# Volume 11, Issue 2

2015

### 1. Mutually Testing Processes

In the standard testing theory of DeNicola-Hennessy one process is considered to be a refinement of another if every test guaranteed by the former is also guaranteed by the latter. In the domain of web services this has been recast, with processes viewed as servers and tests as clients. In this way the standard refinement preorder between servers is determined by their ability to satisfy clients. But in this setting there is also a natural refinement preorder between clients, determined by their ability to be satisfied by servers. In more general settings where there is no distinction between clients and servers, but all processes are peers, there is a further refinement preorder based on the mutual satisfaction of peers. We give a uniform account of these three preorders. In particular we give two characterisations. The first is behavioural, in terms of traces and ready sets. The second, for finite processes, is equational.

### 2. On Reachability for Unidirectional Channel Systems Extended with Regular Tests

"Unidirectional channel systems" (Chambart & Schnoebelen, CONCUR 2008) are finite-state systems where one-way communication from a Sender to a Receiver goes via one reliable and one unreliable unbounded fifo channel. While reachability is decidable for these systems, equipping them with the possibility of testing regular properties on the contents of channels makes it undecidable. Decidability is preserved when only emptiness and nonemptiness tests are considered: the proof relies on an elaborate reduction to a generalized version of Post's Embedding Problem.

### 3. Learning and Designing Stochastic Processes from Logical Constraints

Stochastic processes offer a flexible mathematical formalism to model and reason about systems. Most analysis tools, however, start from the premises that models are fully specified, so that any parameters controlling the system's dynamics must be known exactly. As this is seldom the case, many methods have been devised over the last decade to infer (learn) such parameters from observations of the state of the system. In this paper, we depart from this approach by assuming that our observations are {\it qualitative} properties encoded as satisfaction of linear temporal logic formulae, as opposed to quantitative observations of the state of the system. An important feature of this approach is that it unifies naturally the system identification and the system design problems, where the properties, instead of observations, represent requirements to be satisfied. We develop a principled statistical estimation procedure based on maximising the likelihood of the system's parameters, using recent ideas from statistical machine learning. We demonstrate the efficacy and broad applicability of our method on a range of simple but non-trivial examples, including rumour spreading in social networks and hybrid models of gene regulation.

### 4. Containment for Conditional Tree Patterns

A Conditional Tree Pattern (CTP) expands an XML tree pattern with labels attached to the descendant edges. These labels can be XML element names or Boolean CTPs. The meaning of a descendant edge labelled by A and ending in a node labelled by B is a path of child steps ending in a B node such that all intermediate nodes are A nodes. In effect this expresses the until B, A holds construction from temporal logic.This paper studies the containment problem for CTP. For tree patterns (TP), this problem is known to be coNP-complete. We show that it is PSPACE-complete for CTP. This increase in complexity is due to the fact that CTP is expressive enough to encode an unrestricted form of label negation: ${*}\setminus a$, meaning "any node except an a-node". Containment of TP expanded with this type of negation is already PSPACE-hard. CTP is a positive, forward, first order fragment of Regular XPath. Unlike TP, CTP expanded with disjunction is not equivalent to unions of CTP's. Like TP, CTP is a natural fragment to consider: CTP is closed under intersections and CTP with disjunction is equally expressive as positive existential first order logic expanded with the until operator.

### 5. From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality

C*-algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C*-algebras. At first, this paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C*-algebras. This yields a new probabilistic version of Gelfand duality, involving the "Radon" monad on the category of compact Hausdorff spaces. We then show that the state space functor from C*-algebras to Eilenberg-Moore algebras of the Radon monad is full and faithful. This allows us to obtain an appropriately commuting state-and-effect triangle for C*-algebras.

