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ENGG3190 Logic Synthesis “Multi Level Logic” (Part II) Winter 2014 S. Areibi School of Engineering University of Guelph

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Outline Implicit Don’t CaresImplicit Don’t Cares How they arise due to structureHow they arise due to structure Types and flavors of Don’t CaresTypes and flavors of Don’t Cares –Satisfiability Don’t Cares –Controllability Don’t Cares –Observability Don’t Cares SummarySummary 2

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3 Logic Optimization methods Logic Optimization Multi-level logic (standard cells) Multi-level logic (standard cells) Two-level logic (PLA) Exact (QM) Heuristic (espresso) Heuristic (espresso) Structural (SIS) Structural (SIS) Functional (AC, Kurtis) Functional (AC, Kurtis) Functional (BDD-based) Functional (BDD-based) algebraic Boolean Boolean

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Don’t cares in basic logic design. 1,0 x (allow more flexibility) MLLS don’t care arise naturally and implicitly. Don’t cares Rich source of optimization MLS: Implicit Don’t Cares

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Algebraic model Using the Algebraic model we lose some means of expressing the network. In MLS don’t cares are called “implicit” because they happen naturally. What computational procedure What computational procedure can we use to hunt for the don’t cares?

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What do we know so far about don’t cares? We are free to treat the x or ‘d’ as a ‘0’ or a ‘1’ What is different in Multi-Level Optimization?

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If X, b, y are primary inputs then we cannot say anything yet! by other parts However, If X, b, y are produced by other parts of the network then we might!

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X is not a primary input. It is computed inside a previous node. Now the answer is YES we can say something about impossible patterns. Why?

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There are some impossible patterns: What are the patterns? When can they occur?

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What are the possible patterns that can occur? Recall X=a.b?

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These are obvious. What about remaining patterns?

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We are not actually interested in patterns of a, b, X but interested in the patterns X, b, Y. How can we take the a, b, X patterns and determine if X, b, Y can occur? How can we do this computationally?

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By looking at the Table on the LHS it is clear that some X, b, patterns can happen while others DO NOT!

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The impossible patterns will change the contents of the K-Map!!

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So now we can further optimize the f=Xb +bY + XY in the K-Map by including the don’t cares.

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F = X + bY

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Again we ask if there impossible values for Y given b and c.

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These don’t cares are called: Satisfiability Don’t Cares. Controllability Don’t Cares

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Any other structures that would give rise to Don’t Care Conditions? What happens Previous slides What happens if the nodes preceding a Boolean network are not Primary inputs (i.e., another Node?) Previous slides i. Satisfiability Don’t Cares. ii. Controllability Don’t Cares What happens What happens if the nodes in front of a Boolean network is not a Primary output (i.e., another Node?) Other types of Don’t Cares?? MLS: Implicit Don’t Cares …

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Previously we showed the affect of inputs on F and consequences of getting Don’t Cares (nodes preceding F) How about the new Node Z = f.X.d (behind node f) When does the value of node f affect the primary outputs? When does Z not care about f?

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Note that the table is not in order of Boolean order! For the set X=0, d=0, does the value of f have any affect?

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Where is this leading to??

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Can we use this information to further simplify f?? 0 - - Z is not affected by f(X,b,Y) when X=0, b=-, Y=-

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f = 1 So what does the new network look like?

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Implicit Don’t Cares are very useful to optimize a multi-level logic network!!

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Different Types and flavors of Don’t Cares: Observability Controllability Satisfiability What do they tell us? Are they easy to extract? Can we compute them automatically? Satisfiability MLS: Satisfiability Don’t Cares …

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How will we represent don’t care (DC) patterns at a node? As a Boolean function that makes a 1 when the pattern is impossible This is often called a Don’t Care Cover So, each SDC, CDC, ODC is really just another Boolean function Why do it like this? math Because the math works (!) computational Boolean algebrasolve But more importantly: we can use all the other computational Boolean algebra techniques we learnt (eg, BDDs), to solve for, and manipulate the DC patterns. This turns out to be hugely important to making this practical. Representing DC Patterns …

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SDC is a representation of impossible patterns. SDC x (X,a,b) represents the wire (output of upper function) computational recipe What is a computational recipe to represent SDC x, SDC y, …?

