Abstract:
A methodology to perform binary addition. An operand A and an operand B are presented as input and an operation is performed that respects the laws of the binary addition. The operation is performed with the use of pseudo generate signals, pseudo transmit signals, pseudo half sum signals, pseudo transmit half sum signals, a new-carry and SUM equations. The SUM equation is described for ripple or parallel configurations. All quantities can be used on single bit boundaries, or extensively for any chosen grouping of bits to accommodate chosen technology or grouping so as to facilitate the design and to increase the performance of hardware-implemented adders under the constraints of a varied technology book set. The invention also describes the implementation of a 32-bit adder that requires no more than three logic stages of delay, using a technology that allows up to 3×8 AND-OR books. Its design is achieved with the use of a SUM equation described by the general scheme of the addition and auxiliary functions that reduce the book size needed for the implementation of the sum.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to the design of arithmetic units in digital computer systems and, more particularly, to the design of binary adders. 
     Arithmetic operations in computer systems, such as addition and subtraction, are performed on two operands, each having a given number of bits (e.g., N bits), the operations yielding an N-bit result. 
     Because a great number of arithmetic operations can be performed by high speed digital computers in a relatively short time interval (e.g., one second), the time required to perform any one of such operations can become a significant factor when multiplied by the number of operations per second. Designers are therefore obliged to invent ever faster formulations to perform arithmetic operations. 
     Conventional addition functions, such as carry (or borrow) generate and carry propagate, are described in COMPUTER ARTITHMETIC PRINCIPLES, ARCHITECTURE, AND DESIGN, by Kai Hwang. These functions generally depend on two paths: one that produces carries into a bit position and one that produces a half-sum. The conventional recursive formulas are shown below and are derived as follows: ##EQU1## 
     For every position i a carry is received from position i+1 to be added with A i , B i . In essence: ##EQU2## where C i+1  is the carry bit from position i+1. 
     The maximum value for each individual operand is one. Therefore, MAX [A i  +B i  +C i+1  [=1+1+1=3. Since two bits are required to represent the number three (11, in binary code), a carry to the next position is generated. 
     All combinations are shown below in Table 1. 
     
                       TABLE I______________________________________C.sub.i+1      A.sub.i                B.sub.i    S.sub.i                               C.sub.i______________________________________(1)    0           0     0        0   0(2)    0           0     1        1   0(3)    0           1     0        1   0(4)    0           1     1        0   1(5)    1           0     0        1   0(6)    1           0     1        0   1(7)    1           1     0        0   1(8)    1           1     1        1   1______________________________________ 0 + 0 + 0 = 0 = &gt;S.sub.i = 0, C.sub.i = 0 1 + 1 + 0 = 2 = &gt;S.sub.i = 0, C.sub.i = 1 
    
     The Boolean Equations Of Table 1 
     S i  =1 if and only if the following holds true: 
     
         S.sub.i =(2) or (3) or (5) or (8) 
    
     
         (2)=C.sub.i+1 &#39;A.sub.i &#39;B.sub.i 
    
     where the expression N&#39; indicates the negation of N hereinbelow. 
     
         (2)=1 if and only if C.sub.i+1 =0, A.sub.i =0, and B.sub.i =1 
    
     
         (3)=C.sub.i+1 &#39;A.sub.i B.sub.i 
    
     
         (5)=C.sub.i+1 A.sub.i &#39;B.sub.i &#39; 
    
     
         (8)=C.sub.i+1 A.sub.i B.sub.i 
    
     
         Thus, S.sub.i =C.sub.i+1 &#39;A.sub.i &#39;B.sub.i +C.sub.i+1 &#39;A.sub.i B.sub.i &#39;+C.sub.i+1 A.sub.i &#39;B.sub.i &#39;+C.sub.i+1 A.sub.i B.sub.i 
    
     
         S.sub.i =C.sub.i+1 &#39;[A.sub.i &#39;B.sub.i +A.sub.i B.sub.i &#39;]+C.sub.i+1 [A.sub.i &#39;B.sub.i &#39;+A.sub.i B.sub.i ] 
    
     
         A.sub.i &#39;B.sub.i +A.sub.i B.sub.i &#39;=Exclusive-OR (V) 
    
     
         A.sub.i &#39;B.sub.i &#39;+A.sub.i B.sub.i =Exclusive-OR-Invert 
    
     Thus, 
     
         S.sub.i =C.sub.i+1 [A.sub.i VB.sub.i ]+C.sub.i+1 [A.sub.i VB.sub.i ] 
    
     Similarly, 
     
         S.sub.i =C.sub.i+1 VA.sub.i VB.sub.i 
    
     
         A.sub.1 VB.sub.i is defined as half sum H.sub.i. 
    
     
         S.sub.i =H.sub.i VC.sub.i+1 
    
     
         C.sub.i =(4) or (6) or (7) or (8) 
    
     
         C.sub.i =C.sub.i+1 &#39;A.sub.i B.sub.i +C.sub.i+1 A.sub.i &#39;B.sub.i +C.sub.i+1 A.sub.i B.sub.i &#39;+C.sub.i+1 A.sub.i B.sub.i 
    
     Adding the same term as one that previously appeared does not change a Boolean equation. Thus, 
     
         C.sub.i =C.sub.i+1 &#39;A.sub.i B.sub.i +C.sub.i+1 A.sub.i &#39;B.sub.i +C.sub.i+1 A.sub.i B.sub.i &#39;+C.sub.i+1 A.sub.i B.sub.i +C.sub.i+1 A.sub.i B.sub.i 
    
     
         C.sub.i =[C.sub.i+1 &#39;+C.sub.i+1 ]A.sub.i B.sub.i +C.sub.i+1 [A.sub.i &#39;B.sub.i +A.sub.i B.sub.i &#39;+A.sub.i B.sub.i ] ##EQU3## Thus, 
    
