# Volume 16, Issue 2

2020

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

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. In this Part II, we deal with the cases of all dense and the case of all unbounded linearly ordered sets.
Section: Logic for knowledge representation

### 2. Reachability for infinite time Turing machines with long tapes

Infinite time Turing machine models with tape length $\alpha$, denoted $T_\alpha$, strengthen the machines of Hamkins and Kidder [HL00] with tape length $\omega$. A new phenomenon is that for some countable ordinals $\alpha$, some cells cannot be halting positions of $T_\alpha$ given trivial input. The main open question in [Rin14] asks about the size of the least such ordinal $\delta$. We answer this by providing various characterizations. For instance, $\delta$ is the least ordinal with any of the following properties: (a) For some $\xi<\alpha$, there is a $T_\xi$-writable but not $T_\alpha$-writable subset of $\omega$. (b) There is a gap in the $T_\alpha$-writable ordinals. (c) $\alpha$ is uncountable in $L_{\lambda_\alpha}$. Here $\lambda_\alpha$ denotes the supremum of $T_\alpha$-writable ordinals, i.e. those with a $T_\alpha$-writable code of length $\alpha$. We further use the above characterizations, and an analogue to Welch's submodel characterization of the ordinals $\lambda$, $\zeta$ and $\Sigma$, to show that $\delta$ is large in the sense that it is a closure point of the function $\alpha \mapsto \Sigma_\alpha$, where $\Sigma_\alpha$ denotes the supremum of the $T_\alpha$-accidentally writable ordinals.

### 3. Representing Dependencies in Event Structures

Event structures where the causality may explicitly change during a computation have recently gained the stage. In this kind of event structures the changes in the set of the causes of an event are triggered by modifiers that may add or remove dependencies, thus making the happening of an event contextual. Still the focus is always on the dependencies of the event. In this paper we promote the idea that the context determined by the modifiers plays a major role, and the context itself determines not only the causes but also what causality should be. Modifiers are then used to understand when an event (or a set of events) can be added to a configuration, together with a set of events modeling dependencies, which will play a less important role. We show that most of the notions of Event Structure presented in literature can be translated into this new kind of event structure, preserving the main notion, namely the one of configuration.

### 4. Tight Polynomial Worst-Case Bounds for Loop Programs

In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language - representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication - it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate. A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found. A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid.

### 5. On the incomputability of computable dimension

Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions might be incomputable.

### 6. Register Games

The complexity of parity games is a long standing open problem that saw a major breakthrough in 2017 when two quasi-polynomial algorithms were published. This article presents a third, independent approach to solving parity games in quasi-polynomial time, based on the notion of register game, a parameterised variant of a parity game. The analysis of register games leads to a quasi-polynomial algorithm for parity games, a polynomial algorithm for restricted classes of parity games and a novel measure of complexity, the register index, which aims to capture the combined complexity of the priority assignement and the underlying game graph. We further present a translation of alternating parity word automata into alternating weak automata with only a quasi-polynomial increase in size, based on register games; this improves on the previous exponential translation. We also use register games to investigate the parity index hierarchy: while for words the index hierarchy of alternating parity automata collapses to the weak level, and for trees it is strict, for structures between trees and words, it collapses logarithmically, in the sense that any parity tree automaton of size n is equivalent, on these particular classes of structures, to an automaton with a number of priorities logarithmic in n.

### 7. Cellular Cohomology in Homotopy Type Theory

We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built in stages by attaching spheres of progressively higher dimension, and cellular cohomology defines the groups out of the combinatorial description of how spheres are attached. Our main result is that for finite cell complexes, a wide class of cohomology theories (including the ones defined through Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This result was formalized in the Agda proof assistant.

### 8. On properties of $B$-terms

$B$-terms are built from the $B$ combinator alone defined by $B\equiv\lambda fgx. f(g~x)$, which is well known as a function composition operator. This paper investigates an interesting property of $B$-terms, that is, whether repetitive right applications of a $B$-term cycles or not. We discuss conditions for $B$-terms to have and not to have the property through a sound and complete equational axiomatization. Specifically, we give examples of $B$-terms which have the cyclic property and show that there are infinitely many $B$-terms which do not have the property. Also, we introduce another interesting property about a canonical representation of $B$-terms that is useful to detect cycles, or equivalently, to prove the cyclic property, with an efficient algorithm.

