# Volume 14, Issue 2

2018

### 1. Closed Sets and Operators thereon: Representations, Computability and Complexity

The TTE approach to Computable Analysis is the study of so-called representations (encodings for continuous objects such as reals, functions, and sets) with respect to the notions of computability they induce. A rich variety of such representations had been devised over the past decades, particularly regarding closed subsets of Euclidean space plus subclasses thereof (like compact subsets). In addition, they had been compared and classified with respect to both non-uniform computability of single sets and uniform computability of operators on sets. In this paper we refine these investigations from the point of view of computational complexity. Benefiting from the concept of second-order representations and complexity recently devised by Kawamura & Cook (2012), we determine parameterized complexity bounds for operators such as union, intersection, projection, and more generally function image and inversion. By indicating natural parameters in addition to the output precision, we get a uniform view on results by Ko (1991-2013), Braverman (2004/05) and Zhao & Müller (2008), relating these problems to the P/UP/NP question in discrete complexity theory.

### 2. Monadic Second Order Logic with Measure and Category Quantifiers

We investigate the extension of Monadic Second Order logic, interpreted over infinite words and trees, with generalized "for almost all" quantifiers interpreted using the notions of Baire category and Lebesgue measure.
Section: Automata and logic

### 3. $n$-permutability and linear Datalog implies symmetric Datalog

We show that if $\mathbb A$ is a core relational structure such that CSP($\mathbb A$) can be solved by a linear Datalog program, and $\mathbb A$ is $n$-permutable for some $n$, then CSP($\mathbb A$) can be solved by a symmetric Datalog program (and thus CSP($\mathbb A$) lies in deterministic logspace). At the moment, it is not known for which structures $\mathbb A$ will CSP($\mathbb A$) be solvable by a linear Datalog program. However, once somebody obtains a characterization of linear Datalog, our result immediately gives a characterization of symmetric Datalog.

### 4. A note on first-order spectra with binary relations

The spectrum of a first-order sentence is the set of the cardinalities of its finite models. In this paper, we consider the spectra of sentences over binary relations that use at least three variables. We show that for every such sentence $\Phi$, there is a sentence $\Phi'$ that uses the same number of variables, but only one symmetric binary relation, such that its spectrum is linearly proportional to the spectrum of $\Phi$. Moreover, the models of $\Phi'$ are all bipartite graphs. As a corollary, we obtain that to settle Asser's conjecture, i.e., whether the class of spectra is closed under complement, it is sufficient to consider only sentences using only three variables whose models are restricted to undirected bipartite graphs.

### 5. Van Kampen Colimits and Path Uniqueness

Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, i.e, categories of objects that can be represented as algebras as well as coalgebras, e.g., the category of directed graphs. Multimodeling requires to construct colimits of models, decomposition is given by pullback. Compositionality requires an exact interplay of these operations, i.e., diagrams must enjoy the Van Kampen property. However, checking the validity of the Van Kampen property algorithmically based on its definition is often impossible. In this paper we state a necessary and sufficient yet efficiently checkable condition for the Van Kampen property to hold in presheaf topoi. It is based on a uniqueness property of path-like structures within the defining congruence classes that make up the colimiting cocone of the models. We thus add to the statement "Being Van Kampen is a Universal Property" by Heindel and Soboci\'{n}ski the fact that the Van Kampen property reveals a presheaf-based structural uniqueness feature.

### 6. Domains via approximation operators

In this paper, we tailor-make new approximation operators inspired by rough set theory and specially suited for domain theory. Our approximation operators offer a fresh perspective to existing concepts and results in domain theory, but also reveal ways to establishing novel domain-theoretic results. For instance, (1) the well-known interpolation property of the way-below relation on a continuous poset is equivalent to the idempotence of a certain set-operator; (2) the continuity of a poset can be characterized by the coincidence of the Scott closure operator and the upper approximation operator induced by the way below relation; (3) meet-continuity can be established from a certain property of the topological closure operator. Additionally, we show how, to each approximating relation, an associated order-compatible topology can be defined in such a way that for the case of a continuous poset the topology associated to the way-below relation is exactly the Scott topology. A preliminary investigation is carried out on this new topology.

