Two-Level Minimization

Transcription

1 Two-Level Minimization Logi Ciruits Design Seminars WS2010/2011, Leture 5 Ing. Petr Fišer, Ph.D. Department of Digital Design Faulty of Information Tehnology Czeh Tehnial University in Prague Evropský soiální fond Praha & EU: Investujeme do vaší budounosti

2 Outline Terminology Espresso BOOM FC-Min Exorism PI-SCN-5, ČVUT FIT, Petr Fišer,

3 Terminology Single-output Boolean funtion: f: B n B; B {0, 1} (input) variable Literal variable, negated variable. Ourrene of a variable expression size measure Boolean n-ube produt of literals (produt term) n-dimensional Boolean spae, B n Minterm one point (vertex) in B n Dimension of a ube = log(# of minterms it overs) = n - (# of ube literals) A ube is an impliant of f iff f Cover of f set of ubes (impliants) i representing f, f = i (for ompletely speified funtions) A ube i overs j, i j, iff all minterms inluded in j are inluded in i. PI-SCN-5, ČVUT FIT, Petr Fišer,

4 Cover Cover F of a funtion f Set of ubes i F { 1,..., }; set of ubes overing on-set Note: i f f k k i i1 Example f = b + a F = { b, a } a b PI-SCN-5, ČVUT FIT, Petr Fišer,

5 Cover Irredundant over F of funtion f F { 1,..., k }; f ; F { } f for no ube an be removed from the over Redundant / irredundant ube k i1 i j any j Example f = b + a a b This is REDUNDANT f = b + a + ab PI-SCN-5, ČVUT FIT, Petr Fišer,

6 Cover Prime literal of j Literal that, when deleted from j, F is no longer a over of f Prime ube Cube having all literals prime = prime impliant (PI) Prime over Cover having all ubes prime PI-SCN-5, ČVUT FIT, Petr Fišer,

7 Cover Prime and irredundant over Cannot be simplified by removing literals (ubes) Does not guarantee minimality! Example a b a b PI-SCN-5, ČVUT FIT, Petr Fišer,

8 Cover Essential prime impliant (EPI) 1. Impliant, that must be a part of any minimal over 2. A prime impliant i is essential, if there is a minterm overed by i and no other impliant PI-SCN-5, ČVUT FIT, Petr Fišer,

9 PI-SCN-5, ČVUT FIT, Petr Fišer, Shannon (Boole) Cofator i x i i x i f x f x f ),...,0,..., ( ),...,1,..., ( 1 1 n x n x x x f f x x f f i i 0 1, i i i i x x x x f f f f Cofator f of f by a ube Obtained by ofatoring f by all ube literals of Cofator f xi of f by a variable x i

10 Funtion Inlusion Theorem Theorem: f f 1 Proof: 1) f 1 f f Lemma: x.f = x.f x Assume f = 1, then.f =.f = f PI-SCN-5, ČVUT FIT, Petr Fišer,

11 Inompletely Speified Funtions Funtion uniquely given by on-set (f) off-set (r) d-set (d) F = (f, d, r), where f, d, r are ompletely speified funtions f + d + r = B n Pairwise disjoint: f.r =, f.d =, d.r = PI-SCN-5, ČVUT FIT, Petr Fišer,

12 Inompletely Speified Funtions A ube is an impliant of f iff f + d Cover Completely speified funtion g is a over, iff f g f + d Irredundant, prime similar definitions PI-SCN-5, ČVUT FIT, Petr Fišer,

13 Prime Chek Cube is a prime impliant, iff all literals are prime Any literal removal disturbs the ondition f + d Let l = - { l }, where l is any literal, l. Proedure: Try all literals in for removal (generate l s) Chek for eah l, if it is an impliant If some l is an impliant, is not prime For l be an impliant, the following holds: l l f + d l r, l (f + d) PI-SCN-5, ČVUT FIT, Petr Fišer,

14 Prime Chek How to hek if a ube is an impliant? 1. is an impliant of f, iff f + d f + d (f + d) 1 hek (f + d) for tautology 2. is an impliant of f, iff r = But r = (f + d) ompute off-set of f (if not expliitly provided) 1. tautology heking is NPC 2. omplement of (f + d) may grow exponentially in size PI-SCN-5, ČVUT FIT, Petr Fišer,

15 Irredundant Chek Cube j of over G = { 1, i } of F = (f, d, r) is redundant, iff j ( G-{ j } ) D, where D is a over of d i is redundant iff (( G - { j } ) D ) j 1 again, NPC tautology heking! PI-SCN-5, ČVUT FIT, Petr Fišer,