### 6. Bounding linear head reduction and visible interaction through skeletons

In this paper, we study the complexity of execution in higher-order programming languages. Our study has two facets: on the one hand we give an upper bound to the length of interactions between bounded P-visible strategies in Hyland-Ong game semantics. This result covers models of programming languages with access to computational effects like non-determinism, state or control operators, but its semantic formulation causes a loose connection to syntax. On the other hand we give a syntactic counterpart of our semantic study: a non-elementary upper bound to the length of the linear head reduction sequence (a low-level notion of reduction, close to the actual implementation of the reduction of higher-order programs by abstract machines) of simply-typed lambda-terms. In both cases our upper bounds are proved optimal by giving matching lower bounds. These two results, although different in scope, are proved using the same method: we introduce a simple reduction on finite trees of natural numbers, hereby called interaction skeletons. We study this reduction and give upper bounds to its complexity. We then apply this study by giving two simulation results: a semantic one measuring progress in game-theoretic interaction via interaction skeletons, and a syntactic one establishing a correspondence between linear head reduction of terms satisfying a locality condition called local scope and the reduction of interaction skeletons. This result is then generalized to arbitrary terms by a […]

### 7. Using models to model-check recursive schemes

We propose a model-based approach to the model checking problem for recursive schemes. Since simply typed lambda calculus with the fixpoint operator, lambda-Y-calculus, is equivalent to schemes, we propose the use of a model of lambda-Y-calculus to discriminate the terms that satisfy a given property. If a model is finite in every type, this gives a decision procedure. We provide a construction of such a model for every property expressed by automata with trivial acceptance conditions and divergence testing. Such properties pose already interesting challenges for model construction. Moreover, we argue that having models capturing some class of properties has several other virtues in addition to providing decidability of the model-checking problem. As an illustration, we show a very simple construction transforming a scheme to a scheme reflecting a property captured by a given model.

### 8. Logical strength of complexity theory and a formalization of the PCP theorem in bounded arithmetic

We present several known formalizations of theorems from computational complexity in bounded arithmetic and formalize the PCP theorem in the theory PV1 (no formalization of this theorem was known). This includes a formalization of the existence and of some properties of the (n,d,{\lambda})-graphs in PV1.

### 9. Pure Nash Equilibria in Concurrent Deterministic Games

We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference relations. This includes the purely qualitative framework, where each player has a single omega-regular objective that she wants to satisfy, but also the larger class of semi-quantitative objectives, where each player has several omega-regular objectives equipped with a preorder (for instance, a player may want to satisfy all her objectives, or to maximise the number of objectives that she achieves.)

### 10. Generators and relations for n-qubit Clifford operators

We define a normal form for Clifford circuits, and we prove that every Clifford operator has a unique normal form. Moreover, we present a rewrite system by which any Clifford circuit can be reduced to normal form. This yields a presentation of Clifford operators in terms of generators and relations.

### 11. On Unification Modulo One-Sided Distributivity: Algorithms, Variants and Asymmetry

An algorithm for unification modulo one-sided distributivity is an early result by Tidén and Arnborg. More recently this theory has been of interest in cryptographic protocol analysis due to the fact that many cryptographic operators satisfy this property. Unfortunately the algorithm presented in the paper, although correct, has recently been shown not to be polynomial time bounded as claimed. In addition, for some instances, there exist most general unifiers that are exponentially large with respect to the input size. In this paper we first present a new polynomial time algorithm that solves the decision problem for a non-trivial subcase, based on a typed theory, of unification modulo one-sided distributivity. Next we present a new polynomial algorithm that solves the decision problem for unification modulo one-sided distributivity. A construction, employing string compression, is used to achieve the polynomial bound. Lastly, we examine the one-sided distributivity problem in the new asymmetric unification paradigm. We give the first asymmetric unification algorithm for one-sided distributivity.

