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Relation.tla
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Relation.tla
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----------------------------- MODULE Relation ------------------------------
EXTENDS Naturals, FiniteSets
(***************************************************************************)
(* This module provides some basic operations on relations, represented *)
(* as binary Boolean functions over some set S. *)
(***************************************************************************)
(***************************************************************************)
(* Is the relation R reflexive over S? *)
(***************************************************************************)
IsReflexive(R, S) == \A x \in S : R[x,x]
(***************************************************************************)
(* Is the relation R irreflexive over set S? *)
(***************************************************************************)
IsIrreflexive(R, S) == \A x \in S : ~ R[x,x]
(***************************************************************************)
(* Is the relation R symmetric over set S? *)
(***************************************************************************)
IsSymmetric(R, S) == \A x,y \in S : R[x,y] <=> R[y,x]
(***************************************************************************)
(* Is the relation R asymmetric over set S? *)
(***************************************************************************)
IsAsymmetric(R, S) == \A x,y \in S : ~(R[x,y] /\ R[y,x])
(***************************************************************************)
(* Is the relation R transitive over set S? *)
(***************************************************************************)
IsTransitive(R, S) == \A x,y,z \in S : R[x,y] /\ R[y,z] => R[x,z]
(***************************************************************************)
(* Compute the transitive closure of relation R over set S. *)
(***************************************************************************)
TransitiveClosure(R, S) ==
LET N == Cardinality(S)
trcl[n \in Nat] ==
[x,y \in S |-> IF n=0 THEN R[x,y]
ELSE \/ trcl[n-1][x,y]
\/ \E z \in S : trcl[n-1][x,z] /\ trcl[n-1][z,y]]
IN trcl[N]
(***************************************************************************)
(* Compute the reflexive transitive closure of relation R over set S. *)
(***************************************************************************)
ReflexiveTransitiveClosure(R, S) ==
LET trcl == TransitiveClosure(R,S)
IN [x,y \in S |-> x=y \/ trcl[x,y]]
(***************************************************************************)
(* Is the relation R connected over set S, i.e. does there exist a path *)
(* between two arbitrary elements of S? *)
(***************************************************************************)
IsConnected(R, S) ==
LET rtrcl == ReflexiveTransitiveClosure(R,S)
IN \A x,y \in S : rtrcl[x,y]
=============================================================================
\* Modification History
\* Last modified Sun Jun 14 15:32:47 CEST 2020 by merz
\* Created Tue Apr 26 10:24:07 CEST 2016 by merz