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 can be represented as directed graphs.
Formally, a transition system is a pair (S, →) where S is a set of states and → is a relation of state transitions (i.e., a subset of S × S). 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 relation of labelled transitions (i.e., a subset of S × Λ × S). (p,α,q) ∈ → is written as
and represents 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).
Therefore a labelled state transition system is a F-coalgebra for the functor .
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.
As a mathematical object, an unlabeled transition system is identical with an (unindexed) abstract rewriting system. If we consider the rewriting relation as an indexed set of relations, as some authors do, then a labeled transition system is equivalent to an abstract rewriting system with the indices being the labels. The focus of the study and the terminology are different however. In a transition system one is interested in interpreting the labels as actions, whereas in an abstract rewriting system the focus is on how objects may be transformed (rewritten) into others.