### Term Equational Rewrite Systems and Logics

"... We introduce an abstract general notion of system of equations and rewrites between terms, called Term Equational Rewrite System (TERS), and develop a sound logical deduction system, called Term Equational Rewrite Logic (TERL), to reason about equality and rewriting. Further, we give an analysis of ..."

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We introduce an abstract general notion of system of equations and rewrites between terms, called Term Equational Rewrite System (TERS), and develop a sound logical deduction system, called Term Equational Rewrite Logic (TERL), to reason about equality and rewriting. Further, we give an analysis of algebraic free constructions which together with an internal completeness result may be used to synthetise a complete TERL. Indeed, as an application, we derive a sound and complete equational rewrite

### Abstract TERMGRAPH 2006 Preliminary Version Implementing Nominal Unification

"... Nominal matching and unification underly the dynamics of nominal rewriting. Urban, Pitts and Gabbay gave a nominal unification algorithm which finds the most general solution to a nominal matching or unification problem, if one exists. Later the algorithm was extended by Fernández and Gabbay to deal ..."

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Nominal matching and unification underly the dynamics of nominal rewriting. Urban, Pitts and Gabbay gave a nominal unification algorithm which finds the most general solution to a nominal matching or unification problem, if one exists. Later the algorithm was extended by Fernández and Gabbay to deal with name generation and locality. In this paper we describe first a direct implementation of the nominal unification algorithm, including the extensions, in Maude. This implementation is not efficient (it is exponential in time), but we will show that we can obtain a feasible implementation by using termgraphs.

### Dependent Types for a Nominal Logical Framework

, 2012

"... We present a logical framework based on the nominal approach to representing syntax with binders. First we extend nominal terms, which have a built-in name-abstraction operator and a first-order notion of substitution for variables, with a capture-avoiding substitution operator for names. We then bu ..."

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We present a logical framework based on the nominal approach to representing syntax with binders. First we extend nominal terms, which have a built-in name-abstraction operator and a first-order notion of substitution for variables, with a capture-avoiding substitution operator for names. We then build a dependent type system for this extended syntax

### ON THE MATHEMATICAL SYNTHESIS OF EQUATIONAL LOGICS

"... Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he ..."

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Birkhoff [1935] initiated the general study of algebraic structure. Importantly for our concerns here, his development was from (universal) algebra to (equational) logic. Birkhoff’s starting point was the informal conception of algebra based on familiar concrete examples. Abstracting from these, he introduced the concepts of signature and equational presentation,

### Semantics

"... out of context: nominal absolute denotations for first-order logic and computation ..."

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out of context: nominal absolute denotations for first-order logic and computation

### Creative Commons Attribution License.

, 2010

"... This work is licensed under the ..."

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### unknown title

"... Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness ..."

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Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness

### Unity in nominal equational reasoning: the algebra of

"... equality on nominal sets ..."

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### Contents

, 2009

"... In informal mathematical discourse (such as the text of a paper on theoretical computer science) we often reason about equalities involving binding of object-variables. We find our-selves writing assertions involving meta-variables and capture-avoidance constraints on where object-variables can and ..."

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In informal mathematical discourse (such as the text of a paper on theoretical computer science) we often reason about equalities involving binding of object-variables. We find our-selves writing assertions involving meta-variables and capture-avoidance constraints on where object-variables can and cannot occur free. Formalising such assertions is problematic because the standard logical frameworks cannot express capture-avoidance constraints directly. In this paper we make the case for extending the logic of equality with meta-variables and capture-avoidance constraints, to obtain ‘nominal algebra’. We use nominal techniques that allow for a direct formalisation of meta-level assertions, while remaining close to informal prac-tice. We investigate proof-theoretical properties, we provide a sound and complete semantics in nominal sets, and we compare and contrast our design decisions with other possibilities