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Exploiting Symmetry in SAT-Based Boolean Matching for Heterogeneous FPGA Technology Mapping Yu Hu 1, Victor Shih 2, Rupak Majumdar 2 and Lei He 1 1 Electrical Engineering Dept., UCLA 2 Computer Science Dept., UCLA Presented by Victor Shih Address comments to

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Outline Background and Motivation Review of Standard SAT-based Boolean Matching Proposed Improvements Experimental Results Conclusion and Future Work

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Background FPGA technology mapping Map a design into a network of Programmable Logic Blocks (PLBs) Optimize for area, speed and/or power PLBs containing heterogeneous devices requires Boolean matching (BM) to determine whether function f can be implemented by hardware component H 3A.1 Design, Synthesis and Evaluation of Heterogeneous FPGA with Mixed LUTs and Macro-Gates [Hu, ICCAD’07]

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Example: Boolean Matching (BM) for PLBs Answers a Yes-No question: Can a Boolean function f be implemented in PLB p? If yes, give the configuration bits for all LUTs. f 1 = e*a + c*a + d*a + b*a f 2 = a + b + c + d + e f 1 = (e + c + d + b)*a i.e., f 1 = z*a z = e + c + d + b z e c d b a f1f1 L0 (0000)0 L1 (0001)1 L2 (0010)1 ……… L15(1111)1

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Motivation for SAT-Based PLB BM Application of FPGA PLB Boolean matching Technology mapping Re-synthesis Existing BM algorithms Decomposition-based BM lacks flexibility, i.e., the algorithm is only applicable to selected BLE structure [Cong, TCAD’01] BDD based BM is not scalable (memory explosion) [Ciric, TCAD’03] Fast BM is hard to deal with programmable devices [Wei, ISQED’06] Improvements over Ling’s algorithm obtain 3x speedup [Sean, DAC’06] and 10x speedup [Jason, FPGA’07] SAT-based BM [Ling, DAC’05][Safarpour, DAC’06][Cong, FPGA’07] Introduces extreme flexibility Provide a tradeoff between memory and runtime to deal with complicated BLE structures Still slow, hard to be applied to complex PLBs

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Review: SAT-Based Encoding for BM Encoding non-programmable devices: Requires common/interconnect variables Is a linear time procedure Example: f AND = (x 2 +¬z 1 ) (x 1 +¬z 1 ) (¬x 2 +¬x 1 + z 1 ) f OR = (¬x 3 +g) (¬z 1 +g) (x 3 +z 1 + ¬g) f total = f AND f OR = (x 2 +¬z 1 ) (x 1 +¬z 1 ) (¬x 2 +¬x 1 + z 1 ) (¬x 3 +g) (¬z 1 +g) (x 3 +z 1 + ¬g)

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Review: SAT-Based Encoding for BM Encoding programmable devices: Configuration bits are encoded f LUT =( x 1 + x 2 + ¬L 0 + z 1 ) ( x 1 + x 2 + L 0 + ¬ z 1 ) ( x 1 + ¬ x 2 + ¬L 1 + z 1 ) ( x 1 + ¬ x 2 + L 1 + ¬ z 1 ) (¬ x 1 + x 2 + ¬L 2 + z 1 ) (¬ x 1 + x 2 + L 2 + ¬ z 1 ) (¬ x 1 + ¬ x 2 + ¬L 3 + z 1 ) (¬ x 1 + ¬ x 2 + L 3 + ¬ z 1 )

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Review: SAT-Based Encoding for BM G LUT2 =( x 1 + x 2 + ¬L 0 + z) ( x 1 + x 2 + L 0 + ¬ z) ( x 1 + ¬ x 2 + ¬L 1 + z) ( x 1 + ¬ x 2 + L 1 + ¬ z) (¬ x 1 + x 2 + ¬L 2 + z) (¬ x 1 + x 2 + L 2 + ¬ z) (¬ x 1 + ¬ x 2 + ¬L 3 + z) (¬ x 1 + ¬ x 2 + L 3 + ¬ z) G AND2 = ( x 3 + ¬f ) (¬ x 3 + ¬f ) ( ¬x 3 + ¬z + f ) G = G AND2 · G LUT2 x1x2x3x1x2x3 f SAT: G expand = G[X/000, f/0, z/z 0 ] · G[X/001, f/0, z/z 1 ] G[X/010, f/1, z/z 2 ] · G[X/011, f/0, z/z 3 ] G[X/100, f/1, z/z 4 ] · G[X/101, f/1, z/z 5 ] G[X/110, f/1, z/z 6 ] · G[X/111, f/1, z/z 7 ] Boolean function Configuration bits are encoded as SAT literals The solution of this SAT problem corresponds to the Boolean matching results

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The Problem of Input Permutations [Ling, DAC’05] Virtual MUXes increase runtime exponentially! ? ? ?

