Laboratory 4 Logic forms 4.1 Laboratory work goals Going from Boolean functions to Boolean forms. Logic forms equivalence. Boolean forms simplification. Shannon s theorems. Representation in NAND and NOR logic. 4.2 Theoretical considerations It is named logic form (formula, logic expression) any combination of a finite number of switching variables and constants 0 or 1 reunited through switching operators. It is named Boolean form or Boolean expression a logic form containing only disjunction, conjunction and negation operators. The function generated by evaluating a logic form A(x 1 x n ) for all possible assignments of variables is called switching function attached to form A and is noted f A. It can be noticed that on one hand, there is an infinite number of logic forms that can be generated with variables x 1 x n, and on the other hand, there is a finite number of switching functions that 39
depend on the same set of variables. Because to each logic form it corresponds a unique attached switching function, depending on the same set of variables, it can be concluded that there is an infinite number of logic forms that have a same (only one) switching function attached. It is said that two logic forms A and B, depending on the same set of variables, are equivalent if and only if f A =f B. It is noted A=B. The logic forms that depend on the same set of variables and have the same switching function attached to them form a class of equivalence. Any logic form belongs to only one class of equivalence. The equivalence of logic forms is an equivalence relation thus respecting the three properties: 1) reflexivity (a forms is equivalent to itself); 2) commutativity (if A=B then B=A); 3) transitivity (if A=B and B=C then A=C) The set of logic forms depending on the same set of variables 2 x 1 x n can be divided in 2 n disjunctive classes of equivalence. Determining equivalent forms is extremely important in practice, because it allows the designer to choose for implementation the optimal form given a particular set of imposed conditions (technology, price, size, etc). Determination of various equivalent forms corresponding to a given switching function is performed during the synthesis phase. It is said that a representation form is normal if certain restrictions (norms) were imposed on that representation. 40
A normal form of representation is canonical if the imposed restrictions lead to the uniqueness of representation. The minterms are a canonical form of representation for a minterm function, while the maxterms allow canonical representation of a maxterm function. Let {x 1,,x n } be a set of Boolean variables. It is called P canonical term or minterm any logic product of all n variables, either in direct or complemented forms, that is x & 1 x& 2... x& n. Any logic product not containing all variables is called P term. It is called S canonical term or maxterm any logic sum of all n variables, either in direct or complemented form, that is x & 1 + x& 2 +... + x& n. Any logic sum not containing all variables is called S term. Any switching function (except the constant 0) can be uniquely represented as a logic sum of minterms corresponding to the combinations for which the function is 1. This representation form is called normal disjunctive canonical form. Any completely specified switching function (except the constant 0) can be uniquely represented as a modulo 2 sum of minterms corresponding to the combinations for which the function is 1. This form of representation is also a disjunctive canonical form. Any completely specified switching function (except the constant 1) can be uniquely represented as a logic product of maxterms corresponding to the combinations for which the function is mapped to 0. This representation form is called normal conjunctive canonical form. 41
It is called normal disjunctive form (NDF) any Boolean expression of type P terms logic sum. It is called normal conjunctive form (NCF) any Boolean expression of type S terms logic product. Unlike the canonical forms which are unique for a given function, there is a large number of equivalent NDF and NCF to represent the same function. Because in practice the canonical forms are extremely important, it is necessary to have simple methods to convert NDF and NCF to canonical forms. Algorithm to convert an NDF to the equivalent normal disjunctive canonical form: S1) Starting from an NDF, the missing variables are introduced in each P term as α + α where α is some variable. S2) By applying the distributivity, the parentheses are opened. Each resulting P term will contain all variables the function depends on, thus a minterm. S3) Based on the idempotency property, only one of many identical terms is kept. The resulting form is the normal disjunctive canonical form. Example: Consider the following NDF f(xyz)=xy+ z. f(xyz) = xy (z + z) + (x + x) (y + y) z = = xyz+xy z +xy z +x y z + x y z + x y z = = xyz+xy z +x y z + x y z + x y z = m 7 +m 6 +m 4 +m 2 +m 0 42
Algorithm to convert an NCF to the equivalent normal conjunctive canonical form: S1) Starting from an NCF, the missing variables are introduced in each S term, as α α where α is some variable. S2) By applying the distributivity a new NCF is obtained, in which each resulting S term will contain all variables the function depends on, thus a maxterm. S3) Based on the idempotency property, only one of many identical terms is kept. The resulting form is the normal conjunctive canonical form. Example: Consider the following NCF f(xyz)=(x+y)( x + z ). f(xyz)= (x+y+z z )( x +y y + z ) = =(x+y+z)(x+y+ z )( x +y+ z )( x + y + z ) = M 0 M 1 M 5 M 7 Shannon s expansion theorem Let A(x 1 x n ) be some logic form. This form can be expanded with respect in relation to variable x i, into one of the following two equivalent forms: A(x 1 x i x n ) = x i A(x 1 1 x n )+ xi A(x 1 0 x n ) (4.1) A(x 1 x i x n ) = [x i +A(x 1 0 x n )] [ x i +A(x 1 1 x n )] (4.2) The form (4.1) is called Shannon s expansion theorem disjunctive form while (4.2) is called Shannon s expansion theorem conjunctive form. A(x 1 0 x n ) and A(x 1 1 x n ) are named residual expressions. 43