     
         C.sub.i =A.sub.i B.sub.i +[A.sub.i +B.sub.i ]C.sub.i+1 
    
     A i  B i  is defined as G i  ; A i  +B i  is defined as T i . 
     Thus, 
     
         C.sub.i =G.sub.i +T.sub.i C.sub.i+1 
    
     
         SUM.sub.i =H.sub.i VC.sub.i+1 
    
     
         C.sub.i =G.sub.i +T.sub.i C.sub.i+1 
    
     where V is the Exclusive OR; SUM i  is the SUM of bit i. 
     Executing this formulation is relatively time consuming because it requires the half sum value H i , which is computed immediately, and the term C i+1 , which depends on the calculation of a previous carry (C i+2 ). 
     A better solution, using the same formulation, can be achieved using the carry-look-ahead (CLA) technique. This technique is used to speed up carry propagation in an adder. Carries that are applied to all of the bit positions of an adder can be generated simultaneously by means of additional logic circuitry. 
     While improvements can result using the CLA technique, a carry operation still lies in the critical path. The SUM can be computed either implicitly (i.e., by producing a carry propagated to a previous group of bits and then using a formulation for the SUM that involves that carry) or explicitly (i.e., by producing the carry into bit position that is being considered and using an exclusive OR operation to produce the SUM). Implicit calculations produce the SUM one stage after producing a carry into a group, and explicit calculations produce the SUM two stages after the production of a carry into a group. 
     Accordingly, delays in the critical path are due to the production of carries. Moreover, the SUM requires at least one stage after the creation of the appropriate carry for either implicit or explicit calculation. 
     An improved formulation for the SUM is disclosed in the IBM Journal of Research and Development, Vol. 25, No. 3, pp. 156-166, titled &#34;High-Speed Binary Adder&#34; by Huey Ling. This recursive formulation is described as follows: ##EQU4## with the following definitions: 
     
         H.sub.i =K.sub.i +T.sub.i+1 H.sub.i+1 ; K.sub.i =A.sub.i B.sub.i ; T.sub.i =A.sub.i +B.sub.i. 
    
     The term H i  is not always defined. That is, there are cases in which no value can be attributed to H i . Thus SUM i  in those cases has no meaning and cannot be calculated. 
     When operands A and B are both 32 bits wide, enumerated from 0 to 31 (0 being the most significant bit), 
     
         H.sub.31 =K.sub.31 +T.sub.32 H.sub.32 
    
     In this case, T 32  =A 32  +B 32 , implying that T 32  is outside of the scope of the addition and has no meaning. Consequently H 31  and the SUMs 30 and 31 cannot be computed because they are described as follows: 
     
         SUM.sub.31 =H.sub.31 T.sub.31 &#39;+T.sub.31 H.sub.31 &#39;+K.sub.31 H.sub.32 T.sub.32 
    
     
         SUM.sub.30 =H.sub.30 T.sub.30 &#39;+T.sub.30 H.sub.30 &#39;+K.sub.30 H.sub.31 T.sub.31 
    
     It can also be observed that SUM 31  contains H 32  which is equal to: 
     
         H.sub.32 =K.sub.32 +T.sub.33 H.sub.33 
    
     None of these parameters is defined, so H 32  cannot be computed. 
     Moreover, when a CLA operation occurs, substituting i=30, 29 and 28 results respectively in: 
     
         H.sub.30 =K.sub.30 +T.sub.31 H.sub.31 =K.sub.30 +T.sub.31 K.sub.31 +T.sub.31 T.sub.31 H.sub.32 
    
     
         H.sub.29 =K.sub.29 +T.sub.30 H.sub.30 =K.sub.29 +T.sub.30 K.sub.30 +T.sub.30 T.sub.31 K.sub.31 +T.sub.30 T.sub.31 T.sub.32 H.sub.32 
    
     
         (1) H.sub.28 =K.sub.28 +T.sub.29 H.sub.29 =K.sub.28 +T.sub.29 K.sub.29 +T.sub.29 T.sub.30 K.sub.30 +T.sub.29 T.sub.30 T.sub.31 T.sub.31 +T.sub.29 T.sub.30 T.sub.31 T.sub.32 H.sub.32 
    
     Despite the fact that the aforementioned equations contain quantities with no meaning, which implies that H 28  is undefined as are the SUMs depending thereon; in the aforementioned article by Ling, H 28  is defined and repeatedly used as follows: 
     
         (2) H.sub.28 =K.sub.28 +T.sub.29 H.sub.29 =K.sub.28 +T.sub.29 K.sub.29 +T.sub.29 T.sub.30 K.sub.30 +T.sub.29 T.sub.30 t.sub.31 K.sub.31 T.sub.29 T.sub.30 T.sub.31 T.sub.32 K.sub.32 
    
     Equation (1) is equivalent to equation (2) if and only if: 
     
         T.sub.29 T.sub.30 T.sub.31 T.sub.32 T.sub.33 H.sub.33 =0. 
    
