## Worst and Best Irredundant Sum-of-Products Expressions (2001)

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Venue: | IEEE Trans. Comp |

Citations: | 5 - 0 self |

### BibTeX

@ARTICLE{Sasao01worstand,

author = {Tsutomu Sasao and Jon T. Butler},

title = {Worst and Best Irredundant Sum-of-Products Expressions},

journal = {IEEE Trans. Comp},

year = {2001},

volume = {50},

pages = {935--948}

}

### OpenURL

### Abstract

In an irredundant sum-of-products expression (ISOP), each product is a prime implicant (PI) and no product can be deleted without changing the function. Among the ISOPs for some function f, a worst ISOP (WSOP) is an ISOP with the largest number of PIs and a minimum ISOP (MSOP) is one with the smallest number. We show a class of functions for which the Minato-Morreale ISOP algorithm produces WSOPs. Since the ratio of the size of the WSOP to the size of the MSOP is arbitrarily large when n, the number of variables, is unbounded, the Minato-Morreale algorithm can produce results that are very far from minimum. We present a class of multiple-output functions whose WSOP size is also much larger than its MSOP size. For a set of benchmark functions, we show the distribution of ISOPs to the number of PIs. Among this set are functions where the MSOPs have almost as many PIs as do the WSOPs. These functions are known to be easy to minimize. Also, there are benchmark functions where the fraction of ISOPs that are MSOPs is small and MSOPs have many fewer PIs than the WSOPs. Such functions are known to be hard to minimize. For one class of functions, we show that the fraction of ISOPs that are MSOPs approaches 0 as n approaches infinity, suggesting that such functions are hard to minimize.

### Citations

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Citation Context .... In such a graph, there are n 1 pairs or 2n 2 edges in all. (Replace each pair by an undirected edge. If the directed graph is strongly connected, the undirected graph must be connected. From Harary =-=[14], th-=-ere must be n 1 edges.) Since there are 2n 2 edges in GF , there are 2n 2 PIs in F and, thus, F is a WSOP. (only if) Let F be a WSOP of ST…n; 1†. We show that GF consists of cycles of length 2 onl... |

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Citation Context ...systems use heuristic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PRESTO [4], [33], MINI [15], ESPRESSO =-=[3]-=-, and others [10], [24] produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expressions x1x2_ x2x3_ x1x3 and x1x2_ x1... |

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Citation Context ...tension to Multiple-Output Functions In the case of multiple-output functions, minimization of AND-OR two-level networks or programmable logic arrays (PLAs) can be done using characteristic functions =-=[26, 27, 29]-=-. Definition 4.1 For an n-variable function with m output values, f j (x 1 ;x 2 ;:::;x n )(j =0; 1;:::;m0 1); form an (n+1)-variable two-valued single output function F (x 1 ;x 2 ;:::;x n ;X n+1 ),whe... |

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Citation Context ...Other Functions We also applied Algorithm 5.1 to compare the number of PIs for multiple-output functions. Table 4 shows the distribution of the number of PIs in ISOPs for various arithmetic functions =-=[32]. IN-=-Cn is an n-input n‡1 output function such that the value of the output is x‡1, where x is the value of thes940 IEEE TRANSACTIONS ON COMPUTERS, VOL. 50, NO. 9, SEPTEMBER 2001 input; WGT5 is the sam... |

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Citation Context ...t sum-ofproducts expressions for multiple-valued functions," IEEE International Symposium on Multiple-ValuedLogic, Nova Scotia, Canada, May 28-30, 1997, pp. 55-60. 1 For example, PRESTO [4, 33], =-=MINI [15]-=-, ESPRESSO [3], and others [10, 24] produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expressions x 1sx 2 x 2sx 3sx... |

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Citation Context ... I Introduction Two-level logic minimization is a basic problem in logic synthesis. Although algorithms exist that obtain exact minimum sum-of-products expressions (MSOP) for a large set of functions =-=[7, 8]-=-, the majority of practical systems use heuristic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. 3 This paper is an exte... |

