In theoretical computer science, a state transition system is an abstract machine used in the study of computation. The machine consists of a set 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.
State transition systems coincide mathematically with abstract rewriting systems (as explained further in this article). State transition systems differ however from finite state automata in several ways:
- In a state transition system the set of states is not necessarily finite, or even countable.
- In a state transition system the set of transitions is not necessarily finite, or even countable.
State transition systems can be represented as directed graphs.
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Formal definition
Formally, an unlabelled state transition system is a tuple (S, →) where S is a set (of states) and → ⊆ S × S is a binary relation over S (of transitions). If p, q ∈ S, (p, q) ∈ → is usually written as p → q. This represents the fact that there is a transition from state p to state q.
A labelled transition system is a tuple (S, Λ, →) where S is a set (of states), Λ is a set (of labels) and → ⊆ S × Λ × S is a ternary relation (of labelled transitions). If p, q ∈ S and α ∈ Λ, then (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.
Relation between labelled and unlabelled transition systems.
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.
Comparison with abstract rewriting systems
As a mathematical object, an unlabeled state 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 state transition system is equivalent with an abstract rewriting systems with the indices being the labels. The focus of the study and the terminology are different however. In a state transition system one is interested in interpreting the labels as actions, whereas in an ARS the focus is on how objects may be transformed (rewritten) into others.[1]
See also
References
- ^ Marc Bezem, J. W. Klop, Roel de Vrijer ("Terese"), Term rewriting systems, Cambridge University Press, 2003, ISBN 0521391156. p. 7-8
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