ABC: A System for Sequential Synthesis and Verification Berkeley Logic Synthesis and Verification Group Robert Brayton Alan Mishchenko

Overview Introduction Recent work What and why ABC? ABC fundamentals Areas addressed by ABC Synthesis Technology mapping Verification Contrast with classical methods How is ABC different from SIS? Recent work Speedup Factoring Don’t-care based optimization Scalable sequential synthesis WireMap White boxes

A Plethora of ABCs http://en.wikipedia.org/wiki/Abc ABC (American Broadcasting Company) A television network… ABC (Active Body Control) ABC is designed to minimize body roll in corner, accelerating, and braking. The system uses 13 sensors which monitor body movement to supply the computer with information every 10 ms… ABC (Abstract Base Class) In C++, these are generic classes at the base of the inheritance tree; objects of such abstract classes cannot be created… ABC (supposed to mean “as simple as ABC”) A system for sequential synthesis and verification at Berkeley

Why We Decided to Build ABC SIS Outdated, but many research papers on how a new algorithm beats SIS results Not supported MVSIS Gave us a reason to work on logic synthesis Learned a lot about new methods and better data structures Could see how specializing to binary could provide substantial improvements. ABC Initial intention was to re-implement all algorithms using new data structures (daunting task) Discovered rewriting AIGs P. Bjesse and A. Boralv, "DAG-aware circuit compression for formal verification", Proc. ICCAD ’04, pp. 42-49. Decided to try to keep all transformations fast and scalable No BDDs No SOPs No Espresso BDD

What Is Berkeley ABC? A system for logic synthesis and verification Fast Scalable High quality results (industrial strength) Exploits synergy between synthesis and verification A programming environment Open-source Evolving and improving over time

Design Flow System Specification RTL Verification ABC Logic synthesis Technology mapping Physical synthesis Manufacturing

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Areas Addressed by ABC Combinational synthesis Sequential synthesis AIG rewriting technology mapping resynthesis after mapping Sequential synthesis retiming structural register sweep merging seq. equiv. nodes Formal verification combinational equivalence checking bounded sequential verification unbounded sequential verification equivalence checking using synthesis history

Combinational Synthesis AIG rewriting minimizes the number of AIG nodes without increasing the number of AIG levels Rewriting AIG subgraphs Pre-computing AIG subgraphs Consider function f = abc Rewriting node A a b c A Subgraph 1 b c a A Subgraph 2  a b c Subgraph 1 b c a Subgraph 2 a c b Subgraph 3 Rewriting node B b c a B Subgraph 2 a b c B Subgraph 1 a b c  In both cases 1 node is saved

Technology Mapping Input: A Boolean network (And-Inverter Graph) Output: A netlist of K-LUTs implementing AIG and optimizing some cost function a b c d f e a b c d e f Technology Mapping The subject graph The mapped netlist

Sequential Synthesis Structural register sweep (scleanup) Merge registers with identical drivers Replace stuck-at registers by constants Retiming (dretime) Minimize the number of registers under delay constraints Preserves equivalent initial state Sequential SAT sweeping (scorr) Detecting and merging sequencially equivalent nodes

Formal Verification Equivalence checking D2 D1 Equivalence checking Equivalence checking Takes two designs and makes a miter (AIG) Model checking safety properties Takes design and property and makes a miter (AIG) The goals are the same: to transform AIG until the output is proved constant 0 Breaking News: ABC won a model checking competition at CAV in August 2008 D1 Property checking p

Model Checking Competition

5. ABC 238

Time (sec) ABC # problems solved

Command “dprove” in ABC transforming initial state (“undc”, “zero”) converting into an AIG (“strash”) creating sequential miter (“miter -c”) combinational equivalence checking (“iprove”) bounded model checking (“bmc”) sequential sweep (“scl”) phase-abstraction (“phase”) most forward retiming (“dret -f”) partitioned register correspondence (“lcorr”) min-register retiming (“dretime”) combinational SAT sweeping (“fraig”) for ( K = 1; K  16; K = K * 2 ) signal correspondence (“scorr”) stronger AIG rewriting (“dc2”) sequential AIG simulation interpolation (“int”) BDD-based reachability (“reach”) saving reduced hard miter (“write_aiger”) Preprocessors Combinational solver Fast engines Medium engines Slower Main induction loop Last-gasp engines

