1130 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 Fast Heuristic and Exact Algorithms for Two-Level Hazard-Free Logic Minimization Michael Theobald and Steven M. Nowick Abstract— None of the available minimizers for two-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 two-level hazard-free logic minimizers: ESPRESSO-HF, a heuristic method loosely based on ESPRESSO-II, and IMPYMIN, an exact method based on implicit data structures. Both minimizers can solve all currently available examples, which range up to 32 inputs and 33 outputs. These include examples that have never been solved before. For the more difficult examples that can be solved by other minimizers, our methods are several orders of magnitude faster. As by-products of these algorithms, we also present two additional results. First, we propose a fast new method to check if a hazard-free covering problem can feasibly be solved. Second, we introduce a novel reformulation of the two-level hazard-free logic minimization problem by capturing hazard-freedom constraints within a synchronous function through the addition of new variables. I. INTRODUCTION A SYNCHRONOUS design has been the focus of much recent research activity [10]. In fact, asynchronous design has been applied to several large-scale control and data-path circuits and microprocessors [14], [22], [15], [23], [35], [39], [20], [2]. A number of methods have been developed for the design of asynchronous controllers. Much of the recent work has focused on Petri Net-based methods [18], [1], [16], [5], [40], [6], [7] and burst-mode methods [27], [9], [25], [43], [17], [31], [13]. These two classes of design methods differ in fundamental aspects: the delay model and how the circuit interacts with its environment [10]. Petri Net-based methods typically synthesize circuits to work correctly regardless of gate delays (speed-independent delay model), and the environment is allowed to respond to the circuit’s outputs without timing constraints (input/output mode). In contrast, burst-mode methods synthesize combinational logic to work correctly regardless of gate and wire delays, but the correct sequential operation depends on timing constraints. In particular, the enManuscript received December 30, 1997; revised May 13, 1998. This work was supported by the National Science Foundation under Grant MIP-9501880 and by an Alfred P. Sloan Research Fellowship. This work is an extended version of two recent papers presented at the International Symposium on Advanced Research in Asynchronous Circuits and Systems, March 1998, and the 33rd Design Automation Conference, 1996. This paper was recommended by Associate Editor E. Macii. The authors are with the Department of Computer Science, Columbia University, New York, NY 10027 USA. Publisher Item Identifier S 0278-0070(98)09489-5. vironment must wait for a circuit to stabilize before responding with new inputs (fundamental mode). The focus of this paper is on fundamental-mode asynchronous circuits, such as burst-mode machines. Burst-mode methods have recently been applied to several large and real-world design examples, including a low-power infrared communications chip [19], a second-level cache controller [26], an SCSI controller [41], a differential equation solver [42], and an instruction-length decoder [4]. An important challenge for any asynchronous synthesis method is the development of optimized CAD tools. In synchronous design, CAD packages have been critical to the advancement of modern digital design. In asynchronous design, however, a key constraint is to provide hazard-free logic, i.e., to guarantee the absence of glitches [38]. Much progress has been made in developing hazard-free synthesis methods, including tools for exact two-level hazard-free logic minimization [29], optimal state assignment [12], [31], synthesis for testability [28] and low-power logic synthesis [24]. However, these tools have been limited in handling large-scale designs. In particular, hazard-free two-level logic minimization is a bottleneck in most asynchronous CAD packages. While the currently used Quine–McCluskey-like exact hazard-free minimization algorithm, HFMIN [12], has been effective on small- and medium-sized examples, and is used in several existing CAD packages [27], [9], [25], [43], [17], [13], it has been unable to produce solutions for several large design problems [17], [31]. This limitation has been a major reason for researchers to invent and apply manual as well as automated techniques for partitioning circuits before hazard-free logic minimization can be performed [17]. A. Contributions of This Paper This paper introduces two very efficient two-level logic minimizers for hazard-free multioutput minimization: ESPRESSOHF and IMPYMIN. ESPRESSO-HF is an algorithm to solve the heuristic hazardfree two-level logic minimization problem. The method is heuristic solely in terms of the cardinality of solution. In all cases, it guarantees a hazard-free solution. The algorithm is based on ESPRESSO-II [30], [11], but with a number of significant modifications to handle hazard-freedom constraints. It is the first heuristic method based on ESPRESSO-II to solve the hazard-free minimization problem. ESPRESSO-HF also includes a new and much more efficient algorithm to check for the existence of a hazard-free solution, without generating all prime implicants. 0278–0070/98$10.00 1998 IEEE THEOBALD AND NOWICK: TWO-LEVEL HAZARD-FREE LOGIC MINIMIZATION IMPYMIN is an algorithm to solve the exact hazard-free twolevel logic minimization problem. The algorithm uses an implicit approach that makes use of data structures such as binary decision diagrams (BDD’s) [3] and zero-suppressed (Z)BDD’s [21]. The algorithm is based on a novel theoretical approach to hazard-free two-level logic minimization. The generation of dynamic-hazard-free prime implicants is reformulated as a synchronous prime implicant generation problem. This is achieved by capturing hazard-freedom constraints within a synchronous function by adding new variables. This technique allows us to use an existing method for fast implicit generation of prime implicants. Moreover, our novel approach can be nicely incorporated into a very efficient implicit minimizer for hazard-free logic. In particular, the approach makes it possible to use the implicit set covering solver of SCHERZO [8], the state-of-the-art minimization method for synchronous two-level logic, as a black box. Both ESPRESSO-HF and IMPYMIN can solve all currently available examples, which range up to 32 inputs and 33 outputs. These include examples that have never been previously solved. For examples that can be solved by the currently fastest minimizer HFMIN, our two minimizers are typically several orders of magnitude faster. In particular, IMPYMIN can find a minimum-size cover for all benchmark examples in less than 813 seconds, and ESPRESSO-HF can find very good covers—at most 3% larger than a minimum-size cover—in less than 105 s. ESPRESSO-HF and IMPYMIN are somewhat orthogonal. On the one hand, ESPRESSO-HF is typically faster than IMPYMIN. On the other hand, IMPYMIN 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. B. Paper Organization Section II gives background on circuit models, hazards, and hazard-free minimization. Section III describes the ESPRESSO algorithm for heuristic hazard-free minimization. Section IV introduces a new approach to hazard-free minimization where hazard-freedom constraints are captured by a constructed synchronous function, leading to a new method for computing dynamic-hazard-free prime implicants. Based on the results of Section IV, Section V introduces our new implicit method for exact hazard-free minimization, called IMPYMIN. Section VI presents experimental results and compares our approaches with related work, and Section VII gives conclusions. 1131 B. Multiple-Input Changes Definition II.1: Let and be two minterms. The tranfrom to has start point and end sition cube, point and contains all minterms that can be reached during More formally, is the uniquely a transition from to defined smallest cube that contains and : supercube(A,B). An input transition or multiple-input change from input state (minterm) to is described by transition cube A multiple-input change specifies what variables change 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 change occurs, no further input changes may occur until the circuit has stabilized. In this paper, we consider only transitions where is fully defined; that is, for every C. Function Hazards that does not change monotonically during A function an input transition is said to have a function hazard in the transition. contains a static function Definition II.2: A function if and only if 1) hazard for the input transition from to and 2) there exists some input state such that Definition II.3: A function contains a dynamic function if and only if 1) hazard for the input transition from to and 2) there exist a pair of input states and such that a) and and b) and If a transition has a function hazard, no implementation of the function is guaranteed to avoid a glitch during the transition, assuming arbitrary gate and wire delays [29], [38]. Therefore, we consider only transitions that are free of function hazards.1 D. Logic Hazards A. Circuit Model If is free of function hazards for a transition from input to an implementation may still have hazards due to possible delays in the logic realization. conDefinition II.4: A circuit implementing function tains a static (dynamic) logic hazard for the input transition to minterm if and only if 1) from minterm and 2) for some assignment of delays to gates and wires, the circuit’s output is not monotonic during the transition interval. That is, a static logic hazard occurs if ( ), but the circuit’s output makes an unexpected transition. A dynamic logic hazard occurs if and and but the circuit’s output makes an unexpected transition. This paper considers combinational circuits, which can have arbitrary finite gate and wire delays (an unbounded wire delay model [29]). A pure delay model is assumed as well (see [38]). 