How to write a coequation

There is a large amount of literature on the topic of covarieties, coequations and coequational specifications, dating back to the early seventies. Nevertheless, coequations have not (yet) emerged as an everyday practical specification formalism for computer scientists. In this review paper, we argue that this is partly due to the multitude of syntaxes for writing down coequations, which seems to have led to some confusion about what coequations are and what they are for. By surveying the literature, we identify four types of syntaxes: coequations-as-corelations, coequations-as-predicates, coequations-as-equations, and coequations-as-modal-formulas. We present each of these in a tutorial fashion, relate them to each other, and discuss their respective uses

Nominal Equational Logic

AbstractThis paper studies the notion of “freshness” that often occurs in the meta-theory of computer science languages involving various kinds of names. Nominal Equational Logic is an extension of ordinary equational logic with assertions about the freshness of names. It is shown to be both sound and complete for the support interpretation of freshness and equality provided by the Gabbay-Pitts nominal sets model of names, binding and α-conversion

Resource Combinatory Algebras

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

We give a semantics for the lambda-calculus based on a topological duality theorem in nominal sets. A novel interpretation of lambda is given in terms of adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is necessary)

Ordered models of the lambda calculus

Answering a question by Honsell and Plotkin, we show that there are two equations between lambda terms, the so-called subtractive equations, consistent with lambda calculus but not simultaneously satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the order-incompleteness of lambda calculus, by studying the connection between the notion of absolute unorderability in a specific point and a weaker notion of subtractivity (namely n-subtractivity) for partially ordered algebras. Finally we study the relation between n-subtractivity and relativized separation conditions in topological algebras, obtaining an incompleteness theorem for a general topological semantics of lambda calculus