We present a new multi-valued monitoring approach for linear-time temporal logic that classifies trace prefixes not only according to the existence of correct and erroneous continuations, but also according to the strategic power of the system and its environment to avoid or enforce a violation of the specification. We classify the monitoring status into four levels: (1) the worst case is a vio...

We present a new syntactical proof that first-order Peano Arithmetic with Skolem axioms is conservative over Peano Arithmetic alone for arithmetical formulas. This result – which shows that the Excluded Middle principle can be used to eliminate Skolem functions – has been previously proved by other techniques, among them the epsilon substitution method and forcing. In our proof, we employ Inter...

We give the natural topological model for ¬BD-N, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-N. Also, the natural topological model for ¬BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-N, it is brought out in detail how BD-N fails. keywords: anti-Specker, BD, BD-N, realizability, topol...

These are lecture notes for a tutorial seminar which I gave at a satellite seminar of “Computability and Complexity in Analysis 2004” in Kyoto. The main message of the notes is that computable mathematics is the realizability interpretation of constructive mathematics. The presentation is targeted at an audience which is familiar with computable mathematics but less so with constructive mathema...

We prove general theorems about unique existence of effective subalgebras of classical algebras. The theorems are consequences of standard facts about completions of metric spaces within the framework of constructive mathematics, suitably interpreted in realizability models. We work with general realizability models rather than with a particular model of computation. Consequently, all the resul...

We define a new topos, the Herbrand topos, inspired by the modified realizability topos and our earlier work on Herbrand realizability. We also introduce the category of Herbrand assemblies and characterise these as the ¬¬-separated objects in the Herbrand topos. In addition, we show that the category of sets is included as the category of ¬¬-sheaves and prove that the inclusion functor preserv...

Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the computationally dense ones) are seen to be the ones whose ‘lifts’ to a kind of completion have right adjoints. We characterize topos inclusions corresponding to a general form of relative computability. We characterize p...

We present a new syntactical proof that first-order Peano Arithmetic with Skolem axioms is conservative over Peano Arithmetic alone for arithmetical formulas. This result – which shows that the Excluded Middle principle can be used to eliminate Skolem functions – has been previously proved by other techniques, among them the epsilon substitution method and forcing. In our proof, we employ Inter...

The logical system AF2μ, an extension of second-order predicate calculus with monotone inductive definitions, is presented. Some properties of this system are shown including that it is a good system for extracting programs from proofs by means of a realizability interpretation in the style of Krivine-Parigot, where the realizers are terms of the Curry-system of λ-calculus and the realizability...

We describe a systematic method to build a logic from any programming language described as a Pure Type System (PTS). The formulas of this logic express properties about programs. We define a parametricity theory about programs and a realizability theory for the logic. The logic is expressive enough to internalize both theories. Thanks to the PTS setting, we abstract most idiosyncrasies specifi...