# Volume 9, Issue 2

2013

### 1. Decidable Classes of Tree Automata Mixing Local and Global Constraints Modulo Flat Theories

We define a class of ranked tree automata TABG generalizing both the tree automata with local tests between brothers of Bogaert and Tison (1992) and with global equality and disequality constraints (TAGED) of Filiot et al. (2007). TABG can test for equality and disequality modulo a given flat equational theory between brother subterms and between subterms whose positions are defined by the states reached during a computation. In particular, TABG can check that all the subterms reaching a given state are distinct. This constraint is related to monadic key constraints for XML documents, meaning that every two distinct positions of a given type have different values. We prove decidability of the emptiness problem for TABG. This solves, in particular, the open question of the decidability of emptiness for TAGED. We further extend our result by allowing global arithmetic constraints for counting the number of occurrences of some state or the number of different equivalence classes of subterms (modulo a given flat equational theory) reaching some state during a computation. We also adapt the model to unranked ordered terms. As a consequence of our results for TABG, we prove the decidability of a fragment of the monadic second order logic on trees extended with predicates for equality and disequality between subtrees, and cardinality.

### 2. The degree structure of Weihrauch-reducibility

We answer a question by Vasco Brattka and Guido Gherardi by proving that the Weihrauch-lattice is not a Brouwer algebra. The computable Weihrauch-lattice is also not a Heyting algebra, but the continuous Weihrauch-lattice is. We further investigate the existence of infinite infima and suprema, as well as embeddings of the Medvedev-degrees into the Weihrauch-degrees.

### 3. Infinite sequential Nash equilibrium

In game theory, the concept of Nash equilibrium reflects the collective stability of some individual strategies chosen by selfish agents. The concept pertains to different classes of games, e.g. the sequential games, where the agents play in turn. Two existing results are relevant here: first, all finite such games have a Nash equilibrium (w.r.t. some given preferences) iff all the given preferences are acyclic; second, all infinite such games have a Nash equilibrium, if they involve two agents who compete for victory and if the actual plays making a given agent win (and the opponent lose) form a quasi-Borel set. This article generalises these two results via a single result. More generally, under the axiomatic of Zermelo-Fraenkel plus the axiom of dependent choice (ZF+DC), it proves a transfer theorem for infinite sequential games: if all two-agent win-lose games that are built using a well-behaved class of sets have a Nash equilibrium, then all multi-agent multi-outcome games that are built using the same well-behaved class of sets have a Nash equilibrium, provided that the inverse relations of the agents' preferences are strictly well-founded.

### 4. Two Variable vs. Linear Temporal Logic in Model Checking and Games

Model checking linear-time properties expressed in first-order logic has non-elementary complexity, and thus various restricted logical languages are employed. In this paper we consider two such restricted specification logics, linear temporal logic (LTL) and two-variable first-order logic (FO2). LTL is more expressive but FO2 can be more succinct, and hence it is not clear which should be easier to verify. We take a comprehensive look at the issue, giving a comparison of verification problems for FO2, LTL, and various sublogics thereof across a wide range of models. In particular, we look at unary temporal logic (UTL), a subset of LTL that is expressively equivalent to FO2; we also consider the stutter-free fragment of FO2, obtained by omitting the successor relation, and the expressively equivalent fragment of UTL, obtained by omitting the next and previous connectives. We give three logic-to-automata translations which can be used to give upper bounds for FO2 and UTL and various sublogics. We apply these to get new bounds for both non-deterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). We couple these with matching lower-bound arguments. Next, we look at combining FO2 verification techniques with those for LTL. We present here a language that subsumes both FO2 and LTL, and inherits the model checking properties of both languages. Our results give […]

### 5. Total Representations

Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.

### 6. Time and Parallelizability Results for Parity Games with Bounded Tree and DAG Width

Parity games are a much researched class of games in NP intersect CoNP that are not known to be in P. Consequently, researchers have considered specialised algorithms for the case where certain graph parameters are small. In this paper, we study parity games on graphs with bounded treewidth, and graphs with bounded DAG width. We show that parity games with bounded DAG width can be solved in O(n^(k+3) k^(k + 2) (d + 1)^(3k + 2)) time, where n, k, and d are the size, treewidth, and number of priorities in the parity game. This is an improvement over the previous best algorithm, given by Berwanger et al., which runs in n^O(k^2) time. We also show that, if a tree decomposition is provided, then parity games with bounded treewidth can be solved in O(n k^(k + 5) (d + 1)^(3k + 5)) time. This improves over previous best algorithm, given by Obdrzalek, which runs in O(n d^(2(k+1)^2)) time. Our techniques can also be adapted to show that the problem of solving parity games with bounded treewidth lies in the complexity class NC^2, which is the class of problems that can be efficiently parallelized. This is in stark contrast to the general parity game problem, which is known to be P-hard, and thus unlikely to be contained in NC.

### 7. Bisimulations Meet PCTL Equivalences for Probabilistic Automata

Probabilistic automata (PAs) have been successfully applied in formal verification of concurrent and stochastic systems. Efficient model checking algorithms have been studied, where the most often used logics for expressing properties are based on probabilistic computation tree logic (PCTL) and its extension PCTL^*. Various behavioral equivalences are proposed, as a powerful tool for abstraction and compositional minimization for PAs. Unfortunately, the equivalences are well-known to be sound, but not complete with respect to the logical equivalences induced by PCTL or PCTL*. The desire of a both sound and complete behavioral equivalence has been pointed out by Segala in 1995, but remains open throughout the years. In this paper we introduce novel notions of strong bisimulation relations, which characterize PCTL and PCTL* exactly. We extend weak bisimulations that characterize PCTL and PCTL* without next operator, respectively. Further, we also extend the framework to simulation preorders. Thus, our paper bridges the gap between logical and behavioral equivalences and preorders in this setting.