1 Sequential synthesis methods, which use hazard-free minimization as a substep, typically include constraints in their algorithms, which insure that no transitions with function hazards are generated [27], [43]. II. BACKGROUND The material of this section focuses on hazards and hazardfree logic minimization and is taken from [12] and [29]. For simplicity, we focus on single-output functions. A generalization of these definitions to multioutput functions is straightforward and is described in [12]. 1132 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 E. Conditions for a Hazard-Free Transition We now review conditions to ensure that a sum-of-products is hazard free for a given input transition implementation is the transition (for details, see [29]). Assume that cube corresponding to a function-hazard-free transition from to for a function We say that has a input state transition in cube Lemma II.5: If has a transition in cube then the implementation is free of logic hazards for the input to change from Lemma II.6: If has a transition in cube then the implementation is free of logic hazards for the input is contained in some change from to if and only if (i.e., some product must hold its value at 1 cube of cover throughout the transition). and cases are The conditions for the symmetric. Without loss of generality, we consider only a transition.2 Lemma II.7: If has a transition in cube then the implementation is free of logic hazards for the input change intersecting from to if and only if every cube also contains (i.e., no product may glitch in the middle of transition). a transition from input state Lemma II.8: If has a to that is hazard free in the implementation, then for where the transition every input state is contained in some cube of cover (i.e., subcube subtransition must be free of logic hazards). every transitions and transitions are called static transitions and transitions are called transitions. dynamic transitions. F. Required and Privileged Cubes in Lemma II.6 and the maximal subcubes The cube in Lemma II.8 are called required cubes. Each required to ensure cube must be contained in some cube of cover a hazard-free implementation, as more formally presented in Definition II.9. of specDefinition II.9: Given a function and a set every cube ified function-hazard-free input transitions of corresponding to a transition, and every where is 1 and maximal subcube is a transition, is called a required cube. Lemma II.7 constrains the products that may be included in Each transition cube is called a privileged a cover cube, since no product in the cover may intersect the cube unless also contains its start point. If a product intersects a privileged cube but does not contain its start point, it illegally intersects the privileged cube and may not be included in the cover, as presented more formally in Definition II.10. of specDefinition II.10: Given a function and a set every cube ified function-hazard-free input transitions of corresponding to a transition is called a privileged cube. ! 2A 0 1 transition from transition from B to A: A to B has the same hazards as a 1 !0 Finally, we define a useful special case. A privileged cube is called trivial if the function is only 1 at the start point and is 0 for all other minterms included in the transition cube. In this case, any product that intersects such a privileged cube always covers the start point. All trivial privileged cubes can safely be removed from consideration without loss of information. G. Hazard-Free Covers A hazard-free cover of function is a cover (i.e., set of implicants) of whose AND-OR implementation is hazard free for a given set of specified input transitions. (It is assumed below that the function is defined for all specified transitions; the function is undefined for all other input states.) Theorem II.11 (Hazard-Free Covering [29]): A sum of is a hazard-free cover for function for the products set of specified input transitions if and only if: intersects the OFF-set of ; a) no product of b) each required cube of is contained in some product of ; intersects any (nontrivial) privileged c) no product of cube illegally. Theorem II.11(a) and (c) determines the implicants that may appear in a hazard-free cover of a function called dynamichazard-free (dhf-) implicants. Definition II.12: A dhf-implicant is an implicant that does illegally. A dhfnot intersect any privileged cube of prime implicant is a dhf-implicant contained in no other dhf-implicant. An essential dhf-prime implicant is a dhf-prime implicant that contains a required cube contained in no other dhf-prime implicant. Theorem II.11(b) defines the covering requirement for a must be hazard-free cover of : every required cube of covered, that is, contained in some cube of the cover. Thus, the two-level hazard-free logic minimization problem is to find a minimum cost cover of a function using only dhf-prime implicants where every required cube is covered. The difference between two-level hazard-free logic minimization and the well-know classic two-level logic minimization problem (e.g., solved by Quine–McCluskey algorithm) is that, in the hazard-free case, dhf-prime implicants replace prime implicants as the covering objects, and required cubes replace minterms as the objects to be covered. In general, the covering conditions of Theorem II.11 may not be satisfiable for an arbitrary Boolean function and set of transitions [38], [29]. This case occurs if conditions (b) and (c) cannot be satisfied simultaneously. A hazard-free minimization example is shown in Fig. 1. is a transition; There are four specified transitions. is a it gives rise to one required cube [see part (a)]. transition; it gives rise neither to required cubes nor and are transitions. Each to privileged cubes. of these two transitions gives rise to two required cubes [see (a)] and one privileged cube [see (b)]. A minimum hazardfree cover is shown in part (c). Each required cube is covered, and no product in the cover illegally intersects any privileged cube. In contrast, the cover in part (d) is not hazard free since priv-cube-1 is intersected illegally (highlighted minterm) by THEOBALD AND NOWICK: TWO-LEVEL HAZARD-FREE LOGIC MINIMIZATION 1133 (a) (b) (c) (d) Fig. 1. Two-level hazard-free minimization example. (a) shows the set of required cubes (shaded) and the set of transition cubes (dotted). (b) shows the set of privileged cubes (shaded). (c) shows a minimal hazard-free cover. (d) shows a minimum-cost cover that is not hazard free, since it contains a logic hazard. product In particular, this product may lead to a glitch during transition III. HEURISTIC HAZARD-FREE MINIMIZATION: ESPRESSO-HF A. Overview H. Exact Hazard-Free Minimization Algorithms A single-output exact hazard-free minimizer has been developed by Nowick and Dill [29]. It has recently been extended to hazard-free multivalued minimization3 by Fuhrer et al. [12]. The latter method, called HFMIN, has been the fastest minimizer for exact hazard-free minimization. HFMIN makes use of ESPRESSO-II to generate all prime implicants, then transforms them into dhf-prime implicants, and finally employs ESPRESSO-II’s MINCOV to solve the resulting unate covering problem. Each of the algorithms used in the above three steps is critical, i.e., each has a worst case run time that is exponential. As a result, HFMIN cannot solve several of the more difficult examples. Very recently, Rutten [33], [32] has proposed an alternative exact method. However, his method has yet to be evaluated on difficult examples, e.g., on those that cannot be easily solved by HFMIN (see Section VI-C for more details). 3 It is well known that multioutput minimization can be regarded as a special case of multivalued minimization [30]. The goal of heuristic hazard-free minimization is to find a very good, but not necessarily exactly minimum, solution to the hazard-free covering problem. The basic minimization strategy of ESPRESSO-HF for hazard-free minimization is similar to the one used by ESPRESSO-II. However, we use additional constraints to ensure that the resulting cover is hazard free, and the algorithms are significantly different. One key distinction is in the use of the unate recursive paradigm in ESPRESSO-II, i.e. to decompose operations recursively leading to efficiently solvable suboperations on unate functions. To the best knowledge of the authors, the unate recursive paradigm cannot be applied directly to ESPRESSOII-like heuristic hazard-free minimization. (In [33] and [32], a unate recursive method was proposed, but only for use in the dhf-prime generation step for exact hazard-free minimization; see Section VI-C.) The intuitive reason for this observation is that the operators in ESPRESSO-II manipulate covers. For example, the “many” ON-set minterms (objects to be covered) can typically be stored compactly and manipulated efficiently as an ON-set cover of “a few” cubes. In contrast, required 1134 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 Fig. 2. The ESPRESSO-HF algorithm. cubes cannot be combined into larger cubes without loss of information, which means that the basis for the unate recursive paradigm, i.e., the concept of covers, becomes obsolete. We therefore follow the basic steps of ESPRESSO-II, modified to incorporate hazard-freedom constraints, but without the use of unate recursive algorithms. However, because of the constraints and granularity of the hazard-free minimization problem, high-quality results are still obtained rapidly even for large examples. In this subsection, we describe the basic steps of the algorithm, concentrating on the new constraints that must be incorporated to guarantee a cover to be hazard free. We then describe the individual steps in detail in later subsections. As in ESPRESSO-II, the size of the cover is never increased. In addition, after an initial phase, the cover always represents that is also hazard free. a valid solution, i.e., a cover of Pseudocode for the algorithm is shown in Fig. 2. The first step of ESPRESSO-HF is to read in PLA files specifying a