improve arithmetics efficiency of expression

[hero78119]

Related to
gpu PR: https://github.com/scroll-tech/ceno-gpu/pull/100/
gkr-backend PR scroll-tech/gkr-backend#17

change scope

• add new prove_generic_sumcheck_gpu_v2 and prove with dag expression
• add dag related parameter in prove_generic_sumcheck_gpu and defer flow selection in gpu impl

design rationale

setup dag-expression along with monomial term in vk, so prover can choose optimal strategy accordingly

[hero78119]

circuit stats on dag

Layer Name	num_add	num_mul	max_degree	max_dag_depth	num_scalar
ADD_main	173	87	3	8	44
SUB_main	187	93	3	8	47
AND_main	182	95	2	8	47
OR_main	183	95	2	8	46
XOR_main	182	95	2	8	47
SLL_main	481	405	4	9	107
SRL_main	520	463	4	9	109
SRA_main	528	467	4	9	111
SLT_main	275	232	4	8	59
SLTU_main	271	232	4	8	59
MUL_main	173	87	3	11	47
MULH_main	256	143	3	13	59
MULHSU_main	260	143	3	13	61
MULHU_main	247	137	3	13	59
DIVU_main	576	603	4	13	96
REMU_main	573	603	4	13	96
DIV_main	586	610	4	13	98
REM_main	586	610	4	13	98
ADDI_main	137	71	3	8	41
ANDI_main	145	75	2	8	42
ORI_main	143	75	2	8	41
XORI_main	143	75	2	8	42
SLLI_main	441	382	4	9	102
SRLI_main	479	440	4	9	103
SRAI_main	483	444	4	9	106
SLTI_main	235	214	4	8	57
SLTIU_main	234	214	4	8	57
LUI_main	123	58	2	8	47
AUIPC_main	323	287	3	10	79
BEQ_main	133	84	3	9	40
BNE_main	135	85	3	9	39
BLT_main	243	221	4	9	55
BLTU_main	243	221	4	9	55
BGE_main	248	222	4	9	54
BGEU_main	248	222	4	9	54
JAL_main	95	48	2	8	41
JALR_main	227	176	3	9	67
LW_main	235	168	3	8	69
LHU_main	247	182	3	8	71
LH_main	261	192	3	8	74
LBU_main	276	204	3	9	79
LB_main	292	215	3	9	83
SW_main	244	172	3	8	70
SH_main	253	187	3	8	71
SB_main	306	228	3	9	83
ECALL_HALT_main	86	38	2	8	43
Ecall_Keccak	13178	6758	3	8	2300
weierstrass_add	37156	24177	3	9	1680
weierstrass_double	48042	30842	3	9	1908
weierstrass_add	37170	24177	3	9	1654
weierstrass_double	48036	30842	3	9	1882
weierstrass_decompress	35430	26258	4	9	1357
DYNAMIC_RANGE_18	8	6	2	7	7
DOUBLE_RANGE_DoubleU8	8	6	2	7	8
And_OPS_ROM_TABLE	11	7	2	7	9
Or_OPS_ROM_TABLE	11	7	2	7	9
Xor_OPS_ROM_TABLE	11	7	2	7	9
Ltu_OPS_ROM_TABLE	11	7	2	7	9
RAM_Register_RegTable	10	5	2	7	9
RAM_Memory_StaticMemTable	10	5	2	7	9
RAM_Memory_PubIOTable	8	5	2	7	9
HintsTable_Memory_RAM	10	5	2	7	9
StackTable_Memory_RAM	8	3	2	7	9
HeapTable_Memory_RAM	8	3	2	7	9
LocalRAMTableFinal	13	7	2	7	9
ShardRamCircuit_main	260518	502742	4	8	98805
ECALL_DUMMY_main	190	90	2	8	51
SECP256K1_DECOMPRESS_main	920	435	2	8	157
SHA256_EXTEND_main	3256	1539	2	8	493
BN254_FP_ADD_main	917	435	2	8	157
BN254_FP_MUL_main	924	435	2	8	157
BN254_FP2_ADD_main	1700	803	2	8	269
BN254_FP2_MUL_main	1698	803	2	8	269
PROGRAM	19	11	2	7	13