### 8. Effective Characterizations of Simple Fragments of Temporal Logic Using Carton--Michel Automata

We present a framework for obtaining effective characterizations of simple fragments of future temporal logic (LTL) with the natural numbers as time domain. The framework is based on a form of strongly unambiguous automata, also known as prophetic automata or complete unambiguous Büchi automata and referred to as Carton-Michel automata in this paper. These automata enjoy strong structural properties, in particular, they separate the "finitary fraction" of a regular language of infinite words from its "infinitary fraction" in a natural fashion. Within our framework, we provide characterizations of several natural fragments of temporal logic, where, in some cases, no effective characterization had been known previously, and give lower and upper bounds for their computational complexity.

### 9. Admissibility in Finitely Generated Quasivarieties

Checking the admissibility of quasiequations in a finitely generated (i.e., generated by a finite set of finite algebras) quasivariety Q amounts to checking validity in a suitable finite free algebra of the quasivariety, and is therefore decidable. However, since free algebras may be large even for small sets of small algebras and very few generators, this naive method for checking admissibility in $\Q$ is not computationally feasible. In this paper, algorithms are introduced that generate a minimal (with respect to a multiset well-ordering on their cardinalities) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in Q. In particular, structural completeness (validity and admissibility coincide) and almost structural completeness (validity and admissibility coincide for quasiequations with unifiable premises) can be checked. The algorithms are illustrated with a selection of well-known finitely generated quasivarieties, and adapted to handle also admissibility of rules in finite-valued logics.

### 10. Tree-Automatic Well-Founded Trees

We investigate tree-automatic well-founded trees. Using Delhomme's decomposition technique for tree-automatic structures, we show that the (ordinal) rank of a tree-automatic well-founded tree is strictly below omega^omega. Moreover, we make a step towards proving that the ranks of tree-automatic well-founded partial orders are bounded by omega^omega^omega: we prove this bound for what we call upwards linear partial orders. As an application of our result, we show that the isomorphism problem for tree-automatic well-founded trees is complete for level Delta^0_{omega^omega} of the hyperarithmetical hierarchy with respect to Turing-reductions.

### 11. Unifying the Linear Time-Branching Time Spectrum of Process Semantics

Van Glabbeek's linear time-branching time spectrum is one of the most relevant work on comparative study on process semantics, in which semantics are partially ordered by their discrimination power. In this paper we bring forward a refinement of this classification and show how the process semantics can be dealt with in a uniform way: based on the very natural concept of constrained simulation we show how we can classify the spectrum in layers; for the families lying in the same layer we show how to obtain in a generic way equational, observational, logical and operational characterizations; relations among layers are also very natural and differences just stem from the constraint imposed on the simulations that rule the layers. Our methodology also shows how to achieve a uniform treatment of semantic preorders and equivalences.

### 12. Propositional Dynamic Logic with Converse and Repeat for Message-Passing Systems

The model checking problem for propositional dynamic logic (PDL) over message sequence charts (MSCs) and communicating finite state machines (CFMs) asks, given a channel bound $B$, a PDL formula $\varphi$ and a CFM $\mathcal{C}$, whether every existentially $B$-bounded MSC $M$ accepted by $\mathcal{C}$ satisfies $\varphi$. Recently, it was shown that this problem is PSPACE-complete. In the present work, we consider CRPDL over MSCs which is PDL equipped with the operators converse and repeat. The former enables one to walk back and forth within an MSC using a single path expression whereas the latter allows to express that a path expression can be repeated infinitely often. To solve the model checking problem for this logic, we define message sequence chart automata (MSCAs) which are multi-way alternating parity automata walking on MSCs. By exploiting a new concept called concatenation states, we are able to inductively construct, for every CRPDL formula $\varphi$, an MSCA precisely accepting the set of models of $\varphi$. As a result, we obtain that the model checking problem for CRPDL and CFMs is still in PSPACE.

### 13. Lower Bound on Weights of Large Degree Threshold Functions

An integer polynomial $p$ of $n$ variables is called a \emph{threshold gate} for a Boolean function $f$ of $n$ variables if for all $x \in \zoon$ $f(x)=1$ if and only if $p(x)\geq 0$. The \emph{weight} of a threshold gate is the sum of its absolute values. In this paper we study how large a weight might be needed if we fix some function and some threshold degree. We prove $2^{\Omega(2^{2n/5})}$ lower bound on this value. The best previous bound was $2^{\Omega(2^{n/8})}$ (Podolskii, 2009). In addition we present substantially simpler proof of the weaker $2^{\Omega(2^{n/4})}$ lower bound. This proof is conceptually similar to other proofs of the bounds on weights of nonlinear threshold gates, but avoids a lot of technical details arising in other proofs. We hope that this proof will help to show the ideas behind the construction used to prove these lower bounds.

### 14. Markov two-components processes

We propose Markov two-components processes (M2CP) as a probabilistic model of asynchronous systems based on the trace semantics for concurrency. Considering an asynchronous system distributed over two sites, we introduce concepts and tools to manipulate random trajectories in an asynchronous framework: stopping times, an Asynchronous Strong Markov property, recurrent and transient states and irreducible components of asynchronous probabilistic processes. The asynchrony assumption implies that there is no global totally ordered clock ruling the system. Instead, time appears as partially ordered and random. We construct and characterize M2CP through a finite family of transition matrices. M2CP have a local independence property that guarantees that local components are independent in the probabilistic sense, conditionally to their synchronization constraints. A synchronization product of two Markov chains is introduced, as a natural example of M2CP.