Boolean function and a set of specified functionhazard-free transitions These inputs are used to generate the the set of privileged cubes and their set of required cubes and the OFF-set Generation corresponding start points of these sets is immediate from the earlier lemmas (see also [29]).4 The set can be regarded both as an initial cover of the function and as a set of objects to be covered. Unlike does not in ESPRESSO-II, however, the given initial cover is a cover of general represent a valid solution: while 4 The algorithm does not need an explicit cover for the don’t care set because the operations only require the OFF-set to check if a cube is valid. it is not necessarily hazard free. Therefore, processing begins by first expanding each required cube into the uniquely defined minimum dhf-implicant covering it, or the detection that this is impossible, denoted by “undefined.” The latter case indicates that the hazard-free minimization problem has no solution (see Section III-J). Otherwise, the result is an initial hazard-free and set of objects to be covered cover The next step is to identify essential dhf-implicants using a modified EXPAND step. The algorithm uses a novel approach to identifying equivalence classes of implicants, each of which is treated as a single implicant. Essential implicants, as well as all required cubes covered by them, are then removed from and respectively, resulting in a smaller problem to be solved by the main loop. Before the main loop, the current cover is also made irredundant. Next, as in ESPRESSO-II, ESPRESSO-HF applies the three operators REDUCE, EXPAND, and IRREDUNDANT to the current cover until no further improvement in the size of the cover is possible. Since the result may be a local minimum, the operator LAST_GASP is then applied to find a better solution using a different method. EXPAND uses new hazardfree notions of essential parts and feasible expansion. The other steps differ from ESPRESSO-II as well. At the end, there is an additional step to make the resulting implicants dhf-prime, MAKE_DHF_PRIME, since it is desirable to obtain a cover that consists of dhf-prime implicants. The motivation for this step will be made clear in the remaining subsections. In addition to the steps shown in Fig. 2, our implementation has several optional pre- and postprocessing steps. B. Dhf-Canonicalization of Initial Cover In ESPRESSO-II, the initial cover of a function is provided . This cover is a seed solution, which by its ON-set, is iteratively improved by the algorithm. By analogy, in ESPRESSO-HF, the initial cover is provided by the set of However, unlike ESPRESSO-II, our initial required cubes specification does not in general represent a solution: though is a cover, it is not necessarily hazard free. Therefore, processing begins by expanding each required cube into the uniquely defined minimum dhf-implicant containing it. This expansion represents a canonicalization step, transforming a potentially hazardous initial cover into a hazard-free initial cover Example: Consider the function in the Karnaugh map of specified multiple-input transitions is of Fig. 3. A set transitions, each indicated by arrows. There are two (start point corresponding to a privileged cube: and (start point The initial cover is given by the set of required cubes: This cover is hazardous. In particular, consider the corresponding to the required cube transition from to . Required cube illegally intersects privileged cube since it intersects but does not contain To avoid illegal intersection, must be expanded to the smallest cube that also contains THEOBALD AND NOWICK: TWO-LEVEL HAZARD-FREE LOGIC MINIMIZATION Fig. 3. Canonicalization example. : supercube However, this new now illegally intersects privileged cube cube since it does not contain Therefore, cube in turn must be expanded to the smallest cube containing : supercube The resulting expanded has no illegal intersections and is therefore a cube, dhf-implicant. In this example, is a hazard-free expansion of called a canonical required cube; it can therefore replace in the initial cover. (Note that such a canonicalization is feasible if and only if the hazard-free covering problem has a solution; see Section III-J.) It is easy to see that each required cube has a unique corresponding canonical required cube. Suppose there are two which contain some distinct minimal dhf-implicants, and required cube In this case, we now show that we can construct a dhf-implicant that is smaller than either cube: the and Clearly, implicant intersection of contains Furthermore, if were not a dhf-implicant, then it would intersect some privileged cube illegally, i.e., intersect but not contain its start point However, this would and intersected mean that both original implicants, but at least one of them (say did not contain As a result, would not be a dhf-implicant, since it would illegally intersect thus contradicting our assumption. Therefore, is a dhf-implicant that contains and so and could not have been minimal. In sum, each required cube has a unique corresponding canonical required cube, which contains it. Based on the above discussion, an initial set of required of canonical cubes is replaced by the corresponding set required cubes. This set is then minimized with respect to is a valid hazard-free cover single-cube containment. of the function to be minimized and is used as an initial cover for the minimization process. Interestingly, has a second role as well: it can also be used as the initial set defines a new of objects to be covered. In particular, (not ) must be contained covering problem: each cube of in some dhf-implicant. It is straightforward to show that the two covering problems are equivalent, i.e., a dhf-implicant contains a required cube in if and only if also contains 1135 the corresponding canonical required cube of in To see this, suppose that contained but did not contain the In this case, could not be a canonical required cube of dhf-implicant, since it must illegally intersect at least one of those privileged cubes that caused to be expanded into its canonical required cube. In the above example, any dhf-implicant that contains remust also contain canonical required quired cube Therefore, the hazard-free minimization cube problem is unchanged, but canonical required cubes are now is that it may have smaller used. An advantage of using i.e., being a more efficient representation of the size than are in general larger than problem. Also, since the cubes in the EXPAND operation may be the corresponding ones in sped up. To conclude, the new set of canonical required cubes replaces the original set of required cubes as both i) the initial cover and ii) the set of objects to be covered. Henceforth, the term “set of required cubes” will be used to refer to set We formalize the notion of canonicalization below. Definition III.1: The dhf-supercube of a set of cubes with respect to function and transitions indicated as is the smallest dhf-implicant containing the cubes of The superscript is omitted when it is clear from is computed by the simple the context. algorithm shown in Fig. 4. The canonical required cube of a required cube can now The be defined as the dhf-supercube of the set computation of dhf-supercubes for larger sets will be needed to implement some of the operators presented in the sequel. C. EXPAND In ESPRESSO-II, the goal of EXPAND is to enlarge each implicant of the current cover in turn into a prime implicant. As an implicant is expanded, it may contain other implicants of the cover that can be removed; hence the cover cardinality is reduced. If the current implicant cannot be expanded to contain another implicant completely, then, as a secondary goal, the implicant is expanded to overlap as many other implicants of the current cover as possible. In ESPRESSO-HF, the primary goal is similar: to expand a dhf-implicant of the current cover to contain as many other dhf-implicants of the cover as possible. However, EXPAND in ESPRESSO-HF has two major differences. First, unlike ESPRESSO-II, expansion in some literal (i.e., “raising of entries”) may imply that other expansions be performed. That is, raising of entries is now a binate problem, not a unate problem. In addition, ESPRESSO-HF’s EXPAND uses a different strategy for its secondary goal. By the hazard-free covering theorem, each required cube needs to be contained in some cube of the cover. Therefore, as a secondary goal, an implicant is expanded to contain as many required cubes as possible. We now describe the implementation of EXPAND in ESPRESSO-HF. Pseudocode for the expansion of a single cube is shown in Fig. 5. 1136 Fig. 4. IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 Supercubedhf computation. Fig. 5. Expand (for a cube a). 