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Impact of Virtual MUXes

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Outline Background and Motivation Review of Standard SAT-based Boolean Matching Proposed Improvements Experimental Results Conclusion and Future Work

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Symmetries in Circuits and PLB Functional Symmetries Variable a and variable b are symmetric if swapping a and b does not change the truth table of function F(…, a, …, b, …) E.g.: F(a, b, c) = a·(b + c), where b and c are symmetric General symmetries which consider the permutations of more than two variables can also be explored Architectural Symmetries Structures of certain inputs of a PLB are equivalent E.g.: Inputs of the primary input LUTs of PLBs are symmetric F(b, a, c, d)

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Impact of Considering Symmetries The number of distinct permutations under symmetries decreases substantially Functional symmetries and architecture symmetries independently reduce permutations by 100x

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Overall Algorithm Boolean function Functional symmetry detection Prune via architecture symmetries Non-redundant Permutation Set (NPS) Is NPS empty? Pre-processing target PLB - one-time cost Target architecture Pre-calculate architecture symmetry patterns Architecture symmetry information Characteristic function template Generate characteristic function template Pop permutation p Exit Y N

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Overall Algorithm (cont.) Replicate CNFs of p Solve SAT problem SAT? Return implementable Target architecture Pre-calculate architecture symmetry patterns Architecture symmetry information Characteristic function template Generate characteristic function template Pre-processing target PLB - one-time cost Exit Y Is NPS empty? Pop permutation p Y N N

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Experimental Results Experimental conditions: Tested with Boolean functions in MCNC circuits Target PLB was “ PLB_d ” Used “ miniSAT1.14 ” as SAT solver engine Obtained over 100x speedup compared to the standard approach [Ling ’05 ] Additional 2x achieved using implicant table representation

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Experimental Results Improvement makes SAT-based Boolean matching feasible in the context of technology mapping and re-synthesis

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Other SAT Symmetry Detection Techniques Shatter/Saucy - general symmetry-breaking tool for SAT (version released Sept ‘07) [Aloul, Markov, Sakallah, DAC’03] Augments SAT problem, adding symmetry breaking clauses Same SAT engine used

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Problems in the Proposed Algorithm (Following improvements yet to be published) Conjunctive normal form (CNF) clause generation dominates runtime Generally CNF generation time to SAT solution time is an order of magnitude greater

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Improvement - Iterative Clause Testing miniSAT’s incremental SAT reasoning suggests possible improvement: If any subset of clauses is unsatisfiable, the entire set is unsatisfiable Since SAT testing is relatively inexpensive, we can iteratively test larger and larger subsets to minimize CNF generation Asymptotically, worst-case runtime is equivalent to non-incremental testing – overhead for additional iterations is insignificant Experimentally, this resulted in roughly 3x speedup a + j + d !b + e + s t + a + !f e + !m + !p + !e + g + r + !e … IterationsClauses …

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Improvement – Template Clause-Set Approach miniSAT’s “assumptions” parameter suggests possible improvement: Facilitates parameterization of SAT literals for later resolution We can minimize clause generation time by formulating a set of clauses which is common to all permutations to test, and adding only the clauses specific to each permutation as assumptions Experimentally, this resulted in roughly 10x speedup Note that this approach precludes the iterative clause testing approach

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x1x2x3x1x2x3 f' 000f1f1 001f5f5 010f3f3 011f7f7 100f2f2 101f6f6 110f4f4 111f8f8 Template Clause-Set Approach Detail Example: Every permutation will have the same set of input values, but each will correspond to different outputs x1x2x3x1x2x3 f 000f1f1 001f2f2 010f3f3 011f4f4 100f5f5 101f6f6 110f7f7 111f8f8 x1x2x3x1x2x3 f' 000f1f1 100f2f2 010f3f3 110f4f4 001f5f5 101f6f6 011f7f7 111f8f8 f f' = f ( x 3, x 2, x 1 )f' P 1 = G expand ∙ ¬f 1 ∙ ¬ f 2 ∙ f 3 ∙ ¬ f 4 ∙ f 5 ∙ f 6 ∙ f 7 ∙ f 8 P 2 = G expand ∙ ¬ f 1 ∙ f 2 ∙ ¬ f 3 ∙ f 4 ∙ ¬ f 5 ∙ ¬ f 6 ∙ ¬ f 7 ∙ f 8 … SAT: G expand = G[X/000, f/f 1, z/z 0 ] · G[X/001, f/f 2, z/z 1 ] G[X/010, f/f 3, z/z 2 ] · G[X/011, f/f 4, z/z 3 ] G[X/100, f/f 5, z/z 4 ] · G[X/101, f/f 6, z/z 5 ] G[X/110, f/f 7, z/z 6 ] · G[X/111, f/f 8, z/z 7 ]

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CNF Generation Runtime Breakdown Template clause-set implementation obtains average 10x speedup of CNF generation time

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Conclusions and Future Work An improvement for SAT-based Boolean matching is presented by considering functional and architectural symmetries Over 100x speedup is obtained compared to the standard SAT-based Boolean matching approach Future Work Integrate the improved SAT-based Boolean matcher into heterogeneous FPGA technology mapping phase Perform architecture exploration using our improved technology mapper

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Thanks Exploiting Symmetry in SAT-Based Boolean Matching for Heterogeneous FPGA Technology Mapping Yu Hu, Victor Shih, Rupak Majumdar and Lei He

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