### 12. Deciding the value 1 problem for probabilistic leaktight automata

The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton, are there words accepted with probability arbitrarily close to 1? This problem was proved undecidable recently; to overcome this, several classes of probabilistic automata of different nature were proposed, for which the value 1 problem has been shown decidable. In this paper, we introduce yet another class of probabilistic automata, called leaktight automata, which strictly subsumes all classes of probabilistic automata whose value 1 problem is known to be decidable. We prove that for leaktight automata, the value 1 problem is decidable (in fact, PSPACE-complete) by constructing a saturation algorithm based on the computation of a monoid abstracting the behaviours of the automaton. We rely on algebraic techniques developed by Simon to prove that this abstraction is complete. Furthermore, we adapt this saturation algorithm to decide whether an automaton is leaktight. Finally, we show a reduction allowing to extend our decidability results from finite words to infinite ones, implying that the value 1 problem for probabilistic leaktight parity automata is decidable.

### 13. Thermodynamic graph-rewriting

We develop a new thermodynamic approach to stochastic graph-rewriting. The ingredients are a finite set of reversible graph-rewriting rules called generating rules, a finite set of connected graphs P called energy patterns and an energy cost function. The idea is that the generators define the qualitative dynamics, by showing which transformations are possible, while the energy patterns and cost function specify the long-term probability $\pi$ of any reachable graph. Given the generators and energy patterns, we construct a finite set of rules which (i) has the same qualitative transition system as the generators; and (ii) when equipped with suitable rates, defines a continuous-time Markov chain of which $\pi$ is the unique fixed point. The construction relies on the use of site graphs and a technique of `growth policy' for quantitative rule refinement which is of independent interest. This division of labour between the qualitative and long-term quantitative aspects of the dynamics leads to intuitive and concise descriptions for realistic models (see the examples in S4 and S5). It also guarantees thermodynamical consistency (AKA detailed balance), otherwise known to be undecidable, which is important for some applications. Finally, it leads to parsimonious parameterizations of models, again an important point in some applications.

### 14. Weak bisimulation for coalgebras over order enriched monads

The paper introduces the notion of a weak bisimulation for coalgebras whose type is a monad satisfying some extra properties. In the first part of the paper we argue that systems with silent moves should be modelled coalgebraically as coalgebras whose type is a monad. We show that the visible and invisible part of the functor can be handled internally inside a monadic structure. In the second part we introduce the notion of an ordered saturation monad, study its properties, and show that it allows us to present two approaches towards defining weak bisimulation for coalgebras and compare them. We support the framework presented in this paper by two main examples of models: labelled transition systems and simple Segala systems.

### 15. On absorption in semigroups and $n$-ary semigroups

The notion of absorption was developed a few years ago by Barto and Kozik and immediately found many applications, particularly in topics related to the constraint satisfaction problem. We investigate the behavior of absorption in semigroups and n-ary semigroups (that is, algebras with one n-ary associative operation). In the case of semigroups, we give a simple necessary and sufficient condition for a semigroup to be absorbed by its subsemigroup. We then proceed to n-ary semigroups, where we conjecture an analogue of this necessary and sufficient condition, and prove that the conjectured condition is indeed necessary and sufficient for B to absorb A (where A is an n-ary semigroup and B is its n-ary subsemigroup) in the following three cases: when A is commutative, when |A-B|=1 and when A is an idempotent ternary semigroup.

### 16. Permissive Controller Synthesis for Probabilistic Systems

We propose novel controller synthesis techniques for probabilistic systems modelled using stochastic two-player games: one player acts as a controller, the second represents its environment, and probability is used to capture uncertainty arising due to, for example, unreliable sensors or faulty system components. Our aim is to generate robust controllers that are resilient to unexpected system changes at runtime, and flexible enough to be adapted if additional constraints need to be imposed. We develop a permissive controller synthesis framework, which generates multi-strategies for the controller, offering a choice of control actions to take at each time step. We formalise the notion of permissivity using penalties, which are incurred each time a possible control action is disallowed by a multi-strategy. Permissive controller synthesis aims to generate a multi-strategy that minimises these penalties, whilst guaranteeing the satisfaction of a specified system property. We establish several key results about the optimality of multi-strategies and the complexity of synthesising them. Then, we develop methods to perform permissive controller synthesis using mixed integer linear programming and illustrate their effectiveness on a selection of case studies.