Project 1 Part II: Metaproduct Primes

Transcription

1 Project Part II: Metaproduct Primes What ou know Basic BDD data structure and JAVA implementation A little bit about these things called metaproducts What ou don t know All the tricks with metaproducts Using these to do Prime Implicants R. Rutenbar 2, CMU 8-76, Fall 2 Copright Notice Rob A. Rutenbar, 2 All rights reserved. You ma not make copies of this material in an form without m express permission. R. Rutenbar 2, CMU 8-76, Fall 2 2 Page

2 About Metaproducts Notation was created to support applications where we need to preserve the structure of things like SOP expressions ie, if ou reall WANT to write x + x and ou want to represent and manipulate it as a BDD, what to do? Metaproduct notation Replace each variable x with a pair (, ) If ou see x in a product, then ou get ()() in metaproduct If ou see x in a product, then ou get ()( ) in metaproduct If ou don t see an x or x at all, then ou get ( ) in metaproduct In English is the occurrence variable -> == sas x is here, == sas no x is the sign variable -> == sas x is positive, == negative R. Rutenbar 2, CMU 8-76, Fall 2 3 Metaproduct Example Suppose F(x,) = x + x This is reall just == x, of course Its BDD would be simpl As a metaproduct Assume var ordering was x < Then new ordering is x < < < < < x becomes ()()( ) x becomes ()()()( ) x R. Rutenbar 2, CMU 8-76, Fall 2 4 Page 2

3 Page 3 Metaproduct Example You interpret this b looking at satisfing paths to node There are 2 paths from root to, each makes a product term x is here x is pos is not here. Result: x x is here x is pos is here... is neg... Result: x Put the final products together for final answer: x + x R. Rutenbar 2, CMU 8-76, Fall 2 5 Metaproduct Example What happens if a variable is not present? We alread saw this, but its worth noting In F(x,) = x + x, consider x term There s no in there, but we still have to deal with Rule is: if there s no variable in there, ou still have to include the occurrence variables for the missing original vars, but ou include them in the negative polarit to note their absence So, term x becomes ()()( ) Means no vars in here If this was F(x,,,w) =x, what would happen? We d get ()()( )(r )(rw ) R. Rutenbar 2, CMU 8-76, Fall 2 6

4 Metaproduct Primes Turns out ou can represent Prime Implicants with metaproduct notation x w x w These are all Primes biggest product terms ou can circle in a Kmap for our function These are NOT all Primes not all biggest product terms ou can Circle in this Kmap R. Rutenbar 2, CMU 8-76, Fall 2 7 Metaproduct Primes ike with Boolean functions, problem is SIZE A function can have man man primes wa too man enumerate one b one This is wh representing them with something like a BDD is attractive, since its good at compressing Boolean functions But this is also wh we need a special notation, since we DON T want the BDD to CANGE our function to its canonical form We need to represent it in some SOP form Biq question: how do we find Primes using metaproducts Of course, it s gotta be something recursive, right? R. Rutenbar 2, CMU 8-76, Fall 2 8 Page 4

5 Page 5 Metaproduct Primes There s another Shannon-stle recursive decomposition You start with a BDD in our original variables You end up with a BDD in the (occurrence, sign) metaproduct variables Final BDD represents, as SOP form, A the primes Basic decomposition et P( BDD root of F) = metaproduct BDD for all PRIMES in function F P( x ) = P( * ) P() * P( *) P() * P( *) R. Rutenbar 2, CMU 8-76, Fall 2 9 Metaproduct Primes wh does that work? Roughl speaking this is just the mes check for how to reassemble primes when the get split up during a Shannon decomposition x f(x,,) = x + = 2 primes = f + f = (x + ) + () x f(x,,) = x = prime = f + f = (x) + (x) x x When we factored on, we chopped up the prime like this x x When we factored on, we chopped up the prime like this R. Rutenbar 2, CMU 8-76, Fall 2

6 Metaproduct Primes Reading the decomposition P( x ) = P( * ) If there s no x vars, then get primes in * P() * P( *) If neg x var, get primes in that are NOT in * P() * P( *) If pos x var, get primes in that are NOT in * R. Rutenbar 2, CMU 8-76, Fall 2 Metaproduct Primes et s be a little more careful on the details of BDDs and ops Assumes ou have AND and NOT (ie,! ) on BDDs Assumes P( ) calculated just like ITE, as a top-down recursion Assume var order is fixed: for vas x,,, its: x < < < < <. P( x ) = P( AND[, ] ) AND[ P( ),!P( AND[, ] )] AND[ P( ),!P( AND[, ] )] R. Rutenbar 2, CMU 8-76, Fall 2 2 Page 6

7 Metaproduct Primes: Termination So we know the recursion is: P( x ) = P( AND[, ] ) AND[ P( ),!P( AND[, ] ) AND[ P( ),!P( AND[, ] ) next question: what are the termination conditions for P( )? So, when can we quit, and return a known BDD node answer? Ea case: P( ) = arder case: P( ) = a little mes R. Rutenbar 2, CMU 8-76, Fall 2 3 Metaproduct Primes: Termination Suppose vars are: x,,,w, and we have this recursion P( x ) = P( ) This is just P( ) Intuition This is the product of complements of all occurrence vars later in the order, ie, after In this case: ( )(r )(rw ) = Something that needs more recursion happens here P() means ou re done nothing more at all this prime term P() means ou re done but remember that these vars are absent r rw R. Rutenbar 2, CMU 8-76, Fall 2 4 Page 7

