INFORMATION AND CONTROL 26, 351-368 (1974) A Logic Hazard Detection and Elimination Method* LEVENTE L. M~Tt~ Computer and Automation Institute of the Hungarian Academy of Sciences, Hungary SANTANU DASt Paul H. Henson Research Center, North Electric Co., Delaware, Ohio AND HENRY Y. H. CHUANG* Department of Computer Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15213 This paper is concerned with the detection and elimination of static logic hazards due to single and multiple input changes. T h e method involves only simple manipulation of the Boolean expression corresponding to the logic network. It not only detects the existence of hazards and determines the variable sets with respect to which the hazards exist, but also, for each of these sets, pin-points the subcubes within which each input transition involving exactly the changes of all variables in the set produces the hazardous output. Further, it can determine the terms the absence of which from the expression cause the hazards. Thus, the hazard can be eliminated by including these missing terms in the expression. T h e theoretical foundation of the method is given first followed by an example illustrating the procedure and its effectiveness. T h e procedure is formulated into two different algorithms, one for saving time and the other for saving memory. They have been implemented in F O R T R A N . A related result on prime implicant generation is also included. I. INTRODUCTION S p u r i o u s pulses o c c u r r i n g o n t h e o u t p u t of c o m b i n a t i o n a l s w i t c h i n g circuits d u r i n g a n i n p u t t r a n s i t i o n are called c o m b i n a t i o n a l h a z a r d s . T h e c o m b i n a t i o n a l h a z a r d s [ U n g e r , 1969; B r e d e s o n et al., 1972; Yoeli et al., 1964; E i c h e l b e r g e r , 1965; H u f f m a n , 1957; M c C l u s k y , 1962; M c G h e e , 1969; D a s * Partially supported by the National Science Foundation under Grant GJ-35569. * S. Das and H. Y. H. Chuang were with Computer Systems Laboratory, Washington University, St. Louis. Copyright © 1974 by AcademicPress, Inc. All rights of reproduction in any form reserved. 351 352 ZV~T~, DAS AND CHUANG and Chuang, 1972; Rey, 1974; M~it6 et al., 1973] can be classified with respect to (i) whether the number of the variables changing during the transition is one (single variable) or more (multiple variable), (ii) whether the steady values of the output before and after the transition are identical (static) or different (dynamic), (iii) whether the value of the function for each cell within the smallest subcube which includes the two input states before and after the transition is the same (logic) or not (function). This paper is concerned with the detection and elimination of static logic hazards due to both single and multiple input changes. This problem is not new and there have been a number of attempts [Unger, 1969; B redeson et al., 1972; Yoeli et al., 1964; Eichelberger, 1965; Huffman, 1957; McClusky, 1962; McGhee, 1969] in recent years in finding solution to the problem. Most of the previous works require examination of the existence of hazard for every input transition, some of them even requiring binary to ternary conversion [Yoeli, et al., 1965; Eichelberger, 1965]. The most significant result in this area is due to Eichelberger [Eichelberger, 1965] who first proved that the realization of a Boolean expression would be hazard-free if and only if the expression includes all the prime implicants (implicates) of the function. On the basis of this result, it might appear that hazard detection can be easily effected by generating all the prime implicants (implicates) of the function from the given expression and then noting which are absent in the expression. But in reality, the problem is not that simple. As, for instance, in Figure 1, we see that the omission of the prime implicant x~ gives rise to two variable hazards for transitions between the cells 5 and 12 as well as the cells 4 and 13. But in addition to these, there is a possibility of a single variable hazard between the cells 4 and 5. The generation of all the prime implicants and then comparing with the given expression would detect the omission of the prime implicant x.f. This at once locates the two variable hazards already mentioned but no information about the second hazard, namely the single-variable hazard between the cells 4 and 5, emanates from this exercise. When the number of