16 Multi-Output Funtions No big hange Multi-output funtion f: B n B m ; B {0, 1} PI-SCN-5, ČVUT FIT, Petr Fišer,

17 Multi-Output Funtions Cover F of a funtion f Set of ubes i F { 1,..., k }; f j k j i1 i for j Example f 1 = b + a f 2 = a + b b b a a F = { b, a, a, b } PI-SCN-5, ČVUT FIT, Petr Fišer,

18 Multi-Output Funtions Prime ube (PI) When some literal is removed, it will no longer be an impliant of any funtion from f 1 f m it formerly has been Example f 1 = b + a f 2 = a + b b b a a Primes are: { b, a, a, b, ab } PI-SCN-5, ČVUT FIT, Petr Fišer,

19 SOP Minimization Algorithms Quine-MCluskey Espresso BOOM FC-Min PI-SCN-5, ČVUT FIT, Petr Fišer,

20 Quine-MCluskey Basis of standard SOP minimization proess Produes exat results (user-defined minimum) Method: 1. Generate all PIs (& 1-minterms) 2. Solve the set overing problem (unate over) PI-SCN-5, ČVUT FIT, Petr Fišer,

21 Quine-MCluskey Basis of standard SOP minimization proess Produes exat results (user-defined minimum) Method: 1. Generate all PIs (& 1-minterms) Exponential number of PIs 2. Solve the set overing problem (unate over) NP-hard PI-SCN-5, ČVUT FIT, Petr Fišer,

22 Quine-MCluskey Basis of standard SOP minimization proess Produes exat results (user-defined minimum) Method: 1. Generate all PIs (& 1-minterms) Exponential number of PIs Do not generate provably unneessary PIs Impliit representations of PIs (BDDs, ZDDs) 2. Solve the set overing problem (unate over) NP-hard, not APX, not NPO-C Dominane resolving (waste of time?) Effiient searh spae pruning (better lower bounds) PI-SCN-5, ČVUT FIT, Petr Fišer,

23 Quine-MCluskey Relaxed Method: 1. Generate some PIs 2. Solve the set overing problem approximately Optimum not guaranteed PI-SCN-5, ČVUT FIT, Petr Fišer,

24 Espresso The algorithm: Espresso(f, d, r) { // Some pre-proessing is here do { do { f = Redue(f, d); f = Expand(f, r); f = Irredundant(f, d); } while ( fewer_terms_in(f) ); // Last gasp g = Redue_Gasp(f, r); g = Expand(g, r); f = Irredundant(f+g, d); } while ( G ); // Some post-proessing is here } Main loop PI-SCN-5, ČVUT FIT, Petr Fišer,

25 Espresso - Redue Maximally redue all ubes, so that the over is retained (i.e. no 1-minterm is left unovered) Example a b a b f = b + a + a + b f = ab + a + ab + b PI-SCN-5, ČVUT FIT, Petr Fišer,

26 Espresso - Expand Maximally expand all ubes, so that the over is retained (i.e. no 0-minterm is overed) Example a b a b f = ab + a + ab + b f = ab + a + a + b PI-SCN-5, ČVUT FIT, Petr Fišer,

27 Espresso - Irredundant Remove redundant ubes Example a b a b f = ab + a + a + b f = ab + a + a + b PI-SCN-5, ČVUT FIT, Petr Fišer,

28 Espresso Last Gasp Last gasp: // Redue ubes up to minterms. Cover need not be satisfied g = Redue_Gasp(f, r); // Expand them g = Expand(g, r); // and try to use them in the new over f = Irredundant(f+g, d); There is a hane that brand new ubes will appear A way to get out of a loal minimum PI-SCN-5, ČVUT FIT, Petr Fišer,

29 BOOM Espresso: refines the original funtion over BOOM: produes impliants from srath, the original over serves just as an aid and onstraint Randomized May be run iteratively to arbitrarily improve the solution CD-Searh Impliant Expansion Impliant Redution CP Solution PI-SCN-5, ČVUT FIT, Petr Fišer,

30 BOOM CD-Searh Coverage-Direted Searh Main phase Greedy, randomized algorithm Impliants generated top-down, by reduing universal hyperube (dimension n) literals are added to a ube, until it beomes an impliant Main idea: literals appearing most frequently in the unovered on-set are preferred PI-SCN-5, ČVUT FIT, Petr Fišer,

31 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1):... f(0):... Term in progress: Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

32 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): 6... f(0): 3... Term in progress: Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