     This is because: 
     H 32  =K 32  +T 33  H 33  and equation (1) can be written as: ##EQU5## 
     In conclusion, T 29  T 30  T 31  T 32  T 33  H 33  is undefined, contains parameters not within the scope of the addition, and has no meaning. Consequently this term cannot be assumed ever to be identical to binary 0. 
     The CLA technique can produce redundancies. Specifically, all T i  K i  are equal to K i  and should have been eliminated. Similar conclusions can be applied to any length addition. 
     SUMMARY OF THE INVENTION 
     It is a primary object of the present invention to provide a new set of equations equivalent to the addition operation that can be implemented to any chosen technology. 
     It is a further object of the present invention to provide apparatus for addition that can be implemented in carry-look-ahead circuitry, thus providing adders that are capable of operating in parallel fashion. 
     It is a further object of the invention to create a parallel adder with three logic stages of delay using a maximum 3×8 AND-OR book, where book is defined as one or more cells comprising logical devices capable of computing Boolean expressions. In particular, 3×8 indicates a three-way AND gate and eight-way OR gate configuration. 
     The present invention reduces the delay necessary to execute addition in computer arithmetic units. It is also within the scope of the invention to propose SUM equations that are proven to respect the equivalence with the addition operation, guaranteeing that the recursive formulas are operational. To reduce the delay necessary to calculate the SUM, critical quantities are generated that require less delay than do carries. The SUM is produced in such a way as to require Boolean expressions that can be implemented with the same delay as, or less delay than, delays for the conventional formulations. 
     The formulation of the function in accordance with the present invention is different than the formulation in the prior art. Because pseudo quantities and new-carries can be implemented faster than the carries described by conventional formulations, the new formulation of the function results in faster execution time and preserves the equivalence with the addition. Compared to the aforementioned paper by Ling, the present invention SUM equation contains less terms. Thus, when implemented, it requires smaller technology books, resulting in reduced delay. The new carry is defined for all natural numbers, implying that there are no regions of undefined quantities. Thus the SUM always has a correct addition value. The equations of the present invention produce a new-carry for look ahead with no redundancies to be excluded. Pseudo quantities are defined to improve delay in a CLA scheme, combined with group look ahead to produce defined new-carries. Such pseudo quantities contain no redundancies, so there is no need to delete unwanted quantities, an operation that tends to penalize implementation. 
     New recursive equations are provided herein for the addition function. The implementation of the equations in any chosen technology can be achieved with ripple carry or with any parallel technique. The adder circuitry resulting from implementation comprises pseudo generate signals, pseudo transmit signals, pseudo half sum signals and pseudo transmit half sum signals. The new-carry is defined for any width addition and represents a binary value and a SUM equation described for ripple or parallel configurations. 
     All quantities can be used on single bit boundaries, or extensively for any chosen grouping of bits. All equations can accommodate any chosen technology or grouping so as to facilitate the design and to increase the performance of hardware-implemented adders under the constraints of a varied technology book set. 
     Two operands of a chosen length N are applied to the adder and the end result is the correct binary addition represented in an N-bit result string. In a ripple carry implemented adder, for every bit position, a pseudo transmit half sum signal is created as is a pseudo half sum signal, a new-carry signal and its complement. The ripple new-carry is created with the pseudo generate, pseudo transmit and the previous new-carry. 
     In parallel adders, depending on technology, a SUM equation is chosen that is convenient for the technology constraints. Depending on the width of the adder, a convenient grouping is imposed. For the chosen grouping and the SUM equations, bit pseudo transmit half sums and pseudo half sums are implemented, in addition to group pseudo transit and pseudo generate signals as dictated by the specified SUM and grouping. For every group, the new-carry is derived and implemented, requiring its own group pseudo generate, pseudo transmit signals and a new-carry of some previous group. 
     With respect to the design of a 32-bit parallel adder, a choice is made to change the equations to respect the addition process while shortening the critical path by means of a set of auxiliary functions that take into account the technology constraints. Even with technology constraints of a 3×8 AND-OR maximum book, the altered equations produce a maximum delay of only three logic stages. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     A complete understanding of the present invention may be obtained by reference to the accompanying drawings, when taken in conjunction with the detailed description thereof and in which: 
     FIG. 1 is a schematic block diagram showing the circuitry necessary to produce a pseudo generate signal; 
     FIG. 2 is a schematic block diagram showing the circuitry necessary to produce a pseudo transmit signal; 
     FIG. 3 is a schematic block diagram showing the circuitry necessary to produce a new-carry signal; 
     FIG. 4 is a schematic block diagram showing the circuitry necessary to produce pseudo half sum and pseudo transmit half sum signals; 
     FIGS. 5-7 are schematic block diagrams that show the circuitry necessary to produce a SUM of a given bit of the addition; 
     FIG. 8 is a schematic block diagram showing the circuitry necessary to produce pseudo quantities T for a group of bit-length 1, 2, 3, 4; 
     FIG. 9 is a schematic block diagram showing the circuitry necessary to produce pseudo quantities G* for a group of bit length 1, 2, 3; 
     FIG. 10 is a schematic block diagram showing the circuitry necessary to produce PS 30 , its negation, and the negation of the carry into the adder; 
     FIG. 11 is a schematic block diagram showing the circuitry necessary to produce PS.sub.(22,29) and the negation of the carry into the adder; 
     FIGS. 12-18 are schematic block diagrams that show the circuitry necessary to produce auxiliary functions Y.sub.(10,21), L.sub.(16,22), Z.sub.(11,15), X.sub.(i,9), W.sub.(13,21), R.sub.(6,15), and Q.sub.(i,5), respectively; 
     FIG. 19 produces the pseudo generate quantities G*.sub.(i+1,9) and its complement for i=0; and 
     FIG. 20 produces the SUM for the critical path delay for i=0. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     For ease of comprehension the FIGURES appended hereto are described by Boolean expressions. The symbols involved in the equations are defined hereinbelow. It is assumed that bit positions are labeled 0 for the most significant bit and r for the least significant bit such that r≧0. The subscripts run high order to low. The variables i, m, n belong to the natural numbers and n+1 is the width of the addition. 
     The invention reduces overall delay by creating pseudo quantities and new-carries (PS i ), that are not equivalent or equal to the carries. PS i  requires less hardware and less delay than does the implementation of carries. New SUM equations have been created that maintain equivalence with the addition function by means of pseudo quantities, a new-carry term (PS i ), defined properly for any width addition, and Boolean equations representing the SUM. The new SUM equations can be implemented with only one stage of delay after production of the new-carries. 
     Notation 
     1. N=natural numbers 
     2. V=exclusive OR 
     3. |=undefined. A quantity is defined mathematically to be &#34;undefined&#34; if such a quantity or any logical operation involved with such a quantity has no meaning (i.e., has no value attribute). 
     4. F(.) indicates that F is determined by variables not explicitly stated. 
     FIGS. 1-7 
     FIGS. 1 through 7 can be described by the Boolean equations shown hereinbelow. 
     Referring now to FIG. 1, the pseudo generate signal is described from bit i to bit m, such that i≦m and: 
     