[hero78119]

cache with append only index version

some issue:

1. this routine will panic on ShardRamCircuit for super lengthy expression

Layer Name	num_add	num_mul	max_degree	max_dag_depth	num_scalar
ADD_main	153	74	3	294	44
SUB_main	160	78	3	308	47
AND_main	167	83	2	326	47
OR_main	167	82	2	324	46
XOR_main	168	82	2	326	47
SLL_main	369	306	4	829	107
SRL_main	381	334	4	871	109
SRA_main	389	338	4	885	111
SLT_main	223	154	4	463	59
SLTU_main	222	154	4	462	59
MUL_main	157	76	3	303	47
MULH_main	214	111	3	411	59
MULHSU_main	215	110	3	413	61
MULHU_main	206	104	3	396	59
DIVU_main	397	356	4	889	96
REMU_main	399	356	4	891	96
DIV_main	407	363	4	908	98
REM_main	408	363	4	909	98
ADDI_main	122	61	3	243	41
ANDI_main	136	68	2	271	42
ORI_main	134	68	2	268	41
XORI_main	134	68	2	269	42
SLLI_main	330	288	4	760	102
SRLI_main	345	317	4	805	103
SRAI_main	359	321	4	826	106
SLTI_main	189	140	4	409	57
SLTIU_main	192	140	4	412	57
LUI_main	119	55	2	238	47
AUIPC_main	259	210	3	570	79
BEQ_main	118	73	3	250	40
BNE_main	119	74	3	251	39
BLT_main	191	143	4	412	55
BLTU_main	190	144	4	412	55
BGE_main	194	144	4	415	54
BGEU_main	196	144	4	417	54
JAL_main	94	46	2	195	41
JALR_main	190	137	3	417	67
LW_main	197	130	3	420	69
LHU_main	207	142	3	446	71
LH_main	220	152	3	473	74
LBU_main	230	159	3	497	79
LB_main	244	169	3	526	83
SW_main	206	133	3	433	70
SH_main	212	146	3	454	71
SB_main	257	179	3	549	83
ECALL_HALT_main	83	36	2	169	41
Ecall_Keccak	12589	6318	3	22907	2300
weierstrass_add	32167	20058	3	55841	1680
weierstrass_double	39587	24169	3	67857	1908
weierstrass_add	22728	12577	3	38895	1654
weierstrass_double	28158	15020	3	47253	1882
weierstrass_decompress	21324	13819	4	37878	1357
DYNAMIC_RANGE_18	8	6	2	25	7
DOUBLE_RANGE_DoubleU8	8	6	2	26	8
And_OPS_ROM_TABLE	11	7	2	32	9
Or_OPS_ROM_TABLE	11	7	2	32	9
Xor_OPS_ROM_TABLE	11	7	2	32	9
Ltu_OPS_ROM_TABLE	11	7	2	32	9
RAM_Register_RegTable	10	5	2	28	9
RAM_Memory_StaticMemTable	10	5	2	28	9
RAM_Memory_PubIOTable	8	5	2	26	9
HintsTable_Memory_RAM	10	5	2	28	9
StackTable_Memory_RAM	8	3	2	22	9
HeapTable_Memory_RAM	8	3	2	22	9
LocalRAMTableFinal	13	7	2	35	9

[hero78119]