33 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): f(0): Term in progress: Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

34 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): f(0): Term in progress: Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

35 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): f(0): Term in progress: Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

36 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): f(0): Term in progress: Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

37 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): f(0): Term in progress: (e ) Solution: f = One seleted randomly PI-SCN-5, ČVUT FIT, Petr Fišer,

38 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): ----X f(0): ----X Term in progress: (e ) Solution: f = Forget this part for now PI-SCN-5, ČVUT FIT, Petr Fišer,

39 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): 5---X f(0): 2---X Term in progress: (e ) Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

40 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): 54--X f(0): 23--X Term in progress: (e ) Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

41 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): 543-X f(0): 234-X Term in progress: (e ) Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

42 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): 5432X f(0): 2344X Term in progress: (e ) Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

43 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): 5432X f(0): 2345X Term in progress: (ae ) Solution: f = One seleted randomly PI-SCN-5, ČVUT FIT, Petr Fišer,

44 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): X---X f(0): X---X Term in progress: (ae ) Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

45 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): X4--X f(0): X1--X Term in progress: (ae ) Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

46 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): X42-X f(0): X13-X Term in progress: (ae ) Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

47 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): X422X f(0): X133X Term in progress: (ae ) Solution: f = PI-SCN-5, ČVUT FIT, Petr Fišer,

48 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): X422X f(0): X133X Term in progress: (abe ) Solution: f = abe Seleted PI-SCN-5, ČVUT FIT, Petr Fišer,

49 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): f(0): Term in progress: Solution: f = abe PI-SCN-5, ČVUT FIT, Petr Fišer,

50 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): f(0): Term in progress: (b ) Solution: f = abe One seleted randomly PI-SCN-5, ČVUT FIT, Petr Fišer,

51 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): 2X102 f(0): 3X453 Term in progress: (b d ) Solution: f = abe Seleted PI-SCN-5, ČVUT FIT, Petr Fišer,

52 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): 2X1X2 f(0): 3X4X3 Term in progress: (b d ) Solution: f = abe + b d Seleted PI-SCN-5, ČVUT FIT, Petr Fišer,

53 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): f(0): Term in progress: (a ) Solution: f = abe + b d One seleted randomly PI-SCN-5, ČVUT FIT, Petr Fišer,

54 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): X0100 f(0): X1011 Term in progress: (a d ) Solution: f = abe + b d One seleted randomly PI-SCN-5, ČVUT FIT, Petr Fišer,

55 CD-Searh Example d e a b a X X X 1 X X X X X 1 X X X X Literal ounts: abde f(1): X01X0 f(0): X10X1 Term in progress: (a b d ) Solution: f = abe + b d + a b d One seleted randomly PI-SCN-5, ČVUT FIT, Petr Fišer,

56 BOOM CD-Searh need not produe PIs impliants must be expanded Literals are tried for removal, while the ube is still an impliant Multiple-output funtions: CD-Searh is run for eah output separately impliants must be redued Literals are added, if the produed term impliates additional outputs All produed impliants are put into a ommon impliant pool Covering problem is solved at the end Fast heuristi (LCMC: least-overed, most overing) Fast exat CP solvers (Aura-II) PI-SCN-5, ČVUT FIT, Petr Fišer,

57 BOOM Iterative Proess PI-SCN-5, ČVUT FIT, Petr Fišer,

58 Prime Impliants Literals BOOM Iterative Proess Iterations PI-SCN-5, ČVUT FIT, Petr Fišer,

59 Summarized: BOOM Good for sparse (highly inompletely speified) funtions Computation of andidate literals is O(p), p is the number of terms Good for funtions with many inputs (up to thousands) No EXPTIME(n) algorithm involved Most of algorithms are O(n) Good for PLAs with speified off-set (fr) otherwise the off-set must be omputed, or tautology used Non-deterministi solution quality may be arbitrarily improved, for a ost of runtime mutations in CD-Searh all PIs will be produed in infinite time Bad for funtions with many outputs Group minimization not performed impliitly, like in Espresso Expensive impliant redution phase Bad for funtions speified by many terms Computation of andidate literals is O(p), p is the number of terms Other algorithms are O(p 2 ), p is the number of terms Non-deterministi solution quality is unpreditable If you have a truth table with 500 inputs and 1000 are terms, BOOM is the best hoie PI-SCN-5, ČVUT FIT, Petr Fišer,

60 FC-Min For multi-output funtions Group impliants are generated diretly On-set over is used just as a onstraint, like in BOOM Could be extremely fast Randomized Solution quality may be improved by intensifiation of the searh, for a ost of runtime Very low memory demands no additional term storage involved Good for highly unspeified funtions PI-SCN-5, ČVUT FIT, Petr Fišer,

