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[ | In [[theoretical computer science]] a '''simulation preorder''' is a [[Relation (mathematics)|relation]] between [[state transition system]]s associating systems which behave in the same way in the sense that one system ''simulates'' the other. | ||
Intuitively, a system simulates another system if it can match all of its moves. | |||
The basic definition relates states within one transition system, but this is easily adapted to relate two separate transition systems by building a system consisting of the disjoint union of the corresponding components. | |||
==Formal definition== | |||
Given a [[state transition system|labelled state transition system]] (S, Λ, →), a ''simulation'' relation is a [[binary relation]] R over S (i.e. R ⊆ S × S) such that for every pair of elements p, q ∈ S, if (p,q)∈ R then for all α ∈ Λ, and for all p' ∈ S, | |||
:<math> p \overset{\alpha}{\rightarrow} p' </math> | |||
implies that there is a q' ∈ S such that | |||
:<math> q \overset{\alpha}{\rightarrow} q' </math> | |||
and (p',q') ∈ R. | |||
Equivalently, in terms of relational composition: | |||
:<math>R^{-1}\ ;\ \overset{\alpha}{\rightarrow}\quad {\subseteq}\quad \overset{\alpha}{\rightarrow}\ ;\ R^{-1}</math> | |||
Given two states p and q in S, q ''simulates'' p, written p ≤ q if there is a simulation R such that (p, q) ∈ R. The relation ≤ is a [[preorder]], and is usually called the ''simulation preorder''. It is the largest simulation relation over a given transition system. | |||
Two states ''p'' and ''q'' are said to be ''similar'', written p ≤≥ q, if ''p'' simulates ''q'' and ''q'' simulates ''p''. Similarity is an [[equivalence relation]], but it is coarser than [[bisimilarity]]. | |||
==Similarity of separate transition systems== | |||
When comparing two different transition systems (S', Λ', →') and (S' ', Λ' ', →' '), the basic notions of simulation and similarity can be used by forming the disjoint composition of the two machines, (S, Λ, →) with S = S' ∐ S' ', Λ = Λ' ∪ Λ' ' and → = →' ∪ →' ', where ∐ is the [[disjoint union]] operator between sets. | |||
==See also== | |||
* [[State transition system]]s | |||
* [[Bisimulation]] | |||
* [[Coinduction]] | |||
* [[Operational semantics]] | |||
== References == | |||
# {{Cite conference | |||
| first = David | |||
| last = Park | |||
| year = 1981 | |||
| title = Concurrency and Automata on Infinite Sequences | |||
| booktitle = Proceedings of the 5th GI-Conference, Karlsruhe | |||
| series = [[Lecture Notes in Computer Science]] | |||
| editor = Deussen, Peter | |||
| pages = 167–183 | |||
| volume = 104 | |||
| publisher = [[Springer-Verlag]] | |||
| isbn = 978-3-540-10576-3 | |||
| doi = 10.1007/BFb0017309 | |||
}} | |||
# {{Cite book | |||
| last = Milner | |||
| first = Robin | |||
| title = Communication and Concurrency | |||
| year = 1989 | |||
| publisher = [[Prentice Hall]] | |||
| isbn = 0-13-114984-9 | |||
}} | |||
# {{Cite conference | |||
| last = van Glabbeek | |||
| first = R. J. | |||
| title = The Linear Time – Branching Time Spectrum I: The Semantics of Concrete, Sequential Processes | |||
| booktitle = Handbook of Process Algebra | |||
| url = http://boole.stanford.edu/pub/spectrum1.pdf.gz | |||
| chapter = 1 | |||
| pages = 3–99 | |||
| year = 2001 | |||
| publisher = Elsevier | |||
}} | |||
{{DEFAULTSORT:Simulation Preorder}} | |||
[[Category:Theoretical computer science]] | |||
Revision as of 11:20, 10 January 2014
In theoretical computer science a simulation preorder is a relation between state transition systems associating systems which behave in the same way in the sense that one system simulates the other.
Intuitively, a system simulates another system if it can match all of its moves.
The basic definition relates states within one transition system, but this is easily adapted to relate two separate transition systems by building a system consisting of the disjoint union of the corresponding components.
Formal definition
Given a labelled state transition system (S, Λ, →), a simulation relation is a binary relation R over S (i.e. R ⊆ S × S) such that for every pair of elements p, q ∈ S, if (p,q)∈ R then for all α ∈ Λ, and for all p' ∈ S,
implies that there is a q' ∈ S such that
and (p',q') ∈ R.
Equivalently, in terms of relational composition:
Given two states p and q in S, q simulates p, written p ≤ q if there is a simulation R such that (p, q) ∈ R. The relation ≤ is a preorder, and is usually called the simulation preorder. It is the largest simulation relation over a given transition system.
Two states p and q are said to be similar, written p ≤≥ q, if p simulates q and q simulates p. Similarity is an equivalence relation, but it is coarser than bisimilarity.
Similarity of separate transition systems
When comparing two different transition systems (S', Λ', →') and (S' ', Λ' ', →' '), the basic notions of simulation and similarity can be used by forming the disjoint composition of the two machines, (S, Λ, →) with S = S' ∐ S' ', Λ = Λ' ∪ Λ' ' and → = →' ∪ →' ', where ∐ is the disjoint union operator between sets.
See also
References
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