95 research outputs found
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
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
Confluence of Orthogonal Nominal Rewriting Systems Revisited
Nominal rewriting systems (Fernandez, Gabbay, Mackie, 2004;
Fernandez, Gabbay, 2007) have been introduced as a new framework
of higher-order rewriting systems based on the nominal approach
(Gabbay, Pitts, 2002; Pitts, 2003), which deals with variable
binding via permutations and freshness conditions on atoms.
Confluence of orthogonal nominal rewriting systems has been shown in
(Fernandez, Gabbay, 2007). However, their definition of
(non-trivial) critical pairs has a serious weakness so that the
orthogonality does not actually hold for most of standard nominal
rewriting systems in the presence of binders. To overcome this
weakness, we divide the notion of overlaps into the self-rooted and
proper ones, and introduce a notion of alpha-stability which
guarantees alpha-equivalence of peaks from the self-rooted
overlaps. Moreover, we give a sufficient criterion for uniformity and alpha-stability. The new definition of orthogonality and the
criterion offer a novel confluence condition effectively applicable to many standard nominal rewriting systems. We also report on an
implementation of a confluence prover for orthogonal nominal rewriting systems based on our framework
On the mathematical synthesis of equational logics
We provide a mathematical theory and methodology for synthesising equational
logics from algebraic metatheories. We illustrate our methodology by means of
two applications: a rational reconstruction of Birkhoff's Equational Logic and
a new equational logic for reasoning about algebraic structure with
name-binding operators.Comment: Final version for publication in Logical Methods in Computer Scienc
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
Higher-order port-graph rewriting
The biologically inspired framework of port-graphs has been successfully used
to specify complex systems. It is the basis of the PORGY modelling tool. To
facilitate the specification of proof normalisation procedures via graph
rewriting, in this paper we add higher-order features to the original
port-graph syntax, along with a generalised notion of graph morphism. We
provide a matching algorithm which enables to implement higher-order port-graph
rewriting in PORGY, thus one can visually study the dynamics of the systems
modelled. We illustrate the expressive power of higher-order port-graphs with
examples taken from proof-net reduction systems.Comment: In Proceedings LINEARITY 2012, arXiv:1211.348
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
Equivariant ZFA with Choice: a position paper
We propose Equivariant ZFA with Choice as a foundation for nominal techniques
that is stronger than ZFC and weaker than FM, and why this may be particularly
helpful in the context of automated reasoning.Comment: In ARW 201
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