Analysis of input and output configurations for use in four-valued CCD programmable logic arrays

Transcription

1 nalysis of input and output onfigurations for use in four-valued D programmable logi arrays J.T. utler H.G. Kerkhoff ndexing terms: Logi, iruit theory and design, harge-oupled devies bstrat: s in binary, a multiple-valued programmable logi array (PL) realises a sum-ofproduts expression speified by the user. However, in multiple-valued logi, there are many more operations than in binary, and an important question is the hoie of operations whih provides the greatest number of funtions for a given hip area. n this paper, we analyse various PL onfigurations using operations realised in the peristalti multiple-valued D tehnology. We ompare a multiple-valued D PL implementation with four other proposed designs and show that there is a signifiant differene in hip area required to realise the same set of funtions. The basis of omparison is the set of -valued unary funtions. ntrodution n binary logi, an important iruit is the programmable logi array or PL. ts widespread use is due to a flexibility whih allows the user to speify omplex ombinatorial funtions. Reently, PLs have been suggested for multiple-valued systems [-7]. t is likely that the advantages of the PL will make it an important element in this domain as well. However, the wider hoie of operations assoiated with a radix larger than, makes the determination of whih PL onfiguration to offer the user more diffiult than in binary. Two PL onfigurations were studied in ender et al. [6], where it was shown that the use of the Sum operation at the seond PL level produes more eonomial realisations than with the Max operation over two lasses of funtions. However, the question of whih operations are best suited for general multiple-valued funtions remains open. n implementation of a PL in a radix higher than is desribed in Kerkhoff and utler [7]. The PL is fabriated in D or harge-oupled devie tehnology, and is the first implementation of a multiple-valued PL, not only in D, but in any tehnology. The design is radix independent. That is, no primary iruit hanges are required to aommodate hanges in radix. Only voltage Paper 5E(), first reeived th Deember 985 and in revised form rd Otober 986 J.T. utler is with the Department of Eletrial and omputer Engineering, Naval Postgraduate Shool, Monterey, 99-5, US H.G. Kerkhoff is with the -Tehnology and Eletronis Group, Department of Eletrial Engineering, Twente University of Tehnology, 75OOE Enshede, The Netherlands values applied to the iruits need to be modified. This PL is also the first multiple-valued peristalti D. ll previous multiple-valued D iruits have been surfaehannel D. eause peristalti D has a lower harge-transfer time, there is a orrespondingly higher speed. The fous in Referene 7 is the implementation of a multiple-valued PL with an emphasis on devie and iruit onsiderations. n this paper, we onsider the logi design of the multiple-valued PLs. We examine the question of what PL onfiguration provides the greatest logi apability. The implementation of the PL in Referene 7 is based on the llen-givone algebra [8] with the Max operation replaed by the Sum (Tirumalai and utler []). t requires a iruit whih implements the literal operator. n the implementation of Referene 7, the literal is realised by a pair of subiruits, one realising the stairase-up and the other the stairase-down funtion. However, ertain literal funtions an be realised using only one stairase funtion, suggesting that more areaeffiient PLs an be built using stairase-funtion generators instead of literal-funtion generators. This intuitive notion is quantified in this paper. Speifially, we show that there is a signifiant redution in hip area when suh a substitution is made. Our analysis inludes a total of five PL designs, and we show that the design desribed in Referene 7 ranks midway (rd out of 5) using the hip-area riteria. ll previously implemented D iruits have been in values. This inludes the peristalti D implementation of Referene 7, although it is expeted to work with more values, perhaps 6. eause our analysis requires the use of a speifi radix, we hoose radix. n the next Setion, we introdue the D logi operations, and in Setion, we analyse the various PL onfigurations with respet to the funtions realised. summary of results is given in Setion. D logi operations. onstant Fig. shows the four basi D operations used in the peristalti D PLs desribed here. The first three operations have been fabriated in surfae-hannel D tehnology (Kerkhoff []) in the same way as in peristalti tehnology, while the last operation, Sum, has been realised in surfae-hannel D in a different form than shown here. Fig. la shows a onstant logi-value generator. The square on the left ontaining a blak bar represents a soure of harge whih flows at the lok pulse into a harge well represented by the square labelled p. p is the 68 EE PROEEDNGS, Vol., Pt. E, No., JULY 987