1) Determination of Essential Parts and Update of Local Sets: As in ESPRESSO-II, free entries are maintained to accelerate the expansion [30]. Free entries indicate which literals of an implicant are candidates for removal during the expansion process. To explain this concept in a unified way, the current implicant and its free entries are represented in positional is represented cube notation [30]. As an example, in positional cube notation as 01 10 11 01, i.e. where literal is encoded as 01 (10). Thus, each literal in has a corresponding 0 in the positional cube notation, and changing, or raising, the 0 to 1 corresponds to removing this literal from the implicant. Initially, a free entry is assigned a 1 (0) if the current implicant to be expanded has a 0 (1) in the corresponding position. For the above example, the free entries are: 10 01 00 10. An overexpanded cube is defined as the cube where all its free entries have been raised simultaneously. As in [30], an essential part is one that can never, or always, be raised. However, our definition of essential parts is different from ESPRESSO-II, since a hazard-free cover must be maintained. We determine essential parts in procedure update, described below. First, we determine which entries can never be raised This is achieved by and remove them from that has distance searching for any cube in the OFF-set in this entry, using the same approach as in 1 from ESPRESSO-II. Next, we determine which parts can always be raised, raise This step differs them, and remove them from from ESPRESSO-II. In ESPRESSO-II, a part can always be raised if it is 0 in all cubes of the OFF-set, That is, it is guaranteed that the expanded cube will never intersect the OFF-set. In contrast, in ESPRESSO-HF, we must ensure that an implicant is also hazard free: it cannot intersect the OFF-set, nor can it illegally intersect a privileged cube. Unlike ESPRESSO-II, this is achieved by searching for any column that i) has only 0’s in and ii) where, for each privileged cube in having a will be 1 in this column, the corresponding start point contained by the expanded cube Example: Fig. 1(a) indicates the set of required cubes, which forms an initial hazard-free cover. Consider the (11010101, in positional cube notation). As cube and can in ESPRESSO-II, the zero-entries for literals never be raised, since the cube would intersect the OFF-set. However, after updating the free entries, ESPRESSO-II indicates can always be raised, since the resulting cube that literal will never intersect the OFF-set. In contrast, in ESPRESSO-HF, results in an illegal intersection with privileged raising so it cannot “always be raised.” cube Since hazard-free minimization is somewhat more constrained, the expansion of a cube can be accelerated by the These sets following new operations on two local sets: and is initially assigned are associated with cube the set of privileged (OFF-set) cubes. Both sets are updated as expansion proceeds (in procedure update). where the correspond1) Remove privileged cubes from ing start point is already covered by (since no further checking for illegal intersection is required). to the local OFF2) Move privileged cubes from set if the overexpanded cube does not include set can never be the corresponding start points (since expanded to include these start points, one must avoid intersection with these privileged cubes entirely). to the local OFF-set 3) Move privileged cubes from where , start point intersects the OFF-set ( can never be expanded to include these start points, therefore one must avoid intersection with the cubes entirely). 2) Detection of Feasibly Covered Cubes of : In ESPRESSO -II, a cube in is expanded through a supercube operation. A is said to be feasibly covered by if supercube cube in is an implicant. In ESPRESSO-HF, this definition needs to be modified to insure hazard-free covering, after expansion of cube Definition III.2: A cube in is dhf-feasibly covered by if is defined. This definition insures that the resulting expanded cube, is i) an implicant (does not intersect OFF-set) and ii) is also a dhf-implicant (does not intersect any THEOBALD AND NOWICK: TWO-LEVEL HAZARD-FREE LOGIC MINIMIZATION privileged cube illegally). Effectively, this definition canonicalizes the resulting supercube to produce a dhf-implicant. may properly contain superThat is, since the former may be expanded through a cube series of implications in order to reach the minimum dhfimplicant that contains both and Using this definition, the following is an algorithm to find dhf-feasibly covered cubes of While there are cubes in that are dhf-feasibly covered by iterate the following: cube where is a dhfReplace by feasibly covered cube such that the resulting cube will cover as many cubes of the cover as possible. Covered cubes are then removed, using the “single-cube-containment” operator, thus reducing the cover cardinality. Determine essential parts and update local sets (see above). 3) Detection of Feasibly Covered Cubes of : Once cube can no longer be feasibly expanded to cover any other we still continue to expand it. This is motivated cube of by the hazard-free covering theorem, which states that each required cube needs to be contained in some cube of the cover. Therefore, as a secondary goal, cube is expanded to contain as many required cubes as possible. The strategy used in this substep is similar to that used in the preceding one, i.e., while that are dhf-feasibly covered by cube there are cubes in iterate the following. by where is a dhfReplace feasibly covered required cube such that the resulting cube will cover as many required cubes not already contained in as possible. Covered required cubes are then removed, using the “single-cube-containment” operator. Determine essential parts and update local sets (see above). 4) Constraints on Hazard-Free Expansion: In ESPRESSOII, an implicant is expanded until no further expansion is possible, i.e., until the implicant is prime. Two steps are used: i) expansion to overlap a maximum number of cubes still covered by the overexpanded cube and ii) raising of entries to find the largest prime implicant covering the cube. In ESPRESSO-HF, however, we do not implement these remaining EXPAND steps, based on the following observation. The result of our EXPAND steps (see Sections III-C2 and III-C3) guarantees that a dhf-implicant can never be further expanded to contain additional required cubes. Therefore, by the hazard-free covering theorem, no additional objects (required cubes) can possibly be covered through further expansion. In contrast, in ESPRESSO-II, expansion steps i) and ii) may in fact result in covering of additional ON-set minterms. Because of this distinction, the benefit of further expansion is mitigated. Therefore, in general, our EXPAND algorithm makes no attempt to transform dhf-implicants into dhf-prime implicants. However, since expansion to dhf-primes is important for literal reduction and testability, it is included as a final postprocessing step: MAKE_DHF_PRIME (see Fig. 2). D. Essentials Essential prime implicants are prime implicants that need to be included in any cover of prime implicants. Therefore, it 1137 Fig. 6. Essential example. is desirable to identify them as soon as possible to make the resulting problem size smaller. On the one hand, we know of no efficient solution for identifying the essential dhf-primes using the unate recursion paradigm of ESPRESSO-II. On the other hand, the hazard-free minimization problem is highly constrained by the notion of covering of required cubes, thus allowing a powerful new method to classify essentials as equivalence classes. Example: Consider Fig. 6. The required cube is covered by precisely two dhf-prime implicants: and which cover no other required cubes. Neither nor is an essential dhf-prime, since is covered by both. or (not both) must be included in And yet, clearly, either any cover of dhf-primes. Also, if we assume the standard cost and are of equal cost. function of cover cardinality, Definition III.3: Two dhf-prime implicants are equivalent if they cover the same set of required cubes. An equivalence class of dhf-prime implicants is maximal if its dhf-primes cover a set of required cubes that is not covered by any other equivalence class. A maximal equivalence class of dhf-prime implicants is essential if its dhf-primes cover at least one required cube that is not covered by any other maximal equivalence class. of dhf-primes is In the above example, the set an equivalence class, since both dhf-primes cover the same set In fact, the class is an essential of required cubes equivalence class, since it is the only equivalence class that covers the required cube ESPRESSO-II computes essentials after initial EXPAND and IRREDUNDANT steps. In contrast, ESPRESSO-HF computes essentials as part of a modified EXPAND-step. The new algorithm is outlined as follows. of The algorithm starts with the initial hazard-free cover required cubes. To simplify the presentation, assume that one seed cube is selected and expanded greedily, using EXPAND, This implicant is characterized by the to a dhf-implicant of required cubes that it contains. Dhf-implicant set thus corresponds to an equivalence class of dhf-primes that Since EXPAND guarantees that covers a maximal cover number of required cubes, this equivalence class is also maximal. Moreover, this class is an essential equivalence 1138 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 class if contains some be expanded into any other can be To check if imal equivalence class, a for each required cube required cube which cannot maximal equivalence class. expanded into a different maxsimple pairwise check is used: not covered by determine if is feasible. If no such feasible is called a distinguished required expansion exists for cube, and therefore the equivalence class corresponding to is essential. Otherwise, the process is repeated for every contained in If corresponds to an required cube essential equivalence class, then is removed from the cover. are removed, In addition, all required cubes covered by since it is ensured that they will be covered. This step can result in “secondary essential” equivalence classes. In fact, due to the removal of required cubes, more dhf-prime implicants become equivalent to each other. As a consequence, further equivalence classes may become essential. The procedure iterates until all essentials are identified. The above discussion seems to imply that the essentials step is more or less quadratic in the number of required cubes, i.e., very inefficient. However, by making use of techniques similar to the ones described in the EXPAND Section III-C, e.g., by using an overexpanded cube, the number of necessary calls can be reduced dramatically. Therefore, in practice, essentials can be identified efficiently, and the problem size is usually significantly reduced (see Section VI). E. REDUCE The goal of the REDUCE operator is to set up a cover that is likely to be made smaller by the following EXPAND step. is maximally To achieve this goal, each cube in a cover reduced in turn to a cube such that the resulting set of cubes is still a cover. ESPRESSO-II uses the unate recursive paradigm to maximally reduce each cube. Since ESPRESSO-HF is a required-cube covering algorithm, there is no obvious way to use this paradigm. Fortunately, the hazard-free problem is