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Minimization of AND-OR-EXOR Three Level Networks with AND gate Sharing Hasnain Heickal (SH-223)

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Overview Introduction AND-OR-EXOR networks Objective Preliminary Definitions Properties of EX-SOPs Minimization of EX-SOPs Idea of Minimization Summary Reference

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Introduction Logic networks are usually designed using AND and OR gates (SOP). AND-EXOR networks (EX-SOP) are More compact. Easily testable. Fault tolerant

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AND-OR-EXOR Networks A two input EXOR gate is used. AND gates can be shared or not shared. If not shared an EX-SOP for a function F can be written as F = F a xor F b If shared the EX-SOP can be written as F = (F a + F s ) xor (F b + F s )

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Objective Designing a AND-OR-EXOR three level network. Minimizing the number of products. We will discuss an exact algorithm for minimization.

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Preliminary Definitions τ( F ) Number of products in an expression F. τ(ABC + ABC + AC) = 3 τ(SOP: f ) Number of product in a minimum SOP for f. τ(SOP : (ABC + ABC + AC)) = 2 because it can be minimized as BC + AC.

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Preliminary Definitions τ(EX-SOP NS :f) Number of products in a minimum EX-SOP for f with no product sharing. τ(EX-SOP PS :f) Number of products in a minimum EX-SOP for f with product sharing. A logic function f can represented as f = (f a + g) xor (f b + g)……………………(1) τ(EX-SOP PS : f ) = min{τ(SOP: g ) + τ(SOP: f a ) + τ(SOP: f b )} τ(EX-SOP NS : f ) = min{τ(SOP: f a ) + τ(SOP: f b )} while considering g = 0

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Properties of EX-SOPs On the Karnaugh map of a function, a cell that contains a 1(one) is called a 1-cell and a cell that contains a 0(zero) is called 0-cell. Property 1: In a K-map for an EX-SOP, any 1-cell must be covered by the loop(s) for exactly one SOP. If a 0-cell is covered, then it must be covered by at least one loop from both SOPs. Definition 6: Let g(x) and h(x) be n variable functions. B = {0,1}, if for every a ε B n g(a)=1 satisfies h(a)=1 then g h

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Minimization of EX-SOPs Let g represent the shared products of an EX- SOP of function f. The number of different products in a minimum EX-SOP for f with product sharing is denoted by τ (EX-SOP PS : f : g ). To compute τ (EX-SOP PS : f : g ) using the Eq 1, g is fixed and we choose f a and f b such that Eq 1 satisfies. Thus we have τ (EX-SOP PS : f : g ) = τ (SOP: g ) + min{ τ (SOP: f a ) + τ (SOP: f b ) }

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Minimization of EX-SOPs Lemma 2: The proof of the lemma is out of scope. The proof can be found on the paper [1].

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Idea of Minimization The idea is for 5 of less number of variables. We will try for all possible g and minimize the following Eq for all possible g. We need to use K-map.

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Example Let us consider g = ACD. Possible values of h are ABCD ACD We have to find h that makes minimum. CD A CD AB g 1 11 1111

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Example Lets first try with h = ABCD So K-map for f v h will be CD A CD AB1 1 1 1 1111

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Example Rules for EX-SOP NS Loop 1-cell entries odd number of times. Loop 0-cell entries even number of times. From the K-map we can see f a = B f b = ACD = 2 τ (SOP: g) = 1 Τ (EX-SOP PS : f:g ) = 3 We need to do this for every h. CD A CD AB1 111 1111 fafa fbfb

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Choosing g We can choose g using the following lemma : To obtain minimum EX-SOP of f it is sufficient to consider only the prime implicants of f as shared product of candidate. The proof of this lemma can also be found in the paper [1]. To find the prime implicants of f we can also use K- map.

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Drawbacks Choosing g is very time consuming. We can use Lookup Tables to optimize it. Overall an NP equivalent problem.

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Summary We have seen the algorithm for minimizing AND-OR-EXOR three level networks. We have seen the algorithm for 5 or less variables. There exists algorithm for more variables.

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References D. Debnath and T. Sasao, Minimization of AND-OR-EXOR three level networks with AND gate sharing.

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Thank You

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Digital Logic (Karnaugh Map). Karnaugh Maps Karnaugh maps (K-maps) are graphical representations of boolean functions. One map cell corresponds to a row.

Digital Logic (Karnaugh Map). Karnaugh Maps Karnaugh maps (K-maps) are graphical representations of boolean functions. One map cell corresponds to a row.

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