ABC vs. Other Tools Industrial SIS VIS MVSIS + well documented, fewer bugs - black-box, push-button, no source code, often expensive SIS + traditionally very popular - data structures / algorithms outdated, weak sequential synthesis VIS + very good implementation of BDD-based verification algorithms - not meant for logic synthesis, does not feature the latest SAT-based implementations MVSIS + allows for multi-valued and finite-automata manipulation - not meant for binary synthesis, lacking recent implementations

How Is ABC Different From SIS? Boolean network in SIS a b c d e x y f z Equivalent AIG in ABC a b c d f e x y z AIG is a Boolean network of 2-input AND nodes and invertors (dotted lines)

One AIG Node – Many Cuts Combinational AIG Manipulating AIGs in ABC Each node in an AIG has many cuts Each cut is a different SIS node No a priori fixed boundaries Implies that AIG manipulation with cuts is equivalent to working on many Boolean networks at the same time f a b c d e Different cuts for the same node

Comparison of Two Syntheses “Classical” synthesis Boolean network Network manipulation (algebraic) Elimination Factoring/Decomposition Speedup Node minimization Espresso Don’t cares computed using BDDs Resubstitution Technology mapping Tree based ABC “contemporary” synthesis AIG network DAG-aware AIG rewriting (Boolean) Several related algorithms Rewriting Refactoring Balancing Speedup Node minimization Boolean decomposition Don’t cares computed using simulation and SAT Resubstitution with don’t cares Technology mapping Cut based with choice nodes

Existing Capabilities (2005-2008) Combinational logic synthesis Fast, scalable, good quality Technology mapping with structural choices Cut-based, heuristic, good area/delay, flexible ABC Sequential verification Integrated, interacts with synthesis Sequential synthesis Innovative, scalable, verifiable

Overview Introduction Recent work Summary What is ABC? ABC fundamentals Areas addressed by ABC Synthesis Technology mapping Verification Contrast with classical methods How is ABC different from SIS? Recent work Speedup Factoring Don’t-care based optimization Scalable sequential synthesis WireMap White boxes Summary

Command “speedup” Timing Criticality Critical nodes Critical edges Used by many traditional algorithms Critical edges Used by our algorithm We pre-compute critical edges of critical nodes Reduces computation An edge between critical nodes may not be critical See illustration: edge 13 Primary outputs 4 4 3 3 2 2 1 1 Primary inputs

Delay-Oriented Restructuring Using traditional MUX-restructuring AKA generalized select transform x and y are the critical edge inputs

Overall Algorithm Done only once mapped netlist performSpeedup ( subject graph S, // S is an And-Inverter Graph mapped netlist M, // M was previously derived by tech-mapping of S timing window w, // w is used to detect the critical paths logic depth l, // l is used to detect a logic cone rooted at a node edge count p ) // p limits the number critical edges of the cone { perform timing analysis of M with unit-delay or LUT-library model; pre-compute critical section of M as nodes n such that 0  slack(n)  w; pre-compute timing-critical edges connecting these nodes; for each timing critical node n { find cone C of M that extends l levels down from n; pick the set of timing-critical edges V feeding into C; if the number of edges in V exceeds p, continue; find logic cone C’ in S corresponding to C in M; find variables V’ in S corresponding to V in M; derive cofactors of the function of C’ w.r.t. variables in V’; build multiplexer tree C’’ of the cofactors using variables in V’; add structural choice C’= C’’ to the subject graph S; } return mapped netlist M’ derived by mapping subject graph S with added choices; Done only once

Experimental Results for “speedup” LUT – number of LUTs Lev – number of LUT levels Delay – delay using LUT library Total – total runtime of Baseline Time1 – the runtime of AIG restructuring only Time2 – the total runtime of Speedup Geomean – geometric averages of columns Ratios – ratios of geometric averages

Overview Introduction Recent work Summary What is ABC? ABC fundamentals Areas addressed by ABC Synthesis Technology mapping Verification Contrast with classical methods How is ABC different from SIS? Recent work Speedup Factoring Don’t-care based optimization Scalable sequential synthesis WireMap White boxes Summary

Basic Inner Core Algorithm (DSD) We use a fast disjoint support decomposition (DSD) algorithm as our underlying subroutine follows Bertacco and Damiani, "The disjunctive decomposition of logic functions“, ICCAD '97 but uses heuristics to speed it up no BDDs uses truth tables limit inputs to up to 16 BDD

