Well-structured transition system

In computer science, specifically in the field of formal verification, well-structured transition systems (WSTSs) are a general class of infinite state systems for which many verification problems are decidable, owing to the existence of a kind of order between the states of the system which is compatible with the transitions of the system.

The first definition of a general Well-Structured Transition System (WSTS) was introduced by Alain Finkel in his ICALP 1987 paper titled "A Generalization of the Procedure of Karp and Miller to Well Structured Transition Systems".

WSTS decidability results can be applied to Petri nets, lossy channel systems, and more.

is a quasi-ordering (i.e., a preorder or reflexive, transitive binary relation) such that any infinite sequence of elements

contains an increasing pair

is said to be well-quasi-ordered, or shortly wqo.

For our purposes, a transition system is a structure

is any set (its elements are called states), and

(its elements are called transitions).

In general a transition system may have additional structure like initial states, labels on transitions, accepting states, etc.

made up from a finite control state set

, a data values set

, furnished with a decidable pre-order

is monotonic, i.e. upward compatible, with respect to

) and in addition has a computable set of minima for the set of predecessors of any upward closed subset of

Well-structured systems adapt the theory of well-structured transition systems for modelling certain classes of systems encountered in computer science and provide the basis for decision procedures to analyse such systems, hence the supplementary requirements: the definition of a WSTS itself says nothing about the computability of the relations

Coverability can be decided for any well-structured system, and so can reachability of a given control state, by the backward algorithm of Abdulla et al.[1] or for specific subclasses of well-structured systems (subject to strict monotonicity,[2] e.g. in the case of unbounded Petri nets) by a forward analysis based on a Karp-Miller coverability graph.

The backward algorithm allows the following question to be answered: given a well-structured system and a state

, is there any transition path that leads from a given start state

An intuitive explanation for this question is: if

If a well-quasi-order can be found that models this "containment" of states and which also fulfills the requirement of monotonicity with respect to the transition relation, then this question can be answered.

, one typically considers an upward closed set

The algorithm is based on the facts that in a well-quasi-order

converges after finitely many steps (1).

The algorithm needs to store an upward-closed set

of states in memory, which it can do because an upward-closed set is representable as a finite set of minima.

It starts from the upward closure of the set of error states

and computes at each iteration the (by monotonicity also upward-closed) set of immediate predecessors and adding it to the set

This iteration terminates after a finite number of steps, due to the property (1) of well-quasi-orders.

is in the set finally obtained, then the output is "yes" (a state of