A model of the untyped lambda calculus induces a lambda theory, i.e., a congruence relation on λ-terms closed under ff- and fi-conversion. A semantics (= class of models) of the lambda calculus is incomplete if there exists a lambda theory which is not induced by any model in the semantics. In this paper we introduce a new technique to prove the incompleteness of a wide range of lambda calculus semantics, including the strongly stable one, whose incompleteness had been conjectured by Bastonero-Gouy [6, 7] and by Berline [9]. The main results of the paper are a topological incompleteness theorem and an order incompleteness theorem. In the first one we show the incompleteness of the lambda calculus semantics given in terms of topological models whose topology satisfies a property of connectedness. In the second one we prove the incompleteness of the class of partially ordered models with finitely many connected components w.r.t. the Alexandroff topology. A further result of the paper is a proof of the completeness of the semantics of the lambda calculus given in terms of topological models whose topology is non-trivial and metrizable.

In this paper we describe how to build semantic models that support both nondeterministic choice and probabilistic choice. Several models exist that support both of these constructs, but none that we know of satisfies all the laws one would like. Using domain-theoretic techniques, we show how models can be devised using the "standard model" for probabilistic choice, and then applying modified domain-theoretic models for nondeterministic choice. These models are distinguished by the fact that the expected laws for nondeterministic choice and probabilistic choice remain valid. We also describe some potential applications of our model to aspects of security.

In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λ-theories, the equational incompleteness of lambda calculus semantics, and the λ-theories induced by graph models of lambda calculus.

Traditionally, powerdomains have been used to provide models for various forms of nondeterminism in semantics. We establish a similar analogy between zero nding methods in numerical analysis and powerdomains: Different powerdomain constructions correspond to dierent types of behavior exhibited by numerical methods for zero finding. By combining this observation with the basic quantitative paradigm provided by measurement, a simple and uniform method for analyzing zero finding algorithms is obtained.

We use a subfamily of the Scott-closed sets of a poset to form a local completion of the poset. This is simultaneously a topological analogue of the ideal completion of a poset and a generalization of the sobrification of a topological space. After we show that our construction is the object level of a left adjoint to the forgetful functor from the category of local cpos to the category of posets and Scott-continuous maps, we use this completion to show how local domains can play a role in the study of domain-theoretic models of topological spaces. Our main result shows that any topological space that is homeomorphic to the maximal elements of a continuous poset that is weak at the top also is homeomorphic to the maximal elements of a bounded complete local domain. The advantage is that continuous maps between such spaces extend to Scott-continuous maps between the modeling local domains.