Notes on logic and set theory
A calculus of mobile processes, II
Information and Computation
From λσ to λν: a journey through calculi of explicit substitutions
POPL '94 Proceedings of the 21st ACM SIGPLAN-SIGACT symposium on Principles of programming languages
On the algebraic models of Lambda calculus
Theoretical Computer Science - Modern algebra and its applications
Scrap your boilerplate: a practical design pattern for generic programming
Proceedings of the 2003 ACM SIGPLAN international workshop on Types in languages design and implementation
A Metalanguage for Programming with Bound Names Modulo Renaming
MPC '00 Proceedings of the 5th International Conference on Mathematics of Program Construction
A New Approach to Abstract Syntax Involving Binders
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Semantical Analysis of Higher-Order Abstract Syntax
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Abstract Syntax and Variable Binding
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Semantics of Name and Value Passing
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
Nominal logic, a first order theory of names and binding
Information and Computation - TACS 2001
Theoretical Computer Science
The Lattice of Lambda Theories
Journal of Logic and Computation
Proceedings of the 8th ACM SIGPLAN international conference on Principles and practice of declarative programming
A general mathematics of names
Information and Computation
Capture-avoiding substitution as a nominal algebra
Formal Aspects of Computing
Relational reasoning in a nominal semantics for storage
TLCA'05 Proceedings of the 7th international conference on Typed Lambda Calculi and Applications
Freshness and name-restriction in sets of traces with names
FOSSACS'11/ETAPS'11 Proceedings of the 14th international conference on Foundations of software science and computational structures: part of the joint European conferences on theory and practice of software
Stone duality for nominal Boolean algebras with И
CALCO'11 Proceedings of the 4th international conference on Algebra and coalgebra in computer science
PNL to HOL: From the logic of nominal sets to the logic of higher-order functions
Theoretical Computer Science
Full abstraction for nominal Scott domains
POPL '13 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
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Fraenkel-Mostowski (FM) set theory delivers a model of names and alpha-equivalence. This model, now generally called the 'nominal' model, delivers inductive datatypes of syntax with alpha-equivalence - rather than inductive datatypes of syntax, quotiented by alpha-equivalence. The treatment of names and alpha-equivalence extends to the entire sets universe. This has proven useful for developing 'nominal' theories of reasoning and programming on syntax with alpha-equivalence, because a sets universe includes elements representing functions, predicates, and behaviour. Often, we want names and alpha-equivalence to model capture-avoiding substitution. In this paper we show that FM set theory models capture-avoiding substitution for names in much the same way as it models alpha-equivalence; as an operation valid for the entire sets universe which coincides with the usual (inductively defined) operation on inductive datatypes. In fact, more than one substitution action is possible (they all agree on sets representing syntax). We present two distinct substitution actions, making no judgement as to which one is 'right' - we suspect this question has the same status as asking whether classical or intuitionistic logic is 'right'. We describe the actions in detail, and describe the overall design issues involved in creating any substitution action on a sets universe. Along the way, we think in new ways about the structure of elements of FM set theory. This leads us to some interesting mathematical concepts, including the notions of planes and crucial elements, which we also describe in detail.