This special issue of Mathematical Structures in Computer Science is devoted to the fourteenth Italian Conference on Theoretical Computer Science (ICTCS) held at University of Palermo, Italy, from 9th to 11th September 2013. ICTCS is the conference of the Italian Chapter of the European Association for Theoretical Computer Science and covers a wide spectrum of topics in Theoretical Computer Science, ranging from computational complexity to logic, from algorithms and data structure to programming languages, from combinatorics on words to distributed computing. For this reason, the contributions here included come from very different areas of Theoretical Computer Science. In fact this special issue is motivated by the desire to give people who have presented their ideas at the 14th ICTCS the opportunity to publish papers on their work. Submitted papers have been subject to a careful and severe reviewing process and 11 of them were selected for this special issue.

We are very proud and honoured to dedicate this volume to Corrado Böhm, who has been for us a teacher, mentor, colleague, and friend. But most of all, he was a brilliant role-model to follow.

Resources and their use and consumption form a central part of our life. Many branches of science and engineering are concerned with the question of which given resource objects can be converted into which target resource objects. For example, information theory studies the conversion of a noisy communication channel instance into an exchange of information. Inspired by work in quantum information theory, we develop a general mathematical toolbox for this type of question. The convertibility of resources into other ones and the possibility of combining resources is accurately captured by the mathematics of ordered commutative monoids. As an intuitive example, we consider chemistry, where chemical reaction equations such as

\mathrm{2H_2 + O_2} \lra \mathrm{2H_2O,}

While closely related to both Girard's linear logic and to Deutsch's constructor theory, our framework also produces results very reminiscent of the utility theorem of von Neumann and Morgenstern in decision theory and of a theorem of Lieb and Yngvason on the foundations of thermodynamics.

Concerning pure algebra, our observation is that some pieces of algebra can be developed in a context in which equality is not necessarily symmetric, i.e. in which the equality relation is replaced by an ordering relation. For example, notions like cancellativity or torsion-freeness are still sensible and very natural concepts in our ordered setting.

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.

The second Dagstuhl seminar on coalgebraic logics took place from October 7–12, 2012, in the Leibniz Forschungszentrum Schloss Dagstuhl, following a successful earlier one in December 2009. From the 44 researchers who attended and the 30 talks presented, this collection highlights some of the progress that has been made in the field. We are grateful to Giuseppe Longo and his interest in a special issue in Mathematical Structures in Computer Science.

The International Symposium on Domain Theory (ISDT) is a conference series intended to be a forum for researchers in domain theory and its applications. Topics include topological and logical aspects of domains; categories of domains and powerdomains; continuous posets and their representations; partial orders, lattice theory and metric spaces; types, process algebra and concurrency; non-classical and partial logics; programming language semantics; applications in computer science and mathematics. This conference series was founded by Yingming Liu, Yixiang Chen, Klaus Keimel, and Guo-Qiang Zhang. All ISDT events have taken place in China. The first ISDT was held in Shanghai, October 17–24, 1999; the second ISDT was held in Chengdu, October 22–26, 2001; the third ISDT occurred in Xi'an, China, May 10–14, 2004; the fourth ISDT was held in Changsha, June 2–6, 2006; and the fifth ISDT took place in Shanghai, September 11–14, 2009.

This issue of Mathematical Structures in Computer Science is composed mainly of papers submitted by participants of the Workshop ‘Continuity, Computability, Constructivity: From Logic to Algorithms,’ held in Gregynog, a conference centre of the University of Wales located in the beautiful nature of Mid Wales, in the last week of June 2013. In addition, several colleagues accepted our invitation to contribute to this volume.

We present the no-iteration version of a mixed distributive law of a monad over a comonad in a general 2-category. We present the simplifications that occur in the case of the 2-category Cat.

Theoretical models of recursion schemes have been well studied under the names well-founded coalgebras, recursive coalgebras, corecursive algebras and Elgot algebras. Much of this work focuses on conditions ensuring unique or canonical solutions, e.g. when the coalgebra is well founded.

If the coalgebra is not well founded, then there can be multiple solutions. The standard semantics of recursive programs gives a particular solution, typically the least fixpoint of a certain monotone map on a domain whose least element is the totally undefined function; but this solution may not be the desired one. We have recently proposed programming language constructs to allow the specification of alternative solutions and methods to compute them. We have implemented these new constructs as an extension of OCaml.

In this paper, we prove some theoretical results characterizing well-founded coalgebras, along with several examples for which this extension is useful. We also give several examples that are not well founded but still have a desired solution. In each case, the function would diverge under the standard semantics of recursion, but can be specified and computed with the programming language constructs we have proposed.

We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.

