Definitions Homework. Quine McCluskey Optimal solutions are possible for some large functions Espresso heuristic. Definitions Homework

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1 EECS 33 There be Dragons here Teaher: Offie: Phone: L477 Teh Today s material might at first appear diffiult Perhaps even a bit dry... but follow losely Trust me, if you really get it, there is great depth and beauty here 3 Optimal two-level logi synthesis is N P-omplete Two-level logi minimization Goal: two-level logi realizations with fewest gates and fewest number of gate inputs Upper bound on number of prime impliants grows 3n/n where n is the number of inputs Algebrai Given > 6 inputs, an be intratable Karnaugh map However, there have been advanes in omplete solvers for many funtions Quine MCluskey Optimal solutions are possible for some large funtions Espresso heuristi 5 6 Logi minimization methods Espresso two-level logi minimization heuristi For diffiult and large funtions, solve by heuristi searh Generate only a subset of prime impliants Multi-level logi minimization is also best solved by searh Carefully selet prime impliants in this subset overing on-set The general searh problem an be introdued via two-level minimization Guaranteed to be orret May not be minimal Examine simplified version of the algorithms in Espresso 7 Usually high-quality in pratie 9 Espresso Boolean spae Start with a potentially optimal algorithm Add numerous tehniques for onstraining the searh spae b Use effiient move order to allow pruning If g and h are two Boolean funtions s.t. the on-set of g is a subset of the on-set of h then a Disable baktraking to arrive at a heuristi solver h overs g or... Widely used in industry Still has room for improvement E.g., early reursion termination...g h

2 Redundany in Boolean spae Espresso moves If a formula ontains AB and B, AB B AB is redundant Sometimes redundany is diffiult to observe If f = BC + AB + AC, then AB is redundant fa,b, ab Add a literal to a ube redue Remove a literal from a ube expansion Remove redundant ubes irredundant over 2 4 Espresso moves Irredundant funtions need not be minimal fa,b, ab fa,b, ab Sometimes neessary to inrease ost to esape loal minima Add a literal to a ube redution To later allow expansion in another dimension B C + A C + AB C Redue: AB C + A C + AB C Expand: AB + A C + AB C Irredundant over: AB + A C 5 6 Repeat the following Redue sometimes neessary to ontain ubes within others Another over with fewer terms or fewer literals might exist Shrink prime impliants to allow expansion in another variable 2 An Irredundant Cover is extrated from the expanded primes Similar goals to the Quine-MCluskey prime impliant hart Good performane requires a few triks 3 Expand impliants to their maximum size Impliants overed by an expanded impliant are removed from further onsideration Quality of result depends on order of impliant expansion Heuristi methods used to determine order Repeat sequene Redue, Expand, Irredundant Cover to find alternative prime impliants Keep doing this as long as new overs improve on last solution A number of optimizations are tried, e.g., identify and remove essential primes early in the proess 7 8 Espresso pseudoode Espresso example Proedure EspressoF, D, R : /* F is ON set, D is don t are, R OFF */ 2: R = ComplimentF+D; /* Compute omplement */ 3: F = ExpandF, R; /* Initial expansion */ 4: F = IrredundantF,D; /* Initial irredundant over */ 5: E = EssentialF,D /* Deteting essential primes */ 6: F = F - E; /* Remove essential primes from F */ 7: D = D + E; /* Add essential primes to D */ 8: while CostF keeps dereasing do 9: F = RedueF,D; /* Perform redution, heuristi whih ubes */ : F = ExpandF,R; /* Perform expansion, heuristi whih ubes */ : F = IrredundantF,D; /* Perform irredundant over */ 2: end while 3: F = F + E; 4: return F; 9 fa,b,,d d ab Irredundant but not minimal RedueExpand Irredundant Cover 2

3 Espresso input Espresso output f A, B, C, D = P 4, 5, 6, 8, 9,, 3 + d, 7, 5 Input Meaning.i 4 # inputs.o # outputs.ilb a b d input names.ob f output name.p number of produt terms AB C D = AB C D = AB C D = A B C D = A B C D = A B CD = A B C D = - AB C D = X - AB C D = X - A B C D = X.e end f A, B, C, D = P 4, 5, 6, 8, 9,, 3 + d, 7, 5 Output Meaning.i 4 # inputs.o # outputs.ilb a b d input names.ob f output name.p 3 number of produt terms - A C D = - A B D = AB =.e end ga, B, C, D = AC D + AB D + AB 2 22 Two-level heuristi minimization summary Irredundant over Generating all prime impliants an be too expensive Make inremental hanges: Expand, Redue, and Irredundant Cover to improve over Determining whether inremental hange represents same funtion is diffiult Need to use lever algorithms to speed it up After expansion, it s neessary to remove redundant ubes to reah a loal minimum First, find the relatively essential ubes For eah other ube, hek to see whether it is overed by relatively essential ubes or don t-ares If so, it s totally redundant If not, it s partially redundant Irredundant over Irredundant over Relatively essential ubes must be kept Totally redundant ubes an learly be eliminated A subset of the partially redundant ubes need to be kept Formulate as a unate overing problem We ll ome bak to this in a moment After expansion, it s neessary to remove redundant ubes to reah a loal minimum First, find the relatively essential ubes For eah other ube, hek to see whether it is overed by relatively essential ubes or don t-ares If so, it s totally redundant If not, it s partially redundant Tautology hek for relatively essential ubes Terminology example is a -ube Chek to see whether the union of -ubes and don t-are ubes minus, ofatored by, is a tautology Let A be the set of -ubes Let D be the set of don t-are ubes A D is relatively essential That s it: You an use tautology heking to determine whether a ube is relatively essential Of ourse, an example would make it learer a b f X X X X X X X X Find the relatively essential ubes Find totally redundant ubes Find partially redundant ubes 29 3