Removed zero/one expression

Layer Name	num_add_orig	num_add_opt	Δ add (%)	num_mul_orig	num_mul_opt	Δ mul (%)
ADD_main	173	127	−26.6%	87	80	−8.0%
SUB_main	187	140	−25.1%	93	86	−7.5%
AND_main	182	139	−23.6%	95	92	−3.2%
OR_main	183	139	−24.0%	95	92	−3.2%
XOR_main	182	139	−23.6%	95	92	−3.2%
SLL_main	481	278	−42.2%	405	304	−24.9%
SRL_main	520	292	−43.8%	463	326	−29.6%
SRA_main	528	299	−43.3%	467	329	−29.6%
SLT_main	275	175	−36.4%	232	172	−25.9%
SLTU_main	271	175	−35.4%	232	172	−25.9%
MUL_main	173	126	−27.2%	87	84	−3.4%
MULH_main	256	177	−30.9%	143	138	−3.5%
MULHSU_main	260	179	−31.2%	143	138	−3.5%
MULHU_main	247	171	−30.8%	137	132	−3.6%
DIVU_main	576	335	−41.8%	603	440	−27.0%
REMU_main	573	335	−41.5%	603	440	−27.0%
DIV_main	586	343	−41.5%	610	447	−26.7%
REM_main	586	343	−41.5%	610	447	−26.7%
ADDI_main	137	98	−28.5%	71	65	−8.5%
ANDI_main	145	108	−25.5%	75	73	−2.7%
ORI_main	143	108	−24.5%	75	73	−2.7%
XORI_main	143	108	−24.5%	75	73	−2.7%
SLLI_main	441	245	−44.5%	382	282	−26.2%
SRLI_main	479	259	−45.9%	440	304	−30.9%
SRAI_main	483	266	−44.9%	444	307	−30.8%
SLTI_main	235	146	−37.9%	214	155	−27.6%
SLTIU_main	234	146	−37.6%	214	155	−27.6%
LUI_main	123	94	−23.6%	58	55	−5.2%
AUIPC_main	323	203	−37.1%	287	221	−23.0%
BEQ_main	133	90	−32.3%	84	70	−16.7%
BNE_main	135	90	−33.3%	85	70	−17.6%
BLT_main	243	150	−38.3%	221	160	−27.6%
BLTU_main	243	150	−38.3%	221	160	−27.6%
BGE_main	248	150	−39.5%	222	160	−27.9%
BGEU_main	248	150	−39.5%	222	160	−27.9%
JAL_main	95	73	−23.2%	48	46	−4.2%
JALR_main	227	145	−36.1%	176	139	−21.0%
LW_main	235	152	−35.3%	168	138	−17.9%
LHU_main	247	160	−35.2%	182	149	−18.1%
LH_main	261	168	−35.6%	192	157	−18.2%
LBU_main	276	178	−35.5%	204	166	−18.6%
LB_main	292	187	−36.0%	215	175	−18.6%
SW_main	244	160	−34.4%	172	142	−17.4%
SH_main	253	168	−33.6%	187	155	−17.1%
SB_main	306	201	−34.3%	228	186	−18.4%
ECALL_HALT_main	86	71	−17.4%	38	36	−5.3%
Ecall_Keccak	13178	9297	−29.4%	6758	6610	−2.2%
weierstrass_add	37156	21156	−43.1%	24177	19114	−20.9%
weierstrass_double	48042	24761	−48.4%	30842	23158	−24.9%
weierstrass_decompress	35430	18295	−48.4%	26258	19592	−25.3%
ShardRamCircuit_main	260518	130248	−50.0%	502742	372609	−25.9%
ECALL_DUMMY_main	190	141	−25.8%	90	86	−4.4%
SECP256K1_DECOMPRESS_main	920	697	−24.2%	435	416	−4.4%
SHA256_EXTEND_main	3256	2473	−24.1%	1539	1472	−4.4%
BN254_FP_ADD_main	917	697	−24.0%	435	416	−4.4%
BN254_FP_MUL_main	924	697	−24.6%	435	416	−4.4%
BN254_FP2_ADD_main	1700	1289	−24.2%	803	768	−4.4%
BN254_FP2_MUL_main	1698	1289	−24.1%	803	768	−4.4%
PROGRAM	19	11	−42.1%	11	11	0.0%

[hero78119]