8 Page 8 Primes Example x f(x,,) = x + = 2 primes Appl recursion at root of f() x Ordina BDD with var order: x < < P( x ) = P( ) and note: x = = R. Rutenbar 2, CMU 8-76, Fall 2 5 Primes Example Appling recursion at root of f() P( x ) = P( AND[, ] ) AND[ P( ),!P( AND[, ] )] AND[ P( ),!P( AND[, ] )] R. Rutenbar 2, CMU 8-76, Fall 2 6

9 Page 9 Primes Example Do the obvious simplifications now (just to simplif for this manual example) P( x ) = P( ) AND[ P( ),!P( ) ] This was just P(foo) *!P(foo) = R. Rutenbar 2, CMU 8-76, Fall 2 7 OK, we need to do this one next P( ) = P( AND[, ] ) AND[ P( ),!P( AND[, ] )] AND[ P( ),!P( AND[, ] )] R. Rutenbar 2, CMU 8-76, Fall 2 8

10 Page Again, do obvious simplifications (just for this manual ex) P( ) = P( ) = P( ) P() * stuff = R. Rutenbar 2, CMU 8-76, Fall 2 9 We have to do this one next and its ea P( ) = r P( AND[, ] ) s This is P() = AND[ P( ),!P( AND[, ] )] This is P()* stuff = AND[ P( ),!P( AND[, ] )] This is P()*!P() = P()* = P() R. Rutenbar 2, CMU 8-76, Fall 2 2

11 Page We have to do this one next and its ea P( ) = r s This is P() = product of complements of the vars later in the order than s. Since s is the AST var in the order, the rule is: this is just R. Rutenbar 2, CMU 8-76, Fall 2 2 Return results up recursive call tree P( ) = = P( ) r s Note I m leaving in all the separate and nodes just to simplif the drawing it s a REA BDD, there s onl a single and a single R. Rutenbar 2, CMU 8-76, Fall 2 22

12 Page 2 Return results up recursive call tree P( x ) = P( ) = r s AND[ P( ),!P( )] AND[ P( ), NOT( )] r s R. Rutenbar 2, CMU 8-76, Fall 2 23 This one is next to recurse on AND[ P( ), NOT( )] r s Since we know P() = product of complements of vars below in the order, this is supposed to be: ( )(r ), so we get AND[, NOT( ) ] r r s which is just ordina BDD ops R. Rutenbar 2, CMU 8-76, Fall 2 24

13 Page 3 BDD ops give this AND[, NOT( ) ] r r s = r r s R. Rutenbar 2, CMU 8-76, Fall 2 25 Return results up recursive call tree P( ) = x r s AND[ P( ), NOT( )] r s = r s r r s R. Rutenbar 2, CMU 8-76, Fall 2 26

14 Page 4..and, that s the Final Metaproduct for Prime( ) ook for paths from root to x is not here is here is negative is here is positive r s r r This prime is s R. Rutenbar 2, CMU 8-76, Fall 2 27 Final Primes More paths x is here r s r r s x is negative is not here can ignore sign is not here can ignore sign This prime is x R. Rutenbar 2, CMU 8-76, Fall 2 28

15 Page 5 Final Primes he, there s another one? x is here An unfortunate fact about metaproducts: the can be redundant about primes. You don t get the wrong ones but ou can get right ones several times. r s r r s x is negative is not here can ignore sign is not here This prime is also x R. Rutenbar 2, CMU 8-76, Fall 2 29 Metaproduct Primes So what did we do? x f(x,,) Ordina BDD with var order: x < < x Primes = paths from root to. Can be redundant. x Primes = {x, ) r BDD for Primes( f ) in metaproduct form s r r s R. Rutenbar 2, CMU 8-76, Fall 2 3

16 back to 8-76 Project Part 2 What do we want? We want ou to add P( BDD for function f) as an operator to our JAVA BDD package Do it exactl like we showed here Just like ITE: ou descend the starting BDD for f, and ou recursivel trace out the BDD for P(f) Assume ou have all the vars defined in the right initial order. This means if the real vars are x,, YOU have x,,,,, in order You have 2 basic goals To be able to transform a BDD for function f into Prime(f) To print out some interesting info about these primes R. Rutenbar 2, CMU 8-76, Fall 2 3 Prime( ) Details Things to be careful about Before doing anthing, ou probabl want to build the function: ( )( )(r ) (rlast ) for A our vars. And make an arra of pointers to the nodes, so that when ou need P() = product of complemented occurrence nodes below me ou can just look it up You still need to call FindOrCreateNode()on the 2 new nodes ou make. You want to build and its children first, call FindOrCreatNode(), then finish the recursion on, then call FindOrCreatNode(). Do ou want to do something like a different OPS table for the Prime computation? (It s not required but think about it) You will want to write a printprime routine that walks the paths to the node, and prints out sensible product terms. DO NOT wor about the redundanc issue not our problem. You also want to build a numprimes routine that just prints out the number of paths to the node. Think about it ou don t have to walk them all to do this, it s a ve simple recursion if ou know numprimes(hichild) and numprimes(lochild), and numprimes()= and numprimes()= R. Rutenbar 2, CMU 8-76, Fall 2 32 Page 6

17 Metaproduct Primes: Summa Interesting, sort of funk BDD application Twists the usual interpretation of canonical BDD form around a lot Works fine, a bit arcane (This is a simplification of how people reall do it. There are a bunch of other optimiations to get rid of those redundancies that make it a lot faster. Not worth the grief to go thru them all the violate a lot of BDD rules.) For Project Implement Prime( f ) ook on the /afs/ece/class/ee76/proj directo for more details, and for some info about benchmarks to run Ask TA and Prof questions if there are an issues at all on this one R. Rutenbar 2, CMU 8-76, Fall 2 33 Page 7