         G*.sub.(i,m) =0 if i&gt;n 
    
     
         G*.sub.(i,m) =G.sub.i +G.sub.i+1 +T.sub.i+1 G.sub.i+2 +T.sub.i+1 T.sub.i+2 G.sub.i+3 + . . . +T.sub.i+1 T.sub.i+2 T.sub.i+3 . . . T.sub.m-1 G.sub.m if i≦n 
    
     Referring now also to FIG. 2, the pseudo transmit signal is described from bit i to bit m, T.sub.(i,m), with i≦m: 
     
         T.sub.(i,m) =1                                             if i&gt;n 
    
     
         T.sub.(i,m) =T.sub.i T.sub.i+1 T.sub.i+2 . . . T.sub.m-1 T.sub.m if i≦n 
    
     Referring now also to FIG. 3, the new-carry is described from bit i to bit m, PS.sub.(i,m), such that i≦m: 
     
         PS.sub.(i,m) =Cin                                          if i&gt;n 
    
     
         PS.sub.(i,m) =G*.sub.(i,m) +T.sub.(i+1,m+1) PS.sub.(m+1,z) if i≦n 
    
     where z is a natural number such that m&lt;z and Cin is the carry into the adder. 
     Referring now also to FIG. 4, H i , the pseudo half sum, is described for bit i. 
     
         H.sub.i =|                                        if i&gt;n 
    
     
         H.sub.i =A.sub.i V B.sub.i                                 if i≦n 
    
     and FIG. 4 also describes M i , the pseudo transmit half sum for bit i. 
     
         M.sub.i =|                                        if H.sub.i =| 
    
     
         M.sub.i =H.sub.i V T.sub.i+1                               if H.sub.i ≠| 
    
     Referring now also to FIG. 5, equations are described that will compute the SUM, provided that the proper quantities involved with the expressions are previously calculated. 
     
         SUM.sub.i-1 =M.sub.i-1 PS.sub.(i,m) +H.sub.i-1 PS.sub.(i,m) &#39; 
    
     The equation described by FIG. 5 and the equation described by FIG. 3 are parametrical. The parameters involved are arbitrary (i.e., no law attributes a value to them). The only restriction imposed to the parameters by the definitions is that: 
     
         i≦m&lt;z 
    
     Using the above premises, by imposing different boundaries to the variables present into to the SUM formulas, and by appropriate substitutions, recursions in PS(.) are removed and some are corollaries. 
     Referring now also to FIGS. 6 and 7, the circuitry needed to compute the SUM equations for two of the corollaries are described. 
     FIG. 6 describes the following Boolean equation: ##EQU6## 
     FIG. 7 describes the following equation: ##EQU7## 
     New Recursive Formulas For The Sum 
     The following proves that the aforementioned definitions and SUM expressions will preserve the addition function. 
     LEMMA 1 
     
         G.sub.(i,m) =TiG*.sub.(i,m)                                if i,m≦n 
    
     Proof 
     by definition ##EQU8## but T i  G i  =G i  ##EQU9## 
     LEMMA 2 
     
         G*.sub.(i,z) =G*.sub.(i,m) +T.sub.(i+1,m+1) G*.sub.(m+1,z) such that m&lt;z 
    
     Proof 
     by definition ##EQU10## 
     LEMMA 3 
     
         PS.sub.(m,z) =PS.sub.(m,n) =G*.sub.(m,n) +T.sub.(m+1,n) Cin 
    
     Proof 
     Case 1: if z≧n 
     by definition (FIG. 3) 
     
         PS.sub.(m,z) =G*.sub.(m,z) +T.sub.(m+1,z+1) PS.sub.(z+1,x) with x some N by definition (FIG. 1) and reduction ##EQU11## but G.sub.n+1 +G.sub.n+2 + . . . +T.sub.n+2 . . . T.sub.z-1 G.sub.z =G*.sub.(n+1,z) =0 Thus G*.sub.(m,z) =G*.sub.(m,n) by definition (FIG. 2) ##EQU12## but T.sub.(n+1,z+1) =1 
    
     Thus T.sub.(m+1,z+1) =T.sub.(m+1,n) 
     by definition (FIG. 3) 
     
         PS.sub.(z+1,x) =Cin because z+1&gt;n. 
    
     Thus it can be concluded that 
     
         PS.sub.(m,z) =G*.sub.(m,n) +T.sub.(m+1,n) Cin if z≧n 
    
     Case 2: z&lt;n 
     
         (0.0) PS.sub.(m,z) =G*.sub.(m,z) +T.sub.(m+1,z+1) PS.sub.(z+1,X) such that: 
    
     
         (0.1) z&lt;X≦n or (0.2) X&gt;n 
    
     subcase 1 
     (0.1) implies that there exists some succession of natural numbers such that 
     
         z&lt;X&lt;X.sub.0 &lt;X.sub.1 &lt;X.sub.2 &lt; . . . &lt;X.sub.j &lt;X.sub.i such that X.sub.i =n. 
    
     By substitutions and by definition 2 (0.1), ##EQU13## By applying lemma 2 as many times as needed: 
     
         PS.sub.(m,z) =PS.sub.(m,n) 
    
     Thus the lemma holds for subcase 1. 
     subcase 2 
     (0.2) implies that by substitution (0.0) can be written as: 
     
         PS.sub.(m,z) =G*.sub.(m,z) +T.sub.(m+1,z+1) G*.sub.(z+1,n) +T.sub.(m+1,z+1) T.sub.(z+2,n) PS.sub.(n+1,X) 
    
     By lemma 2 and the definition of PS(.) 
     
         PS.sub.(m,z) =G*.sub.(m,n) +T.sub.(m+1,n) Cin 
    
     Thus lemma 3 holds true for subcase 2. 
     Theorem 1 
     The following set of equations is equivalent to the addition. 
     