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RECALL gate consistency functions We will use the complement of gate consistency

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Anything that makes this function a 1 is an impossible pattern for node X !!

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1 1 1 0

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SDCs are associated with every internal wire in the Boolean Logic Network SDCs explain impossible patterns of input to, and output of, each node. SDCs are easy to compute!! But SDCs alone are not the Don’t Cares used to simplify nodes We use SDCs to build CDCs, which give impossible patterns at input nodes SDCs: Summary …

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Controllability Don’t Cares are: cannot happen at inputs Patterns that cannot happen at inputs of a network. Computing Controllability Computing Controllability Don’t Cares (CDC). Detailed Example. Controllability MLS: Controllability Don’t Cares …

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Can Compute to Simplify Boolean Networks

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Why Does This Work?

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Step #1: Calculate the SDCs of f which is needed by CDC of f.

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What about SDC on primary inputs?

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computation is becoming simpler So our computation is becoming simpler since we can ignore the primary inputs.

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When you perform the computation you only include variables from internally computed nodes. Only arrows coming from bubbles So if we do that then …

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CONSENSUS ALSO CALLED CONSENSUS of F wrt Xi RECALL

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CDC = The Universal quantification with respect to `b’ of the following quantity. Universal Quantification = Co-Factored to b = 1 AND Co-Factored to b = 0. If you do the Boolean Algebra, what do we get?

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If X=0, then both ‘a’ and ‘b’ have to be 0, But if ‘a’ and ‘b’ are both 0 then Y has to be a ‘0’ That is why XY=01 is impossible. So we can use XY = 01 as a don’t care, and further simplify F using this pattern. If X=0 then Y cannot be 1

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Accounting for don’t care conditions from the Primary Inputs will further simplify the Function f. How …

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CDCs give impossible patterns at input to node F – use as DCs Impossible because of the network structure of the nodes feeding node F. CDCs can be computed mechanically from SDCs on wires input to F SDCs Internal local CDCs: computed just from SDCs on wires into F. External global CDSs: DC patterns at network input, can be included too. But CDCs still not all the Don’t Cares available to simplify nodes CDCs derived from the structure of nodes “in front of” node F. We need to look at DCs that derive from nodes “in back of” node F. These are nodes between the output of F and primary outputs of overall network. CDCs: Summary …

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complete We will complete our discussion on don’t cares. Observability don’t cares Observability don’t cares are: mask the output patterns that mask the output of a node. Similar computational techniques will be used. Observability MLS: Observability Don’t Cares …

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ODC F = Patterns of (a,b) that ZF’s make Z insensitive to F’s Value

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Give examples of circuits when an input can be a don’t care??

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f = x’, df/dx = f x xor f x’ f x =0, f x’ =1 df/dx = 0 xor 1 = 1 f = x.y, f x = y and f x’ =0, df/dx = y xor 0 = y What makes this function == 1? We must make y=1, if you change x f will change f = x+y, f x =1, f x’ =y df/dx = 1 xor y = y’, what make this function == 1? We must make y=0, Any change on x will change f f = x xor y, f x = y’, f x’ =y, df/dx = y xor y’ = 1, RECALL

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Opposite Insensitive We want the Opposite: want F Insensitive to X RECALL

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Once you compute the Boolean Difference and complement it, this will give you the pattern that when applied to Z will not depend on F. What are the steps or recipe?

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F is an XOR Gate After getting ODC F (a,b) We can simply replace XOR with OR gate!!

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If F affects even one of the outputs then we cannot use ODC on F to simplify the expressions.

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CS1Q Computer Systems Lecture 8

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