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Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 mbia University New York, NY 10027CUCS-001-98AbstractNone of tilt available minimize™ for 2-level hazard-free logic minimization can synthesize very large circuits. This limitation has forced researchers to resort to manual and automated circuit partitioning techniques. This paper introduces two new cucs-001-98 2-levcl logic minirnize.rs: Esi’BESSO-HF. a heuristic method which is loosely based on ESi’BESSO-H. and 1.MPYM1N, an exact method based on implicit dcucs-001-98
ata structures.lỉoth minimizers can solve all currently available examples, which range up Io 32 inputs and 33 outputs. These include examples that haFast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 these algorithms, we also present two additional results. First, we introduce a fast new algorithm to check if a hazard-free covering problem can feasibly be solved. Second, we introduce a novel formulation of the 2-level hazard-free logic minimization problem by capturing hazard-freedom constraint cucs-001-98 s within a synchronous function by adding new variables.■ This work was supported by NSF under Giant no. MIP-9501880 and by an Alfred I’. Sloan Researcucs-001-98
ch Fellowship. The presented work is an extended version of two recent conference papers (32. 31]1IIntroductionAsynchronous design has been the focus Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 [11. 18. 12, 19. 2. 30. 34, 15, 1|.A number of methods have been developed for the design of hazard-free controllers 22. 20. 37. 13. 27]. These methods have been applied to several large and realistic design examples, including a low-power infrared communications chip [14 . a S(*cond-level cache-co cucs-001-98 ntroller [21], a SCSI controller [35\ a differential equation solver [36], anil an instruction length decoder [4].An important aspect of these methodscucs-001-98
is the development of optimized CAI) tools. In synchronous design. CAI) packages have been critical to the advancement of modern digital design. In aFast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 . 27] and synthesis-for-testability [24]. However, these tools have been limited in handling large-scale designs.In particular, hazard-free 2-level logic minimization is an important step in all the above-mentioned CAI) tools. However, while the currently used Quine-McCluskey-like exact hazard-free cucs-001-98 minimization algorithm. Hemin [10]. has been effective on small- anil medium-sized examples, it has been unable to produce solutions for several largecucs-001-98
design problems [13, 27]. This limitation has been a major reason for researchers to invent and apply manual as well as automated techniques for partFast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 ient 2-level hazard-free logic minimizers for multi-output minimization: Espresso-HF and 1MPYMIN.ESPRESSO-HF is an algorithm to solve the heuristic hazard-free two-level logic minimization problem. The method is heuristic solely in terms of the cardinality of solution. In all cases, it guarantees a cucs-001-98 hazard-free solution. The algorithm is based on Espresso-II[26. 9]. but with a number of significant modifications to handle hazard-freedom constraintcucs-001-98
s. It is the first heuristic method based on Espresso-11 to solve the hazard-free2minimization problem. Espresso-HF also includes a new and much more Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 ct hazard-free two-level logic minimization problem, rhe algorithm uses an implicit approach which makes use of data structures such as BDDs [3] and zero-suppressed BDDs [17]. The algorithm is based on a novel theoretical approach IO hazard-free two-level logic minimization. We reformulate the gener cucs-001-98 ation of dynamic-hazard-free prime implicants as a synchronous prime implicant generation problem. This is achieved by incorporating hazard-freedom cocucs-001-98
nstraints within a synchronous function by adding new variables. This technique allows to leverage off an existing method for fast implicit generationFast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 icular, the approach makes it possible to use the implicit set covering solver of Scherzo (8. C), 5, 7 . the stati'-of-the-art minimization method for synchronous twOrlevel logic, as a black box.Both Espresso-HF and Impymin can solve all currently available examples, which range up to 32 inputs and cucs-001-98 33 outputs, These include examples that have never been previously solved. For examples that can be solved by the currently fastest minimizer Hemin oucucs-001-98
r two minimizersare typically several orders of magnitude faster. In particular, Impymin can find a minimum-size cover for all benchmark examples in lFast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 MPYMIN are somewhat orthogonal. On the one hand Espresso-HF is typically faster than Impymin. On the other hand. ĨMPYMIN computes a cover of minimum size, whereas Espresso-HF is not guaranteed to find a minimum cover but typically does find a cover of very good quality.Paper OrganizationThe paper is cucs-001-98 organized as follows. Section 2 gives background on circuit models, hazards and hazard-free minimization. Section 3 describes the Espresso-HF algoritcucs-001-98
hm for heuristic hazard-free minimization. Section I introduces a new approach to hazard-free minimization where hazard-freedom constraints are capturFast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 n 4. Section 5 introduces our new implicit method for exact hazard-free minimization, called IMI’YMIN. Section 6 presents experimental results and compares our approaches with related work, and Section 7 gives conclusions. Background information on BDD, ZBDDs. and implicit logic minimization can be cucs-001-98 found in the appendix.2 BackgroundThe material of this section focuses on hazards and hazard-free logic minimization, and is taken from [10] and [25,cucs-001-98
23]. Idr simplicity, we focus on single-output functions. A generalization of these definitions to multi-output functions is straightforward, and is dFast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 del [25]). A pure delay model is assumed as well (see [33]).2.2Multiple-Input ChangesDefinition 2.1 Let. .4 and B be two mintenns. The transition cube. Ị.4. /1. from .4 to B has start point .4 and end point B. and contains ail mintenns that can be reached during a transition from .4 to B. More forma cucs-001-98 lly, if .4 and B are described by products, with i-th literals .4j and Bi, respectively, then the i-th literal for the product, oft = [-4, B] is the Bcucs-001-98
oolean function A, + B, (alternatively, A, B is the uniquely defined smallest cube that contains .4 and 13: supercube(A.B)/ .4n input transition or muFast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science Colum cucs-001-98 hange value and what the corresponding starting and ending values are. Input variables are assumed to change simultaneously. (Equivalently, since inputs may be skewed arbitrarily by wire delays, inputs can be assumed to change monotonically in any order and at any time.) Once a multiple-input cucs-001-98 Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization*Michael Theobald Steven M. NowickDepartment of Computer Science ColumGọi ngay
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