In the distributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective strings x, y ∈ {0,1}n are at the Hamming distanceH(x, y) = 0 or H(x, y) = $\frac{n}{2}$. Buhrman et al. (STOC' 98) proved that the exact quantum communication complexity of this problem is O(log n) while the deterministic communication complexity is Ω(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed $\frac{n}{2}$ ⩽ k ⩽ n, whether H(x, y) = 0 or H(x, y) = k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds if k is an even such that $\frac{1}{2}$n ⩽ k < (1 − λ)n, where 0 < λ < $\frac{1}{2}$ is given. We also deal with a promise version of the well-known disjointness problem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.

Activities such as clinical investigations (CIs) or financial processes are subject to regulations to ensure quality of results and avoid negative consequences. Regulations may be imposed by multiple governmental agencies as well as by institutional policies and protocols. Due to the complexity of both regulations and activities, there is great potential for violation due to human error, misunderstanding, or even intent. Executable formal models of regulations, protocols and activities can form the foundation for automated assistants to aid planning, monitoring and compliance checking. We propose a model based on multiset rewriting where time is discrete and is specified by timestamps attached to facts. Actions, as well as initial, goal and critical states may be constrained by means of relative time constraints. Moreover, actions may have non-deterministic effects, i.e. they may have different outcomes whenever applied. We present a formal semantics of our model based on focused proofs of linear logic with definitions. We also determine the computational complexity of various planning problems. Plan compliance problem, for example, is the problem of finding a plan that leads from an initial state to a desired goal state without reaching any undesired critical state. We consider all actions to be balanced, i.e. their pre- and post-conditions have the same number of facts. Under this assumption on actions, we show that the plan compliance problem is PSPACE-complete when all actions have only deterministic effects and is EXPTIME-complete when actions may have non-deterministic effects. Finally, we show that the restrictions on the form of actions and time constraints taken in the specification of our model are necessary for decidability of the planning problems.

Multivalued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees).

A more general understanding is possible, if the category-theoretic properties of multivalued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multivalued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch).

Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multivalued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multivalued functions.

A natural question in the λ-calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in λ-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are λ-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) λ-theory where the range property is violated. A second, more complex example of a non-r.e. λ-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term.

We propose the new concept of Krivine ordered combinatory algebra ($\mathcal{^KOCA}$) as foundation for the categorical study of Krivine's classical realizability, as initiated by Streicher (2013).

We show that $\mathcal{^KOCA}$'s are equivalent to Streicher's abstract Krivine structures for the purpose of modeling higher-order logic, in the precise sense that they give rise to the same class of triposes. The difference between the two representations is that the elements of a $\mathcal{^KOCA}$ play both the role of truth values and realizers, whereas truth values are sets of realizers in $\mathcal{AKS}$s.

To conclude, we give a direct presentation of the realizability interpretation of a higher order language in a $\mathcal{^KOCA}$, which showcases the dual role that is played by the elements of the $\mathcal{^KOCA}$.