### 7. Study of Behaviours via Visitable Paths

Around 2000, J.-Y. Girard developed a logical theory, called Ludics. This theory was a step in his program of Geometry of Interaction, the aim of which being to account for the dynamics of logical proofs. In Ludics, objects called designs keep only what is relevant for the cut elimination process, hence the dynamics of a proof: a design is an abstraction of a formal proof. The notion of behaviour is the counterpart in Ludics of the notion of type or the logical notion of formula. Formally a behaviour is a closed set of designs. Our aim is to explore the constructions of behaviours and to analyse their properties. In this paper a design is viewed as a set of coherent paths. We recall or give variants of properties concerning visitable paths, where a visitable path is a path in a design or a set of designs that may be traversed by interaction with a design of the orthogonal of the set. We are then able to answer the following question: which properties should satisfy a set of paths for being exactly the set of visitable paths of a behaviour? Such a set and its dual should be prefix-closed, daimon-closed and satisfy two saturation properties. This allows us to have a means for defining the whole set of visitable paths of a given set of designs without closing it explicitly, that is without computing the orthogonal of this set of designs. We finally apply all these results for making explicit the structure of a behaviour generated by constants and multiplicative/additive […]

### 8. Analyzing Timed Systems Using Tree Automata

Timed systems, such as timed automata, are usually analyzed using their operational semantics on timed words. The classical region abstraction for timed automata reduces them to (untimed) finite state automata with the same time-abstract properties, such as state reachability. We propose a new technique to analyze such timed systems using finite tree automata instead of finite word automata. The main idea is to consider timed behaviors as graphs with matching edges capturing timing constraints. When a family of graphs has bounded tree-width, they can be interpreted in trees and MSO-definable properties of such graphs can be checked using tree automata. The technique is quite general and applies to many timed systems. In this paper, as an example, we develop the technique on timed pushdown systems, which have recently received considerable attention. Further, we also demonstrate how we can use it on timed automata and timed multi-stack pushdown systems (with boundedness restrictions).

### 9. A Categorical Approach to Syntactic Monoids

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category $\mathcal D$. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott ($\mathcal D=$ sets), the syntactic ordered monoids of Pin ($\mathcal D =$ posets), the syntactic semirings of Polák ($\mathcal D=$ semilattices), and the syntactic associative algebras of Reutenauer ($\mathcal D$ = vector spaces). Assuming that $\mathcal D$ is a commutative variety of algebras or ordered algebras, we prove that the syntactic $\mathcal D$-monoid of a language $L$ can be constructed as a quotient of a free $\mathcal D$-monoid modulo the syntactic congruence of $L$, and that it is isomorphic to the transition $\mathcal D$-monoid of the minimal automaton for $L$ in $\mathcal D$. Furthermore, in the case where the variety $\mathcal D$ is locally finite, we characterize the regular languages as precisely the languages with finite syntactic $\mathcal D$-monoids.

### 10. Uniqueness of directed complete posets based on Scott closed set lattices

In analogy to a result due to Drake and Thron about topological spaces, this paper studies the dcpos (directed complete posets) which are fully determined, among all dcpos, by their lattices of all Scott-closed subsets (such dcpos will be called $C_{\sigma}$-unique). We introduce the notions of down-linear element and quasicontinuous element in dcpos, and use them to prove that dcpos of certain classes, including all quasicontinuous dcpos as well as Johnstone's and Kou's examples, are $C_{\sigma}$-unique. As a consequence, $C_{\sigma}$-unique dcpos with their Scott topologies need not be bounded sober.

### 11. Weihrauch-completeness for layerwise computability

We introduce the notion of being Weihrauch-complete for layerwise computability and provide several natural examples related to complex oscillations, the law of the iterated logarithm and Birkhoff's theorem. We also consider hitting time operators, which share the Weihrauch degree of the former examples but fail to be layerwise computable.

### 12. The language of Stratified Sets is confluent and strongly normalising

We study the properties of the language of Stratified Sets (first-order logic with $\in$ and a stratification condition) as used in TST, TZT, and (with stratifiability instead of stratification) in Quine's NF. We find that the syntax forms a nominal algebra for substitution and that stratification and stratifiability imply confluence and strong normalisation under rewrites corresponding naturally to $\beta$-conversion.