more constrained, making it possible to use an efficient enumerative approach based on required cubes. Our REDUCE algorithm is as follows. The algorithm reduces each cube in the cover in order. In particular, a cube is reduced to the smallest dhf-implicant that covers all required cubes that are uniquely covered by (i.e., contained is the in no other cube of the cover ). That is, if set of required cubes that are uniquely covered by then is replaced by Note that the outcome of this algorithm depends on the order in which the cubes of the cover are processed. Suppose is reduced before and that and cover some required cube but no other cube of covers If is reduced to a cannot be reduced to a cube that does not cover then cube that does not cover minterms) may in fact be irredundant with respect to covering of required cubes. Therefore, as in REDUCE, our approach is required-cube based. Considering the Hazard-Free Covering Theorem, it is straightforward that IRREDUNDANT can be reduced to a covering problem of the cubes in by the cubes in That is, the problem reduces to a minimum-covering problem of i) required cubes using ii) dhf-implicants in the current cover. In practice, the number of required cubes and cover cubes usually make the covering problem manageable. ESPRESSO-II’s MINCOV can be used to solve this covering problem exactly or heuristically (using its heuristic option). G. Last Gasp The inner loop of ESPRESSO-HF may lead to a suboptimal local minimum. The goal of LAST_GASP is to use a different approach to attempt to reduce the cover size. In ESPRESSOis independently reduced to the smallest II, each cube cube containing all minterms not covered by any other cube In contrast, ESPRESSO-HF computes, for each the of smallest dhf-implicant containing all required cubes that are not covered by any other cube in As in ESPRESSO-II, cubes that can actually be reduced by Each such this process are added to an initially empty set is then expanded in turn with the goal to cover at using the operator, least one other cube of Finally, and, if achieved, the expanded cube is added to with the hope the IRREDUNDANT operator is applied to to escape the above-mentioned local minimum. H. Make-dhf-prime The cover being constructed so far does not necessarily consist of dhf-primes. It is usually desirable to expand each dhf-implicant of the cover to make it dhf-prime as a last step. This can be achieved by a modified EXPAND step. A simple greedy algorithm will expand an implicant to a dhf-prime: while dhf-feasible, raise a single entry of I. Pre- and Postprocessing Steps ESPRESSO-HF includes optional pre- and postprocessing steps. In particular, the efficiency of ESPRESSO-HF depends very much on the size of the ON-set and OFF-set covers that are given to it. Thus, ESPRESSO-HF includes an optional preprocessing step that uses ESPRESSO-II to find covers of smaller size for the initial ON-set and OFF-set.5 ESPRESSO-HF also includes a postprocessing step to reduce the literal count of a cover, similar to ESPRESSO-II’s MAKE_SPARSE. J. Existence of a Hazard-Free Solution F. Irredundant As indicated earlier, for certain Boolean functions and sets of transitions, no hazard-free cover exists [29]. The currently used exact hazard-free minimization method HFMIN is only able to decide if a hazard-free solution exists after generating ESPRESSO-II uses the unate recursive paradigm to find an irredundant cover. In our case, we cannot employ the same algorithm, since a “redundant cover” (according to covering of and OFF-set are necessary to form the initial set of required cubes More important, the OFF-set is used to check if a cube expansion is valid (see Fig. 4). 5 ON-set Q: THEOBALD AND NOWICK: TWO-LEVEL HAZARD-FREE LOGIC MINIMIZATION 1139 Based on this approach, we present a new implicit method for exact two-level hazard-free logic minimization in Section V. A. Overview and Intuition Fig. 7. Existence example. all dhf-prime implicants. A hazard-free solution does not exist if and only if the dhf-prime implicant table includes at least one required cube not covered by any dhf-prime implicant. However, since the generation of all dhf-primes may very well be infeasible6 for even medium-sized examples, it is important to find an alternative approach. We now introduce a new theorem to check for the existence of a hazard-free solution, without the need to generate all dhfprime implicants. This theorem leads directly to a fast and simple algorithm that is incorporated into ESPRESSO-HF. Theorem III.4: Given a function and a set of specified a solution of the function-hazard-free input transitions of two-level hazard-free logic minimization problem exists if and is defined for each required cube only if The proof is immediate from the discussion in Section III-B. Example: Consider the Boolean function in Fig. 7, with four specified input transitions. To check for existence of a hazard-free solution, we compute for each required cube Except for it holds since no privileged cube is that intersected illegally. To compute note that privileged cube is intersected illegally, i.e. Since now intersects privileged cube we get , leading does not exist directly to the fact that intersects the OFF-set. Thus, a hazard-free cover because does not exist for this example. IV. A NOVEL APPROACH TO INCORPORATING HAZARDFREEDOM CONSTRAINTS WITHIN A SYNCHRONOUS FUNCTION After having discussed the heuristic hazard-free minimization problem in the previous section, we will now shift our discussion to the exact hazard-free minimization problem. We begin by presenting a novel technique that recasts the dhf-prime implicant generation problem as a prime generation problem for a new synchronous function, with extra inputs. 6 This refers to “explicit representations”; we will show later that “implicit representations” very often are feasible. We first give a simple overview of our entire method. Details and formal definitions are provided in the remaining sections. Our approach is to recast the generation of dhfinto the prime implicants of an asynchronous function generation of prime implicants for a new synchronous function Here, hazard-freedom constraints are incorporated into the function by adding extra inputs. (The exact definition of is given in Section IV-B.) An overview of the method is best illustrated by a simple example. Example: Consider Fig. 8. The Karnaugh map in part defined over three variables A represents a function The shaded area corresponds to the only (the second privileged cube nontrivial privileged cube of [101, 100] is trivial; see Section II-F). We now define a shown in part B. Function is new synchronous function That is, obtained from by adding a single new variable is defined over four variables: In general, to generate one new -variable is added for each nontrivial . Next, the prime implicants of the privileged cube in synchronous function are computed (shown in part B as ovals). Finally, a simple filtering procedure is used to filter out those prime implicants that correspond to those in that intersect the privileged cube illegally. The remaining prime implicants of are shown in part C. We then “delete” the -dimension from the prime implicants and obtain the (part D). entire final set of dhf-prime implicants of Our approach is motivated by the fact that dhf-primeimplicants are more constrained than prime implicants of the same function. While prime implicants are maximal implicants that do not intersect the OFF-set of the given function, dhfprime-implicants, in addition, must also not intersect privileged cubes illegally. Thus, there are two different kinds of constraints for dhf-prime-implicants: “maximality” constraints and “avoidance of illegal intersections” constraints. Our idea is to unify these two types of constraints, i.e., to transform the avoidance constraints into maximality constraints so that dhf-primes can be generated in a uniform way. Intuitively, this unification can be achieved by adding auxiliary variables, i.e., by lifting the problem into a higher dimensional Boolean space. In summary, the big picture is as follows. The definition of ensures that all dhf-prime implicants of (dhf-Prime(f,T)) can be easily obtained from the set of prime implicants of While may also include certain products that are nonhazard free, these latter ones are filtered out easily using a postprocessing step. B. The Auxiliary Synchronous Function We now explain how the synchronous function is derived. is a single-output For simplicity, assume for now that function. Suppose is defined over the set of variables and that the set of transitions gives rise to the set of 1140 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 (a) (d) (b) (c) Fig. 8. Example for recasting prime generation. (a) shows the function (f ; T ) whose dhf-primes are to be computed. (b) shows the auxiliary synchronous function g and its primes. (c) shows primes of g that do not intersect illegally. (d) shows the final dhf-primes of (f; T ) after deleting the z1 variable. nontrivial privileged cubes The ; that idea is to define a function over is, one new variable is added per privileged cube. Formally, is defined as follows: Function is the product of and some function that depends on the new inputs. The intuition behind the definition of is half of the domain, is defined as while that, in the half of the domain, is defined as but with in the “filled in” with all 0’s (i.e., is the th privileged cube “masked out”). Example: As an example, Fig. 8(a) shows a Boolean funcwith privileged cube (highlighted in gray). tion Fig. 8(b) shows the corresponding new function with added In the half, function is identical to In variable half, is identical to except that is 0 throughout the which corresponds to the privileged cube the entire cube in the original function In particular, function where as is defined C. Prime Implicants of Function To understand the role of function we consider its prime implicants We start by considering a function that has only one Let be any implicant of the function privileged cube that is contained in the plane of Since