How hard is it to verify flat affine counter systems with the finite monoid property ?

We study several decision problems for counter systems with guards defined by convex polyhedra and updates defined by affine transformations. In general, the reachability problem is undecidable for such systems. Decidability can be achieved by imposing two restrictions: (i) the control structure of the counter system is flat, meaning that nested loops are forbidden, and (ii) the set of matrix powers is finite, for any affine update matrix in the system. We provide tight complexity bounds for several decision problems of such systems, by proving that reachability and model checking for Past Linear Temporal Logic are complete for the second level of the polynomial hierarchy $\Sigma^P_2$, while model checking for First Order Logic is PSPACE-complete

A Decidable Fragment of Second Order Logic With Applications to Synthesis

We propose a fragment of many-sorted second order logic called EQSMT and show that checking satisfiability of sentences in this fragment is decidable. EQSMT formulae have an exists^*forall^* quantifier prefix (over variables, functions and relations) making EQSMT conducive for modeling synthesis problems. Moreover, EQSMT allows reasoning using a combination of background theories provided that they have a decidable satisfiability problem for the exists^*forall^* FO-fragment (e.g., linear arithmetic). Our decision procedure reduces the satisfiability of EQSMT formulae to satisfiability queries of exists^*forall^* formulae of each individual background theory, allowing us to use existing efficient SMT solvers supporting exists^*forall^* reasoning for these theories; hence our procedure can be seen as effectively quantified SMT (EQSMT) reasoning

Topological Approximate Dynamic Programming under Temporal Logic Constraints

In this paper, we develop a Topological Approximate Dynamic Programming (TADP) method for planningin stochastic systems modeled as Markov Decision Processesto maximize the probability of satisfying high-level systemspecifications expressed in Linear Temporal Logic (LTL). Ourmethod includes two steps: First, we propose to decompose theplanning problem into a sequence of sub-problems based on thetopological property of the task automaton which is translatedfrom the LTL constraints. Second, we extend a model-freeapproximate dynamic programming method for value iterationto solve, in an order reverse to a causal dependency of valuefunctions, one for each state in the task automaton. Particularly,we show that the complexity of the TADP does not growpolynomially with the size of the product Markov DecisionProcess (MDP). The correctness and efficiency of the algorithmare demonstrated using a robotic motion planning example.Comment: 8 pages, 6 figures. Accepted by 58th Conference on Decision and Contro

Soft Linear Logic and Polynomial Complexity Classes

AbstractWe describe some results inspired to Lafont's Soft Linear Logic (SLL) which is a subsystem of second-order linear logic with restricted rules for exponentials, correct and complete for polynomial time computations. SLL is the basis for the design of type assignment systems for lambda-calculus, characterizing the complexity classes PTIME, PSPACE and NPTIME. PTIME is characterized by a type assignments system where types are a proper subset of SLL formulae. The characterization consists in the fact that a well typed term can be reduced to normal form by a number of beta-reductions polynomial in its lenght, and moreover all polynomial time functions can be computed by well typed terms. PSPACE is characterized by a type assignment system obtained from the previous one, by extending the set of types by a type for booleans, and the lambda-calculus by two boolean constants and a conditional constructor. The system assigns types to terms in such a way that the evaluation of programs (closed terms of type boolean) can be performed carefully in polynomial space. Moreover all polynomial space decision problems can be computed by terms typable in this system. In order to characterize NPTIME we extend the lambda-calculus by a nondeterministic choice operator, and the system by a rule for dealing with this new term constructor

Large-Scale Solution Approaches for Healthcare and Supply Chain Scheduling

This research proposes novel solution techniques for two real world problems. We first consider a patient scheduling problem in a proton therapy facility with deterministic patient arrivals. In order to assess the impacts of several operational constraints, we propose single and multi-criteria linear programming models. In addition, we ensure that the strategic patient mix restrictions predetermined by the decision makers are also enforced within the planning horizon. We study the mathematical structures of the single criteria model with strict patient mix restrictions and derive analytical equations for the optimal solutions under several operational restrictions. These efforts lead to a set of rule of thumbs that can be utilized to assess the impacts of several input parameters and patient mix levels on the capacity utilization without solving optimization problems. The necessary and sufficient conditions to analytically generate exact efficient frontiers of the bicriteria problem without any additional side constraint are also explored. In a follow up study, we investigate the solution techniques for the same patient scheduling problem with stochastic patient arrivals. We propose two Markov Decision Process (MDP) models that are capable of tackling the stochasticity. The second problem of interest is a variant of the parallel machine scheduling problem. We propose constraint programming (CP) and logic-based Benders decomposition algorithms in order to make the best decisions for scheduling nonidentical jobs with time windows and sequence dependent setup times on dissimilar parallel machines in a fixed planning horizon. This problem is formulated with (i) maximizing total profit and (ii) minimizing makespan objectives. We conduct several sensitivity analysis to test the quality and robustness of the solutions on a real life case study

Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.Comment: Revised versio

Real-time and Probabilistic Temporal Logics: An Overview

Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, model checking, etc. for each logic analysed. We also provide a comparison of features of the temporal logics discussed