We prove the conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL). The conservation theorem states that the property of having infinite reductions (here infinite chains of cut elimination steps) is preserved by non-erasing steps. This turns the quest for strong normalisation (SN) into one for non-erasing weak normalisation (WN), and indeed we use this result to prove SN of simply typed DiLL (with promotion). Along the way to the theorem we achieve a number of additional results having their own interest, such as a standardisation theorem and a slightly modified system of nets, DiLL∂ϱ.

We consider a notion of approximation for terms of de Groote–Saurin Λμ-calculus. Then, we introduce an intersection type assignment system for that calculus which is invariant under subject conversion. The type assignment system also induces a filter model, which is an extensional Λμ-model in the sense of Nakazawa and Katsumata. We then establish the approximation theorem, stating that a type can be assigned to a term in the system if and only if it can be assigned to same of its approximations.

We extend the simply-typed lambda-calculus with a mechanism for dynamic and incremental rebinding of code. Fragments of open code which can be dynamically rebound are values. Differently from standard static binding, which is done on a positional basis, rebinding is done on a nominal basis, that is, free variables in open code are associated with names which do not obey α-equivalence. Moreover, rebinding is incremental, that is, just a subset of names can be rebound, making possible code specialization, and rebinding can even introduce new names. Finally, rebindings, which are associations between names and terms, are first-class values, and can be manipulated by operators such as overriding and renaming. We define a type system in which the type for a rebinding, in addition to specify an association between names and types (similarly to record types), is also annotated. The annotation says whether or not the domain of the rebinding having this type may contain more names than the ones that are specified in the type. We show soundness of the type system.

Tissue P systems are a class of bio-inspired computing models motivated by biochemical interactions between cells in a tissue-like arrangement. Tissue P systems with cell division offer a theoretical device to generate an exponentially growing structure in order to solve computationally hard problems efficiently with the assumption that there exists a global clock to mark the time for the system, the execution of each rule is completed in exactly one time unit. Actually, the execution time of different biochemical reactions in cells depends on many uncertain factors. In this work, with this biological inspiration, we remove the restriction on the execution time of each rule, and the computational efficiency of tissue P systems with cell division is investigated. Specifically, we solve subset sum problem by tissue P systems with cell division in a time-free manner in the sense that the correctness of the solution to the problem does not depend on the execution time of the involved rules.

In this paper, we study the relationship between ordering structures on semirings and semiring-induced valuation algebras. We show that a semiring-induced valuation algebra is a complete (resp. continuous) lattice if and only if the semiring is complete (resp. continuous) lattice with respect to the reverse order relation on semirings. Furthermore, a semiring-induced information algebra is compact, if the dual of the semiring is an algebraic lattice.

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.

In Interactive realizability for second-order Heyting Arithmetic with EM1 and SK1 (the excluded middle and Skolem axioms restricted to Σ10-formulas), realizers are written in a classical version of Girard's System F. Since the usual reducibility semantics does not apply to such a system, we introduce a constructive forcing/reducibility semantics: though realizers are not computable functionals in the sense of Girard, they can be forced to be computable. We apply this semantics to show how to extract witnesses for realizable Π20-formulas. In particular, a constructive and efficient method is introduced. It is based on a new ‘(state-extending-continuation)-passing-style translation’ whose properties are described with the constructive forcing/reducibility semantics.

Induction is a well-established proof principle that is taught in most undergraduate programs in mathematics and computer science. In computer science, it is used primarily to reason about inductively defined datatypes such as finite lists, finite trees and the natural numbers. Coinduction is the dual principle that can be used to reason about coinductive datatypes such as infinite streams or trees, but it is not as widespread or as well understood. In this paper, we illustrate through several examples the use of coinduction in informal mathematical arguments. Our aim is to promote the principle as a useful tool for the working mathematician and to bring it to a level of familiarity on par with induction. We show that coinduction is not only about bisimilarity and equality of behaviors, but also applicable to a variety of functions and relations defined on coinductive datatypes.

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.

We consider computation with real numbers that arise through a process of physical measurement. We have developed a theory in which physical experiments that measure quantities can be used as oracles to algorithms and we have begun to classify the computational power of various forms of experiment using non-uniform complexity classes. Earlier, in Beggs et al. (2014 Reviews of Symbolic Logic7(4) 618–646), we observed that measurement can be viewed as a process of comparing a rational number z – a test quantity – with a real number y – an unknown quantity; each oracle call performs such a comparison. Experiments can then be classified into three categories, that correspond with being able to return test results

$$\begin{eqnarray*} z < y\text{ or }z > y\text{ or }\textit{timeout},\\ z < y\text{ or }\textit{timeout},\\ z \neq y\text{ or }\textit{timeout}. \end{eqnarray*} $$