         (2.1) SUM.sub.i-1 =M.sub.i-1 PS.sub.(i,m) +H.sub.i-1 PS.sub.(i,m) &#39; 
    
     
         (2.2) PS.sub.(i,m) =G*.sub.(i,m) +T.sub.(i+1,m+1) PS(m+1,z) 
    
     Proof 
     (2.1) implies: ##EQU14## 
     Case 1: i&gt;n+1 
     if i&gt;n+1 by definition (FIG. 4) 
     
         SUM.sub.i-1 =| 
    
     Case 2: i=n+1 
     by definitions (FIGS. 2 and 3) 
     
         T.sub.i =1 and PS.sub.(i,m) =Cin 
    
     Thus T i  PS.sub.(i,m) =Cin and SUM n  =H n  V Cin 
     Thus the set of equations 2.1 and 2.2 is equivalent to the addition for the least significant bits. 
     Case 3: i&lt;n+1 
     It must be proved that the carry to the bit position i-1, C i  is equal to T i  PS.sub.(i,m) with PS.sub.(i,m) defined by (2.2). This is because in this case H i-1  is always defined to be equal to the half sum. 
     Given that C i  =G.sub.(i,n) +T.sub.(i,n) Cin, prove that: 
     
         (1) G.sub.(i,n) +T.sub.(i,n) Cin=T.sub.i PS.sub.(i,m) 
    
     By lemma 3 PS.sub.(i,m) =G*.sub.(i,n) +T.sub.(i+1,n) Cin ##EQU15## but T i  G*.sub.(i,n) =G.sub.(i,n) (lemma 1) and by definition 
     
         T.sub.i T.sub.(i+1,n) =T.sub.(i,n) 
    
     Thus G.sub.(i,n) +T.sub.(i,n) Cin=T i  PS.sub.(i,m) 
     Equations 2.1 and 2.2 are parametrical. The parameters involved are arbitrary (i.e., no law attributes a value to them). The only restriction imposed on the parameters by the foregoing definitions is that: 
     
         i≦m≦z 
    
     Using the above premises, imposing different boundaries to the variables present into to the SUM formulas, and by appropriate substitutions, recursions in PS(.) can be removed and some are reported as the following corollaries: 
     Corollary 1 (removal of one recurrence) 
     The following set of equations is equivalent to the addition. ##EQU16## (2.4) PS.sub.(m+1,z) =G*.sub.(m+1,z) +T.sub.(m+2,z+1) PS.sub.(z+1,k) 
     Proof 
     By direct substitution of (2.2) into (2.1) and imposing i≦m&lt;z&lt;k. 
     Corollary 2 (removal of two recurrences) 
     The following set of equations is equivalent to the addition. ##EQU17## 
     
         (2.6) PS.sub.(z+1,k) =G*.sub.(z+1,k) +T.sub.(z+2,k+1) PS.sub.(k+1,x) 
    
     Proof 
     Same procedure as corollary 1 with the imposition i≦m&lt;z&lt;k&lt;x. 
     Corollary 3 (removal of three recurrences) 
     The following set of equations is equivalent to the addition. ##EQU18## 
     
         (2.8) PS.sub.(k+1,x) =G*.sub.(k+1,x) +T.sub.(k+2,x+1) PS.sub.(x+1,v) 
    
     Proof 
     Same procedure as corollary 1 with the imposition i≦m&lt;z&lt;k&lt;x&lt;v. 
     THEOREM 2 
     For every removal of recursion: 
     (1) the resulting expressions have two terms that are recursive, namely one that contains the new-carry and one that contains its complement; 
     (2) there are two OR terms added in respect to the previous removal; 
     (3) the terms that contain recursion are maximum width ANDs; and 
     (4) if an `m`-way was the maximum-way AND in the previous removal then an `m+1`-way AND is the maximum-way AND in the current removal. 
     Proof by Induction 
     Theorem 1 and corollary 1 imply that the base of the induction holds true (i.e., for i=0 and i=1). 
     Suppose that it is true for the removal i. In that case: 
     (1) if F i  (.) is a Boolean expression not recursive and R i  (.),Y i  (.), G i  *(.) and T i  (.) are not recursive terms, then for the ith removal: 
     
         (2.9) SUM.sub.(i-1) =F.sub.i (.)+R.sub.i (.)PS.sub.i (.)+Y.sub.i (.)PS.sub.i (.)&#39; 
    
     
         (2.10) PS.sub.i (.)=G.sub.i *(.)+T.sub.i (.)PS.sub.i+1 (.) 
    
     (2) if there were n-2 terms in the previous removal then (2.9) contains n terms; 
     (3) the terms R i  (.)PS i  (.) and Y i  (.)PS i  (.)&#39; are m-way ANDs such that m is maximum-way in (2.9); and 
     (4) the maximum-way AND in the previous removal was m-1. 
     It must be proven that the theorem holds for i+1. By substituting (2.10) into (2.9) implies: 
     
         SUM.sub.i =F.sub.i (.)+R.sub.i (.)G.sub.i *(.)+R.sub.i (.)T.sub.i (.)PS.sub.i+1 (.)+Y.sub.i (.)G.sub.i *(.)&#39;T.sub.i (.)&#39;+Y.sub.i (.)G.sub.i *(.)&#39;PS.sub.i+1 (.)&#39; 
    