the plane is defined as also corresponds to an implicant of Now, consider the expansion of into the plane of function There are two possibilities: either i) can expand plane or ii) cannot expand into the plane. into plane means that In case i), expansion of into the is identical to in the expanded region. Therefore, does in the original function (if not intersect privileged cube in the plane, and it did, would have all 0’s in expansion would be impossible). In case ii), expansion into THEOBALD AND NOWICK: TWO-LEVEL HAZARD-FREE LOGIC MINIMIZATION the plane is impossible. In this case, must intersect in function has all 0’s in In summary, may or may not be able to expand from into planes. Expansion can occur precisely in the original if does not intersect the privileged cube function, since function is identically defined as in both planes “outside” the privileged cube. Expansion cannot occur because in the if intersects the privileged cube plane, the privileged cube is filled in entirely with 0’s. of in Example: Consider the minterm of Fig. 8(b), which corresponds to the minterm can be expanded into the plane into the prime implicant (shaded oval). Intuitively, the expansion is of : does not intersect the privileged cube, i.e., possible since which corresponds to the privileged cube of the cube However, the implicant the original function (oval with thick dark border) of cannot be expanded into the plane: it intersects the privileged cube, and therefore plane is filled with the corresponding region in the 0’s. Note that prime generation is an expansion process that continues until no further expansion is possible. Let us now consider the general case, i.e., where may have more than one privileged cube. We show that the support variables of each prime of precisely define which privileged cubes are intersected by the corresponding implicant in Let be any prime implicant of Here, is a positive or negative -literal.7 However, can only be a negative -literal. The reason is that is a negative unate function in -variables (by the definition of ), and therefore prime implicants of will never include the restriction of to the positive -literals. We indicate by -literals, i.e., Note that is an implicant of by the definition of We now show that the presence, or absence, of literals in prime implicant indicates precisely which privileged cubes If includes literal then intersects are intersected by privileged cube in function To see this, note that since is prime, clearly cannot be further expanded into the plane. As a result, must intersect privileged cube in the original function On the other hand, if does not include then does not intersect Intuitively, the primes are maximal in two senses: they are maximally expanded in or maximally nonintersecting of privileged cubes, in some combination, which is explicitly indicated by the set of support of the primes. In sum, the key observation is that the set of support of a prime implicant of immediately indicates which privileged in cubes are intersected by the corresponding implicant This observation will be critical in obtaining the final set of dhf-prime implicants of 7 Note that q may not depend on all of the x-variables: some may not appear here. 1141 D. Transforming into - Once is computed, can be directly computed. The important insight for this computation fall into three classes is that the prime implicants of Each prime is with respect to a specific privileged cube distinguished based on if and how it intersects the privileged in i.e., based on the intersection of with : cube that do not intersect the Class 1) prime implicants does not intersect ; privileged cube, i.e., Class 2) prime implicants that intersect the privileged intersects and contains cube legally, i.e., its start point; Class 3) prime implicants that intersect the privileged intersects but does not cube illegally, i.e., contain the start point. Prime implicants that fall into Classes 2) and 3) (i.e., intersects some privileged cube) can be immediately identified by the observation of the previous subsection. Those that fall into Class 3) can then be identified, and removed, using a contains the start simple containment check: determine if point of each intersected privileged cube. can therefore be computed as The set Filter out all prime follows. Start with the set implicants that fall in Class 3) with respect to the first privileged cube. Then, filter out all prime implicants that fall in Class 3) with respect to the second privileged cube, and so on. Finally, one obtains a set such that each of its elements is if restricted to the -variables. a valid dhf-implicant of The reason is that, first, all primes of are implicants of if restricted to -variables, and second, the filtering removed any element that intersected any privileged cube illegally. Therefore, the set only includes dhf-implicants. In fact, it also This fact will be contains all dhf-prime-implicants of proved in the next subsection. and its prime Example: Fig. 8(b) shows function implicants, Fig. 8(c) shows the result of filtering out primes that illegally In intersect regions corresponding to privileged cubes in (oval with thick dark border) falls into Class this case, 3) with respect to : it is deleted since it has a -literal, i.e., intersects the region corresponding to privileged cube and, in addition, does not contain the start point However, (shaded oval) falls into Class 1): it is not deleted since it does not have a -literal and therefore does not intersect the region corresponding to the privileged The remaining two primes and cube fall into Class 2): they intersect the region corresponding and also contain the start point. Fig. 8(d) shows the to result of step 3, which deletes the -literals in each cube. which is the final set We obtain Note that the introduction of the -variable which is not ensures that the dhf-implicant of since it is contained by the prime a prime implicant of is nevertheless generated. implicant 1142 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 E. Formal Characterization of in Terms of Function - In this subsection, based on the above discussion, we present the main result of this section: a new formal characterization of We use the following notations. and denote the positive and negative cofactors of with respect to respectively. denotes an operator on a set of variable cubes that removes all -literals of each cube. As an example, 8 The -operator on a set of cubes (single-cube-containment) removes those cubes contained in other cubes. Let Theorem IV.1: Given be the set of nontrivial privileged cubes,9 and be the set of corresponding start points. Define Then the set - can be expressed as follows: (1) includes implicants Intuition: The set that do not intersect the privileged cube The set includes implicants of that legally intersect i.e., contain the corresponding start The operation ensures that only those implicants point remain that are legal with respect to all privileged cubes, removes implicants i.e., that are dhf-implicants. The contained in other implicants to yield the final set of dhfprime-implicants. is also Proof: “ ” [any product in contained in (1)]. . Then does not intersect any Let privileged cube illegally, i.e., for each privileged cube, it holds that either contains the corresponding start point or does not intersect the privileged cube at all. intersects legally and does not Suppose —i.e., is an implicant of ; intersect is an implicant of then is a prime implicant of because of the following. results in a cube that is not an 1) Removing (any) and hence not an implicant of implicant of 2) Removing (any) positive or negative literal (of ) results in a cube such that its restriction to the , is not a dhf-prime implicant. Thus literals, either intersects the OFF-set of or intersects for some privileged cube , and is therefore of 8 RemZ can formally be expressed by existential quantification over z variables, i.e., RemZ (P ) = fx 2 fx1 ; x1 ; x2 ; x2 ; 1 1 1 ; xn ; xn g3 j9z 2 3 fz1 ; z 1 ; z2 ; z 2 ; 1 1 1 ; z ; z g : xz 2 P g: l 9 In l the theorem, pi denotes the complement function of p1 = x1 x2 x4 : Then, p1 = x1 x2 x4 = x1 + x2 + x4 : pi : Example: no longer an implicant of In either case, is not an implicant of Thus, for each is by construction in at least one of or Therefore, is contained in the intersection of those sets. cannot be filtered out by the -operator since Also, by construction all cubes contained in (1) are dhf-implicants. Thus, is contained in (1). “ ” (any product contained in (1) is also contained in dhf-Prime(f)). We show that is not Assume contained in (1). is a dhf-implicant that is strictly contained in Case i) some dhf-prime implicant. Then is filtered out -operator and therefore not because of the contained in (1). Case ii) is not a dhf-implicant. By construction, all cubes contained in (1) are dhf-implicants: the intersection ensures that each cube is valid with respect to all privileged cubes, i.e., for each privileged cube, the cube either does not intersect it or else contains the start point. Thus, cannot be contained in (1). F. Multioutput Case For simplicity of presentation only, it was assumed that is a single-output function. However, it is well known [34] that multioutput logic minimization can be reduced to single-output minimization. Based on this theorem, the above characterization carries over in a straightforward way to multioutput functions. All examples given later in the experimental results section are multioutput functions. V. EXACT HAZARD-FREE MINIMIZATION: IMPYMIN Based on the ideas of the previous section, we are now able to present a new exact implicit minimization algorithm for multioutput two-level hazard-free logic. A. Implicit Two-Level Logic Minimization: SCHERZO We first briefly review the state-of-the-art synchronous exact two-level logic minimization algorithm, called SCHERZO [8], which forms a basis of our new hazard-free implicit minimization method. SCHERZO has two significant differences from classic minimization algorithms like the well-known Quine–McCluskey algorithm: • SCHERZO uses data structures like BDD’s [3] and ZBDD’s [21] to represent Boolean functions and sets of products very efficiently. Thus, the complexity of the minimization problem is shifted, and the cost of the cyclic core computation10 is now independent of the number of products (e.g., the number of prime implicants) that are manipulated. 