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Citation Context ...f an SOP is the number of PIs in the SOP. The size of a CSOP, WSOP, and MSOP of function f is denoted as …CSOP : f†, …WSOP : f†, and …MSOP : f†, respectively. The following is well known. =-=Theorem 2.1 [13], [22-=-]. For any switching function of n variables, …MSOP : f† 2n 1 . This upper bound is firm. For example, the exclusive OR function, fEXORˆx1 x2 ... xn, has 2n 1 minterms, all of which are PIs, and,... |

31 |
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Citation Context ...ons, we know of no study that shows how many ISOPs exist with various number of product terms. Although various methods to generate all the ISOPs for a logic function are known [22], [12], [21], [6], =-=[35]-=-, [25], no experimental results have been reported. Experiments are computationally intensive even for functions with a small number of variables. However, we can obtain the statistical properties of ... |

21 |
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Citation Context ...ULTIPLE-OUTPUT FUNCTIONS In the case of multiple-output functions, minimization of AND-OR two-level networks or programmable logic arrays (PLAs) can be done using characteristic functions [26], [27], =-=[29]. Definition 4.1. For-=- an n-variable function with m output values, fj…x1;x2; ...;xn†…jˆ0; 1; ...;m 1†; form an …n‡1†-variable two-valued single output function F…x1;x2; ...;xn;Xn‡1†, where xi is a bin... |

15 |
Logic design of digital systems
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Citation Context ...istic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PRESTO [4], [33], MINI [15], ESPRESSO [3], and others =-=[10]-=-, [24] produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expressions x1x2_ x2x3_ x1x3 and x1x2_ x1x3_ x1x2_ x1x3 ar... |

12 |
McBOOLE: A new procedure for exact logic minimization
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Citation Context ... I Introduction Two-level logic minimization is a basic problem in logic synthesis. Although algorithms exist that obtain exact minimum sum-of-products expressions (MSOP) for a large set of functions =-=[7, 8]-=-, the majority of practical systems use heuristic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. 3 This paper is an exte... |

11 | Multiple-valued decomposition of generalized Boolean functions and the complexity of programmable logic arrays - Sasao - 1981 |

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Recursive Operators for Prime Implicant and Irredundant Normal Form Determination
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Citation Context ...lasses of functions, where the ratio of the WSOP size (number of PIs) to the MSOP size is arbitrarily large when the number of variables is unbounded. We show that the Minato-Morreale algorithm [19], =-=[20]-=- produces WSOPs for this class. We also show an n-variable multiple-output function whose MSOP size is at most 2n and whose WSOP size is at least 2n . . T. Sasao is with the Department of Computer Sci... |

10 | An application of multiple-valued logic to a design of programmable logic arrays
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Citation Context ...N TO MULTIPLE-OUTPUT FUNCTIONS In the case of multiple-output functions, minimization of AND-OR two-level networks or programmable logic arrays (PLAs) can be done using characteristic functions [26], =-=[27], [29]. Definition 4.-=-1. For an n-variable function with m output values, fj…x1;x2; ...;xn†…jˆ0; 1; ...;m 1†; form an …n‡1†-variable two-valued single output function F…x1;x2; ...;xn;Xn‡1†, where xi is... |

9 |
Fast Generation of prime-irredundant covers from binary decision diagrams
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Citation Context ...show classes of functions, where the ratio of the WSOP size (number of PIs) to the MSOP size is arbitrarily large when the number of variables is unbounded. We show that the Minato-Morreale algorithm =-=[19]-=-, [20] produces WSOPs for this class. We also show an n-variable multiple-output function whose MSOP size is at most 2n and whose WSOP size is at least 2n . . T. Sasao is with the Department of Comput... |

9 | Simplest normal truth functions - Nelson - 1955 |

8 |
Arithmetic ternary decision diagrams and their applications
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Citation Context ...r general functions, the numberofminterms and PIs are very large. Thus, we use an ROBDD (reduced ordered binary decision diagram) to represent the function, and a Prime TDD (Ternary decision diagram) =-=[31]-=- to represent the set of all the PIs. In the Prime TDD for f,each path from the root node to the constant1node corresponds to a PI for f . WealsouseaROBDD to represent the Petrick function. While ther... |