     Thus by the i+1 removal: 
     (1) of all terms involved in SUM, two are recursive; 
     (2) the number of terms involved is n+2; and 
     (3) given the fact that Y i  (.)PS i  (.) and R i  (.)PS i  (.) were maximum-way ANDs and the fact that F i  (.) remains unchanged, the maximum width ANDs are the terms Y i  (.)G i  *(.)&#39;T i  (.)&#39;, R i  (.)T i  (.)PS i+1  (.) and Y i  (.)G i  *(.)&#39;PS i+1  (.)&#39; which includes the two recursive terms. Because the width of the AND was m, all the above quantities are `m+1`-way ANDs. 
     Therefore the theorem holds for every removal of a recursion. 
     Theorem 3 
     The equation that computes the SUM is (i+2)×2(i+1) AND-OR function i being equal to the number of the removed recurrences. 
     Proof by Induction 
     
         for i=0 
    
     The theorem holds because (2.1) is a 2×2 AND-OR. 
     Suppose it is true for i. Prove that, for i+1, the SUM is a (i+3)×2(i+2) AND-OR. 
     Given that the theorem holds for i and because of theorem 2 the removal of the i+1 recursion implies that the maximum-way AND will be i+2+1=i+3 and the maximum-way OR 2(i+1)+2=2(i+2). Thus the SUM is a (i+3)×2(i+2) AND-OR and the theorem holds true for all the removals of recursion. 
     General Comparison Between the Schemas 
     While it is true that given a technology any schema may be implemented with one or more algorithms, one fact remains unique: the expressions that describe a function from its positive and/or negative primary inputs will implicitly or explicitly determine the width of the gates and consequently the equivalent books needed for an optimal implementation of the function. Because actual technologies do not allow arbitrary width gates and/or books, it can be inferred that such expressions directly or indirectly will determine the corresponding logic levels in an optimal implementation of the given function with an appropriate algorithm for any chosen technology. Thus, since in each logic level there is associated a delay and a number of cells (two of the major concerns in logic design), the &#34;primary&#34; expressions determine performance and area. 
     The above premises imply that, if two formulations exist for a function (e.g., addition) then a comparison of their AND-OR equations formulated using only the &#34;primary&#34; terms can certainly be used as measure of performance and cost in terms of area. 
     One of the objects of this invention is to shorten the critical paths of the adder and therefore to prove the superiority of the present invention with respect to conventional processes. It is sufficient to prove that the expressions involved are inherently better to implement independent of the technology and algorithms. This can be achieved by proving that the width of the &#34;primary&#34; expressions involved in the critical paths are of a lesser magnitude than the ones involved in conventional methods of addition and that the area required to produce the critical expressions in the present method is smaller. That is because even if it may be the case that the final formulation of the SUM as described by this new schema may require more hardware, this hardware will be involved with the non-critical paths. Thus, if the width of the &#34;primary&#34; expressions and the area involved in the critical paths are less, then the critical paths have been improved. 
     For the addition function both schemas imply that the delay of the critical paths are the ones that contain recursion, namely C i  as described by (1.b) and PS.sub.(i,m) as described by (2.2), while the other expressions that determine the SUM can be computed immediately. 
     Theorem 4 
     If # is the number of OR terms needed to produce PS.sub.(i,m) starting from the primary inputs, then 2#-1 terms are needed to produce C i . 
     Proof 
     By lemma 3 
     
         PS.sub.(i,m) =G*.sub.(i,n) +T.sub.(i+1,n) Cin 
    
     Let g,t be the OR terms needed respectively for G*.sub.(i,n) and T.sub.(i+1,n) Cin 
     
         Then #=g+t 
    
     Given that C i  =G.sub.(i,n) +T.sub.(i,n) Cin and because of lemma 1: 
     
         G.sub.(i,n) =T.sub.i G*.sub.(i,n) =G.sub.i +T.sub.i G.sub.(i+1,n) 
    
     but G*.sub.(i,n) =G i  +G.sub.(i+1,n) 
     Thus G.sub.(i+1,n) has g-1 OR terms. Given that T i  has two OR terms: 
     
         G.sub.(i,n) has 1+2(g-1)=2g-1 terms. 
    
     Given that Cin will not produce an extra OR term and by definition T.sub.(i,n) =T i  T.sub.(i+1,n) then: T.sub.(i,n) has 2t OR terms and C i  has 2(g+t)-1=2#-1 OR terms. 
     Theorem 5 
     PS.sub.(i,m) contains exactly 2.sup.(n-i+1) OR terms if fully expanded. 
     Proof 
     G*.sub.(i,n) contains: 
     
         1+1+2+2.sup.2 +2.sup.3 + . . . +2.sup.(n-i-1) =1+2.sup.(n-i) -1=2.sup.(n-i) OR terms. 
    
     And T.sub.(i+1,n) contains: 2.sup.(n-i) OR terms. 
     Given that Cin does not produce OR terms and by lemma 3 PS.sub.(i,m) =PS.sub.(i,n) =G*.sub.(i,n) +T.sub.(i+1,n) Cin 
     Thus PS.sub.(i,m) contain 2.sup.(n-i) +2.sup.(n-i) =2.sup.(n-i+1) OR terms. 
     Theorem 6 
     PS.sub.(i,m) contains: one two-way AND, 2.sup.(n-i) (n-i+1)-way ANDs and 2.sup.(r-2) r-way ANDs with r getting all the values between and including 2 and n-i+1 if fully expanded. 
     Proof 
     The definition G*.sub.(i,m) implies that there is at least one two-way AND in PS.sub.(i,n) --the one due to the G i  term. 
     The terms in G*.sub.(i,m) are a geometrical series after G i  and the definition of the T&#39;s dictates that the width due to the T terms be equal to the corresponding power of 2. Then if r-way corresponds to the jth power of 2 in the series the definition of G determines that r=j+2. For each power there will be 2.sup.(r-2) r-way ANDs in PS.sub.(i,n). Given that 0≦j≦n-i-1, then 2≦j+2≦n-i-1+2 and r belongs to N such that 2≦r≦n-i+1. 
     Finally, because of the definition of T&#39;s, there are 2.sup.(n-1) OR terms in T.sub.(i+1,n) Cin. Given that Cin will add an extra input to the ANDs, there exist 2.sup.(n-i) (n-i+1)-way ANDs in PS.sub.(i,n). 
     Theorem 7 
     There are 2.sup.(n-i+1) (n-i+2)-way ANDs and 2.sup.(r-2) r-way ANDs with r getting all the values between and including 2 and n-i+2 in C i . 
     Proof 
     By the same reasoning as in the previous theorem and realizing that the terms in G.sub.(i,n) are a geometrical series beginning from bit i and the fact that C i  contains the term T.sub.(i,n) instead of T.sub.(i+1,n). 
     Theorem 8 
     There are exactly 2.sup.(n-i+2) -1 OR terms in C i . 
     Proof 
     Follows immediately from theorems 4 and 5. 
     Theorem 9 
     C i  requires more area than PS.sub.(i,m). 
     Proof 
     Follows immediately from theorems 5, 6 and 8. 
     Theorem 10 
     The width of the gates involved to produce C i  from its primary inputs is of a greater magnitude than the ones that produce PS(.) 
     Proof 
     Follows immediately from theorems 4, 5, 6, and 7. 
     Theorems 4 through 10 determine that the methodology proposed by this study will contain in its critical paths less hardware as well as less &#34;primary&#34; terms in its expressions than do the traditional formulation of the critical paths. 
     If a full carry-look-ahead is not permitted (which is the case in certain technologies) then the grouping in the factorization process will be bigger in PS(.) than in C i . Therefore, PS(.) ripples less. Since rippling is directly translated to delay, the proposed schema results in implementations of faster adders than conventional adders. 
     One other advantage for the methodology herein described in that the corollaries explicitly state, depending on technology, that the last logic level can be used more efficiently. That is, the paths can be broken into more than one, thus removing the dependencies of PS(.) into the bit that the SUM is calculated. Nevertheless the Cin can be used on the first logic level to produce PS(.) for high order bits. 
     FIGS. 8-20 
     FIGS. 8-20 refer to equations discussed hereinbelow that describe the critical path of SUM 0 , and by extension, every other path. This demonstrates the design of an adder of three logic stages with the use of a maximum 3×8 AND-OR book. The diagrams represent the design of the present invention with simple gates that can be transformed to appropriate more complex books by any person skilled in the art of computer system design. 
     Referring now also to FIG. 8, circuits are described that are necessary to compute the following Boolean described equations: 
     
