Rodney R. Howell

Algorithms and complexity concerning the preemptive scheduling of periodic, real-time tasks on one processor

We investigate the preemptive scheduling of periodic, real-time task systems on one processor. First, we show that when all parameters to the system are integers, we may assume without loss of generality that all preemptions occur at integer time values. We then assume, for the remainder of the paper, that all parameters are indeed integers. We then give, as our main lemma, both necessary and sufficient conditions for a task system to be feasible on one processor. Although these conditions cannot, in general, be tested efficiently (unless P=NP), they do allow us to give efficient algorithms for deciding feasibility on one processor for certain types of periodic task systems. For example, we give a pseudo-polynomial-time algorithm for synchronous systems whose densities are bounded by a fixed constant less than 1. This algorithm represents an exponential improvement over the previous best algorithm. We also give a polynomial-time algorithm for systems having a fixed number of distinct types of tasks. Furthermore, we are able to use our main lemma to show that the feasibility problem for task systems on one processor is co-NP-complete in the strong sence. In order to show this last result, we first show the Simultaneous Congruences Problem to be NP-complete in the strong sense. Both of these last two results answer questions that have been open for ten years. We conclude by showing that for incomplete task systems, that is, task systems in which the start times are not specified, the feasibility problem is ∑2p-complete.

Feasibility problems for recurring tasks on one processor

Abstract We give a comprehensive summary of our recent research on the feasibility problems for various types of hard-real-time preemptive task systems on one processor. We include results on periodic, sporadic, and hybrid task systems. While many of the results herein have appeared elsewhere, this is the first paper presenting a holistic view of the entire problem.

Relationships between glycogen storage disease and tophaceous gout

Abstract Two patients with glucose-6-phosphatase deficiency glycogen storage disease and tophaceous gout have been studied in an effort to establish the possible relationships between the primary defect in carbohydrate metabolism and their hyperuricemia and gouty arthritis. The present studies support the view that lacticacidemia is a contributing factor to the hyperuricemia of these patients. However, hypoglycemia and ketonemia probably also contribute to the urate retention. The finding of an increased rate of urate biosynthesis in the one patient whose de novo purine biosynthesis was greater than normal points to an additional mechanism for the hyperuricemia found in this type of glycogen storage disease.

Echocardiographic abnormalities in the mucopolysaccharide storage diseases.

The mucopolysaccharide storage diseases express themselves clinically with a wide variety of abnormalities, including growth and mental retardation, skeletal abnormalities, clouded corneas, nerve compression syndromes, upper airway obstruction and cardiovascular involvement, to name the most common. In most cases the cause of early death is cardiorespiratory failure secondary to cardiovascular involvement and upper airway obstruction. The findings of cardiac ultrasound examination in 29 children, adolescents and young adults are presented. In addition to the previously well-described abnormalities of the mitral and aortic valves in several types of mucopolysaccharide storage disease, we report patchy involvement in some cases, 3 instances of asymmetric septal hypertrophy not previously reported in mucopolysaccharide storage diseases, cardiac involvement in half of our patients with Sanfilippo syndrome and a lack of age-related severity of cardiac involvement even within the specific syndromes.

A taxonomy of fairness and temporal logic problems for Petri nets

In this paper, we define a temporal logic for reasoning about Petri nets. We show the model checking problem for this logic to be PTIME equivalent to the Petri net reachability problem. Using this logic and two refinements, we show the fair nontermination problem to be PTIME equivalent to reachability for several definitions of fairness. For other versions of fairness, this problem is shown to be either PTIME equivalent to the boundedness problem or highly undecidable. In all, 24 versions of fairness are examined.

On Non-preemptive Scheduling of Recurring Tasks Using Inserted Idle Times

We consider the problem of non-preemptively scheduling periodic and sporadic task systems on one processor using inserted idle times. For periodic task systems, we prove that the decision problem of determining whether a periodic task system is schedulable for all start times with respect to the class of algorithms using inserted idle times is NP-hard in the strong sense, even when the deadlines are equal to the periods. We then show that if there exists a polynomial time scheduling algorithm which correctly schedules a periodic task system T whenever T is feasible for all start times, then P = NP. We also prove that with respect to the same class of algorithms, the problem of determining whether there exist start times for which a periodic task system is feasible is also NP-hard in the strong sense even when the deadlines are equal to the periods. The second part of the paper concentrates on sporadic task systems and inserted idle times. It seems reasonable to suppose that to insert idle times properly, knowledge of future releases of tasks is required. Thus, inserted idle times should not be expected to have much use in scheduling sporadic task systems. We provide a formal basis for these intuitions by proving that if a sporadic task system is schedulable by an online algorithm that uses inserted idle times, then it is schedulable by an online algorithm that does not use inserted idle times. We also prove that there cannot exist an optimal on-line inserted idle time algorithm for scheduling sporadic task systems, even if the deadlines correspond to the minimum separation time between successive releases of the same task. We conclude by considering the amount of look-ahead needed to schedule sporadic tasks correctly.