7 |
Unate Truth Functions
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Citation Context ...ibutions of ISOPs. That is, some functions haveanarrow distribution, where the WSOP is nearly or exactly the same size as the MSOP. These tend to be easy to minimize. For example, for unate functions =-=[17] and parit-=-y functions, there is exactly one ISOP.Such functions are classified as "trivial" in the Berkeley PLA Benchmark Set (e.g. ALU1, BCD, DIV3, CLP1, CO14, MAX46, NEWPLA2, NEWBYTE, NEWTAG, and RY... |

4 |
Logic Design and Switching Theory, WileyInterscience Publication
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Citation Context ... f is a product that implies f such that the deletion of any literal from the product results in a new product that does not imply f. Definition 2.4. A complete sum-of-products expression (CSOP) [2], =-=[22]-=- of a function f is the SOP of all PIs of f. Definition 2.5. An irredundant sum-of-products expression (ISOP) is an SOP where each product is a PI and no PI can be deleted without changing the functio... |

4 |
Generation of prime implicants from subfunctions and a unifying approach to the covering problem
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Citation Context ...e know of no study that shows how many ISOPs exist with various number of product terms. Although various methods to generate all the ISOPs for a logic function are known [22], [12], [21], [6], [35], =-=[25]-=-, no experimental results have been reported. Experiments are computationally intensive even for functions with a small number of variables. However, we can obtain the statistical properties of ISOPs ... |

4 |
Advanced Logical Circuit Design Techniques
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Citation Context ...the majority of practical systems use heuristic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PRESTO [4], =-=[33]-=-, MINI [15], ESPRESSO [3], and others [10], [24] produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expressions x1x2... |

4 |
Palmini-fats Boolean minimizer for personal computers
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Citation Context ...r multiple-valued functions," IEEE International Symposium on Multiple-ValuedLogic, Nova Scotia, Canada, May 28-30, 1997, pp. 55-60. 1 For example, PRESTO [4, 33], MINI [15], ESPRESSO [3], and ot=-=hers [10, 24]-=- produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expressions x 1sx 2 x 2sx 3sx 1 x 3 and x 1sx 2 x 1sx 3sx 1 x 2s... |

4 |
A remark on minimal polynomials of Boolean functions
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Citation Context ...ng S n f0;1;:::;n02g =(x 1 x 2 x 1 x 3 111x n01 x n )(x 1sx 2sx 1sx 3 111x n01sx n ): (End of Example) We are interested in the sizes of the CSOP, an MSOP, and a WSOP for ST(n; k). Voight and Wegener =-=[36]-=- consider the CSOP and WSOP sizes for general symmetric functions, stating expressions and outlining a proof. Our next result gives the CSOP,MSOP, and WSOP sizes for ST(n; k) functions. A complete pro... |

3 | TTL Data Book for Design Engineers - Instruments, Inc, et al. - 1976 |

3 |
Switching Theory for Logic Synthesis,Kluwer
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Citation Context ...her Functions We also applied Algorithm 5.1 to compare the number of PIs for multiple-output functions. Table 6.3 shows the distribution of the number of PIs in ISOPs for various arithmetic functions =-=[32]-=-. INCn is an n-input n+ 1 output function, such that the value of the output is x+ 1, where x is the value of the input; WGT5 is the same as RD53, a 5-input 3-output function, where the output is a bi... |

2 |
ªMINI: A Heuristic Approach for Logic Minimization,º
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- 1974
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Citation Context ...y of practical systems use heuristic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PRESTO [4], [33], MINI =-=[15]-=-, ESPRESSO [3], and others [10], [24] produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expressions x1x2_ x2x3_ x1x... |

2 |
A state-machine synthesizer --SMS
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Citation Context ... worst and best sum-ofproducts expressions for multiple-valued functions," IEEE International Symposium on Multiple-ValuedLogic, Nova Scotia, Canada, May 28-30, 1997, pp. 55-60. 1 For example, PR=-=ESTO [4, 33]-=-, MINI [15], ESPRESSO [3], and others [10, 24] produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expressions x 1sx ... |