10 A set covering problem can be reduced in size by repeated elimination of essential elements and application of dominance relations. The remaining set covering problem (if any) is called the cyclic core. THEOBALD AND NOWICK: TWO-LEVEL HAZARD-FREE LOGIC MINIMIZATION • SCHERZO includes new algorithms that operate on these data structures. The motivation is that the logic minimization problem can be considered as a set covering problem over a lattice. More specifically, both the covering objects and the objects to be covered are subsets of the lattice of all Boolean products (over the set of literals). A new cyclic core computation algorithm then uses two and which operate on and endomorphisms , respectively, to capture dominance relations and to compute the fixpoint C, which can be shown to be isomorphic to the cyclic core. Algorithm: Input: Output: SCHERZO Boolean function . All minimum two-level implementations of . of Prime(f) (the set of all 1) Compute the ZBDD or covering objects). Here, is prime implicants of given as a BDD. of the set of ON-set minterms 2) Compute the ZBDD of (i.e., the objects to be covered). 3) Solve the implicit set covering problem (Note that “ ” replaces “ ” usually used to describe the relation between the two sorts of objects of a covering problem, since our set covering problem is considered over a lattice, as explained above.) B. Implicit Two-Level Hazard-Free Logic Minimization: IMPYMIN Nowick/Dill reduced two-level hazard-free logic minimization to a unate covering problem (see Section II) where each required cube must be covered by at least one dhf-prime implicant. As with synchronous logic minimization in SCHERZO, hazard-free logic minimization can also be considered over the lattice of the set of products (over the set of literals). The major difference from synchronous two-level logic minimization is the setting up of the covering problem. In particular, a method efficiently, is needed that computes the set preferably in an implicit manner. To do so, we use the of Section IV. The new characterization of algorithm is as follows. Algorithm: Input: Output: IMPYMIN Boolean function , set of input transitions . All minimum hazard-free two-level . implementations of of 1) Compute the ZBDD 2) Compute the ZBDD of [set of re]. quired cubes of 3) Solve the implicit unate set covering problem 1143 We now explain each of the steps in detail. : 1) Computation of the ZBDD of Suppose that is given as a BDD (if is given as a set of cubes, we first compute its BDD). From the BDD representing we can easily compute a BDD for the auxiliary synchronous using an existing function and then the ZBDD of we recursive algorithm [8]. From the ZBDD of can then compute the final ZBDD of using Theorem IV.1. It remains to show that the necessary and for operations, these steps can be implemented efficiently on ZBDD’s. : Assuming that positive and neg• Computing ative literal nodes of the same variable are always adjacent in the ZBDD, we only need to traverse the ZBDD We replace each node labeled with a of variable by the result of the following operation. We compute the set union of the two successors corresponding to those products that include positive literal and to those The resulting ZBDD products that do not depend on may actually include nonprimes, i.e., cubes contained in other cubes. However, these cubes are filtered out by (see below). : Analogously. • Computing the ZBDD of : deletes all • Computing the ZBDD of literals in the ZBDD. We traverse the ZBDD, and at each - or -literal, we replace the corresponding node with the ZBDD corresponding to the union of the two successors. • Single-Cube Containment (SCC): The last task, the ap-operator, which removes cubes plication of the contained in other cubes, is actually not performed in this step, since it is automatically handled in Step 3 of the algorithm. To summarize, based on Theorem IV.1 we can compute the in an implicit manner. covering objects, : From the set 2) Computation of the ZBDD of the set of required cubes can easily be of input transitions computed (see [29]). This set can be stored as a ZBDD. 3) Solving the Implicit Covering Problem: The implicit set can then be solved analcovering problem ogously to Step 3 of SCHERZO, i.e., passed directly to the unate set covering solver of SCHERZO. C. A Note on the Efficiency of IMPYMIN IMPYMIN appends -variables in dhf-prime generation during the construction of the synchronous function It is worth pointing out that the algorithm does not become unattractive even in cases where many -variables are necessary. Such cases typically arise when there are many dynamic transitions, and hence many privileged cubes. In practice, the addition of many -variables does not necessarily imply that the BDD for will be much larger than the BDD for (see Section VI-D). Experimental results also indicate that IMPYMIN has significantly better run time than existing asynchronous methods on large examples. It also performs hazard-free logic minimization nearly as efficiently as synchronous logic minimization for 1144 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 many examples. One reason is that the new characterization of the set of dhf-prime implicants, presented in Section IV, makes it possible to use state-of-the-art synchronous techniques for implicit prime generation and implicit set covering solving (see Section VI-D for a detailed discussion). VI. EXPERIMENTAL RESULTS AND COMPARISON WITH RELATED WORK Prototype versions of our two new minimizers ESPRESSOHF11 and IMPYMIN were run on several well-known benchmark circuits [12], [37] on an UltraSPARC 140 workstation (Memory: 89 MB real/230 MB virtual). A. Comparison of Exact Minimizers: IMPYMIN Versus HFMIN The table in Fig. 9 compares our new exact minimizer IMPYMIN with the currently fastest available exact minimizer, HFMIN, by Fuhrer et al. [12]. For smaller problems, HFMIN is faster. It should be noted, though, that our implementation is not yet optimized.12 However, the bottleneck of HFMIN becomes clearly visible already for medium-sized examples. For sd-control and stetson-p2, IMPYMIN is more than 3 times faster; for the benchmark pscsi-pscsi, it is more than 15 times faster. For very large examples, IMPYMIN outperforms HFMIN by a large factor. While HFMIN cannot solve stetson-p1 within 20 h, IMPYMIN can solve it in just 813 s. The superiority of implicit techniques becomes very apparent for the benchmark cache-ctrl. While HFMIN gives up (after many minutes of run time) because the 230 MB of virtual memory is exceeded, our method can minimize the benchmark in just 301 s. B. Comparison of Our New Methods: IMPYMIN Versus ESPRESSO-HF Fig. 10 compares our two new minimizers ESPRESSO-HF and IMPYMIN. Besides run time and size of solution, the table also reports the number of essentials (for ESPRESSO-HF) and the number of variables that need to be added (for IMPYMIN). The two minimizers are somewhat orthogonal. On the one hand, IMPYMIN 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. In particular, ESPRESSO-HF finds always a cover that is at most 3% larger than the minimum cover size. It is worth pointing out that many examples were very positively influenced by our new notion of essentials. Quite a few examples can be minimized by just the essentials step, resulting in a guaranteed minimum solution; e.g., dram-ctrl and pe-send-ifc. 11 Our implementation is not a simple modification of the ESPRESSO-II code. We do not reuse any ESPRESSO-II code. The reason is that while we use the same set of main operators—EXPAND, REDUCE, IRREDUNDANT—the algorithms that implement these operators, as explained in detail in Section III, are actually very different from ESPRESSO-II. 12 Our BDD package is still very inefficient. In particular, it includes a static (i.e., not a dynamic) hashtable. The hashtable for small examples is unnecessarily large. In fact, the run time is completely dominated by initializing the hashtables. If we use an appropriate-sized hashtable for smaller examples, experiments indicate that IMPYMIN can solve the small examples as fast as HFMIN. Fig. 9. Comparison of exact hazard-free minimizers (#c—number of cubes in minimum solution, time—run time in seconds). On the other hand, ESPRESSO-HF is typically faster than IMPYMIN. However, since neither tool has been highly optimized for speed, we think it is very important to analyze the intrinsic advantages and disadvantages of the two methods. Intuitively, both methods overcome the three bottlenecks of HFMIN—prime implicant generation, transformation of prime implicants to dhf-prime implicants, and solution of the covering problem—each of which is solved by an algorithm with exponential worst case behavior. However, the way in which ESPRESSO-HF and IMPYMIN overcome these bottlenecks is very different. Whereas IMPYMIN uses implicit data structures (but still follows some of the same basic steps as HFMIN), ESPRESSO-HF follows a very different approach. Thus, the two methods are orthogonal in their approach to overcome these bottlenecks. Moreover, while ESPRESSO-HF is faster than IMPYMIN on all of our examples, this does not mean that this is necessarily true for other examples. In this context, it is important to note that very often the role data structures like BDD’s play in obtaining efficient implementations of CAD algorithms is misunderstood. Using BDD’s, many CAD problems can now be solved much faster than before the inception of BDD’s. However, the naive approach of taking an existing CAD algorithm and augmenting it with BDD’s does not necessarily lead to a good tool (see discussion in [8]). In particular, it is not easily possible to simply augment ESPRESSO-HF or HFMIN with BDD’s to obtain a high-quality tool. Rather, a