Graph rewriting

Terminology

• Normal form
• Confluent
• Terminating

Terminating and confluent system

When a GRS is terminating and confluent, we have the equivalence of the four strategies:

  Onf (S) ≃ Iter(S) ≃ Pick (Iter (S)) ≃ Iter (Pick (S))

They all compute one graph which is the unique normal form. In this case, the Onf strategy should be used as it is the more efficient one.

Terminating system

When a GRS is terminating, we have the equivalence of the two strategies:

  Onf (S) ≃ Pick (Iter (S))

Again, prefer the more efficient Onf.

See below for an example of a non-terminating system where the equivalence does not hold.

Example of non-terminating rewriting system

The following code described a non-terminating rewriting system:

package S {
  rule B2A { pattern { e: N -[B]-> M } commands { del_edge e; add_edge N -[A]-> M } }
  rule B2C { pattern { e: N -[B]-> M } commands { del_edge e; add_edge N -[C]-> M } }
  rule C2B { pattern { e: N -[C]-> M } commands { del_edge e; add_edge N -[B]-> M } }
  rule C2D { pattern { e: N -[C]-> M } commands { del_edge e; add_edge N -[D]-> M } }
}

strat iter { Iter (S) }
strat pick_iter { Pick (Iter (S)) }
strat iter_pick { Iter (Pick (S)) }
strat onf { Onf (S) }

Each rule replaces one edge label with another. For example, the rule B2A removes an edge with a B label and adds one with an A label.

Let G_A, G_B, G_C and G_D be the four graphs with two nodes and one edge labelled A, B, C and D respectively.

The schema below shows how the four rules act on these four graphs:

Applying S to G_B

• The strategy Iter (S) applied to G_B produces the two graphs G_A and G_D (i.e. the two normal forms).
• The strategy Pick (Iter (S)) applied to G_B may produce (unpredictably):
  • the graph G_A
  • the graph G_D
• The strategy Iter (Pick (S)) applied to G_B may produce (unpredictably):
  • the graph G_A
  • the graph G_D
  • the empty set
• The strategy Onf (S) applied to G_B may lead to (unpredictably):
  • the output of the graph G_A
  • the output of the graph G_D
  • a non-terminating execution (in practice Grew tries to detect these cases and raises an error after a given number of rule applications)

In the last three cases, the output is unpredictable, but multiple executions with the same input data will give the same output. However, if the order of the rules in the package S is changed, the behaviour may be different.