2 |
Introduction to Switching and Automata Theory,McGraw-Hill
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- 1965
(Show Context)
Citation Context ...e of an SOP is the number of PIs in the SOP. The size of a CSOP, WSOP, and MSOP of function f is denotedas(CSOP : f), (WSOP : f), and (MSOP : f),respectively. The following is well known. Theorem 2.1 =-=[13, 22]-=- For any switching function of n variables, (MSOP : f) 2 n01 . This upper bound is firm. For example, the exclusive OR function, f EXOR = x 1 8x 2 8:::8x n , has 2 n01 minterms, all of which are PIs, ... |

2 |
Fast generation of prime-irredundantcovers from binary decision diagrams
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(Show Context)
Citation Context ...asses of functions, where the ratio of the WSOP size (number of PIs) to that of the MSOP size is arbitrarily large when the number of variables is unbounded. Weshow that the Minato-Morreale algorithm =-=[19, 20]-=- produces WSOPs for this class. We also showann-variable multiple-output function whose MSOP size is at most 2n and whose WSOP size is at least 2 n . We also show an algorithm that produces all ISOPs ... |

1 |
ªComputer-Aided Minimization Procedure for Boolean Functions,º
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(Show Context)
Citation Context ...65] is identical to ST…9; 3†. It has 1,680 PIs, …WSOP : ST…9; 3†† 148, and…MSOP : ST…9; 3††ˆ84. POP [9], a PRESTO-type [4], [33] logic minimization algorithm, produced an ISOP wit=-=h 148 products. CAMP [1] pro-=-duced an ISOP with 130 PIs, while MINI [15] did well, producing 85 PIs. Table 3 shows the distribution of ISOPs to the number of PIs in an ISOP for ST…n; 1† for 3 n 7. This data was obtained by Al... |

1 |
ªCanonical Expressions in Boolean Algebra,º dissertation
- Blake
- 1937
(Show Context)
Citation Context ...ction f is a product that implies f such that the deletion of any literal from the product results in a new product that does not imply f. Definition 2.4. A complete sum-of-products expression (CSOP) =-=[2]-=-, [22] of a function f is the SOP of all PIs of f. Definition 2.5. An irredundant sum-of-products expression (ISOP) is an SOP where each product is a PI and no PI can be deleted without changing the f... |

1 |
ªA State-Machine SynthesizerÐSMS,º
- Brown
(Show Context)
Citation Context ...[8], the majority of practical systems use heuristic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PRESTO =-=[4]-=-, [33], MINI [15], ESPRESSO [3], and others [10], [24] produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expression... |

1 |
ªA Theorem on Trees,º Quarterly
- Cayley
(Show Context)
Citation Context ...ected tree with n 1 edges. Thus, there are 2…n 1† PIs in a WSOP for ST…n; 1†. This proves Theorem 3.2. It follows that the number of WSOPs is the number of undirected trees on n labeled nodes.=-= Cayley [5] in 1889 sho-=-wed that this number is nn 2 . tu Theorem 7.4. The number of ISOPs for ST…n; 1† with n‡1 PIs is 1 2 n 1 2 Proof. From Theorem 7.1, an ISOP of ST…n; 1† that has n‡1 PIs corresponds to a min... |

1 |
ªComputing Irredundant Normal Forms Abbreviated Presence Functions,º
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- 1965
(Show Context)
Citation Context ...unctions, we know of no study that shows how many ISOPs exist with various number of product terms. Although various methods to generate all the ISOPs for a logic function are known [22], [12], [21], =-=[6]-=-, [35], [25], no experimental results have been reported. Experiments are computationally intensive even for functions with a small number of variables. However, we can obtain the statistical properti... |

1 |
ªTwo-Level Logic Minimization: An Overview,º Integrated
- Coudert
- 1994
(Show Context)
Citation Context ...NTRODUCTION TWO-LEVEL logic minimization is a basic problem in logic synthesis. Although algorithms exist that obtain the exact minimum sum-of-products expressions (MSOP) for a large set of functions =-=[7]-=-, [8], the majority of practical systems use heuristic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PREST... |

