DATA STRUCTURES FOR LOGIC OPTIMIZATION

Transcription

1 DATA STRUCTURES FOR LOGIC OPTIMIZATION Outlne Revew of Boolean algera. c Govann De Mchel Stanford Unversty Representatons of logc functons. Matrx representatons of covers. Operatons on logc covers. Background Functon f(x 1,x 2,...,x,...,x n ). Cofactor of f wth respect to varale x : f x f(x 1,x 2,...,1,...,x n ). Cofactor of f wth respect to varale x : f x f(x 1,x 2,...,0,...,x n ). Functon:f = a + c + ac Cofactors: f a = + c f a = c Boole s expanson theorem: Expanson: f = af a + a f a = a( + c)+a c f(x 1,x 2,...,x,...,x n ) = x f x +x f x

2 Background Background Functon f(x 1,x 2,...,x,...,x n ). Functon f(x 1,x 2,...,x,...,x n ). Postve unate n x when: f x f x Negatve unate n x when: f x f x A functon s postve/negatve unate when postve/negatve unate n all ts varales. Boolean dfference of f w.r.t. varale x : f/ x f x f x. Consensus of f w. r. to varale x : C x f x f x. Smoothng of f w. r. to varale x : S x f x + f x. Generalzed expanson f = a + c + ac Gven: A Boolean functon f. c a Orthonormal set of functons: φ, =1, 2,...,k. (a) () Then: f = k φ f φ (c) (d) Where f φ s a generalzed cofactor. The Boolean dfference f/ a = f a f a = c + c. The consensus C a = f a f a = c. The smoothng S a = f a + f a = + c. The generalzed cofactor s not unque, ut satsfes: f φ f φ f + φ

3 Generalzed expanson theorem Functon f = a + c + ac. Bass: φ 1 = a and φ 2 = a +. Bounds: a f φ1 1 Gven: Two functons f and g. Orthonormal set of functons: φ, =1, 2,...,k. Boolean operator. a c + a c f φ2 a + c + ac. Cofactors: f φ1 =1 andf φ2 = a c + a c. f = φ 1 f φ1 + φ 2 f φ2 = a1+(a + )(a c + a c) = a + c + ac Then: f g = k φ (f φ g φ ) Corollary: f g = x (f x g x )+x (f x g x ) Matrx representatons of logc covers The postonal cue notaton Used n logc mnmzers. Dfferent formats. Usually one row per mplcant. Symols: 0,1,*. (and other) Encodng scheme: Operatons: Intersecton AND Unon OR

4 f = a d + a + a + ac d Cofactor computaton Cofactor of α w.r. to β. Vod when α does not ntersect β. a a a n + n Cofactor of a set C = {γ } w.r. to β: Set of cofactors of γ w.r. to β. f = a + a Cofactor w.r. to 01 11: Frst row vod. Second row Cofactor f a = Multple-valued-nput functons Input varales can have many values. Representatons: Lterals: set of vald values. Sum of products of lterals. Extenson of postonal cue notaton. Key fact: Multple-output nary-valued functons represented as mv sngle-output functons.

5 Operatons on logc covers 2-nput, 3-output functon: f 1 = a + a f 2 = a f 3 = a + a Recursve paradgm: Expand aout a mv-varale. Apply operaton to cofactors. Merge results. Mv representaton: Unate heurstcs: Operatons on unate functons are smpler. Select varales so that cofactors ecome unate functons. Tautology Recursve tautology Check f a functon s always TRUE. Recursve paradgm: Expand aout a mv-varale. If all cofactors are TRUE then functon s a tautology. Unate heurstcs: If cofactors are unate functons addtonal crtera to determne tautology. Faster decson. TAUTOLOGY: the cover has a row of all 1s. (Tautology cue). NO TAUT.: the cover has a column of 0s. (A varale that never takes a value). TAUTOLOGY: the cover depends on one varale, and there s no column of 0s n that feld. When a cover s the unon of two sucovers, that depend on dsont susets of varales, then check tautology n oth sucovers.

6 ac c a a f = a + ac + a c + a c a a c Select varale. Cofactor w.r.to s: Select varale a Cofactor w.r.to a s Tautology. Cofactor w.r.to a s: No column of 0 Tautology. Cofactor w.r.to s: Functon s a TAUTOLOGY. Contanment f = a + ac + a c a a c ac a Theorem: A cover F contans an mplcant α ff F α s a tautology. Consequence: Contanment can e verfed y the tautology algorthm. Check coverng of c C(c) = Take the cofactor: Tautology c s contaned y f.

7 Complementaton Recursve paradgm: f = x f x + x f x Termnaton rules The cover F s vod. Hence ts complement s the unversal cue. Steps: Select varale. Compute cofactors. Complement cofactors. Recur untl cofactors can e complemented n a straghtforward way. The cover F has a row of 1s. Hence F s a tautology and ts complement s vod. The cover F conssts of one mplcant. Hence the complement s computed y De Morgan s law. All the mplcants of F depend on a sngle varale, and there s not a column of 0s. The functon s a tautology, and ts complement s vod. Unate functons ac a a Theorem: f = a + ac + a c a a c If f e postve unate: f = f x + x f x. If f e negatve unate: f = x f x + f x. Select nate varale a. Consequence: Compute cofactors: Complement computaton s smpler. F a s a tautology, hence F a s vod. One ranch to follow n the recurson. Heurstc: Select varales to make the cofactors unate. F a yelds:

8 (2) (3) Select unate varale. Compute cofactors: RECURSIVE SEARCH F a s a tautology, hence F a s vod. a a F a = and ts complement s Re-construct complement: ntersected wth C( ) = yelds F = TAUT a COMP = φ F = TAUT a COMP = φ F a = c COMP = c ntersected wth C(a) = yelds Complement: F = Summary Matrx orented representaton: Used n two-level logc mnmzer. May e wasteful of space (sparsty). Good heurstcs ted to ths representaton. Effcent Boolean manpulaton explots recurson.