Disjoint Support Decomposition (DSD) (Simple Disjunctive Decomposition) Theorem 1 [Ashenhurst 1959]. For a completely specified Boolean function, there is a unique maximal DSD (up to the complementation of inputs and outputs and factoring of ANDs/ORs and XORs). E C D A B G x1 x2 x3 x4 x5 H F D a c 1

Non-Disjoint Decomposition Definition: A function F has an ( ) -decomposition if it can be written as where ( ) is a partition of the variables x and D is a single output function. H D a c b The variables in the set b are called the shared variables. The variables a are called the bound set and c the free set. 1

Non-Disjoint Decomposition Theorem 2: A function has an - decomposition if and only if each of the cofactors of F with respect to has a DSD structure in which the variables are in a separate sub-tree. E C D A B G x4 x5 x1 x2 x3 X Z W Y x4 x5 x1 x2

Application of Factoring (uses Theorem 2) Rewriting a k-LUT mapped circuit. For each LUT, and each cut of no more than 16 inputs, express the output of the LUT as truth table in terms of the cut variables – F(x) Find variables b such that its cofactors are support reducing we exhaustively look for up to two variables in the b set Take the best (a,b) set and decompose F=H(D(a,b),b,c) Recursively decompose H and D if they do not fit into a k-LUT. If improvement, replace LUTs in cut with its new decomposition. Experimental results later

Overview Introduction Recent work Summary What is ABC? ABC fundamentals Areas addressed by ABC Synthesis Technology mapping Verification Contrast with classical methods How is ABC different from SIS? Recent work Speedup Factoring Don’t-care based optimization Scalable sequential synthesis WireMap White boxes Summary

Windowing a Node in the Network for Don’t-Care Computation Boolean network (k-LUT mapped circuit) Definition A window for a node in the network is the context in which the don’t-cares are computed A window includes n levels of the TFI m levels of the TFO all re-convergent paths captured in this scope Window with its PIs and POs can be considered as a separate network Window POs Window PIs n = 3 m = 3

Care Set Representation “Miter” constructed for the window POs If output is 1 then we care … Window Window Same window with inverter f Window f x x s

Resubstitution Resubstitution considers a node in a Boolean network and expresses it using a different set of fanins X X Computation can be enhanced by use of don’t cares

Resubstitution with Don’t-Cares Consider all or some nodes in Boolean network. For each node Create window Select possible fanin nodes (divisors) For each candidate subset of divisors Rule out some subsets using simulation Check resubstitution feasibility using SAT Compute resubstitution function using interpolation A low-cost by-product of completed SAT proofs Update the network if there is an improvement

Resubstitution with Don’t Cares Given: node function F(x) to be replaced care set C(x) for the node candidate set of divisors {gi(x)} for re-expressing F(x) Find: A resubstitution function h(y) such that F(x) = h(g(x)) on the care set SPFD Theorem: Function h exists if and only if every pair of care minterms, x1 and x2, distinguished by F(x), is also distinguished by gi(x) for some i C(x) F(x) g1 g2 g3 C(x) F(x) g1 g2 g3 h(g) F’(x)

Checking Resubstitution using SAT F Miter for resubstitution check h(g) SPFD theorem in practice Note use of care set, C. Resubstitution function exists if and only if SAT problem is unsatisfiable. An h(g) is obtained by interpolation

Experimental Results

Overview Introduction Recent work Summary What is ABC? ABC fundamentals Areas addressed by ABC Synthesis Technology mapping Verification Contrast with classical methods How is ABC different from SIS? Recent work Speedup Factoring Don’t-care based optimization Scalable sequential synthesis WireMap White boxes Summary

The Main Idea Consider registers and nodes of a design Detect candidate equivalences in this set using random/guided simulation Prove candidates by K-step induction Merge the resulting equivalences This is a subset of sequential synthesis with Practical advantages (does not move registers, etc) Scales to large designs Offers substantial improvements Comes with a verification guarantee

Base Case Inductive Case Candidate equivalences: {A,B}, {C,D} ? D C SAT-2 ? Proving internal equivalences in a topological order in frame K A B SAT-1 ? D C SAT-4 ? PIk A B SAT-3 PI1 ? C D D C SAT-2 A ? Assuming internal equivalences to in uninitialized frames 0 through K-1 B A B SAT-1 PI1 PI0 C D Initial state A Proving internal equivalences in initialized frames 0 through K-1 B PI0 Symbolic state

Dynamic Partitioning (register correspondence) B A = D C = B’ A’ = ? Illustration for two candidate equiv. classes: {A,B}, {C,D} Partition 1 B A D C B’ A’ D’ C’ One time-frame of the design B A = D C = D’ C’ = ? Partition 2

Academic Benchmarks Columns “Baseline”, “Reg Corr” and “Sig Corr” show geometric means.