4 Deteting relatively essential ubes Reursive pivoting? How to determine whether a ube is fully overed by other and don t-are ubes? Could deompose everything to minterm anonial form Reall that there may be 2 n minterms, given n variables Deomposition is a bad idea Exponential Could also reursively pivot on variables if inlusion fails XX X, X,,, Lets us terminate reursion as soon as ube is overed by single other ube, e.g., X X However, even with pruning, this is still slow in pratie Worst-ase time omplexity? 3 32 Definition: Cofator by variable Definition: Cofator by ube, usage f x = f, x 2,...,x n f x = f, x 2,...,x n Note that it s ommutative, f x x2 = f x2 x Given that is a ube, and literals l,l 2,...l n, ofatoring the funtion by the ube is equivalent to sequentially ofatoring by all ube literals, i.e., f = f l,l2,...ln f f = A tautology is a funtion that is always true A ube is less than or equal to a funtion, i.e., is fully overed by the funtion, if and only if the funtion ofatored by the ube is a tautology Problem onversion Conversion benefits Thus, we have taken the problem Determine whether a ube,, is overed by a set of -ubes, A, or don t-are, D, ubes. and onverted it to Determine whether a set of -ubes, A, and don t-are ubes, D, ofatored by ube is a tautology. Cofatoring eliminates variables, speeding analysis Tautology is a straight-forward and well-understood problem However, tautology heking is not easy Could pivot on all variables......but this is too slow Example? Unate funtions Unate funtions fx, x 2,...,x n is monotonially inreasing in x if and only if x 2,...,x n : f,x 2,...,x n f,x 2,...,x n fx, x 2,...,x n is monotonially dereasing in x if and only if x 2,...,x n : f,x 2,...,x n f,x 2,...,x n A funtion that is neither monotonially inreasing or monotonially dereasing in x is non-monotoni in x A funtion that is monotonially inreasing or monotonially dereasing in x is unate in x A funtion that is unate in all its variables is unate a b Unate 37 38

5 Unate overs Identifying unate overs is easy Unate funtions are diffiult to identify A over is unate as long as the omplemented and unomplimented literals for the same variable do not both appear Identifying unate overs is easy a b f X X X X San the olumns for the presene of a and Note, some unate funtions an have non-unate overs Unate overs always express a unate funtion 39 4 Unate over tautology heking Fast tautology heking using unate overs A unate over is a tautology if and only if it ontains a, i.e., XXX Think of it this way: There is some point or ube in the input spae of the funtion at whih all ubes interset Thus, the only way to have a tautology is for one of the ubes to be a tautology Thus, it s trivial to hek unate overs for tautology Searh for a tautology ube Given that C is a over ontaining ubes omposed of variables x,x 2,...,x n TautologyC if C is unate then if C ontains a -row then Return true else Return TautologyC x TautologyC x 4 42 Fast tautology heking using unate overs Example of unate ofatoring What if the over isn t unate? Can still aelerate If over C is unate in a variable, x, then fator out x C = x F x 2,...,x n + F 2 x 2,...,x n or C = x F x 2,...,x n + F 2 x 2,...,x n Cover unate only in a a = b a = + b + b Notie anything nie about this? Fast tautology heking using unate overs Example of unate ofatoring Assume C = x F x 2,...,x n + F 2 x 2,...,x n Then the C x ofator is F x 2,...,x n + F 2 x 2,...,x n and the C x ofator is F 2 x 2,...,x n Cover unate only in a a = b a = + b + b F = + b F 2 = b 45 46

6 Fast tautology heking using unate overs final version Clearly, C x = F x 2,...,x n + F 2 x 2,...,x n C x = F 2 x 2,...,x n C x C x Therefore we need only onsider C x for tautology heking, signifiantly simplifying the problem, i.e., if C x is a tautology, then C x is obviously also a tautology. Given that C is a over ontaining ubes omposed of variables x,x 2,...,x n TautologyC if C is unate then if C ontains a -row then Return true else Return TautologyC x Summary: Fast tautology heking More ompliated example Identify unate overs No olumns with s and zeros If unate, san for an XXX row If not unate, ofator on preferable unate variable Only need to onsider unomplemented or unomplemented ofator Why? F 2 F + F 2 a b f X X X X X X Relatively essential hek for X? Full hek on Two-level heuristi minimization summary Espresso summary Generating all prime impliants an be too expensive Make inremental hanges: Expand, Redue, and Irredundant Cover Determining whether inremental hange represents same funtion is too expensive Use ofatoring to onvert it to a tautology hek Use unateness to make the tautology hek fast in most ases Redue: Allows expansion in another diretion, get out of loal minima Expand: Dereases omplexity, in pratie bloking matrix used for expansion. Searh with would also work but would be slower in most ases. Irredundant over: Remove redundant ubes Tautology hek used in many plaes, gave example of use in Irredundant over use We have only srathed the surfae! 5 52 CAD Questions Computer Aided Design of Integrated Ciruits and Systems Also alled Eletronis Design Automation EDA Without it, omputers wouldn t work What is the unate overing problem? Where have we seen it used? What an tautology heking be used for? How do we make it fast? 53 54

7 Next leture: Implementation tehnologies Reading assignment PALs, PLAs MUX, DEMUX review Steering logi M. Morris Mano and Charles R. Kime. Logi and Computer Design Fundamentals. Prentie-Hall, NJ, fourth edition, 28 Chapter 4 M. Morris Mano and Charles R. Kime. Web supplements to Logi and Computer Design Fundamentals. Prentie-Hall, NJ. VLSI Programmable Logi Devies, doument 55 57