4 research outputs found
Relational Graph Models at Work
We study the relational graph models that constitute a natural subclass of
relational models of lambda-calculus. We prove that among the lambda-theories
induced by such models there exists a minimal one, and that the corresponding
relational graph model is very natural and easy to construct. We then study
relational graph models that are fully abstract, in the sense that they capture
some observational equivalence between lambda-terms. We focus on the two main
observational equivalences in the lambda-calculus, the theory H+ generated by
taking as observables the beta-normal forms, and H* generated by considering as
observables the head normal forms. On the one hand we introduce a notion of
lambda-K\"onig model and prove that a relational graph model is fully abstract
for H+ if and only if it is extensional and lambda-K\"onig. On the other hand
we show that the dual notion of hyperimmune model, together with
extensionality, captures the full abstraction for H*
Effective lambda models versus recursively enumerable lambda theories
A longstanding open problem is whether there exists a non-syntactical model of the untyped
位-calculus whose theory is exactly the least 位-theory 位尾. In this paper we investigate the
more general question of whether the equational/order theory of a model of the untyped
位-calculus can be recursively enumerable (r.e. for short). We introduce a notion of effective
model of 位-calculus, which covers, in particular, all the models individually introduced in the
literature. We prove that the order theory of an effective model is never r.e.; from this it
follows that its equational theory cannot be 位尾 or 位尾畏. We then show that no effective model
living in the stable or strongly stable semantics has an r.e. equational theory. For Scott鈥檚
semantics, we investigate the class of graph models and prove that no order theory of a
graph model can be r.e., and that there exists an effective graph model whose
equational/order theory is the minimum among the theories of graph models. Finally, we
show that the class of graph models enjoys a kind of downwards L篓owenheim鈥揝kolem
theorem
Effective lambda models versus recursively enumerable lambda theories
A longstanding open problem is whether there exists a non-syntactical model of the untyped
位-calculus whose theory is exactly the least 位-theory 位尾. In this paper we investigate the
more general question of whether the equational/order theory of a model of the untyped
位-calculus can be recursively enumerable (r.e. for short). We introduce a notion of effective
model of 位-calculus, which covers, in particular, all the models individually introduced in the
literature. We prove that the order theory of an effective model is never r.e.; from this it
follows that its equational theory cannot be 位尾 or 位尾畏. We then show that no effective model
living in the stable or strongly stable semantics has an r.e. equational theory. For Scott鈥檚
semantics, we investigate the class of graph models and prove that no order theory of a
graph model can be r.e., and that there exists an effective graph model whose
equational/order theory is the minimum among the theories of graph models. Finally, we
show that the class of graph models enjoys a kind of downwards L篓owenheim鈥揝kolem
theorem
Effective lambda models versus recursively enumerable lambda theories
A longstanding open problem is whether there exists a non-syntactical model of the untyped
位-calculus whose theory is exactly the least 位-theory 位尾. In this paper we investigate the
more general question of whether the equational/order theory of a model of the untyped
位-calculus can be recursively enumerable (r.e. for short). We introduce a notion of effective
model of 位-calculus, which covers, in particular, all the models individually introduced in the
literature. We prove that the order theory of an effective model is never r.e.; from this it
follows that its equational theory cannot be 位尾 or 位尾畏. We then show that no effective model
living in the stable or strongly stable semantics has an r.e. equational theory. For Scott鈥檚
semantics, we investigate the class of graph models and prove that no order theory of a
graph model can be r.e., and that there exists an effective graph model whose
equational/order theory is the minimum among the theories of graph models. Finally, we
show that the class of graph models enjoys a kind of downwards L篓owenheim鈥揝kolem
theorem