feat: Update SMT Circuit class and add gate relaxation functionality

[Sarkoxed]

This pr adds gate relaxation functionality to Circuit class. It substitutes subcircuits with their "arithmetic" equivalents.
Also it adds util directory.

Util

• default_model and default_model_single will output pretty formatted result of Solver work
• smt_timer will measure the amount of time that it took to solve something
• base4 provides you with accumulators and base4 decomposition of a number

Terms

• Added isBitVector() method that foreshadows future BitVector api

Subcircuits

• functions in subcircuits.cpp now return new structure: CircuitProps.
• CircuitProps consists of
  • start_gate indicates where to start subcircuit comparison
  • num_gates tells a pure subcircuit size
  • idxs and and gate_idxs indicate where to search for the values that are used in optimized operation

Also added tests to confirm that all the indices

Circuit

• Added members bool optimizations and map<uint32_t, bool> optimized to keep track of optimized out variables that should not be considered by solver
• Added functions handle_range_constraint, handle_logic_constraint. They perform binary search to indicate the bit_length of specific operation and apply optimization to subcircuit
• Added cache for subcircuits to avoid recomputing them each time

Tests check that

• all the assert_equal'd variables handled properly during circuit initialization
• range_constraints are optimized properly

[AztecBot]

Benchmark results

Metrics with a significant change:

• note_successful_decrypting_time_in_ms (8): 196 (+31%)
• note_trial_decrypting_time_in_ms (8): 102 (+85%)

Detailed results

All benchmarks are run on txs on the Benchmarking contract on the repository. Each tx consists of a batch call to create_note and increment_balance, which guarantees that each tx has a private call, a nested private call, a public call, and a nested public call, as well as an emitted private note, an unencrypted log, and public storage read and write.

This benchmark source data is available in JSON format on S3 here.

Values are compared against data from master at commit aa90f6ed and shown if the difference exceeds 1%.

L2 block published to L1

Each column represents the number of txs on an L2 block published to L1.

Metric	8 txs	32 txs	64 txs
l1_rollup_calldata_size_in_bytes	5,668	18,820	36,356
l1_rollup_calldata_gas	66,400	239,032	469,808
l1_rollup_execution_gas	667,081	953,942	1,337,165
l2_block_processing_time_in_ms	1,327	4,953 (+1%)	9,073 (-4%)
note_successful_decrypting_time_in_ms	⚠️ 196 (+31%)	540 (-1%)	932 (-5%)
note_trial_decrypting_time_in_ms	⚠️ 102 (+85%)	55.1 (-10%)	63.9 (-44%)
l2_block_building_time_in_ms	18,320 (-2%)	68,382 (-4%)	136,455 (-3%)
l2_block_rollup_simulation_time_in_ms	8,231 (-1%)	28,621 (-6%)	56,943 (-4%)
l2_block_public_tx_process_time_in_ms	10,068 (-2%)	39,703 (-3%)	79,415 (-2%)

L2 chain processing

Each column represents the number of blocks on the L2 chain where each block has 16 txs.

Metric	5 blocks	10 blocks
node_history_sync_time_in_ms	13,860 (-2%)	26,572 (+1%)
note_history_successful_decrypting_time_in_ms	1,316 (-3%)	2,468 (+3%)
note_history_trial_decrypting_time_in_ms	136 (+31%)	156 (+27%)
node_database_size_in_bytes	19,030,096	35,778,640
pxe_database_size_in_bytes	29,859	59,414

Circuits stats

Stats on running time and I/O sizes collected for every circuit run across all benchmarks.

Circuit	circuit_simulation_time_in_ms	circuit_input_size_in_bytes	circuit_output_size_in_bytes
private-kernel-init	284 (-2%)	44,370	27,700
private-kernel-ordering	216 (-2%)	52,324	14,326
base-parity	1,861 (-4%)	128	311
base-rollup	721 (-3%)	177,083	925
root-parity	1,445 (-12%)	1,244	311
root-rollup	68.8	4,487	789
private-kernel-inner	639 (-1%)	73,229	27,700
public-kernel-app-logic	450 (-1%)	35,262	28,215
public-kernel-tail	176 (-1%)	40,926	28,215
merge-rollup	8.28	2,696	925

Tree insertion stats

The duration to insert a fixed batch of leaves into each tree type.

Metric	1 leaves	16 leaves	64 leaves	128 leaves	512 leaves	1024 leaves	2048 leaves	4096 leaves	32 leaves
batch_insert_into_append_only_tree_16_depth_ms	9.93 (-1%)	16.0 (-1%)	N/A	N/A	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_16_depth_hash_count	16.8	31.6	N/A	N/A	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_16_depth_hash_ms	0.581 (-1%)	0.495	N/A	N/A	N/A	N/A	N/A	N/A	N/A
batch_insert_into_append_only_tree_32_depth_ms	N/A	N/A	46.0	72.1	230	442 (-1%)	865	1,699 (-2%)	N/A
batch_insert_into_append_only_tree_32_depth_hash_count	N/A	N/A	96.0	159	543	1,055	2,079	4,127	N/A
batch_insert_into_append_only_tree_32_depth_hash_ms	N/A	N/A	0.472	0.446	0.422 (+1%)	0.415 (-1%)	0.412	0.408 (-2%)	N/A
batch_insert_into_indexed_tree_20_depth_ms	N/A	N/A	54.2 (+1%)	106	333 (-1%)	657	1,306	2,573 (-1%)	N/A
batch_insert_into_indexed_tree_20_depth_hash_count	N/A	N/A	104	207	691	1,363	2,707	5,395	N/A
batch_insert_into_indexed_tree_20_depth_hash_ms	N/A	N/A	0.485 (+1%)	0.478	0.455	0.453	0.453	0.449 (-1%)	N/A
batch_insert_into_indexed_tree_40_depth_ms	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	60.9 (-1%)
batch_insert_into_indexed_tree_40_depth_hash_count	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	109
batch_insert_into_indexed_tree_40_depth_hash_ms	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	0.534

Miscellaneous

Transaction sizes based on how many contract classes are registered in the tx.

Metric	0 registered classes
tx_size_in_bytes	22,014

Transaction processing duration by data writes.

Metric	0 new note hashes	1 new note hashes
tx_pxe_processing_time_ms	4,018	1,737 (-3%)

Metric	0 public data writes	1 public data writes
tx_sequencer_processing_time_ms	12.5 (-1%)	1,254 (-1%)

[Rumata888]

Please address my comments

[Rumata888]

LGTM