Well Structured Transition Systems with History

We propose a formal model of concurrent systems in which the history of a computation is explicitly represented as a collection of events that provide a view of a sequence of configurations. In our model events generated by transitions become part of the system configurations leading to operational semantics with historical data. This model allows us to formalize what is usually done in symbolic verification algorithms. Indeed, search algorithms often use meta-information, e.g., names of fired transitions, selected processes, etc., to reconstruct (error) traces from symbolic state exploration. The other interesting point of the proposed model is related to a possible new application of the theory of well-structured transition systems (wsts). In our setting wsts theory can be applied to formally extend the class of properties that can be verified using coverability to take into consideration (ordered and unordered) historical data. This can be done by using different types of representation of collections of events and by combining them with wsts by using closure properties of well-quasi orderings.Comment: In Proceedings GandALF 2015, arXiv:1509.0685

Interprocedural Reachability for Flat Integer Programs

We study programs with integer data, procedure calls and arbitrary call graphs. We show that, whenever the guards and updates are given by octagonal relations, the reachability problem along control flow paths within some language w1* ... wd* over program statements is decidable in Nexptime. To achieve this upper bound, we combine a program transformation into the same class of programs but without procedures, with an Np-completeness result for the reachability problem of procedure-less programs. Besides the program, the expression w1* ... wd* is also mapped onto an expression of a similar form but this time over the transformed program statements. Several arguments involving context-free grammars and their generative process enable us to give tight bounds on the size of the resulting expression. The currently existing gap between Np-hard and Nexptime can be closed to Np-complete when a certain parameter of the analysis is assumed to be constant.Comment: 38 pages, 1 figur

Push-Down Automata with Gap-Order Constraints

International audienceWe consider push-down automata with data (Pdad) that operate on variables ranging over the set of natural numbers. The conditions on variables are defined via gap-order constraint. Gap-order constraints allow to compare variables for equality, or to check that the gap between the values of two variables exceeds a given natural number. The messages inside the stack are equipped with values that are natural numbers reflecting their “values”. When a message is pushed to the stack, its value may be defined by a variable in the program. When a message is popped, its value may be copied to a variable. Thus, we obtain a system that is infinite in two dimensions, namely we have a stack that may contain an unbounded number of messages each of which is equipped with a natural number. We present an algorithm for solving the control state reachability problem for Pdad based on two steps. We first provide a translation to the corresponding problem for context-free grammars with data (Cfgd). Then, we use ideas from the framework of well quasi-orderings in order to obtain an algorithm for solving the reachability problem for Cfgds