nominal Logic is an extension of first-order predicate logic in which term-formers can bind names in their arguments, which gives PNL much of the expressive power of higher-order logic, but models and derivations of PNL are first- order in character.

### PNL to HOL: From the logic of nominal sets to the logic of higher-order functions

- PhilosophyTheor. Comput. Sci.
- 2012

### Semantic nominal terms

- Philosophy
- 2009

Nominal terms use a characteristic combination of features: Two levels of variable: atoms a and unknowns X. Freshness conditions a#X and permutations π. An abstraction [a]r with a distinctive…

### Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free

- MathematicsThe Journal of Symbolic Logic
- 2012

By operations on models it is shown how to relate completeness with respect to permissive-nominal models to completenesswith respect to nominal models with finite support, using an infinite generalisation of nominal atoms-abstraction.

### Permissive-nominal logic

- PhilosophyPPDP
- 2010

Permissive-Nominal Logic is an extension of first-order logic where term-formers can bind names in their arguments, which allows for direct axiomatisations with binders, such as the ∀l-quantifier and the λ-binder of the lambda-calculus.

### Nominal Anti-Unification

- Computer ScienceRTA
- 2015

An algorithm that computes nominal anti-unification, which relies on a subalgorithm that constructively decides equivariance between two terms-in-context, and proves soundness and completeness properties of both algorithms and analyzes their complexity.

### Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms

- Computer ScienceInf. Comput.
- 2010

### Extensions of nominal terms

- Linguistics
- 2011

This thesis studies two major extensions of nominal terms, with λ-abstraction over nominal unknowns and atoms, and an extension with an arguably better theory of freshness and α-equivalence, to extend nominal terms so that they can be used for metaprogramming on proofs, programs, etc.

### Nominal Unification and Matching of Higher Order Expressions with Recursive Let

- Computer ScienceFundam. Informaticae
- 2022

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time, and it is proved that there is a guessing strategy for nominal unity with letrec and atom-variable that is a trade-off between exponential growth and non-determinism.

## References

SHOWING 1-10 OF 54 REFERENCES

### Permissive nominal terms and their unication

- Linguistics
- 2009

We introduce permissive nominal terms, and their unication. Nominal terms are one way to extend rst-order terms with binding. However, they lack some useful properties of rst- and higher-order terms:…

### Nominal Logic: A First Order Theory of Names and Binding

- Computer ScienceTACS
- 2001

Nominal Logic is introduced, a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping and for freshness of names, from which a notion of binding can be derived.

### Nominal Unification from a Higher-Order Perspective

- PhilosophyTOCL
- 2012

It is shown that nominal unification can be quadratically reduced to a particular fragment of higher-order unification problems: higher- order pattern unification, and it is proved that the translation preserves most generality of unifiers.

### Completeness and Herbrand theorems for nominal logic

- PhilosophyJournal of Symbolic Logic
- 2006

This paper reviews nominal logic and shows why finite-support models are insufficient both in theory and practice, and identifies the class of models with respect to which nominal logic is complete: ideal-supported models in which the supports of values are elements of a proper ideal on the set of names.

### Semantic nominal terms

- Philosophy
- 2009

Nominal terms use a characteristic combination of features: Two levels of variable: atoms a and unknowns X. Freshness conditions a#X and permutations π. An abstraction [a]r with a distinctive…

### Capture-avoiding substitution as a nominal algebra

- PhilosophyFormal Aspects of Computing
- 2007

A special feature of this method is the use of nominal techniques, which give us access to a stronger assertion language, which includes so-called ‘freshness’ or ‘capture-avoidance’ conditions.

### Nominal (Universal) Algebra: Equational Logic with Names and Binding

- Computer ScienceJ. Log. Comput.
- 2009

The case is made for extending the logic of equality with meta-variables and capture-avoidance constraints, to obtain ‘nominal algebra’ and investigate proof-theoretical properties, and provide a sound and complete semantics in nominal sets.

### Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms

- Computer ScienceInf. Comput.
- 2010