2 apaity of the well. The harge soure fills the well ompletely, produing a quantity of harge equal to a logi p. The blak bar to the right of this well is a loked transfer gate, whih transfers the well ontents to the right and out of the onstant generator. For example, if the well apaity is a logi, p =, and the onstant generator produes a logi at its output. Fig. ll p - -HD asi D operations a onstant (ost = ) b Multithreshold (ost = ) nhibit (ost = 8) d Metal summer (ost = ) i±. Multithreshold Fig. \b shows a multithreshold or fixed-overflow iruit. t the lok pulse, the input harge at x is transferred into the well labelled p. f this harge exeeds the well apaity p, the exess flows right and into the next well. This has a apaity of logi, as indiated by the three orner braes. t the next lok pulse, the ontents of this well are transferred out aross the loked transfer gate, represented by the vertial blak bar, beoming the multithreshold output. The multithreshold iruit produes x p at the output if x > p and otherwise.. nhibit The inhibit iruit is shown in Fig. \. t has two inputs, a primary input x and a ontrol input y. When a logi appears at y, the logi value at the primary input x flows into the well labelled p. When the ontrol input has a nonzero logi value, this flow is bloked, and a logi appears at the inhibit output at the right. For y =, x passes to the output if x ^ p and p passes to the output when x > p. Thus, the inhibit ats as a swith. The inhibit shown in Fig. \ is a simplified version of that introdued in Referene for surfae-hannel Ds, where there is another output whih is x when y >.. Sum Fig. \d shows the Sum operation. The metal line stores the harge whih is dumped from inputs x and y at the lok pulse. The harge adds, reating a voltage whih is proportional to the total harge. n the PL onfiguration desribed below, the metal summers are the PL olumns. n the surfae-hannel D implementations [], the Sum operation is performed in a well rather than a metal line..5 ost fators The above four D operations are used in the PL onfigurations to be desribed later. To ompare onfigurations, we use relative ost fators of D operations, speifially those proposed in Referene. This ost is primarily a measure of the hip area oupied by the operation. For example, the onstant generator requires muh less hip area than the inhibit, and thus has a orrespondingly smaller relative ost. Other fators whih determine the ost are the number of different powersupply and data lines, as well as the sensitivity of the realised funtion to proess and voltage variations. The osts are integers and are shown in Fig.. nalysis of realised funtions. PL omponents n this setion, we analyse the funtions realised by the proposed PL designs. The PL struture has two basi omponents: (a) input onfigurations logi required between primary PL inputs and the PL olumns (b) output onfigurations logi required between the PL olumns and the PL output. Two input onfigurations, stairase and step, and two output onfigurations, omplement and inhibit, will be onsidered. Taken together, there are ombinations. These are ompared to the onfiguration ombination used in the multiple-valued PL implementation of Referene 7, whih onsists of a literal-funtion input onfiguration and an inhibit output onfiguration. Thus, the five ombinations are nput onfiguration (i) stairase funtions (ii) stairase funtions (iii) step funtions (iv) step funtions (v) literal funtions. nput onfigurations Output onfiguration omplement or diret inhibit or diret omplement or diret inhibit or diret inhibit always [7].. Stairase-funtion generator: The first type of input onfiguration is shown in Fig. a. t onsists of a onstant metal sum me>r generator^ *. x -y y-o- Fig. a b P i i X "", r -- i. X y a= a= a = PL input onfiguration: stairase-funtion generator b= b=l b= fixed overflow driven by a primary input x. The fixed overflow has one well with a programmable apaity given by the value of the lower-ase letter in the well and another well whih ollets the overflow. t a lok pulse, the ontents of this latter well are applied to a metal summer, whih orresponds to one olumn of the PL. Fig. a shows two input onfigurations. n the top onfiguration, x is applied as the input to the fixed overflow. n the one below, the omplement of y, y, is applied. mplementation of the omplement is straightforward. t is obtained by interhanging the onnetions to eletrodes on the D substrate of the devie whih realises the iruit shown above it (p., Referene ). Fig. b shows that the funtion realised by eah input EE PROEEDNGS, Vol., Pt. E, No., JULY