Monomial vs Dag add/mul

Layer Name	DAG num_add	Monomial num_add	DAG num_mul	Monomial num_mul	DAG max_degree	DAG max_dag_depth	DAG num_scalar
ADD_main	127	24	80	51	3	7	43
SUB_main	140	24	86	51	3	7	46
AND_main	139	28	92	57	2	7	46
OR_main	139	28	92	57	2	7	45
XOR_main	139	28	92	57	2	7	46
SLL_main	278	107	304	286	4	7	106
SRL_main	292	116	326	317	4	7	108
SRA_main	299	116	329	317	4	7	110
SLT_main	175	56	172	149	4	7	58
SLTU_main	175	56	172	149	4	7	58
MUL_main	126	25	84	54	3	8	46
MULH_main	177	36	138	83	3	10	58
MULHSU_main	179	36	138	83	3	10	60
MULHU_main	171	36	132	83	3	10	58
DIVU_main	335	150	440	450	4	10	95
REMU_main	335	150	440	450	4	10	95
DIV_main	343	150	447	450	4	10	97
REM_main	343	150	447	450	4	10	97
ADDI_main	98	20	65	43	3	7	40
ANDI_main	108	24	73	49	2	7	41
ORI_main	108	24	73	49	2	7	40
XORI_main	108	24	73	49	2	7	41
SLLI_main	245	100	282	272	4	7	101
SRLI_main	259	109	304	303	4	7	102
SRAI_main	266	109	307	303	4	7	105
SLTI_main	146	52	155	141	4	7	56
SLTIU_main	146	52	155	141	4	7	56
LUI_main	94	16	55	33	2	7	46
AUIPC_main	203	68	221	184	3	9	78
BEQ_main	90	27	70	64	3	8	39
BNE_main	90	27	70	64	3	7	38
BLT_main	150	53	160	144	4	8	54
BLTU_main	150	53	160	144	4	8	54
BGE_main	150	53	160	144	4	7	53
BGEU_main	150	53	160	144	4	7	53
JAL_main	73	13	46	27	2	7	40
JALR_main	145	45	139	114	3	7	66
LW_main	152	44	138	110	3	7	68
LHU_main	160	49	149	123	3	7	70
LH_main	168	51	157	128	3	7	73
LBU_main	178	55	166	138	3	7	78
LB_main	187	57	175	143	3	7	82
SW_main	160	44	142	110	3	7	69
SH_main	168	49	155	124	3	7	70
SB_main	201	58	186	146	3	7	82
ECALL_HALT_main	71	8	36	17	2	7	41
Ecall_Keccak	9297	2784	6610	5595	3	8	2298
weierstrass_add	21156	4511	19114	11599	3	8	1679
weierstrass_double	24761	5296	23158	13697	3	8	1907
weierstrass_decompress	18295	5331	19592	14681	4	8	1356
DYNAMIC_RANGE_18	5	3	6	7	2	6	6
HeapTable_Memory_RAM	6	1	3	3	2	5	8
LocalRAMTableFinal	6	5	6	11	2	5	7
ShardRamCircuit_main	6003	956	11780	2778	4	8	1724
ECALL_DUMMY_main	141	32	86	65	2	7	50
SECP256K1_DECOMPRESS_main	697	153	416	307	2	7	156
SHA256_EXTEND_main	2473	537	1472	1075	2	7	492
BN254_FP_ADD_main	697	153	416	307	2	7	156
BN254_FP_MUL_main	697	153	416	307	2	7	156
BN254_FP2_ADD_main	1289	281	768	563	2	7	268
BN254_FP2_MUL_main	1289	281	768	563	2	7	268
PROGRAM	11	8	11	17	2	6	12

[hero78119]

Example

StackTable_Memory_RAM zero_expr 0 + WitIn(1) * (0 + Challenge(2) * (2 + WitIn(0) * Challenge(1) + 0*Challenge(1)^2 + 0*Challenge(1)^3 + 0*Challenge(1)^4 + Challenge(0) + (-1)))

StackTable_Memory_RAM monomial term [Term { scalar: (2*Challenge(2) + 1 * Challenge(2) * 0*Challenge(1)^2 + 1 * Challenge(2) * 0*Challenge(1)^3 + 1 * Challenge(2) * 0*Challenge(1)^4 + 1 * Challenge(2) * Challenge(0) + -1*Challenge(2)), product: [W[1]] }, Term { scalar: (0 + 1 * Challenge(2) * Challenge(1)), product: [W[0], W[1]] }]

