Decidability in the logic of subsequences and supersequences

We consider first-order logics of sequences ordered by the subsequence ordering, aka sequence embedding. We show that the \Sigma_2 theory is undecidable, answering a question left open by Kuske. Regarding fragments with a bounded number of variables, we show that the FO2 theory is decidable while the FO3 theory is undecidable

On shuffle products, acyclic automata and piecewise-testable languages

We show that the shuffle L \unicode{x29E2} F of a piecewise-testable language $L$ and a finite language $F$ is piecewise-testable. The proof relies on a classic but little-used automata-theoretic characterization of piecewise-testable languages. We also discuss some mild generalizations of the main result, and provide bounds on the piecewise complexity of L \unicode{x29E2} F

On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

We investigate the ordinal invariants height, length, and width of well quasi orders (WQO), with particular emphasis on width, an invariant of interest for the larger class of orders with finite antichain condition (FAC). We show that the width in the class of FAC orders is completely determined by the width in the class of WQOs, in the sense that if we know how to calculate the width of any WQO then we have a procedure to calculate the width of any given FAC order. We show how the width of WQO orders obtained via some classical constructions can sometimes be computed in a compositional way. In particular, this allows proving that every ordinal can be obtained as the width of some WQO poset. One of the difficult questions is to give a complete formula for the width of Cartesian products of WQOs. Even the width of the product of two ordinals is only known through a complex recursive formula. Although we have not given a complete answer to this question we have advanced the state of knowledge by considering some more complex special cases and in particular by calculating the width of certain products containing three factors. In the course of writing the paper we have discovered that some of the relevant literature was written on cross-purposes and some of the notions re-discovered several times. Therefore we also use the occasion to give a unified presentation of the known results

Verifying nondeterministic probabilistic channel systems against ω\omega-regular linear-time properties

Lossy channel systems (LCSs) are systems of finite state automata that communicate via unreliable unbounded fifo channels. In order to circumvent the undecidability of model checking for nondeterministic LCSs, probabilistic models have been introduced, where it can be decided whether a linear-time property holds almost surely. However, such fully probabilistic systems are not a faithful model of nondeterministic protocols. We study a hybrid model for LCSs where losses of messages are seen as faults occurring with some given probability, and where the internal behavior of the system remains nondeterministic. Thus the semantics is in terms of infinite-state Markov decision processes. The purpose of this article is to discuss the decidability of linear-time properties formalized by formulas of linear temporal logic (LTL). Our focus is on the qualitative setting where one asks, e.g., whether a LTL-formula holds almost surely or with zero probability (in case the formula describes the bad behaviors). Surprisingly, it turns out that -- in contrast to finite-state Markov decision processes -- the satisfaction relation for LTL formulas depends on the chosen type of schedulers that resolve the nondeterminism. While all variants of the qualitative LTL model checking problem for the full class of history-dependent schedulers are undecidable, the same questions for finite-memory scheduler can be solved algorithmically. However, the restriction to reachability properties and special kinds of recurrent reachability properties yields decidable verification problems for the full class of schedulers, which -- for this restricted class of properties -- are as powerful as finite-memory schedulers, or even a subclass of them.Comment: 39 page

Use of Functional Electrical Stimulation for Functional Mobility of a Pediatric Patient with Cerebral Palsy: A Case Report

Background: Children with cerebral palsy (CP) have varying impairments of motor control and muscle tone, impacting functional mobility. One physical therapy intervention for children with limited functional mobility is utilizing functional electrical stimulation (FES) as an intervention to facilitate movement. FES cycling and FES assisted tasks are becoming more readily studied and utilized as an intervention for this population of patients. To date, there are few studies that evaluate the use of FES interventions combined with land based interventions in children with spastic quadriplegic CP. Purpose: Discuss the use of FES cycling and aided functional activities as an intervention for a pediatric patients with spastic quadriplegic CP. Intervention: During a four-week timeframe, a five-year-old boy with spastic quadriplegic CP, participated in physical therapy sessions one time per week utilizing FES cycling and FES assisted functional activities and one time per week utilizing adaptive cycling and non-FES assisted activities. The RT300 FES cycling machine was utilized, as well as the portable SAGE controller component of the RT300 for functional tasks of supine bridging, sit to stand, and static standing. Outcome Measures: Data points from the RT300 were utilized to track objective changes between each session of FES cycling. Qualitative data comparing the level of physical assistance required with each functional activity was utilized to track changes between FES and non-FES sessions each week. Discussion: FES cycling and FES assisted activities may provide feasible and well-tolerated physical activity interventions for children with spastic quadriplegic CP. A multi-modal intervention approach with FES and non-FES activities may provide variety and opportunities for motor learning in the pediatric setting