variables involved in the expression is large, many such hazards will not be detected. Of course, if we were interested only in the elimination of the hazards, the information about these undetected hazards is not at all necessary. Unfortunately, in practice, due to "don't care" situations, we might be interested in removing only some of the hazards as the transitions giving rise to the other hazards could be just "don't care" situations. Thus, this simple approach based on the generation of all the prime implicants is virtually L O G I C HAZARD D E T E C T I O N AND E L I M I N A T I O N YZ~ x 0 oo 4 0t ( oo L I A II 01 < II I0 12 8 I3 ~ ( ~ ¢ 3 15 2 '4 353 I I tl hO 10 w ~ f (xw~ z~w) FIG. 1. Karnaugh map and logic realization of the expression ~ + w:f + xz. useless when we are interested in knowing the exact locations of all possible hazards. Moreover, the generation of prime implicants [Nelson, 1957; Slage et al., 1970; Das and Khabra, 1972; McClusky, 1956; Su, 1969; Roth, 1958; Tison, 1967] of a function from a given arbitrary expression is not as simple as it might appear on the face of it. Motivated by all these considerations, we have devised an algebraic approach, which is very direct and involves only simple manipulation of the Boolean expression, corresponding to the logic network. The method is also very powerful. It not only detects the existence of hazards and determines the variable sets with respect to which the hazards exist, but also, for each of these sets, pin-points the subcubes within which each input transition involving exactly the changes of all variables in the set produces the hazardous output. Further, it can determine the terms whose absence from the expression cause the hazards. Thus, the 354 MATt~, DAS AND CHUANG hazards can be eliminated by including these missing terms in the expression. Two different algorithms have been developed from one basic idea and both of them have been implemented in F O R T R A N [M~it6 et al., 1973]. T h e first algorithm is intended to save computing time, while the second is for saving memory space. T h e second one has also the flexibility that instead of detecting the hazards with respect to all the variables, we can specify the variables we are interested in. As a result, the hazards with respect to only these variables will be detected. 2. THEORETICAL FOUNDATIONS Terms with complementary literals play a key role in our hazard detection and elimination technique. T h e first definition characterizes such terms: DEFINITION 1. A product containing M variables ( M > 0) both complemented and uncomplemented is called an unstable term, u m . The part of u m which includes only those variables appearing both complemented and uncomplemented is called the unstable p a r t , U M , of UM • T h e other part of UM is called its stable p a r t , S M . Thus u M = U m S M . When u M ~ - U M , SM--1. Definition 2 introduces two special kinds of subcubes which are closely related with unstable terms. DEFINITION 2. A subcube for which the variables of UM are not assigned is called an M - s u b c u b e of U m . Those M-subcubes of UM for which S m = 1 are called the acceptable subcubes of u u . Here M refers to the dimensionality of the unstable part as well as the subcube. There are some input transitions, related to an unstable term, in which we are particularly interested. DEFINITION 3. A transition between any two input states which represent a diagonal of an M-subcube of U M is called a bridge transition of U M , and the transition is said to span this M-subcube. To illustrate the concepts of the previous definitions, an example is presented. For a function of four variables, x, y , z , w , the unstable term u 2 = x y y w w consists of the unstable part U~ = y y w w and the stable part S 2 = 2. The four 2-subcubes of U2 are 0-0-, 0-1-, 1-1-, 1-0-. Since $2 is not identically equal to 1, not all the 2-subcubes of U~ are acceptable subcubes of ua. The two acceptable subcubes of u 2 are 0-0-, 0-1-. I 1 --~ I~ and I~ ~ I1, LOGIC HAZARD DETECTION AND E L I M I N A T I O N Z W"~,Y oo ol II 355 I0 oo OI I0 J Z W'~., ' O0 OI I0 O0 OI 11 \/ /\ \/ /\ \/ /',, \/ /\ I0 (b) Fie. 2. (a) The 2-subcubes of U2 = yajwff~ (framed) and the acceptable subcubes of u2 = 2y~w~ (striped). (b) The bridge transitions of U2 = y~w*5. where 11 = 0100 and I S 0001 are two of the bridge transitions of U 2 . Figure 2 shows the relevant subeubes and transitions. Next, we analyze the role which a single