It can be noticed that the residual expressions depend at most on n-1 variables, being simpler than the initial form. This constitutes one of the biggest advantages in using this expansion theorem. Algorithm to convert an NDF to NAND logic representation S1) The NDF is complemented twice. S2) DeMorgan theorem is applied such that there are obtained only negated logic products. S3) Each complemented logic product is represented as a NAND operator. S4) If necessary, the complementations are replaced with NAND operators. Example: It is considered the following NDF A(x,y,z)=xy+yz+xz and the requirement is to represent it with NAND operators. A(x,y,z)= xy + yz + xz = xy yz xz =(x y) (y z) (x z) Algorithm to convert an NDF to NOR logic representation S1) The NCF is complemented twice. S2) DeMorgan theorem is applied such that there are obtained only negated logic sums. S3) Each complemented logic sum is represented as a NOR operator. S4) If necessary, the complementations are replaced with NOR operators. Example: It is considered the following NCF A(x,y,z)=(x+y)(y+z)(x+z) and the requirement is to represent it with NOR operators. 44
A(x,y,z)= (x + y)(y + z)(x + z) = x + y + y + z + x + z =(x y) (y z) (x z) 4.3 Lab activity progress It is given the function f(x,y,z)=σ(0,4,5,7) and it is required: Determination of disjunctive canonical form; Determination of conjunctive canonical form; Determination of a normal disjunctive form as simple as Determination of a normal conjunctive form as simple as The two normal forms must be implemented on the test It is given the logic form F(x,y,z)= (x y z) ( (. It will be experimentally determined the switching function attached to this logic form. It will be determined the equivalent normal disjunctive canonical form. It will be determined a normal disjunctive form as simple as possible and it will be implemented on the test board. It will be experimentally checked its equivalence with the initial form. It is considered the logic form G(u,x,y,z =(x y) (u + + +. It is required determination of a normal disjunctive form utilising the Shannon s expansion theorem. 45
The obtained expression must be simplified, using the theorems of the Boolean algebra. It will be proven experimentally the equivalence of the normal disjunctive form with the initial form. It will be determined the NAND logic form and it will be experimentally proven the equivalence with the previous forms. The last exercise is to be repeated for the switching function (,,, = + + + 4.4 Proposed problems 1. It is considered the switching function f(x,y,z,t) = Σ(0,1,3,5,7,10,14,15) and it is required: a) Determination of the disjunctive canonical form; b) Determination of the conjunctive canonical form; c) Determination of a normal disjunctive form as simple as d) Determination of a normal conjunctive form as simple as 2. It is given the logic form f(u,x,y,z =(xy+u) (u x + +(. a) It is required determination of a normal disjunctive form utilising the Shannon s expansion theorem. b) The obtained expression must be simplified, using the theorems of the Boolean algebra. c) It will be proven experimentally the equivalence of the normal disjunctive form with the initial form. 46
d) It will be determined the NAND logic form and it will be experimentally proven the equivalence with the previous forms. 3. It is considered the function f(x,y,z,u)= Σ (1,2,6,7,11,13,15) and it is required: a) Determination of the disjunctive canonical form; b) Determination of the conjunctive canonical form; c) Determination of a normal disjunctive form as simple as d) Determination of a normal conjunctive form as simple as 4. It is considered the logic form f(u,x,y,z =(u y+x y) (u + +(. a) It is required determination of a normal conjunctive form utilising the Shannon s expansion theorem. b) The obtained expression must be simplified, using the theorems of the Boolean algebra. c) It will be proven experimentally the equivalence of the normal conjunctive form with the initial form. d) It will be determined the NOR logic form and it will be proven the equivalence with the previous forms. 5. It is considered the function f(x,y,z,t)= Σ (0,1,2,7,9,11,14) and it is required: a) Determination of the disjunctive canonical form; b) Determination of the conjunctive canonical form; 47
c) Determination of a normal disjunctive form as simple as d) Determination of a normal conjunctive form as simple as 6. It is considered the logic form f(,,, = + + +( ( a) It is required determination of a normal disjunctive form utilising the Shannon s expansion theorem. b) The obtained expression must be simplified, using the theorems of the Boolean algebra. c) It will be proven experimentally the equivalence of the normal disjunctive form with the initial form. d) It will be determined the NAND logic form and it will be proven the equivalence with the previous forms. 7. It is considered the function f(x,y,z,u)= Σ (0,2,4,9,11, 13,14,15) and it is required: a) Determination of the disjunctive canonical form; b) Determination of the conjunctive canonical form; c) Determination of a normal disjunctive form as simple as d) Determination of a normal conjunctive form as simple as 8. It is considered the logic form f(u,x,y,z =(u y xy uz) (ux + +(. a) It is required determination of a normal disjunctive form utilising the Shannon s expansion theorem. 48
b) The obtained expression must be simplified, using the theorems of the Boolean algebra. c) It will be proven experimentally the equivalence of the normal disjunctive form with the initial form. d) It will be determined the NAND logic form and it will be proven the equivalence with the previous forms. 9. It is considered the function f(x,y,z,t)= Σ (1,2,4,8,9,10,14) and it is required: a) Determination of the disjunctive canonical form; b) Determination of the conjunctive canonical form; c) Determination of a normal disjunctive form as simple as d) Determination of a normal conjunctive form as simple as 10. It is considered the logic form f(,,, =( + + +( ( a) It is required determination of a normal conjunctive form utilising the Shannon s expansion theorem. b) The obtained expression must be simplified, using the theorems of the Boolean algebra. c) It will be proven experimentally the equivalence of the normal conjunctive form with the initial form. d) It will be determined the NOR logic form and it will be proven the equivalence with the previous forms. 49