The dynamics of a language describes how programs are executed. The most important way to define the dynamics of a language is by the method of structural dynamics, which defines a transition system that inductively specifies the step–by–step process of executing a program. Another method for presenting dynamics, called contextual dynamics, is a variation of structural dynamics in which the transition rules are specified in a slightly different way. An equational dynamics presents the dynamics of a language by a collection of rules defining when one program is definitionally equivalent to another.

Transition Systems

A transition system is specified by the following four forms of judgment:

1. s state, asserting that s is a state of the transition system.

2. s final, where s state, asserting that s is a final state.

3. s initial, where s state, asserting that s is an initial state.

4. s ⟼ s', where s state and s' state, asserting that state s may transition to state s'.

In practice, we always arrange things so that no transition is possible from a final state: if s final, then there is no s' state such that s ⟼ s'. A state from which no transition is possible is stuck.Whereas all final states are, by convention, stuck, theremay be stuck states in a transition system that are not final. A transition system is deterministic iff for every state s there exists at most one state s' such that s ⟼ s'; otherwise, it is non-deterministic.

A transition sequence is a sequence of states s0, …, sn such that s0 initial, and si ⟼ si+1 for every 0 ≤ i < n. A transition sequence is maximal iff there is no s such that sn ⟼ s, and it is complete iff it is maximal and sn final. Thus, every complete transition sequence is maximal, but maximal sequences are not necessarily complete. The judgment s ↓ means that there is a complete transition sequence starting from s, which is to say that there exists s' final such that s ⟼*s'.