new theoretical formulation was needed on the characterization of dhf-prime implicants (see Section IV-E), on which the new exact implicit minimizer could be based. C. Comparison with Rutten’s Work An interesting alternative approach to our new characterization of dhf-prime implicants (see Section IV-E) was recently proposed by Rutten et al. [33], [32] as part of an exact hazard-free minimization algorithm. Their new algorithm to computing dhf-prime implicants is very different from ours. THEOBALD AND NOWICK: TWO-LEVEL HAZARD-FREE LOGIC MINIMIZATION 1145 Fig. 10. Comparison of the heuristic hazard-free minimizer ESPRESSO-HF, the exact hazard-free minimizer IMPYMIN, the heuristic minimizer ESPRESSO-II, and the exact minimizer SCHERZO. (#c—number of cubes in solution, time—run time in seconds, #e—number of essentials, #v—number of added variables, BDD f =g —BDD sizes without/with added variables). Their approach follows a divide-and-conquer paradigm. In particular, the dhf-prime generation problem is split into three subproblems with respect to a splitting variable. The first (second, third) subproblem generates those dhf-prime implicants that have a positive literal (negative literal, don’tcare literal) for the splitting variable. The underlying idea why this approach may be efficient is that it allows one to determine illegal intersections of privileged cubes already during the splitting phase (see [33] for details), which can significantly reduce the recursion tree and lead fast to terminal cases. In the merging phase of the divide-and-conquer approach, the solutions to the subproblems are then combined. However, it is worth pointing out that a major difference between our work and Rutten’s is that his approach is not based on implicit representations, while ours is. Furthermore, while Rutten’s work is promising, it has not been fully evaluated so far. In particular, he only presented run times for functions that are significantly smaller than those that can be handled by our two new methods. To be precise, on the examples he reports, his own reimplementation of the existing HFMIN tool never takes more than a few seconds. Thus, Rutten evaluates his approach (and admittedly shows improvement) only on examples that can already easily be solved by existing algorithms. In contrast, as shown in the previous subsection, our new methods are more powerful, since they can solve examples efficiently that cannot be solved by HFMIN within several hours of run time. D. Comparison of Synchronous Versus Asynchronous Minimization We now compare our two new tools for two-level hazardfree minimization, ESPRESSO-HF and IMPYMIN, with the two corresponding state-of-the-art tools for two-level nonhazardfree minimization, ESPRESSO-II and SCHERZO. The table in Fig. 10 compares both run time and cardinality of solution for all four minimizers. The table also indicates the number of identified essentials for the two heuristic minimizers, ESPRESSO-II and ESPRESSO-HF. Finally, for IMPYMIN, it re- ports the number of added variables and their impact on BDD size. The run-time comparison indicates that, although our tools are not implemented as efficiently as their synchronous counterparts, they are comparably fast. Interestingly, our tools are actually faster than the synchronous tools for the two largest examples, cache-ctrl and stetson-p1. For our set of benchmarks, this result seems to indicate that the more constrained asynchronous problem, which is to minimize a function without hazards for a set of transitions may be easier than the corresponding synchronous problem, which is to minimize the same function without any specified input transitions and without hazard-free constraints. The comparison in terms of cardinality of solution indicates an increase in the asynchronous case compared with the synchronous case. In an earlier comparison [29], it was observed that the logic overhead for the asynchronous case was never greater than 6%. In contrast, in our table, there is a large variation in overhead, ranging from 0% (stetson-p3) to 60% (ssci-trcv-bm). The increase in overhead is due to the fact that we now report on significantly more complex problems: while [29] only performed single-output minimization, we do multioutput minimization (on many of the same circuit examples), including for functions ranging up to 32 inputs and 33 outputs. However, it is important to note that this table should not be used to draw general conclusions regarding how much logic overhead asynchronous designs incur due to the necessity to avoid hazards. Our benchmark functions have been generated by asynchronous synthesis methods, i.e., these functions do not really make much sense in a synchronous system. On the one hand, functions derived from asynchronous FSM’s must have function-hazard-free input changes and critical race-free state changes, unlike those derived from synchronous FSM’s. On the other hand, asynchronous FSM’s are typically specified in a more controlled environment, with more don’t cares. A truly fair comparison on this interesting point is much beyond the scope of paper. 1146 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 11, NOVEMBER 1998 Fig. 10 also compares the number of identified essentials, using both hazard-free and nonhazard-free algorithms. ESPRESSO-HF’s new formulation of essential equivalence classes typically allows many more essentials to be identified than in ESPRESSO-II. For example, in cache-ctrl, ESPRESSO-HF identifies 50 essentials (out of an exact minimum cover of 97 cubes), while ESPRESSO-II identifies only seven essentials (out of an exact minimum cover of 80 cubes). Thus, ESPRESSO-HF makes positive use of hazard-freedom constraints to obtain a very strong formulation of essentials, which has a positive impact on both run time and quality of solution. Finally, the table indicates that adding (sometimes many) variables in IMPYMIN does not lead to an explosion in terms of BDD size. To incorporate hazard-freedom constraints, into IMPYMIN (unlike SCHERZO) transforms the BDD of The table, which compares the BDD of auxiliary function the corresponding BDD sizes for the same BDD package and variable ordering, indicates that adding variables for this transformation increases the BDD size, even for large examples only by a small factor, which is typically about two. Thus, the BDD size of auxiliary function is not much larger than the BDD size of VII. CONCLUSIONS We have presented two new minimization methods for multioutput two-level hazard-free logic minimization: ESPRESSOHF, a heuristic method based loosely on ESPRESSO-II, and IMPYMIN, an exact method based on implicit data structures. Both tools can solve all examples that we have available. These include several large examples that could not be minimized by previous methods.13 In particular, both tools can solve examples that cannot be solved by the currently fastest minimizer, HFMIN. On the more difficult examples that can be solved by HFMIN, ESPRESSO-HF and IMPYMIN are typically orders of magnitude faster. Although ESPRESSO-HF is a heuristic minimizer, it almost always obtains an exactly minimum-size cover. ESPRESSO-HF also employs a new fast method to check for the existence of a hazard-free solution, which does not need to generate all dhf-prime implicants. IMPYMIN performs exact hazard-free logic minimization nearly as efficiently as synchronous logic minimization by incorporating state-of-the-art techniques for implicit prime generation and implicit set covering solving. IMPYMIN is based on the new idea of incorporating hazard-freedom constraints within a constructed synchronous function by adding extra inputs. We expect that the proposed technique may very well be applicable to other hazard-free optimization problems, too. ACKNOWLEDGMENT The authors would like to thank O. Coudert for very helpful discussions and for his immense help with the experiments. 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New York: IEEE Computer Society Press, 1992, pp. 346–350. Michael Theobald received the Diplom degree in computer science from Johann Wolfgang GoetheUniversität, Frankfurt/Main, Germany, in 1994. He currently is pursuing the Ph.D. degree in computer science at Columbia University, NY. His research interests include synchronous and asynchronous circuits, computer-aided digital design, logic synthesis, formal verification, efficient algorithms and data structures, and combinatorial optimization. Mr. Theobald received the Honorable Mention Award at the 1997 International Conference on VLSI Design. He was a Best Paper Finalist at the 1998 IEEE Async Symposium. Steven M. Nowick received the B.A. degree from Yale University, New Haven, CT, and the Ph.D. degree in computer science from Stanford University, Stanford, CA, in 1993. His Ph.D. dissertation introduced an automated synthesis method for locally clocked asynchronous state machines, and he formalized the asynchronous specification style called “burst mode.” He currently is an Associate Professor of computer science at Columbia University, NY. His research interests include asynchronous circuits, computer-aided digital design, low-power and high-performance digital systems, logic synthesis, and formal verification of finite-state concurrent systems. Dr. Nowick received an NSF Faculty Early Career (CAREER) Award (1995), an Alfred P. Sloan Research Fellowship (1995), and an NSF Research Initiation Award (RIA) (1993). He received a Best Paper Award at the 1991 International Conference on Computer Design and was a Best Paper Finalist at the 1993 Hawaii International Conference on System Sciences and at the 1998 Async Symposium. He was Program Committee Cochair of the IEEE Async-94 Symposium and is Program Committee Cochair of the forthcoming IEEE Async-99 Symposium. He is a member of several international program committees, including ICCAD, ICCD, ARVLSI, IWLS, and Async. He is also Guest Editor of a forthcoming special issue of the PROCEEDINGS OF THE IEEE on asynchronous circuits and systems (February 1999).

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