### 9. Synthesis of Orchestrations and Choreographies: Bridging the Gap between Supervisory Control and Coordination of Services

We present a number of contributions to bridging the gap between supervisory control theory and coordination of services in order to explore the frontiers between coordination and control systems. Firstly, we modify the classical synthesis algorithm from supervisory control theory for obtaining the so-called most permissive controller in order to synthesise orchestrations and choreographies of service contracts formalised as contract automata. The key ingredient to make this possible is a novel notion of controllability. Then, we present an abstract parametric synthesis algorithm and show that it generalises the classical synthesis as well as the orchestration and choreography syntheses. Finally, through the novel abstract synthesis, we show that the concrete syntheses are in a refinement order. A running example from the service domain illustrates our contributions.

### 10. Trace Refinement in Labelled Markov Decision Processes

Given two labelled Markov decision processes (MDPs), the trace-refinement problem asks whether for all strategies of the first MDP there exists a strategy of the second MDP such that the induced labelled Markov chains are trace-equivalent. We show that this problem is decidable in polynomial time if the second MDP is a Markov chain. The algorithm is based on new results on a particular notion of bisimulation between distributions over the states. However, we show that the general trace-refinement problem is undecidable, even if the first MDP is a Markov chain. Decidability of those problems was stated as open in 2008. We further study the decidability and complexity of the trace-refinement problem provided that the strategies are restricted to be memoryless.

### 11. Completeness of the ZX-Calculus

The ZX-Calculus is a graphical language for diagrammatic reasoning in quantum mechanics and quantum information theory. It comes equipped with an equational presentation. We focus here on a very important property of the language: completeness, which roughly ensures the equational theory captures all of quantum mechanics. We first improve on the known-to-be-complete presentation for the so-called Clifford fragment of the language - a restriction that is not universal - by adding some axioms. Thanks to a system of back-and-forth translation between the ZX-Calculus and a third-party complete graphical language, we prove that the provided axiomatisation is complete for the first approximately universal fragment of the language, namely Clifford+T. We then prove that the expressive power of this presentation, though aimed at achieving completeness for the aforementioned restriction, extends beyond Clifford+T, to a class of diagrams that we call linear with Clifford+T constants. We use another version of the third-party language - and an adapted system of back-and-forth translation - to complete the language for the ZX-Calculus as a whole, that is, with no restriction. We briefly discuss the added axioms, and finally, we provide a complete axiomatisation for an altered version of the language which involves an additional generator, making the presentation simpler.

### 12. Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that $\mathsf{NP}$ is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k \geq 1$ it is consistent with theory $T$ that computational class ${\mathcal C} \not \subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one of the pairs: $T = \mathsf{T}^1_2$ and ${\mathcal C} = \mathsf{P}^\mathsf{NP}$, $T = \mathsf{S}^1_2$ and ${\mathcal C} = \mathsf{NP}$, $T = \mathsf{PV}$ and ${\mathcal C} = \mathsf{P}$. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory $\mathsf{PV}$ already formalizes sophisticated arguments, such as a proof of the PCP Theorem. These […]

### 13. Forward Analysis for WSTS, Part III: Karp-Miller Trees

This paper is a sequel of "Forward Analysis for WSTS, Part I: Completions" [STACS 2009, LZI Intl. Proc. in Informatics 3, 433-444] and "Forward Analysis for WSTS, Part II: Complete WSTS" [Logical Methods in Computer Science 8(3), 2012]. In these two papers, we provided a framework to conduct forward reachability analyses of WSTS, using finite representations of downward-closed sets. We further develop this framework to obtain a generic Karp-Miller algorithm for the new class of very-WSTS. This allows us to show that coverability sets of very-WSTS can be computed as their finite ideal decompositions. Under natural effectiveness assumptions, we also show that LTL model checking for very-WSTS is decidable. The termination of our procedure rests on a new notion of acceleration levels, which we study. We characterize those domains that allow for only finitely many accelerations, based on ordinal ranks.

### 14. Failure of Normalization in Impredicative Type Theory with Proof-Irrelevant Propositional Equality

Normalization fails in type theory with an impredicative universe of propositions and a proof-irrelevant propositional equality. The counterexample to normalization is adapted from Girard's counterexample against normalization of System F equipped with a decider for type equality. It refutes Werner's normalization conjecture [LMCS 2008].