Distributed Asynchronous Games With Causal Memory are Undecidable

We show the undecidability of the distributed control problem when the plant is an asynchronous automaton, the controllers use causal memory and the goal of the controllers is to put each process in a local accepting state

A Unifying Framework for Deciding Synchronizability

Several notions of synchronizability of a message-passing system have been introduced in the literature. Roughly, a system is called synchronizable if every execution can be rescheduled so that it meets certain criteria, e.g., a channel bound. We provide a framework, based on MSO logic and (special) tree-width, that unifies existing definitions, explains their good properties, and allows one to easily derive other, more general definitions and decidability results for synchronizability

On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D \cup \{u,v\}$ is at most $\alpha$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $\alpha = 1$ by Ding et al. For any constant $\alpha > 1$, we give an $O(n^{1-\frac{1}{\alpha}}(\log n)^{\frac{1}{\alpha}})$-approximation algorithm. When $\alpha \geq 5$, we give an $O(\sqrt{n}\log n)$-approximation algorithm. Finally, we prove that, when $\alpha =2$, unless $NP \subseteq DTIME(n^{poly\log n})$, for any constant $\epsilon > 0$, the problem admits no polynomial-time $2^{\log^{1-\epsilon}n}$-approximation algorithm, improving upon the $\Omega(\log n)$ bound by Du et al. (albeit under a stronger hardness assumption)

Making Metric Temporal Logic Rational

We study an extension of MTL in pointwise time with regular expression guarded modality Reg_I(re) where re is a rational expression over subformulae. We study the decidability and expressiveness of this extension (MTL+Ureg+Reg), called RegMTL, as well as its fragment SfrMTL where only star-free rational expressions are allowed. Using the technique of temporal projections, we show that RegMTL has decidable satisfiability by giving an equisatisfiable reduction to MTL. We also identify a subclass MITL+UReg of RegMTL for which our equisatisfiable reduction gives rise to formulae of MITL, yielding elementary decidability. As our second main result, we show a tight automaton-logic connection between SfrMTL and partially ordered (or very weak) 1-clock alternating timed automata

Deciding Reachability for Piecewise Constant Derivative Systems on Orientable Manifolds

© 2019 Springer-Verlag. This is a post-peer-review, pre-copyedit version of a paper published in Reachability Problems: 13th International Conference, RP 2019, Brussels, Belgium, September 11–13, 2019, Proceedings. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-030-30806-3_14A hybrid automaton is a finite state machine combined with some k real-valued continuous variables, where k determines the number of the automaton dimensions. This formalism is widely used for modelling safety-critical systems, and verification tasks for such systems can often be expressed as the reachability problem for hybrid automata. Asarin, Mysore, Pnueli and Schneider defined classes of hybrid automata lying on the boundary between decidability and undecidability in their seminal paper ‘Low dimensional hybrid systems - decidable, undecidable, don’t know’ [9]. They proved that certain decidable classes become undecidable when given a little additional computational power, and showed that the reachability question remains unsolved for some 2-dimensional systems. Piecewise Constant Derivative Systems on 2-dimensional manifolds (or PCD2m) constitute a class of hybrid automata for which decidability of the reachability problem is unknown. In this paper we show that the reachability problem becomes decidable for PCD2m if we slightly limit their dynamics, and thus we partially answer the open question of Asarin, Mysore, Pnueli and Schneider posed in [9]

Towards Blackbox Identity Testing of Log-Variate Circuits

Derandomization of blackbox identity testing reduces to extremely special circuit models. After a line of work, it is known that focusing on circuits with constant-depth and constantly many variables is enough (Agrawal,Ghosh,Saxena, STOC\u2718) to get to general hitting-sets and circuit lower bounds. This inspires us to study circuits with few variables, eg. logarithmic in the size s. We give the first poly(s)-time blackbox identity test for n=O(log s) variate size-s circuits that have poly(s)-dimensional partial derivative space; eg. depth-3 diagonal circuits (or Sigma wedge Sigma^n). The former model is well-studied (Nisan,Wigderson, FOCS\u2795) but no poly(s2^n)-time identity test was known before us. We introduce the concept of cone-closed basis isolation and prove its usefulness in studying log-variate circuits. It subsumes the previous notions of rank-concentration studied extensively in the context of ROABP models

Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth Three Circuits

We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following: 1. There exists an explicit n-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2^{Omega(n)} size. 2. Any multilinear depth three circuit computing IMM_{n,d} (the iterated matrix multiplication polynomial formed by multiplying d, n * n symbolic matrices) has n^{Omega(d)} size. IMM_{n,d} can be easily computed by a poly(n,d) sized ROABP. 3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n,d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n^{Omega(d)}. This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] between these two models. The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field