State transition system

In theoretical computer science, a transition system is a concept used in the study of computation. It is used to describe the potential behavior of discrete systems. It consists of states and transitions between states, which may be labeled with labels chosen from a set; the same label may appear on more than one transition. If the label set is a singleton, the system is essentially unlabeled, and a simpler definition that omits the labels is possible.

Transition systems coincide mathematically with abstract rewriting systems (as explained further in this article) and directed graphs. They differ from finite state automata in several ways:

Transition systems can be represented as directed graphs.

Formally, a transition system is a pair (S, →) where S is a set of states and → is a set of state transitions (i.e., a subset of S × S). The fact that there is a transition from state p to state q, i.e. (p, q) ∈ →, is written as p → q.

A labelled transition system is a tuple (S, Λ, →) where S is a set of states, Λ is a set of labels and → is a set of labelled transitions (i.e., a subset of S × Λ × S). The fact that (p,α,q) ∈ → is written as

This represents the fact that there is a transition from state p to state q with label α. Labels can represent different things depending on the language of interest. Typical uses of labels include representing input expected, conditions that must be true to trigger the transition, or actions performed during the transition. Labelled transitions systems were originally introduced as named transition systems.

If, for any given p and α, there exists only a single tuple (p,α,q) in →, then one says that α is deterministic (for p). If, for any given p and α, there exists at least one tuple (p,α,q) in →, then one says that α is executable (for p).

There are many relations between these concepts. Some are simple, such as observing that a labelled transition system where the set of labels consists of only one element is equivalent to an unlabelled transition system. However, not all these relations are equally trivial.

...
Wikipedia