unstable term plays in logic 0 hazards. LEMMA 1. Given two input states I 1 and I 2 differing in the value of M variables. A n unstable term UM can have a 0-1-0 sequence of values during the tramition from 11 to I~ iff the M-subcube spanned by this transition is an acceptable subeube of UM . Proof. Obviously the stable values of uM before and after the transition 356 ~TI}, DAS AND CHUANG are 0. If the M-subcube spanned by the transition from I x to I 2 is an acceptable subcube of UM, then during the transition the stable part of UM remains identically 1, while aU the variables involved in the unstable part change simultaneously. Due to stray delays it could happen that for a short moment every changing variable and its complement assume 1-value simultaneously resulting in a momentary 1-value for UM. Thus, the "if" part of the lemma is proved. If the M-subcube spanned by this transition is not an acceptable subcube of UM but still is an M-subcube of its unstable part, then the stable part of UM has to be identically 0 during the transition. If the M-subcube is not an M-subcube of the unstable part of UM, then at least one of the variables in the unstable part does not change during the transition. In neither case can the 0-1-0 sequence occur. Thus, the "only if" part of the lemma is also proved. Q.E.D. Now that we have established the relation between a logic 0 hazard and an unstable term, we go on to our first theorem which gives the necessary and sufficient conditions for the existence of logic 0 hazard in the realization of a Boolean expression based on this relationship. THEOREM 1. A sum-of-products expression E, free of N-variable logic 0 hazard ( N < M), has an M-variable logic 0 hazard within an M-subcube if/= (1) (2) (3) there is an unstable term Uu = UMSM in E; the value of E is Ofor each cell within the M-subcube; the M-subcube is an acceptable subcube of UM . Proof. The "if" part of the theorem follows from the proof of Lemma 1. The existence of an unstable term uK for some K is necessary since no sum of stable terms can cause logic 0 hazard. If K < M, then uK cannot produce M-variable logic hazard in an M-subcube since no M-subcube can contain bridge transition of UK. If K > M and UK produces an M-variable logic 0 hazard, then uK also causes a K variable logic 0 hazard in a K-subcube contained by the M-subcube and this contradicts the given hypothesis. Thus K = M, i.e., condition (1) is necessary. The necessity of condition (2) and (3) follow from the definition of logic hazard and from Lemma 1, respectively. Thus the "only if" part of the theorem is established. Q.E.D. It can be seen that this theorem shows not only how to detect the existence of logic 0 hazard but also how to pin-point the subcubes within which any bridge-transition produces the hazardous output. By now it is clear that Theorem 1 is based on unstable terms in sum-ofproducts expressions. But it is very likely that the logic designer will delete LOGIC HAZARD DETECTION AND E L I M I N A T I O N 357 the unstable terms from his expressions during the transformation and simplification process, thus eliminating the logic 0 hazard. However, this does not diminish the usefulness of the theorem. It can be conveniently used to detect logic 0 hazards in product-of-sums expressions or in mixed expressions. These expressions can be transformed to sum-of-products form through logic-hazard preserving transformations and the resulting sum-of-products expressions can be examined using Theorem 1 to detect possible logic 0 hazards. Unger [1969] has shown that the application of associative, distributive and DeMorgan laws are static hazard preserving. It is possible that the sum-of-products expression resulting from the application of these transformations will contain large number of terms. The next two lemmas are concerned with some additional hazard-preserving transformations which could be used to simplfy the sum-of-products expression. LEMMA 2. The following transformations on a sum-of-products expression will not affect its logic 0 hazard behavior: (1) any simplification involving only stable terms and resulting in sum-ofproducts form; (2) collecting terms with common unstable part, factoring out the unstable part, and then simplifying the resulting sum of the stable parts. Proof. Since neither transformation changes the conditions of Theorem 1, the resulting expression preserves the logic 0 hazard behavior of the original expression. Q.E.D. LEMMA 