Some complexity bounds for problems concerning finite and 2-dimensional vector addition systems with states

Abstract In this paper, we analyse the complexity of the reachability, containment, and equivalence problems for two classes of vector addition systems with states (VASSs): finite VASSs and 2-dimensional VASSs. Both of these classes are known to have effectively computable semilinear reachibility sets (SLSs). By giving upper bounds on the sizes of the SLS representations, we achieve upper bounds on each of the aforementioned problems. In the case of the finite VASSs, the SLS representation is simply a listing of the reachability set; therefore, we derive a bound on the norm of any reachable vector based on the dimension, number of states, and amount of increment caused by any move in the VASS. The bound we derive shows an improvement of two levels in the primitive recursive hierarchy over results previously obtained by McAloon (1984), thus answering a question posed by Clote (1986). We then show this bound to be optimal. We feel that the techniques we use in deriving our upper bounds represent an original approach to the problem, and since they yield improvements over previous results, we feel these techniques may have applications to other problems. In the case of 2-dimensional VASSs, we analyse an algorithm given by Hopcroft and Pansiot (1979) that generates an SLS representation of the reachability set. Specifically, we show that the algorithm operates in 2 2 cln nondeterministic time, where l is the length of the binary representation of the largest integer in the VASS, n is the number of transitions, and c is some fixed constant. We also give examples for which this algorithm will take 2 2 dln nondeterministic time for some positive constant d . Finally, we give a method of determinizing the algorithm in such a way that it requires no more than 2 2 cln deterministic time. From this upper bound and special properties of the generated SLSs, we derive upper bounds of Dtime (2 2 cln ) for the three problems mentioned above.

Problems concerning fairness and temporal logic for conflict-free Petri nets

Abstract In this paper, we examine the complexity of the fair nontermination problem for conflict-free Petri nets under several definitions of fairness. For each definition of fairness, we are able to show the problem to be complete for either NP, PTIME, or NLOGSPACE. We then address the question of whether these results extend to the more general model checking problem with respect to the temporal logic for Petri nets introduced by Suzuki. Since many of the model checking problems concerning finite state systems can be reduced to a version of the fair nontermination problem, it would seem plausible that the model checking problem for conflict-free Petri nets would be decidable. However, it turns out that unless the logic is severely restricted, model checking is undecidable for conflict-free Petri nets. In particular, the problem is undecidable even when formulas are of the form Gƒ (“invariantly ƒ”) where ƒ contains no temporal logic operators. On the other hand, we show that model checking for conflict-free Petri nets is NP-complete for L(F, X)—the logic restricted to the operators F (eventually), X (next time), ∧, and ∨, with negations allowed only on the predicates.

Completeness results for conflict-free vector replacement systems

Abstract In this paper, we give completeness results for the reachability, containment, and equivalence problems for conflict-free vector replacement systems (VRSs). We first give an NP algorithm for deciding reachability, thus giving the first primitive recursive algorithm for this problem. Since Jones, Landweber, and Lien have shown this problem to be NP-hard, it follows that the problem is NP-complete. Next, we show as our main result that the containment and equivalence problems are ∏ 2 P -complete, where ∏ 2 P is the set of all languages whose complements are in the second level of the polynomial-time hierarchy. In showing the upper bound, we first show that the reachability set has a semilinear set (SLS) representation that is exponential in the size of the problem description, but which has a high degree of symmetry. We are then able to utilize in part a strategy introduced by Huynh (concerning SLSs) to complete our upper bound proof.

Familial recurrent rhabdomyolysis due to carnitine palmityl transferase deficiency

Muscle carnitine palmityltransferase (CPT) activity was very low (0 to 14 per cent of controls) in two brothers with a syndrome of recurrent rhabdomyolysis and myoglobulinuria. In isolated muscle mitochondria the majority (87.5 per cent) of total measurable CPT enzyme activity could be attributed to external membrane CPT with severe deficiency of inner membrane CPT. By contrast, control mitochondria demonstrated a 1:1 distribution of external membrane CPT to inner membrane CPT. Thus, myoglobinuria may be due to a genetic defect of lipid metabolism in skeletal muscle, with inner membrane CPT deficiency presenting the same clinical features as external membrane CPT deficiency.