1. 1 Exact Two-Level Minimization of Hazard-Free Logic with Multiple Input Changes Montek Singh Tue, Oct 16, 2007

2. 2Definitions  Let P={0,1} and B={0,1,*}.  Boolean function: f: P n  B  minterm: an element in the domain P n of function ON-set: set of minterms for which f=1 ON-set: set of minterms for which f=1 OFF-set: set of minterms for which f=0 OFF-set: set of minterms for which f=0 DC (don’t-care)-set: set of minterms for which f=* DC (don’t-care)-set: set of minterms for which f=*  literal: each variable, x i, has three corresponding literals: x i, x i and x i *. literal x i =1 iff variable x i = 1 literal x i =1 iff variable x i = 1 literal x i =1 iff variable x i = 0 literal x i =1 iff variable x i = 0 literal x i * =1 iff variable x i = * literal x i * =1 iff variable x i = *  product: Boolean product (AND) of literals  cube: set of minterms representable by a product  sum-of-products: Boolean sum (OR) of products it contains a minterm if some product contains the minterm it contains a minterm if some product contains the minterm

3. 3 Definitions (contd.)  product Y contains product X (X  Y) if the cube for X is a subset of the cube for Y intersection of products X and Y is the set of minterms contained in the intersection of the corresponding cubes intersection of products X and Y is the set of minterms contained in the intersection of the corresponding cubes  an implicant of a function: product term that contains no OFF-set minterm prime implicant (PI): implicant contained in no other implicant prime implicant (PI): implicant contained in no other implicant essential prime implicant: PI containing an ON-set minterm contained in no other PI essential prime implicant: PI containing an ON-set minterm contained in no other PI  cover of f : a sum-of-products which contains: all of the ON-set minterms all of the ON-set minterms none of the OFF-set minterms none of the OFF-set minterms may include some DC-set minterms may include some DC-set minterms cost function for a cover: each implicant has the same cost cost function for a cover: each implicant has the same cost  sometimes literal-count can be used as a secondary cost  2-level logic minimization problem: find a minimum-cost cover of a function

4. 4Background:  Circuit and Delay Model arbitrary fine gate and wire delays arbitrary fine gate and wire delays each wire = connection + attached delay element each wire = connection + attached delay element each gate = instantaneous Boolean operator + delay element attached to its output wire each gate = instantaneous Boolean operator + delay element attached to its output wire  “unbounded wire delay model” (all delays attached to wires)  Delays are “pure” delays propagation of waveform, but does not alter it delays propagation of waveform, but does not alter it conservative: does not filter out glitches conservative: does not filter out glitches  Delay Assignment assign fixed, finite delay values to every gate and wire assign fixed, finite delay values to every gate and wire

5. 5 Background: Multiple-Input Changes  transition cube: cube with start point and end point transition cube [A,B], has start point A and end point B transition cube [A,B], has start point A and end point B contains all minterms that can be reached during transition from A to B contains all minterms that can be reached during transition from A to B i -th literal for the product of [A,B] is A i +B i i -th literal for the product of [A,B] is A i +B i  x + x = x*  open transition cube [A,B): [A,B] – B  multiple-input change or input transition: [A,B] inputs change concurrently, in any order and at any time inputs change concurrently, in any order and at any time inputs change monotonically inputs change monotonically fundamental mode: once inputs change, no further input changes may occur until the circuit has stabilized fundamental mode: once inputs change, no further input changes may occur until the circuit has stabilized  transition interval: t I ≤ t ≤ t F  transition is… static: if f(A) = f(B) static: if f(A) = f(B) dynamic: if f(A) ≠ f(B) dynamic: if f(A) ≠ f(B)

6. 6Hazards communication must be hazard-free! communication must be hazard-free! special design challenge = “hazard-free synthesis” special design challenge = “hazard-free synthesis”  Hazards: potential “glitches” on wire clean signals hazardous signals clock tick no problem for clocked systems

7. 7 Function Hazards  A function f has a function hazard iff there exists an input transition for which f does not change monotonically there exists an input transition for which f does not change monotonically  f contains a static function hazard for input transition from A to C iff: f(A) = f(C), and f(A) = f(C), and there exists some input state B  [A,C] such that f(A) ≠ f(B). there exists some input state B  [A,C] such that f(A) ≠ f(B).  f contains a dynamic function hazard for input transition from A to D iff: f(A) ≠ f(D) f(A) ≠ f(D) there exists a pair of input states B and C (A ≠ B, C ≠ D) such that there exists a pair of input states B and C (A ≠ B, C ≠ D) such that  B  [A,D] and C  [B,D], and  f(B) = f(D) and f(A) = f(C)  if a transition has a function hazard, no implementation of the function is guaranteed to avoid glitches during transition, assuming arbitrary gate and wire delays

8. 8 Logic Hazards  If f is free of function hazards, yet, for an input transition [A,B], has a non-monotonic change at the output, for some delay assignment static logic hazard in [A,B]: if f(A) = f(B) static logic hazard in [A,B]: if f(A) = f(B) dynamic logic hazard [A,B]: if f(A) ≠ f(B) dynamic logic hazard [A,B]: if f(A) ≠ f(B)

9. 9 2-Level Hazard-Free Logic Minimization Given: A Boolean function f, and a set, T, of specified function- hazard-free (static and dynamic) input transitions of f A Boolean function f, and a set, T, of specified function- hazard-free (static and dynamic) input transitions of fFind: A minimum-cost cover of f whose AND-OR implementation is free of logic hazards for every input transition t  T A minimum-cost cover of f whose AND-OR implementation is free of logic hazards for every input transition t  T

10. 10 Conditions for a Hazard-Free Transition  0  0: implementation is free of logic hazards  1  1: implementation is free of logic hazards iff … [A,B] is contained in some cube of cover C [A,B] is contained in some cube of cover C [A,B] is called a required cube [A,B] is called a required cube  1  0: implementation is free of logic hazards iff … every cube c  C intersecting [A,B] also contains A every cube c  C intersecting [A,B] also contains A [A,B] is called a privileged cube [A,B] is called a privileged cube  0  1: symmetric conditions as 1  0 above

11. 11 Hazard-Free Covers: Existence Hazard-Free Covering Theorem:  A sum-of-products C is a hazard-free cover of f iff: no cube of C intersects the OFF-set of f no cube of C intersects the OFF-set of f each required cube of f is contained in some cube of cover C each required cube of f is contained in some cube of cover C no cube of C intersects any privileged cube illegally no cube of C intersects any privileged cube illegally