61 FC-Min Algorithm 1. Find Retangle Cover of the on-set (therefore FC-Min) Determine impliants, their number, but not their struture (literals) 2. Validate the over using the on-set over Derive the struture of impliants and validate them 3. Expand impliants PI-SCN-5, ČVUT FIT, Petr Fišer,

62 { FC-Min Example Find Cover phase y0-y4 PLA: x 0 - x 4 y 0 - y Rows Columns t 1 {4, 6, 8} {y 3, y 4 } t 2 {1, 2, 7} {y 1, y 2 } t 3 {8, 9} {y 0, y 2 } t 4 {3} {y 1, y 3 } t 5 {0, 1} {y 0, y 1 } t 6 {4, 7} {y 2, y 4 } PI-SCN-5, ČVUT FIT, Petr Fišer,

63 FC-Min Impliant generation Main Idea: When a term (ube) should over a partiular output vetor (set of olumns), the orresponding input vetor must be ontained in this ube Thus the minimum term for t i must be onstruted as a minimum superube of all the input vetors orresponding to rows of t i PI-SCN-5, ČVUT FIT, Petr Fišer,

64 FC-Min Example impliant generation t 1 overs 4, 6 and PI-SCN-5, ČVUT FIT, Petr Fišer,

65 FC-Min Example impliant generation Impliants: t 1 : t 2 : t 3 : t 4 : t 5 : t 6 : SOP Forms: y 0 = t 3 + t 5 = x 0 x 2 x 3 ' + x 0 x 2 ' x 4 y 1 = t 2 + t 4 = x 2 'x 3 ' + x 0 ' x 1 x 2 x 3 x 4 y 2 = t 2 + t 3 + t 6 = x 2 'x 3 ' + x 0 x 2 x 3 ' + x 0 ' x 1 ' y 3 = t 1 + t 4 = x 1 'x 2 + x 0 ' x 1 x 2 x 3 x 4 y 4 = t 1 + t 6 = x 1 'x 2 + x 0 ' x 1 ' PI-SCN-5, ČVUT FIT, Petr Fišer,

66 FC-Min Impliant generation & validation Sounds nie. But what if the generated superube intersets the off-set (i.e. is not an impliant)? impliant is not valid, must be reomputed Options: 1. Reompute the whole retangle over 2. Chek an impliant for validity immediately after it is produed in the FC phase, generate a new one, if not valid FC and impliant generation phases are interleaved Retangle over generation algorithm is randomized and driven by depth fator, termination is guaranteed (singular ase of 1-term vetor) PI-SCN-5, ČVUT FIT, Petr Fišer,

67 FC-Min Impliant Expansion Superubes may be further expanded Like in BOOM attempting for literal removal, while the ube is still an impliant PI-SCN-5, ČVUT FIT, Petr Fišer,

68 FC-Min Summarized Good for funtions with many outputs or, better, many visible group impliants Bad for single-output funtions Here the impliant generation is performed purely adho Many inputs do not matter All algorithms are O(n) Number of terms, expliit off-set speifiation Like in BOOM PI-SCN-5, ČVUT FIT, Petr Fišer,

69 BOOM-II Combination of BOOM and FC-Min CD-Searh produes PIs BOOM START FC:BOOM FC-Min FC-Min produes group impliants Expensive Impliant Redution phase is substituted by FC-Min BOOM (CD-S, IE, IR) FC-Min (FC, FI, IE) NO STOP? YES CP Solution END PI-SCN-5, ČVUT FIT, Petr Fišer,

70 SOP Minimization Referenes W.V. Quine, The problem of simplifying truth funtions, Amer. Math. Monthly, 59, No.8, 1952, pp E.J. MCluskey, Minimization of Boolean funtions, The Bell System Tehnial Journal, 35, No. 5, Nov. 1956, pp R.K. Brayton et al., Logi minimization algorithms for VLSI synthesis, Boston, MA, Kluwer Aademi Publishers, 1984, 192 pp. O. Coudert and J.C. Madre. Impliit and Inremental Computation of Primes and Essential Primes of Boolean funtions, Pro. of 29th DAC, Anaheim CA, USA, June 1992, pp O. Coudert, Two-level logi minimization: an overview, Integration, the VLSI journal, 17-2, pp , Ot O. Coudert, J.C. Madre and H. Fraisse. A New Viewpoint on Two-Level Logi Minimization, Pro. of 30th DAC, Dallas TX, USA, June 1993, pp O. Coudert, Doing two-level logi minimization 100 times faster, Pro. of the sixth annual ACM-SIAM symposium on Disrete algorithms, 1995, pp P. Fišer and J. Hlavička, BOOM - A Heuristi Boolean Minimizer, Computers and Informatis, Vol. 22, 2003, No. 1, pp P. Fišer and H. Kubátová, Boolean Minimizer FC-Min: Coverage Finding Proess, Pro. 30th Euromiro Symposium on Digital Systems Design (DSD'04), Rennes (FR), , pp P. Fišer, H. Kubátová, Two-Level Boolean Minimizer BOOM-II, Pro. 6th Int. Workshop on Boolean Problems (IWSBP'04), Freiberg, Germany, , pp PI-SCN-5, ČVUT FIT, Petr Fišer,