[hero78119]

Build dag from monomial term with common-subtree-extraction

Layer Name	Monomial Add	DAG Add	Monomial Mul	DAG Mul	Mul Reduction (%)
ADD_main	24	24	51	25	50.98% ↓
SUB_main	24	24	51	25	50.98% ↓
AND_main	28	28	57	29	49.12% ↓
OR_main	28	28	57	29	49.12% ↓
XOR_main	28	28	57	29	49.12% ↓
SLL_main	107	107	286	108	62.24% ↓
SRL_main	116	116	317	117	63.09% ↓
SRA_main	116	116	317	117	63.09% ↓
SLT_main	56	56	149	57	61.74% ↓
SLTU_main	56	56	149	57	61.74% ↓
MUL_main	25	25	54	26	51.85% ↓
MULH_main	36	36	83	37	55.42% ↓
MULHSU_main	36	36	83	37	55.42% ↓
MULHU_main	36	36	83	37	55.42% ↓
DIVU_main	150	150	450	151	66.44% ↓
REMU_main	150	150	450	151	66.44% ↓
DIV_main	150	150	450	151	66.44% ↓
REM_main	150	150	450	151	66.44% ↓
ADDI_main	20	20	43	21	51.16% ↓
ANDI_main	24	24	49	25	48.98% ↓
ORI_main	24	24	49	25	48.98% ↓
XORI_main	24	24	49	25	48.98% ↓
SLLI_main	100	100	272	101	62.87% ↓
SRLI_main	109	109	303	110	63.69% ↓
SRAI_main	109	109	303	110	63.69% ↓
SLTI_main	52	52	141	53	62.41% ↓
SLTIU_main	52	52	141	53	62.41% ↓
LUI_main	16	16	33	17	48.48% ↓
AUIPC_main	68	68	184	69	62.50% ↓
BEQ_main	27	27	64	28	56.25% ↓
BNE_main	27	27	64	28	56.25% ↓
BLT_main	53	53	144	54	62.50% ↓
BLTU_main	53	53	144	54	62.50% ↓
BGE_main	53	53	144	54	62.50% ↓
BGEU_main	53	53	144	54	62.50% ↓
JAL_main	13	13	27	14	48.15% ↓
JALR_main	45	45	114	46	59.65% ↓
LW_main	44	44	110	45	59.09% ↓
LHU_main	49	49	123	50	59.35% ↓
LH_main	51	51	128	52	59.38% ↓
LBU_main	55	55	138	56	59.42% ↓
LB_main	57	57	143	58	59.44% ↓
SW_main	44	44	110	45	59.09% ↓
SH_main	49	49	124	50	59.68% ↓
SB_main	58	58	146	59	59.59% ↓
ECALL_HALT_main	8	8	17	9	47.06% ↓
Ecall_Keccak	2784	2784	5595	2788	50.17% ↓
weierstrass_add	4511	4511	11599	4512	61.10% ↓
weierstrass_double	5296	5296	13697	5297	61.33% ↓
weierstrass_decompress	5331	5331	14681	5332	63.68% ↓
DYNAMIC_RANGE_18	3	3	7	4	42.86% ↓
And_OPS_ROM_TABLE	4	4	9	5	44.44% ↓
StackTable_Memory_RAM	1	1	3	2	33.33% ↓
ShardRamCircuit_main	956	956	2778	970	65.09% ↓
ECALL_DUMMY_main	32	32	65	33	49.23% ↓
SECP256K1_DECOMPRESS_main	153	153	307	154	49.84% ↓
SHA256_EXTEND_main	537	537	1075	538	49.95% ↓
BN254_FP_ADD_main	153	153	307	154	49.84% ↓
BN254_FP_MUL_main	153	153	307	154	49.84% ↓
BN254_FP2_ADD_main	281	281	563	282	49.91% ↓
BN254_FP2_MUL_main	281	281	563	282	49.91% ↓
PROGRAM	8	8	17	9	47.06% ↓