Abstract We present a logical system CFP (Concurrent Fixed Point Logic) that supports the extraction of nondeterministic and concurrent programs that are provably total and correct. CFP is an intuitionistic first-order logic with inductive and coinductive definitions extended by two propositional operators, $$B|_{A}$$ B | A (restriction, a strengthening of implication) and $${\mathbf {\downdownarrows }}(B)$$ ⇊ ( B ) (total concurrency). The source of the extraction are formal CFP proofs, the target is a lambda calculus with constructors and recursion extended by a constructor Amb (for McCarthy’s amb) which is interpreted operationally as globally angelic choice and is used to implement nondeterminism and concurrency. The correctness of extracted programs is proven via an intermediate domain-theoretic denotational semantics. We demonstrate the usefulness of our system by extracting a nondeterministic program that translates infinite Gray code into the signed digit representation. A noteworthy feature of our system is that the proof rules for restriction and concurrency involve variants of the classical law of excluded middle that would not be interpretable computationally without Amb.

A dyadic subbase S of a topological space X is a subbase consisting of a countable collection of pairs of open subsets that are exteriors of each other. If a dyadic subbase S is proper, then we can construct a dcpo DS in which X is embedded. We study properties of S with respect to two aspects. (i) Whether the dcpo DS is consistently complete depends on not only S itself but also the enumeration of S. We give a characterization of S that induces the consistent completeness of DS regardless of its enumeration. (ii) If the space X is regular Hausdorff, then X is embedded in the minimal limit set of DS. We construct an example of a Hausdorff but non-regular space with a dyadic subbase S such that the minimal limit set of DS is empty.

This special issue of Mathematical Structures in Computer Science is composed mainly of papers submitted by participants of the Dagstuhl Seminar on Computing with Infinite Data: Topological and Logical Foundations. The workshop took place in the Schloss Dagstuhl - Leibniz Center for Informatics in the first half of October 2011.

Every compact metric space $X$ is homeomorphically embedded in an $\omega$ -algebraic domain $D$ as the set of minimal limit (that is, non-finite) elements. Moreover, $X$ is a retract of the set $L(D)$ of all limit elements of $D$ . Such a domain $D$ can be chosen so that it has property M and finite-branching, and the height of $L(D)$ is equal to the small inductive dimension of $X$ . We also show that the small inductive dimension of $L(D)$ as a topological space is equal to the height of $L(D)$ for domains with property M. These results give a characterisation of the dimension of a space $X$ as the minimal height of $L(D)$ in which $X$ is embedded as the set of minimal elements. The domain in which we embed an $n$ -dimensional compact metric space $X$ ( $n \leq \infinity$ ) has a concrete structure in that it consists of finite/infinite sequences in $\{0,1,\bot\}$ with at most $n$ copies of $\bot$ .