1 |
ªMcBOOLE: A New Procedure for Exact Logic Minimization,º
- Dagenais, Agrawal, et al.
- 1986
(Show Context)
Citation Context ...UCTION TWO-LEVEL logic minimization is a basic problem in logic synthesis. Although algorithms exist that obtain the exact minimum sum-of-products expressions (MSOP) for a large set of functions [7], =-=[8]-=-, the majority of practical systems use heuristic logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PRESTO [4]... |

1 |
ªA Design System for the PLA-Based
- , Hofmann, et al.
- 1985
(Show Context)
Citation Context ...e results of the Minato-Morreale algorithm. The 9SYM (or SYM9) [11], [15] function shown in [3, p. 165] is identical to ST…9; 3†. It has 1,680 PIs, …WSOP : ST…9; 3†† 148, and…MSOP : ST��=-=�9; 3††ˆ84. POP [9]-=-, a PRESTO-type [4], [33] logic minimization algorithm, produced an ISOP with 148 products. CAMP [1] produced an ISOP with 130 PIs, while MINI [15] did well, producing 85 PIs. Table 3 shows the distri... |

1 |
ªThe Problem of Simplifying Logical Expressions,º
- Dunham, Fridshal
- 1959
(Show Context)
Citation Context ... k, and r. Table 2 shows the number of PIs in the MSOP and the WSOP of ST…n; k† r , as well as the total number of PIs. Shown also are the results of the Minato-Morreale algorithm. The 9SYM (or SY=-=M9) [11], [15] function shown-=- in [3, p. 165] is identical to ST…9; 3†. It has 1,680 PIs, …WSOP : ST…9; 3†† 148, and…MSOP : ST…9; 3††ˆ84. POP [9], a PRESTO-type [4], [33] logic minimization algorithm, produced... |

1 |
ªIrredundant Disjunctive and Conjunctive Forms of a Boolean Function,º
- Ghazala
- 1957
(Show Context)
Citation Context ...gle-output functions, we know of no study that shows how many ISOPs exist with various number of product terms. Although various methods to generate all the ISOPs for a logic function are known [22], =-=[12]-=-, [21], [6], [35], [25], no experimental results have been reported. Experiments are computationally intensive even for functions with a small number of variables. However, we can obtain the statistic... |

1 |
ªA Survey and Assessment of Progress in Switching Theory and Logical Design in the Soviet Union,º
- Kautz
- 1966
(Show Context)
Citation Context ...t …WSOP : f† 2n 1 , for any f. Indeed, Meo [18] conjectured this in the mid 1960s. However, a counterexample was published in Russian in 1962 by Yablonski [37] (which was reported in English by Ka=-=utz [16] in -=-1966). Specifically, Yablonski showed: Theorem 2.2 [37]. There exists a switching function on n variables where …WSOP : f† > 2n 1 . by showing an ISOP for S7 f0;1;3;4;6;7g with 70 PIs. This is six... |

1 |
ªUnate Truth Functions,º IRE Trans
- McNaughton
- 1961
(Show Context)
Citation Context ...utions of ISOPs. That is, some functions have a narrow distribution, where the WSOP is nearly or exactly the same size as the MSOP. These tend to be easy to minimize. For example, for unate functions =-=[17] a-=-nd parity functions, there is exactly one ISOP. Such functions are classified as ªtrivialº in the Berkeley PLA Benchmark Set (e.g., ALU1, BCD, DIV3, CLP1, CO14, MAX46, NEWPLA2, NEWBYTE, NEWTAG, and ... |

1 |
ªOn the Synthesis of Many-Variable Switching Functions,º Networks and Switching Theory
- Meo
- 1968
(Show Context)
Citation Context ...s 2n 1 minterms, all of which are PIs, and, so, …MSOP : fEXOR†ˆ2n 1 . Further, …WSOP : fEXOR†ˆ2n 1 , there being only one ISOP. It is tempting to believe that …WSOP : f† 2n 1 , for any f=-=. Indeed, Meo [18]-=- conjectured this in the mid 1960s. However, a counterexample was published in Russian in 1962 by Yablonski [37] (which was reported in English by Kautz [16] in 1966). Specifically, Yablonski showed: ... |

