By the axiom of foundation, the restriction of the membership relation to the transitive closure of " A " is a well-founded relation.
32.
In Conway's model, " a " is reflexive and transitive, whence its reflexive transitive closure should be " a ".
33.
In terms of the implication graph of the 2-satisfiability instance, Krom's inference rule can be interpreted as constructing the transitive closure of the graph.
34.
Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and path from " x " to y.
35.
A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second-order logic with the addition of a transitive closure operator.
36.
It turns out that first-order logic is lacking in expressive power : it cannot express certain types of Boolean queries, e . g . queries involving transitive closure.
37.
Here \ vdash _ M ^ * represents the reflexive and transitive closure of the step relation \ vdash _ M meaning any number of consecutive steps ( zero, one or more ).
38.
An example of a non-transitive relation with a less meaningful transitive closure is " " x " is the day of the week after " y " ".
39.
One useful reformulation of this problem is that PH = PSPACE if and only if second-order logic over finite structures gains no additional power from the addition of a transitive closure operator.
40.
The transitive closure of a bean is defined in terms of the serialization protocol for the Java programming language, that is, the fields that would be stored by serializing the bean instance.