         T.sub.(i,i+3) =(A.sub.(i) +B.sub.(i))(A.sub.(i+1) +B.sub.(i+1))(A.sub.(i+2) +B.sub.(i+2))+ 
    
     
         (A.sub.(i+3) +B.sub.(i+3)) 
    
     
         T.sub.(i) =A.sub.(i) +B.sub.(i) 
    
     
         T.sub.(i,i+1) =(A.sub.(i) +B.sub.(i))(A.sub.(i+1) +B.sub.(i+1)) 
    
     
         T(i,i+2)=(A.sub.(i) +B.sub.(i))(A.sub.(i+1) +B.sub.(i+1))(A.sub.(i+2) +B.sub.(i+2)) 
    
     Referring now also to FIG. 9, the following Boolean expressions are described: 
     
         G*.sub.(i) =A.sub.(i) B.sub.(i) 
    
     
         G*.sub.(i,i+1) =A.sub.(i) B.sub.(i) +A.sub.(i+1) B.sub.(i+1) 
    
     
         G*.sub.(i,i+2) =A.sub.(i) B.sub.(i) +A.sub.(i+1) B.sub.(i+1) +A.sub.(i+1) A.sub.(i+2) B.sub.(i+2) + 
    
     
         B.sub.(i+1) A.sub.(i+2) B.sub.(i+2) 
    
     Referring now also to FIG. 10, the Boolean expression for the production of PS 30  and its complement is described. The complement has been produced by the negation of the PS 30  and if implemented that way the logic stages will be increased. Also, similar freedoms have been allowed in other places including the two previous FIGURES. This facilitates comprehension and diminishes the complexity of the FIGURES. 
     The Boolean expression for PS 30  is: 
     
         PS.sub.(30) =A.sub.(30) B.sub.(30) +A.sub.(31) B.sub.(31) +A.sub.(31) Cin+B.sub.(31) Cin 
    
     Referring now also to FIG. 11, circuitry for PS.sub.(22,29) and its complement is computed as follows: 
     
         PS.sub.(22,29) =G*.sub.(22,24) +T.sub.(23,25) G*.sub.(25,27) +T.sub.(23,25) T.sub.(26,28) G*.sub.(28,29)+T.sub.(23,26) T.sub.(27,30) PS.sub.(30) 
    
     FIGS. 12-18 compute the Boolean expressions that calculate the auxiliary function, and described as shown hereinbelow. ##EQU19## 
     Referring now also to FIG. 19, the Boolean expression is described that computes the pseudo generate signal for i=0: 
     
         G*.sub.(i+1,9) =G*.sub.(i+1,3) +T.sub.(i+2,4) G*.sub.(4,6) +T.sub.(i+2,4) T.sub.(5,7) G*.sub.(7,9) 
    
     Referring now also to FIG. 20, the computation of SUM i , with i belonging to the critical sum is described. 
     For i=0 then: 
     
         SUM.sub.(i) =M.sub.(i) G*.sub.(i+1,9) +Q.sub.(i,5) R.sup.(6,15) + 
    
     
         Q.sub.(i,5) T.sub.(6,12) W.sub.(13,21) + 
    
     
         Q.sub.(i,5) T.sub.(6,22) PS.sub.(22,29) + 
    
     
         H.sub.(i) T.sub.(i+2,10) &#39;G*.sub.(i+1,9) &#39;+ 
    
     
         X.sub.(i,9) Z.sub.(11,15) + 
    
     
         X.sub.(i,9) G*.sub.(10,15) &#39;L.sub.(16,22) + 
    
     
         X.sub.(i,9) Y.sub.(10,21) PS.sub.(22,29) &#39; 
    
     The Implementation 
     For the following section, notational conventions are as follows: 
     (1) &#39; is the inverse or NOT of the expression 
     (2) (a+b) is &#34;a&#34; logical OR &#34;b&#34; 
     (3) (ab) is &#34;a&#34; logical AND &#34;b&#34; 
     The critical path of the three stage adder is described as follows. The SUM of bit 0 requires maximum add time. That is, it lies on the critical path. Therefore, the design of SUM 0  dictates the maximum delay of the adder implementation. By applying corollary 3, SUM 0  can be expressed as follows: 
     For i=0, 
     