Industrial Benchmarks In case of multiple clock domains, optimization was applied only to the domain with the largest number of registers.

Reasons for Large Improvements Redundancy introduced by HDL compilers Early logic duplication by the designer Accidental sequential redundancies Sequential redundancies present due to reuse of design components that had more functionality than needed

Overview Introduction Recent work Summary What is ABC? ABC fundamentals Areas addressed by ABC Synthesis Technology mapping Verification Contrast with classical methods How is ABC different from SIS? Recent work Speedup Factoring Don’t-care based optimization Scalable sequential synthesis WireMap White boxes Summary

Motivation Fewer pin-to-pin connections should make the design easier to place and route Newer FPGAs allow two outputs per LUT Thus fewer pin-to-pin connections should produce a mapping that “packs” better into dual-output LUTs 50

Area Recovery Overview Perform delay-optimal mapping Recover area off critical paths Area-flow (global view) Chooses cuts with better logic sharing Exact local area (local view) New idea: Cut-based area recovery algorithms can be extended to minimize edges (pin-to-pin connections) Both are important Note that area is the same as single-output LUT count 51

WireMap Algorithm Perform delay-optimal mapping Recover area off critical paths Area-flow (global view) Break ties with minimum edge flow Exact local area (local view) Break ties with exact local edge count Note that area is the same as single-output LUT count 52

Experimental Setup WireMap implemented in ABC Compared WireMap against two algorithms in ABC Baseline – basic mapping with area recovery Mapping with Structural Choices – mapping with area recovery for several netlists produced by synthesis WireMap was implemented on top of mapping with choices Used VPR to place/route design for wirelength and critical path delays Used maximum cardinality matching to pack single-output LUTs into dual-output LUTs using Compared against the best ABC mapping algorithm (MSC) MSC – keeps multiple snapshots of a given netlist as different synthesis algorithms are tried. Some nodes will have multiple structures that implement them. Cut based mapping algorithm will thus have parallel structures to enumerate – more cut choices leads to better results. Packing was done by finding the maximum number of possible pairings for the LUTs. Not placement or timing driven. VPR experiment done to show impact of reducing # of connections to routability of the design. 53

Results Summary Comparing WireMap against the best mapping with structural choices in ABC WireMap results: Reduction in edges by 9.3% Reduction in dual-output LUT count by 9.4%, compared to mapping with choices Single-output LUT count only reduced by 1.3% Reduction in wire length by 8.5% Reduction in power by 20% WireMap edge and dual-output LUT count due to edge reduction algorithm, not to LUT count reduction. 54

Overview Introduction Recent work Summary What is ABC? ABC fundamentals Areas addressed by ABC Synthesis Technology mapping Verification Contrast with classical methods How is ABC different from SIS? Recent work Speedup Factoring Don’t-care based optimization Scalable sequential synthesis WireMap White boxes Summary

Comb and Seq Boxes o3 b o4 c a o1 o2 n8 n1 n6 n3 n7 n4 n2 Comb box FF1 FF3 FF4 FF5 FF Comb box

Treating Boxes as Black Seq box n1 n4 n3 n7 n6 n8 a o1 o2 FF1 FF3 FF4 FF5 FF Comb box For simplicity, boxes can be treated as “black”. Thus box outputs become inputs to the rest of the logic and box inputs become outputs. Delay and logic information is lost.

Treating Boxes as White c o3 o4 n2 Seq box n1 n4 n3 n7 n6 n8 a o1 o2 FF1 FF3 FF4 FF5 FF Comb box Example: Nodes o1 and o3 may be equivalent in the design, but this equivalence cannot be detected if the boxes are treated as black. Solution: Consider logic inside white boxes for synthesis, but keep it unchanged during synthesis and mapping.

Future Work ABC Improving AIG-based synthesis and mapping Integrating synthesis/ mapping/retiming Co-developing synthesis and verification Integrating synthesis with place and route ABC Creating special configurable design flows Supporting emerging technologies

To Learn More Visit ABC webpage http://www.eecs.berkeley.edu/~alanmi/abc Read recent papers http://www.eecs.berkeley.edu/~alanmi/publications Send email alanmi@eecs.berkeley.edu brayton@eecs.berkeley.edu