3. Given a sum-of-products expression which may or may not contain unstable terms. Application of the absorption law to the expression is logic hazard preserving. Proof. (1) With respect to logic 1 hazards, the statement of the lemma follows from the fact that no application of the absorption law can change the prime implicants existing in the expression. (2) For logic 0 hazards we consider the following cases: (i) A stable term absorbs another stable term. The proof for this case follows directly from Lemma 2. (ii) A stable term absorbs an unstable term. The stable term generates at least one "1" within each acceptable subcube of the unstable term. Therefore with respect to this unstable term, condition (2) of Theorem 1 358 M~TI~, DAS AND CHUANG is never satisfied. Thus, ks omission will not change the logic 0 hazard behavior of the expression. (iii) An unstable term absorbs another unstable term. Whenever the term which can be absorbed emits a spurious 0-1-0 pulse, the term which can absorb it will also emit such a pulse. (iv) An unstable term cannot absorb any stable term. Hence the lemma is proved. Q.E.D. Theoretically, we have by now sufficient tools for detecting logic 0 hazards Of any expression. However, some practical difficulties arise when one attempts to use Theorem 1. In order to make our hazard detection techniques more practical, we define a new operation called _D-operation (deletion operation). DEFINITION 4. T h e operation _D(E/UM) on an expression E with respect to an unstable part UM consists of the following two steps: (i) delete all the unstable terms from E (Should all terms be deleted, the result would be 0), (ii) delete all the literals of Um from the remaining expression and the result is the function represented by the final expression (Should any terms be completely deleted, the result would be 1). For instance, if E = 2~z + 2~w + x z ~ + ~ z ~ + y~z + x2w~, U 1 = y~z, then _D(E/U1) = ~2z + 2~w + x z ~ + ~ z ~ + yye + xx~w = xz + X~o + z~. T h e following lemma shows the implication of the _D=operation. LEMMA 4. Given a sum-of-products expression E and unstable part UM. The function _D(E/UM) covers an M-subcube of UM iff the M-subcube contains at least one 1 of E. Proof. T h e function _D(E/UM) does not contain any variable of UM. Therefore, it can not have different values within any M-subcube of UM. Moreover, deleting a variable from a sum-of-products expression can not change the value of the function from 1 to 0. Hence, each M-subcube of UM which contains at least one 1 of E is covered by _D(E/UM). Any M-subcube of UM containing only O's of E can not be covered by _D(E/UM), since the values of E in the M-subeube do not depend on the variables of UM. Q.E.D. Figure 3 illustrates how _D(E/U1) covers 1-subcubes of U 1 . Now we are ready to give a more practical form of Theorem 1. LOGIC HAZARD DETECTION AND ELIMINATION ZW~ Y E" oo Ol II O0 I 0 °t° Ol I 0 II I 0 I0 [ I 0 359 I0 °l 0 0 0 0 j I 1 I t _D(EIup: ZW~ OO O1 II I0 O0 0 0 0 0 01 I I 0 0 II I I 0 0 10 I I I I FIG. 3. The effect of the _D-operation. THEOREM 2. Given a sum-of-products expression E containing unstable terms, and suppose one of its unstable term is u M = UMSM . Then the expression E contains an M variable logic 0 hazard with respect to the variables of UM within the M-subcubes of UM not covered by the function H, where H = D(E/UM) + SM. Proof. T h e existence of UM implies that condition (1) as well as condition (2) of T h e o r e m 1 are all covered by _D(E/UM). By the definition of acceptable subcubes (Definition 2), the M-subcubes which do not satisfy condition (3) of T h e o r e m 1 are covered by SM. Therefore, each uncovered M-subcube of U M satisfies all the conditions of T h e o r e m 1, and the logic 0 hazard exists in these M-subcubes. Q.E.D. 360 M£T~, DAS AND CHUANG It is obvious that expression E implies H (i.e., H is always 1 whenever E is 1). Since the value of H does not depend on the variables of Urn, H keeps its value constant during any bridge transition of Um . By Theorem 2, H is identically 0 during the hazard producing bridge transitions of UM . Therefore, "AND"ing H with E would eliminate the logic 0 hazard in the M-subcubes not covered by H. It can be seen that the creation of function H is easy to mechanize and this can serve as the basis of a logic-0 hazard detection algorithm. Further, this same theorem can also be used to detect logic 1 hazard of an expression by inverting it, considering the