_{Statistics for linear cofactor relationships in MCNC benchmarks}

_{Classical symmetries and single variable symmetries are common in Boolean functions. The table below gives the number of non-skew and skew linear cofactor relationships in each of the linear cofactor classes, for all MCNC benchmark circuits.}

Name	Ins	Outs	LCC1		LCC2		LCC3		LCC4
			Non-skew	Skew	Non-skew	Skew	Non-skew	Skew	Non-skew	Skew
9symml	9	1	0	0	36	0	0	0	0	0
alu2	10	6	27	3	22	7	7	5	11	5
alu4	14	8	39	3	42	7	11	5	37	1
apex6	135	99	1725	39	2948	45	194	29	712	13
apex7	49	37	752	114	2701	51	98	47	255	15
b1	3	4	8	2	5	7	2	8	1	3
b9	41	21	219	156	545	17	37	23	14	1
C1355	41	32	0	0	256	0	0	0	0	0
C17	5	2	4	8	10	2	6	2	0	0
C1908	33	25	6	62	2002	158	0	2	1354	2
C2670	233	140	495	524	3338	309	71	126	9607	13
C3540	50	22	21	101	1663	229	8	3	4551	0
C432	36	7	0	17	212	0	0	27	0	0
C499	41	32	0	0	256	0	0	0	0	0
C5315	178	123	2288	413	11685	830	37322	77	37242	48
C7552	207	108	7	3480	9112	3898	44604	157	44595	57
c8	28	18	61	24	457	41	41	24	192	25
cc	21	20	191	5	210	9	33	7	7	3
cht	47	36	342	0	218	0	90	0	39	0
cm138a	6	8	0	360	360	0	0	120	0	0
cm150a	21	1	0	40	84	0	0	0	120	0
cm151a	12	2	22	22	70	0	0	0	56	0
cm152a	11	1	0	0	24	0	0	0	28	0
cm162a	14	5	9	72	139	0	3	4	32	0
cm163a	16	5	30	56	138	0	10	4	32	0
cm42a	4	10	0	180	180	0	0	60	0	0
cm82a	5	3	1	1	15	32	1	1	10	4
cm85a	11	3	26	36	62	4	10	0	4	0
cmb	16	4	396	396	792	0	132	132	0	0
comp	32	3	34	2	68	2	34	2	16	2
cordic	23	2	5	5	115	0	0	0	24	0
count	35	16	0	384	2394	0	0	16	288	0
cu	14	11	396	13	399	3	110	3	6	1
decod	5	16	480	0	480	0	160	0	0	0
des	256	245	2049	286	8400	517	329	164	4463	125
dalu	75	16	32	0	5010	0	0	0	6207	0
example2	85	66	3193	101	3189	77	578	61	316	35
f51m	8	8	4	4	9	76	4	4	28	1
frg1	28	3	7	21	108	7	1	7	43	1
frg2	143	139	6170	487	11461	95	291	76	1290	18
i1	25	16	111	43	371	32	36	17	3	4
i8	133	81	85	81	3690	81	7	3	2524	3
i9	88	63	0	0	770	0	0	0	483	0
k2	45	45	1681	0	4750	0	325	0	3348	0
lal	26	19	132	309	632	16	32	73	40	4
majority	5	1	0	8	10	0	0	0	0	0
mux	21	1	40	0	84	0	0	0	120	0
my_adder	33	17	1	1	204	1110	1	1	528	18
pair	173	137	3196	1586	15906	43	226	291	3249	10
parity	16	1	0	0	240	480	0	0	120	0
pcle	19	9	207	8	495	8	97	8	80	8
pcler8	27	17	229	16	994	32	103	32	88	16
pm1	16	13	176	147	340	3	32	31	1	1
rot	135	107	1658	355	5756	192	141	126	433	16
sct	19	15	96	135	373	10	22	47	40	4
t	5	2	4	0	10	0	6	0	0	0
t481	16	1	0	8	24	2	0	2	80	0
tcon	17	16	24	24	16	32	24	24	8	16
term1	34	10	589	117	580	2	39	11	80	4
too_large	38	3	234	0	502	0	0	0	2	0
ttt2	24	21	303	2	649	12	26	21	47	10
unreg	36	16	48	16	160	16	48	16	80	16
vda	17	39	471	0	1629	0	119	0	409	0
x1	17	39	872	497	1996	13	98	64	90	2
x2	17	39	471	109	1629	2	119	24	409	2
x3	17	39	471	39	1629	45	119	29	409	13
x4	17	39	471	86	1629	41	119	46	409	15
z4ml	7	4	1	1	22	57	1	1	21	5
Total			30610	11005	114305	8652	85927	2063	124681	540
Ratio			8.1%	2.9%	30.3%	2.3%	22.8%	0.6%	33%	0.1%

_{The following conclusions are evident from the table:}

_{· Linear cofactor relationships exist in all MCNC benchmarks.}

_{· 30% of the LCRs are classical and single variable symmetries.}

_{· More than 90% of the LCRs  are non-skew.}