Two Level Logic Minimization in VLSI and Advanced Digital Design - Prof. Vincent Mooney, Study notes of Electrical and Electronics Engineering

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ECE 3060 VLSI and Advanced Digital Design Lecture 10 Two Level Logic Minimization ECE 3060 Lecture 10–2 Motivation • We will study modern techniques for manipulating and minimizing boolean functions • Issue: Tractibility of minimization problem for large number of variables • Exact methods • Heuristic methods • Issue: Representation of boolean expressions in a form conducive to boolean operations • Implicant tables • Binary decision diagrams • Issue: Manipulation of realistic multilevel networks • Graph representations • multilevel minimization • Technology mapping ECE 3060 Lecture 10–5 Definitions • Boolean variable: • Boolean literal: or • Product or cube: product of literals • Implicant: product term implying a value of a function (usually TRUE) • binary hypercube in the boolean space • Minterm: product using all input variables implying a value of a function (usually TRUE) • vertex in the boolean space a B∈ a a ECE 3060 Lecture 10–6 Tabular Representations • Truth table • list of all minterms of a function • Implicant table or cover • list of all implicants of a function sufficient to define the function • Comment: • Implicant tables are smaller in size • Example Cover 001 10 *11 11 101 01 11* 11 x ab ac+= y ab bc ac+ += abc xy ECE 3060 Lecture 10–7 Cube Representation • • + + + 000 100 110 010 001 011 111 c a b 101 F abc abc abc abc abc+ + + += F = ab bc ac ab ECE 3060 Lecture 10–10 Minimal or Irredundant Cover • Cover of the function that is not a proper superset of another cover • no implicant can be dropped • local optimum 000 100 110 001 011 111 c a b 101 010 ECE 3060 VLSI and Advanced Digital Design Lecture 10 Two Level Logic Minimization ECE 3060 Lecture 10–2 Motivation • We will study modern techniques for manipulating and minimizing boolean functions • Issue: Tractibility of minimization problem for large number of variables • Exact methods • Heuristic methods • Issue: Representation of boolean expressions in a form conducive to boolean operations • Implicant tables • Binary decision diagrams • Issue: Manipulation of realistic multilevel networks • Graph representations • multilevel minimization • Technology mapping ECE 3060 Lecture 10–5 Definitions • Boolean variable: • Boolean literal: or • Product or cube: product of literals • Implicant: product term implying a value of a function (usually TRUE) • binary hypercube in the boolean space • Minterm: product using all input variables implying a value of a function (usually TRUE) • vertex in the boolean space a B∈ a a ECE 3060 Lecture 10–6 Tabular Representations • Truth table • list of all minterms of a function • Implicant table or cover • list of all implicants of a function sufficient to define the function • Comment: • Implicant tables are smaller in size • Example Cover 001 10 *11 11 101 01 11* 11 x ab ac+= y ab bc ac+ += abc xy ECE 3060 Lecture 10–7 Cube Representation • • + + + 000 100 110 010 001 011 111 c a b 101 F abc abc abc abc abc+ + + += F = ab bc ac ab ECE 3060 Lecture 10–10 Minimal or Irredundant Cover • Cover of the function that is not a proper superset of another cover • no implicant can be dropped • local optimum 000 100 110 001 011 111 c a b 101 010 ECE 3060 Lecture 10–11 Minimal Cover with respect to single-implicant containment • no implicant is contained by any other implicant • weak local optimum 000 100 110 010 001 011 111 c a b 101 ECE 3060 Lecture 10–12 Logic Minimization • Exact methods: • compute minimum cover • often intractable for large functions • based on Quine-McCluskey method • Heuristic methods: • Compute minimal cover (possibly minimum) • There are a large variety of methods and programs • academic: MINI, PRESTO, ESPRESSO (UCBerkeley) • industry: Synopsys, Cadence, Mentor Graphics, Zuken ECE 3060 Lecture 10–15 Prime Implicant Table • Rows: minterms • Columns: prime implicants • Exponential size: for a function • minterms • up to prime implicants f :Bn B→ 2n 3n n⁄ ECE 3060 Lecture 10–16 Example • Function: • Choose cover by selecting a set of implicants which cover all minterms. f abc abc abc abc abc+ + + += • Primes: • Implicant Table: Label PIs 00* *01 1*1 11* α β γ δ Minterms Primes 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 α β γ δ abc abc abc abc abc ECE 3060 Lecture 10–17 Cube Representation 000 100 110 001 011 111 c a b 101 010 000 100 110 001 011 111 101 010 (a) prime implicants (b) minimum cover ECE 3060 Lecture 10–20 Matrix representation • View implicant table of some function as Boolean matrix: • the ith minterm is covered by the jth prime implicant • The (Boolean) selection vector selects which prime implicants will be in the cover. • To cover , find an which satisfies • i.e. select enough columns to cover all rows • To find a minimum cover, minimize cardinality of , i.e. the number of nonzero entries of . f A aij 1=( ) ⇒ x f x yi 1 i∀≥ Ax y= x x ECE 3060 Lecture 10–21 Example • The magnitude of indicates the number of prime implicants which cover the ith minterm. yi 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 1 2 1 1 1 = ECE 3060 Lecture 10–22 Branch and Bound Algorithm • Exact algorithm, but not polynomial time. • First step: • Remove Essential Prime Implicants (EPIs) which are columns incident to one (or more) row(s) with a single 1 in them. • Modify by removing the column and incident rows • Example: rows 1 and 5 from previous matrix becomes A 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 A= 1 1 ECE 3060 Lecture 10–25 Branching Algorithm • Remove essential primes from consideration. • Perform a depth-first search of remaining covers. • Bounding algorithm used to prune search tree. α C∈ β C∈ α C∉ β C∉ γ C∉γ C∈ ECE 3060 Lecture 10–26 Branch and Bound Algorithm EXACT_COVER( , , ) { /* is current best estimate */ Reduce matrix and update corresponding ; Calculate current_estimate for this branch; /* we don’t cover this */ if (current_estimate >= | |) return ( ); if ( has no rows) return ( ); Select a branching column ; ; /* this changes element in */ = after deleting column and rows incident to ; = EXACT_COVER( , , ); if ; ; = after deleting column ; = EXACT_COVER( , , ); if ; return ( ); } A x b b A x b b A x c xc 1= c x à A c c x̃ à x b x̃ b<( ) b x̃̃= xc 0= à A c x̃ à x b x̃ b<( ) b x̃= b ECE 3060 Lecture 10–27 Example • Consider , , There are no essential primes, and no row or column dominance. • Denote the implicants and the minterms as • Choose (i.e. ) A 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 = x x1 x2 x3 x4 x5 = b 1 1 1 1 1 = P j μi P1 c 1=