restrictions of the following theorem. THEOREM 3. Given a sum-of-products expression E not containing unstable terms, if we obtain its inverse if, in sum-of-products form through the use of only the associative, distributive and DeMorgan laws and simplify it by using only the idempotent and absorption laws, then the expression E and E interchange 1 and 0 hazards. Proof. It has been proved by Unger [1969] that the above inversion procedure interchanges 1 and 0 hazards. By Lemmas 2 and 3, simplification by idempotent and absorption laws does not change the logic 0 hazard behavior of an expression. Thus the theorem is proved. Q.E.D. Since for the detection of logic 1 hazard in E we have to apply Theorem 2 to E, we will be concerned with the inverse of H (denoted by R) in our procedure. Since E implies H, R will imply E. Furthermore, R keeps its value 1 during the hazardous bridge transitions of UM. Therefore, "OR"ing R with E will eliminate the logic 1 hazard within the M-subcubes of UM not covered by H (i.e., covered by R). The following theorem (and corollary)shows that inclusion of R in the sum-of-products expression (H in the product-of-sums expression) will not result in any new hazard originally not present in the expression. THEOREM 4. Given two sum-of-products expressions E and R. I f R "implies" E (i.e., E is always 1 whenever R is 1), then the expression E'produced by "OR"ing E and R will not have any logic 1 hazard not present in E. Proof. Let us consider two input states I 1 and I2 where E*(I1) -~ E*(I2) = 1. If there is no single term in E* having 1 for both 11 and I2 (i.e., the transition from I 1 to I 2 has a 1 hazard), then E can not have such a term too. Thus E* can not have logic 1 hazards not present in E. Q.E.D. Obviously the dual of this theorem can be stated as follows: LOGIC HAZARD DETECTION AND ELIMINATION 361 COROLLARY. Given two product-of-sums expressions E and H. If E implies H (i.e., H is always 1 whenever E is 1), then the expression E* produced by " A N D " i n g E and H will not have any logic 0 hazard not present in E. After inclusion of all the R's (or H's) corresponding to the unstable terms of/2 (or E), we will have to repeat the whole procedure to see if higher-order logic hazards (hazards involving bigger sets of variables, than already detected), are still there. The repetition of the procedure has to be continued till all the hazards have been detected and eliminated. Let us show by an example how to use these theorems for logic hazard detection and elimination. Suppose we would like to know whether or not the following expression has logic hazards: E = 2 ~ + ~y + y w + x 2 + xyw. By (1) of Theorem 1, E does not have any logic 0 hazard since it is a sum-ofproducts expression not containing any unstable term. T o detect its logic 1 hazards we take the inverse of E, taking into consideration the restrictions of Theorem 3, resulting in : E = g y z + gyw + x z ~ + y z ~ + y y z + xxww. Any logic 0 hazard in E~ corresponds to a logic 1 hazard in E within the same subcubes with respect to the same variables. Let us first examine the effect of the unstable t e r m y y z . This term contains one variable both complemented and uncomplemented. Thus, u 1 = y y z has an unstable part U 1 ~ y y and a stable part S 1 = z. By Theorem 2,/2 has a logic 0 hazard with respect to the variable y within the 1-subcubes of y y not covered by the function H = -D(E/U1) -{- S1 = 2-~- .o~-~ W, i.e., within the 1-subcube 1-11 for the bridge transitions 11 ~-+ I 2 o f y ~ where 11 = 1011 and I 2 = 1111. T h e other unstable term is u 2 ---- x 2 w ~ with U~ = x x w w and S 2 = 1. In this case H = y + z. Thus, /2 has a two variable logic 0 hazard with respect to x and w within the subcube -10-, for the bridge transitions I~ +-+ 14 and 15+-+16, where I 3 = 0100, 14 = 1101, 15 = 0101, and 16 ---- 1100. Now we have no more unstable terms in/2. Let us eliminate the hazards already detected. Using Theorem 4 we create the following expression E* having the same truth table as E, and containing the terms of both H ' s as well as those of E. E* ~ 2 ~ + 2y -]-yw + x 2 + xa]w - / x z w + y2. T h e inclusion of these terms of Er's into E* eliminates the hazards previously 362 MATI~, DAS AND CHUANG detected. However, E* still may have some other logic 1 hazards. So let us repeat the whole procedure. Inverting E* and examining the unstable terms of E* shows that only one of them produces hazard. This unstable term is U s ~ y