Analysis and Application of Transition Systems Based on Petri Nets and Relation Matrices to Business Process Management

Algorithm 1

The generation algorithm of the alignment transition system between the event log and the process model.

	Input: Petri net model N1=(P1, T1; F1, α1, mi,1, mf,1), and relation matrix LRM[S][S − {s0}].
	Output: alignment transition system TS = (S, M, T).
	Initialize: S ⟵ ∅, M ⟵ ∅, T ⟵ ∅, S^start ⟵ {(mi,1, s0)}, S^end ⟵ ∅.
(1)	S ⟵ S^start; n ⟵ 1;
(2)	WHILE (n ≤ |S|) DO
(3)	(mj, sx)⟵S[n];
(4)	IF(mj = mf,1 AND ) THEN
(5)	S^end ⟵ S^end ∪ ;
(6)	END IF
	//judge the current state to be the final state;
(7)	FOR (all sy ∈ ∂set(LRM^row)) DO
(8)	IF (LRM[sx][sy] ≠ #) THEN
(9)	M ⟵ M ∪ {(LRM[sx][sy],>>)};
(10)	S ⟵ S ∪ {(mj, sy)};
(11)	T ⟵ T ∪ {(mj, sx), (LRM[sx][sy], >>), (mj, sy)};
(12)	END IF
(13)	END FOR
	//the following log moves, related new states, and transitions that may be generated;
(14)	IF(mj ≠ mf,1) THEN
(15)	FOR(all tk ∈ T1) DO
(16)	IF( ∈ mj) THEN
(17)	M ⟵ M ∪ {(>>, tk)};
(18)	mj[tk > my;
(19)	S ⟵ S ∪ {(my, sx)};
(20)	T ⟵ T ∪ {(mj, sx), (>>, tk), (my, sx)};
(21)	END IF
(22)	END FOR
(23)	END IF
	//the following model moves, related new states, and transitions that may be generated;
(24)	FOR((all sy ∈ ∂set(LRM^row)) AND (all tk ∈ T1)) DO
(25)	IF ( ∈ mj) AND (LRM[sx][sy] = α(tk)) THEN
(26)	M ⟵ M ∪ {(α(tk), tk)};
(27)	mj [tk > my;
(28)	S ⟵ S ∪ {(my, sy)};
(29)	T ⟵ T ∪ {(mj, sx), (α(tk), tk), (my, sy)};
(30)	END IF
(31)	END FOR
	//the following synchronous moves, related new states, and transitions that may be generated;
(32)	n ⟵ n + 1;
(33)	END WHILE
	//delete the cycles with cost 0 in the transition system;
(34)	FOR (all cycles with cost 0 in TS) Do
(35)	Delete all the edges with cost 0;
(36)	FOR(all nodes in the cycle) DO
(37)	FOR(all nodes have no out edge) DO
(38)	Delete nodes;
(39)	Set the parents of nodes to be nodes;
(40)	END FOR
(41)	END FOR
(42)	END FOR
(43)	RETURN TS;