         SUM.sub.i =M.sub.(i) G*.sub.(i+1,9) +M.sub.(i) T.sub.(i+2,10) G*.sub.(10,15) + 
    
     
         M.sub.(i) T.sub.(i+2,16) G*.sub.(16,21) +M.sub.(i) T.sub.(i+2,22) PS.sub.(22,29) + 
    
     
         H.sub.(i) T.sub.(i+2,10) &#39;G*.sub.(i+1,9) &#39;+ 
    
     
         H.sub.(i) G*.sub.(i+1,9) &#39;T.sub.(11,16) &#39;G*.sub.(10,15) &#39;+ 
    
     
         H.sub.(i) G*.sub.(i+1,9) &#39;G*.sub.(10,15) &#39;G*.sub.(16,21) &#39;T.sub.(17,22) &#39;+ 
    
     
         H.sub.(i) G*.sub.(i+1,9) &#39;G*.sub.(10,15) &#39;G*.sub.(16,21) &#39;PS.sub.(22,29) &#39; 
    
     It is obvious that the SUM 0  as described previously cannot be implemented in one stage by using a 3×8 AND-OR book. However it can be rewritten as follow for i=0: 
     
         SUM.sub.i =M.sub.(i) G*.sub.(i+1,9) +Q.sub.(i,5) R.sub.(6,15) + 
    
     
         Q.sub.(i,5) T.sub.(6,12) W.sub.(13,21) +Q.sub.(i,5) T.sub.(6,22) PS.sub.(22,29) + 
    
     
         H.sub.(i) T.sub.(i+2,10) &#39;G*.sub.(i+1,9) &#39;+X.sub.(i,9) Z.sub.(11,15) + 
    
     
         X.sub.(i,9) G*.sub.(10,15) &#39;L.sub.16,22) + 
    
     
         X.sub.(i,9) Y.sub.(10,21) PS.sub.(22,29) &#39; 
    
     with: 
     
         Q.sub.(i,5) =M.sub.(i) T.sub.(i+2,5) 
    
     
         R.sub.(6,15) =T.sub.(6,10) G*.sub.(10,15) 
    
     
         W.sub.(13,21) =T.sub.(13,16) G*.sub.(16,21) 
    
     
         X.sub.(i,9) =H.sub.(i) G*.sub.(i+1,9) &#39; 
    
     
         Z.sub.(11,15)=T.sub.(11,16) &#39;G*.sub.(10,15) &#39; 
    
     
         L.sub.(16,22) =G*.sub.(16,21) &#39;T.sub.(17,22)&#39; 
    
     
         Y.sub.(10,21) =G*.sub.(10,15) &#39;G*.sub.(16,21) &#39; 
    
     The transformed SUM 0  can be implemented in one stage in a 3×8 AND-OR book, assuming that all quantities included in the equations have been implemented previously. 
     Such quantities can be re-written as follows for i=0: ##EQU20## 
     In addition to those quantities, group transmit signals and their complements must be implemented. However, such implementation is not critical since they can be computed by AND and NAND logic. Also, the group pseudo generates (G*) must be implemented. However, the groups involved are smaller than G*.sub.(i+1,9). Therefore, smaller books, or books the same size as those used for G*.sub.(i+1,9) are required for implementation. 
     The previous equations can be computed in one stage in a 3×8 AND-OR book, assuming that all quantities included in the equations have been implemented previously. 
     Such quantities can be expressed as follows: 
     
         T.sub.(i,i+3) =(A.sub.(i) +B.sub.(i)) (A.sub.(i+1) +B.sub.(i+1)) (A.sub.(i+2) +B.sub.(i+2)) (A.sub.(i+3) +B.sub.(i+3)) 
    
     
         T.sub.(i) =A.sub.(i) +B.sub.(i) 
    
     
         T.sub.(i,i+1) =(A.sub.(i) +B.sub.(i)) (A.sub.(i+1) +B.sub.(i+1)) 
    
     
         T.sub.(i,i+2) =(A.sub.(i) +B.sub.(i)) (A.sub.(i+1) +B.sub.(i+1)) (A.sub.(i+2) +B.sub.i+2)) 
    
     
         G*.sub.(i) =A.sub.(i) B.sub.(i) 
    
     
         G*.sub.(i,i+1) =A.sub.(i) B.sub.(i) +A.sub.(i+1) B.sub.(i+1) 
    
     
         G*.sub.(i,i+2) =A.sub.(i) B.sub.(i) +A.sub.(i+1) B.sub.(i+1) +A.sub.(i+1) A.sub.(i+2) B.sub.(i+2) +B.sub.(i+1) A.sub.(i+2) B.sub.(i+2) 
    
     
         PS.sub.(30) =A.sub.(30) B.sub.(30) +A.sub.(31) B.sub.(31) +A.sub.(31) Cin+B.sub.(31) Cin 
    
     
         PS.sub.(+) &#39;=(A.sub.(30) B.sub.(30) +A.sub.(31) B.sub.(31) +A.sub.(31) Cin+B.sub.(34) Cin)&#39; 
    
     
         H.sub.(i) =A.sub.(i) &#39;B.sub.(i) +A.sub.(i) B.sub.(i) &#39; for all i 
    
     
         H.sub.(i) &#39;=(A.sub.(i) &#39;B.sub.(i) +A.sub.(i) B.sub.(i) &#39;)&#39; for all i 
    
     
         Cin&#39;=(Cin)&#39; 
    
     all of the previous quantities can be implemented, as needed, from the primary inputs to the adder. Therefore, three stages are needed to produce SUM 0  and, consequently, all other sums of the 32-bit adder. 
     Since other modifications and changes varied to fit particular operating requirements and environments will be apparent to those skilled in the art, the invention is not considered limited to the 32-bit adder example chosen for purposes of disclosure, and covers all changes and modifications which do not constitute departures from the true spirit and scope of this invention.