y z z with the unstable part U s = y y z z and the stable part S~ = 1. I n this case, H = ~ + ~. Thus, the hazardous subcube is 1--1, while the hazardous bridge transitions are 17 *-+/8 and 19 ~ 110 , where 17 = 1001, 18 ~- 1111, 19 = 101l, and 110 = 1101. Including the term H = x w into a new expresssion together with E* and simplifying it by the permissible transformations of T h e o r e m 2, we get E * * = 5 ~ q- 2y q- y w q- x~, q- y 5 q- xw. Applying the same procedure to E** we find no more logic hazards in E**, E: Z W ~ Y oo ol oo (, /7 o ,A o tl \1 , I I0 0 I 10 -'-,_1 o o 01 II I0 I o o (a) O0 E**: O0 OI II 10 (b) FIG. 4. (a) Karnaugh map of expression E showing transitions with logic 1 hazards. (b) Karnaugh map of expression E**. All the logic 1 hazards of E are eliminated. 363 LOGIC HAZARD DETECTION AND ELIMINATION and so E** is a logic hazard free expression of E. Figure 4(a) shows the Karnaugh map of E, while Fig. 4(b) shows that of E**. One can see the logic 1 hazards of E, and how they are eliminated in E**. 3. ALGORITHMS FOR LOGIC HAZARD DETECTION AND ELIMINATION Two different algorithms have been developed based on the hazard detection and elimination procedure just demonstrated. The first algorithm is intended to save computing time, while the second is for saving memory space. The important parts of the two algorithms are discussed in this section. I EXPAND F ) ~ I L YES ~.YES iF- I GETTHE NEXTUM I I FINDSI~I ~ i I I J P~ P+R I I OUTPUT~A.DUM L FIG. 5. Flowchart of the Time Saving Algorithm for Logic Hazard Detection and Elimination. 643/2614-5 364 MATIn, DAS AND CHUANG The flowchart of the Time Saving Algorithm for Logic Hazard Detection and Elimination is shown in Fig. 5. The algorithm is concerned with the logic 1 hazards of E, where E is a sum-of-products expression. The expression F is the inverse of E, and is in product-of-sums form. While executing the block "Expand F," the restrictions of Theorem 3 should be taken into consideration. In the step "Get the next U m ," the unstable part of the next unstable term is chosen. Furthermore, in the step "Find S M ," S M is the sum of the stable parts of all unstable terms having UM in common. It is obvious that the steps as well as the organization of this algorithm are very similar to the procedure shown in the previous example. It turns out that this algorithm is also easy to use when one uses pencil and paper. In Fig. 6 we show the flowchart of the Memory Saving Algorithm for Logic COMPOSE ANOTHER UM yES N° PARTITION E INTO G AND H TRANSFORM G INTO Q ¢ NO I R----" [ I OTPOTANOOM I 1 E+R I FIG. 6. Flowchart of the Memory Saving Algorithm for Logic Hazard Detection and Elimination. LOGIC HAZARD DETECTION AND E L I M I N A T I O N 365 Hazard Detection and Elimination. In this algorithm, V is the set of variables with respect to which logic 1 hazards of the sum-of-products expression E are to be detected, and V can be defined beforehand. If not defined, V is the whole set of variables in E. The maximum number of variables in V which can change simultaneously (Mm) is also defined initially, and its value, if not defined, is the number of variables in V. In this algorithm, F is the inverse of E in sum-of-products form with all unstable terms deleted. F need not be directly derived from E. It would suffice if F is any expression which represents the inverse function of E. The block "Compose another UM" consists of the following four steps: Step 1. Choose a set of M variables from V (M = 1 initially). Step 2. If all possible combinations of M variables in V have already been chosen, then increase M by 1 and go to Step 3; else go to Step 4. Step 3. If M is greater than Mm, then exit because UM does not exit; else go back to Step 1. Step 4. Create UM b y " A N D " i n g all variables chosen in Step 1 with their inverses, and then exit. In the block "Partition E into G and H," each term of E containing any variable of UM goes to G, the remaining to H. The block "Transform G into Q" means the execution of the following procedure: Get the dual of G in product-of-sums form. Expand it by applying the distribution law successively. After each multiplication, drop out every term which is not able to absorb UM, then apply the idempotent and absorption laws to the remaining terms. The set of terms in the final result is called Q. In the time-saving algorithm, all the logic 1 hazards of E are detected in one run. On the other hand, in the memory saving algorithm, only