### 13. A Dichotomy for First-Order Reducts of Unary Structures

Many natural decision problems can be formulated as constraint satisfaction problems for reducts $\mathbb{A}$ of finitely bounded homogeneous structures. This class of problems is a large generalisation of the class of CSPs over finite domains. Our first result is a general polynomial-time reduction from such infinite-domain CSPs to finite-domain CSPs. We use this reduction to obtain new powerful polynomial-time tractability conditions that can be expressed in terms of the topological polymorphism clone of $\mathbb{A}$. Moreover, we study the subclass $\mathcal{C}$ of CSPs for structures $\mathbb{A}$ that are reducts of a structure with a unary language. Also this class $\mathcal{C}$ properly extends the class of all finite-domain CSPs. We apply our new tractability conditions to prove the general tractability conjecture of Bodirsky and Pinsker for reducts of finitely bounded homogeneous structures for the class $\mathcal{C}$.

### 14. Computability in Basic Quantum Mechanics

The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space $\mathcal{H}$. In terms of the Hilbert lattice $\mathcal{L}$ of closed linear subspaces of $\mathcal{H}$ the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch's Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category $\mathbf{QCB}_0$ which is equivalent to the category $\mathbf{AdmRep}$ of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann's Spectral Theorem.

### 15. An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part I)

There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. A recent result by Balbiani, Goranko, and Sciavicco presented an explicit two-sorted point-interval temporal framework in which time instants (points) and time periods (intervals) are considered on a par, allowing the perspective to shift between these within the formal discourse. We consider here two-sorted first-order languages based on the same principle, and therefore including relations, as first studied by Reich, among others, between points, between intervals, and inter-sort. We give complete classifications of its sub-languages in terms of relative expressive power, thus determining how many, and which, are the intrinsically different extensions of two-sorted first-order logic with one or more such relations. This approach roots out the classical problem of whether or not points should be included in a interval-based semantics.

### 16. On Sub-Propositional Fragments of Modal Logic

In this paper, we consider the well-known modal logics $\mathbf{K}$, $\mathbf{T}$, $\mathbf{K4}$, and $\mathbf{S4}$, and we study some of their sub-propositional fragments, namely the classical Horn fragment, the Krom fragment, the so-called core fragment, defined as the intersection of the Horn and the Krom fragments, plus their sub-fragments obtained by limiting the use of boxes and diamonds in clauses. We focus, first, on the relative expressive power of such languages: we introduce a suitable measure of expressive power, and we obtain a complex hierarchy that encompasses all fragments of the considered logics. Then, after observing the low expressive power, in particular, of the Horn fragments without diamonds, we study the computational complexity of their satisfiability problem, proving that, in general, it becomes polynomial.

### 17. An estimation for the lengths of reduction sequences of the $\lambda\mu\rho\theta$-calculus

Since it was realized that the Curry-Howard isomorphism can be extended to the case of classical logic as well, several calculi have appeared as candidates for the encodings of proofs in classical logic. One of the most extensively studied among them is the $\lambda\mu$-calculus of Parigot. In this paper, based on the result of Xi presented for the $\lambda$-calculus Xi, we give an upper bound for the lengths of the reduction sequences in the $\lambda\mu$-calculus extended with the $\rho$- and $\theta$-rules. Surprisingly, our results show that the new terms and the new rules do not add to the computational complexity of the calculus despite the fact that $\mu$-abstraction is able to consume an unbounded number of arguments by virtue of the $\mu$-rule.

### 18. SOS-based Modal Decomposition on Nondeterministic Probabilistic Processes

We propose a method for the decomposition of modal formulae on processes with nondeterminism and probability with respect to Structural Operational Semantics. The purpose is to reduce the satisfaction problem of a formula for a process to verifying whether its subprocesses satisfy certain formulae obtained from the decomposition. To deal with the probabilistic behavior of processes, and thus with the decomposition of formulae characterizing it, we introduce a SOS-like machinery allowing for the specification of the behavior of open distribution terms. By our decomposition, we obtain (pre)congruence formats for probabilistic bisimilarity, ready similarity and similarity.

### 19. Extensional Semantics for Higher-Order Logic Programs with Negation

We develop an extensional semantics for higher-order logic programs with negation, generalizing the technique that was introduced in [Bezem99,Bezem01] for positive higher-order programs. In this way we provide an alternative extensional semantics for higher-order logic programs with negation to the one proposed in [CharalambidisER14]. As an immediate useful consequence of our developments, we define for the language we consider the notions of stratification and local stratification, which generalize the familiar such notions from classical logic programming. We demonstrate that for stratified and locally stratified higher-order logic programs, the proposed semantics never assigns the unknown truth value. We conclude the paper by providing a negative result: we demonstrate that the well-known stable model semantics of classical logic programming, if extended according to the technique of [Bezem99,Bezem01] to higher-order logic programs, does not in general lead to extensional stable models.