71 ESOP Minimization The on-set must be overed all 1 s = = = = = = 0 1 s must be overed by an odd number of ubes, 0 s by even number of ubes PI-SCN-5, ČVUT FIT, Petr Fišer,

72 ESOP Minimization Why ESOP synthesis / minimization (and why not)? XOR is really powerful XOR-based iruits are well testable XOR gate is big But not too muh. 2-NAND = 4 transistors, 2-XOR = 6 transistors and XOR-based iruits sometimes have muh less gates than only NAND-based ones Not many well-known ESOP minimization algorithms Where is the problem? ESOP is not as intuitive as SOP What proesses produe ESOPs? PI-SCN-5, ČVUT FIT, Petr Fišer,

73 ESOP Minimization SOP: a b + b + a + ab a b a b ESOP: ab a b PI-SCN-5, ČVUT FIT, Petr Fišer,

74 ESOP Minimization SOP: ab + a b + a b + ab a b a b ESOP: a b PI-SCN-5, ČVUT FIT, Petr Fišer,

75 PI-SCN-5, ČVUT FIT, Petr Fišer, ESOP Minimization Mostly based on iterative ube transformations (Boolean): x x x x x x x x f f x f x f x f f x f x f f f x f x f f x f x f where : expansion Davio Negative : expansion (Reed - Muller) Davio Positive : expansion Shannon : expansion Shannon xy y xy x x x x 1 1 1

76 EXORCISM Based on these fats: 1. Two idential ubes may be added to ESOP, without hanging the funtion 1 2 n = 1 2 n 2. Distane-1 ubes may be merged ab ab = a + Exorlink operation Generalization of several simple ube transformations Replaing two ubes by a set of ubes, without hanging the funtion Distane-k Exorlink Distane of proessed ubes is k Produes k! ube groups, k ubes in eah PI-SCN-5, ČVUT FIT, Petr Fišer,

77 EXORCISM Algorithm esop HeuristiMinimization( fun F, iterations it ) { esop Cover = GenerateStartingCover( F ); for ( i = 0; i < iterations; i++ ) { ResetCubePairs( Cover ); do { do { Cover = AgressiveMinimization(Cover); } while (there is improvement); Cover = LastGaspMinimization(Cover); } while (there is improvement); } Cover = RefinementMinimization(Cover); return Cover; } [Mishhenko, 2001] PI-SCN-5, ČVUT FIT, Petr Fišer,

78 Referenes ESOP Minimization D. E. Muller. Appliation of Boolean algebra to swithing iruit design and to error detetion. IRE Trans. on Eletron. Comp. Vol. EC-3, pp. 6-12, D. Brand, T. Sasao, "Minimization of AND-EXOR Expressions Using Rewrite Rules", IEEE Transations on Computers, vol. 42, pp , N. Song, M. Perkowski, "EXORCISM-MV-2: Minimization of Exlusive Sum of Produt Expressions for Multiple-Valued Input Inompletely Speified Funtions, ISMVL 1993, pp A. Mishhenko, M. Perkowski, "Fast Heuristi Minimization of Exlusive-Sums-of- Produts" In International Workshop on Reed-Muller expansions in iruit design, 2001, pp N. Song, Minimization of Exlusive Sum of Produt Expressions for Multi-Valued Input Inompletely Speified Funtions, M.S. Thesis. EE Dept. Portland State University.Portland, OR, S. Stergiou, K. Daskalakis, G. Papakonstantinou, "A fast and effiient heuristi ESOP minimization algorithm" In Proeedings of the 14th ACM Great Lakes symposium on VLSI, Boston, MA, USA, 2004, pp PI-SCN-5, ČVUT FIT, Petr Fišer,