the logic 1 hazards with respect to all possible combinations of the set of variables specified in V are detected. Consequently, the latter algorithm is more flexible as it can use the information given by the designer more effectively. However, when all possible logic 1 hazards are to be detected, this algorithm has to make an exhaustive search and therefore is slower than the time saving one. But the space saving algorithm trades off execution time in favor of less memory space. The same two algorithms, with reinterpretation of the expressions E and F, can be used to detect and eliminate logic 0 hazard. Both algorithms have been implemented in FORTRAN. The result of one of these programs implementing the time saving algorithm, TSHAZARD, for the earlier example is shown in Fig. 7. The details of the implementations as well as the program listing are given in [Mfit6 et al., 1973). 366 1VL~TE, DA8 AND CHUANG EXPRESSION T O BE CORRECTED: -TZ~W + -TXY + WY + ~ZX + WX~Y HAZARD W I T H I N T H E SUBCUBE: -~ZY W I T H RESPECT T O T H E VARIABLES: WX HAZARD W I T H I N T H E SUBCUBE: ZWX W I T H RESPECT T O T H E VARIABLES: Y HAZARD W I T H I N T H E SUBCUBE: WX W I T H RESPECT T O T H E VARIABLES: ZY EXPRESSION CORRECTED: -TZ~ W + ~XY FIG. 7. + WY + ~ZX + ~ZY + WX Output of TSHAZARD program, where -TA denotes A. 4. A RELATED RESULT ON PRIME IMPLICANT GENERATION A related result which is significant by itself is the following theorem. This is similar to one by Nelson [1954] which is often used as the basic principle for prime implicant generation (Slage et al., 1970; Das and Khabra, 1972). We include this theorem here mainly because our proof not only is simple and conceptually easy to understand, but also provides a close relationship between hazard and prime implicant. THEOREM 5. Given a sum-of-products expression not containing unstable terms. I f we take its inverse E (or dual E 9) through the use of associative, distributive and DeMorgan laws, and simplify it by dropping out the unstable terms and then using the absorption law, then F (or E 9) contains all its prime implicants. Proof. Since E does not have any kind of logic 0 hazard (Theorem 1), it follows from Theorem 3 that E (or E 9) does not contain any kind of logic 1 hazard. 1 Eichelberger has proved [Eichelberger, 1965] that a sum-ofproducts realization of a function is logic 1 hazard free iff it contains all the prime implicants of the function. But, since E (or E D) is logic 1 hazard free, it has to contain all its prime implicants. Q.E.D. 1 It is obvious that Theorem 3 remains valid after replacing the inverse /~ with the dual E D. LOGIC HAZARD DETECTION AND ELIMINATION 367 5. CONCLUSIONS Logic 0 hazards and terms wkh complementary literals are closely related. Their relation is analyzed first and then a logic 0 hazard detection procedure is developed based on this relation. A set of logic hazard preserving transformations have been developed and these give a wider generality to the method, making it usable for both logic 0 and logic 1 hazard detection. Two logic hazard detection and elimination algorithms have been developed using the same basic idea, but with the first one emphasizing the reduction of the execution time and the second one the memory space. Both algorithms are implemented in F O R T R A N , although F O R T R A N obviously is not efficient for implementing such algorithms which involve mostly symbol manipulation. ACKNOWLEDGMENTS Mr. M~it6 is grateful to the Hungarian Academy of Sciences, the National Academy of Sciences (USA) and the Computer Systems Laboratory of Washington University, St. Louis, for supporting his research. RECEIVED:July 30, 1973; REVISED:July 16, 1974 REFERENCES BREDESON, J. AND HULINA,P. (1972), Elimination of static and dynamic hazards for multiple input changes in combinational switching circuits, Inform. Contr. 20, 114-124. DAS, SANTANUAND CHUANG,Y. H. (1972), "A New Hazard Detection Technique," Technical Memorandum No. 157, Computer Systems Laboratory, Washington University, St. Louis. DAS, S. AND KHABRA,N. (1972), Clanse-column table approach for generating all the prime implicants of switching functions, IEEE Trans. Computers C-21, 1239-1246. EICHELBERGER,E. (1965), Hazard detection in combinational and sequential switching circuits, I B M J. Res. Develop. 9, 90-99. HUFFMAN,D. (1957), The design and use of hazard-free switching networks, J. Assoc. Comput. 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