1 |
ªDetermination of Irredundant Normal Forms of a Truth Function by Iterated Consensus of the Prime Implicants,º
- Jr
- 1960
(Show Context)
Citation Context ...tput functions, we know of no study that shows how many ISOPs exist with various number of product terms. Although various methods to generate all the ISOPs for a logic function are known [22], [12], =-=[21]-=-, [6], [35], [25], no experimental results have been reported. Experiments are computationally intensive even for functions with a small number of variables. However, we can obtain the statistical pro... |

1 | ªSimplest Normal Truth Functions,º - Nelson - 1954 |

1 |
ªPalmini-Fats Boolean Minimizer for
- Nguyen, Perkowski, et al.
- 1987
(Show Context)
Citation Context ...logic minimization algorithms. These produce irredundant sum-of-products expressions (ISOPs) that are not necessarily minimum. For example, PRESTO [4], [33], MINI [15], ESPRESSO [3], and others [10], =-=[24]-=- produce nonminimum ISOPs. An ISOP is the OR of prime implicants (PIs) such that deleting any PI changes the function. For example, two expressions x1x2_ x2x3_ x1x3 and x1x2_ x1x3_ x1x2_ x1x3 are both... |

1 |
ªMultiple-Valued Minimization for PLA Optimization,º
- Rudell, Sangiovanni-Vincentelli
- 1987
(Show Context)
Citation Context ...TENSION TO MULTIPLE-OUTPUT FUNCTIONS In the case of multiple-output functions, minimization of AND-OR two-level networks or programmable logic arrays (PLAs) can be done using characteristic functions =-=[26], [27], [29]. Definit-=-ion 4.1. For an n-variable function with m output values, fj…x1;x2; ...;xn†…jˆ0; 1; ...;m 1†; form an …n‡1†-variable two-valued single output function F…x1;x2; ...;xn;Xn‡1†, where... |

1 |
ªTernary Decision Diagrams and Their Applications,º Representation of Discrete Functions
- Sasao
- 1996
(Show Context)
Citation Context ... general functions, the number of minterms and PIs are very large. Thus, we use an ROBDD (reduced ordered binary decision diagram) to represent the function and a Prime_TDD (Ternary decision diagram) =-=[31]-=- to represent the set of all the PIs. In the Prime_TDD for f, each path from the root node to the constant 1 node corresponds to a PI for f. We also use an ROBDD to represent the Petrick function. Whi... |

1 |
ªA Remark on Minimal
- Voight, Wegener
- 1989
(Show Context)
Citation Context ...†…x1_ x2_ _xn† ST…n; 2†ˆS n f2;3;...;ng Sn f0;1;...;n 2g ˆ…x1x2_ x1x3_ _xn 1xn† …x1x2_ x1x3_ _xn 1xn†: We are interested in the sizes of the CSOP, an MSOP, and a WSOP for ST…n; k=-=†. Voight and Wegener [36] con-=-sider the CSOP and MSOP sizes for general symmetric functions, stating expressions, and outlining a proof. Our next result gives the CSOP, MSOP, and WSOP sizes for ST…n; k† functions. A complete p... |

1 |
ªThe Problem of Bounding the Length of Reduced Disjunctive Normal forms,º Prob
- Yablonski
- 1962
(Show Context)
Citation Context ... only one ISOP. It is tempting to believe that …WSOP : f† 2n 1 , for any f. Indeed, Meo [18] conjectured this in the mid 1960s. However, a counterexample was published in Russian in 1962 by Yablon=-=ski [37] (wh-=-ich was reported in English by Kautz [16] in 1966). Specifically, Yablonski showed: Theorem 2.2 [37]. There exists a switching function on n variables where …WSOP : f† > 2n 1 . by showing an ISOP ... |