3 onfiguration is a stairase funtion in whih the maximum logi value depends on the apaity of the programmable well. n speifying the PL funtion, the user hooses whether the input is omplemented or not. This determines whether a stairase-down or stairase-up funtion, respetively, is produed. lso shown in Fig. a is a onstant generator in whih the output is applied to the metal summer. Thus, the output of the metal summer is the sum of stairase funtions and possibly a onstant. The value of the onstant is determined by the user, and there is exatly one onstant generator for eah PL olumn... Step-funtion generator: Fig. a shows a more omplex PL input onfiguration. s in the previous Fig. b a onstant generator ri metal summer DH S El»-» >> - PL input onfiguration: step-funtion generator X onfiguration, the input or its omplement drives a fixedoverflow iruit. However, in this ase, the output well drives the sense input of an inhibit iruit whih, in turn, produes the input to the metal summer. For example, in the input onfiguration driven by x, if x ^ b, all of the input harge resides in the well labelled b and none flows to the overflow well. Sine there is no harge there, the flow of harge from the soure well of the inhibit is not bloked, and the well labelled d fills to its apaity d. The metal summer, in this ase, reeives a logi d. Fig. b shows that the funtions realised are step funtions in whih the height and transition point are determined by well apaities d and b, respetively. f the input is applied unomplemented, a step-down funtion is generated, while a omplemented input produes a step-up funtion. lso shown in Fig. a is a onstant generator, whih is applied to the metal summer. Thus, the metal-summer output is the sum of step funtions and a onstant. s in the ase of the stairase-funtion generator, the value of the onstant is programmable and there is one onstant generator per olumn... Literal generator: The literal funtion in -valued logi [8] is the unary funtion = if a k^x k otherwise ny multiple-valued logi funtion f(x u x,..., x n ) an be realised as a sum-of-produts: The realisation of the literal funtion an be ahieved using two opies of the stairase-funtion generator, one with a diret input and the other with a omplemented input. The pair is alled a literal generator, and is shown in Fig.. The output funtion is the sum of an inreasing and a dereasing stairase funtion. f the two nonzero parts of the funtion do not overlap, the resulting sum is for a range of values of x. s it happens, this range orresponds to the range of values of x k, a k ^ x k ^ b k, where the literal funtion is. literal generator is used with an inhibit iruit at the olumn output to produe the produt term in the summation of eqn.. The proess by whih this is aomplished will be explained later... osts of input onfigurations: The basis on whih we ompare the various PLs is the relative ost Fig. a metal summer 7 N PL input onfiguration: literal generator fator desribed in Setion. The ost for eah input onfiguration is the sum of the osts of the omponent operators. s mentioned previously, the ost is primarily a measure of the hip area oupied. The stairase-funtion generator onsists of a multithreshold operator with a ost of. Sine omplementation is obtained so inexpensively, the stairase-funtion generator is viewed as having the same ost, regardless of whether or not the omplementer is present. The step-funtion generator onsists of one fixed overflow and one inhibit and has a total ost of. The literal generator onsists of two opies of the stairasefuntion generator and has a ost of 8.. Output onfigurations.. omplement: Fig. 5a shows the omplement iruit at the output of the PL olumn. This iruit realises y, where y is the input and is ordinary subtration. The omplement an be realised with a fixed overflow, inhibits, onstant generators, and an adder []. The dotted line in Fig. 5a shows that the metal summer from the input onfigurations an be used to diretly drive the PL metal summer. Thus, the user an hoose the sum of input funtions either diretly or after it has been applied to a omplement. lso, Fig. 5a shows a onstant generator, whih an supply a onstant to be summed at the PL output. This is p, the apaity of the onstant-generator output well, a parameter whih an be speified by the user.,..., X n ) _, u=l where a b = min(a, b), is a trunated (to ) arithmeti sum, p u e {,, }, and q is the number of produt terms []... nhibit: n Fig. 5b, the inhibit iruit is used at the olumn output. The output of the inhibit is either or/, where / is the apaity of the inhibit output well. is produed when the input from the summer is any nonzero value, and/is produed when the input is O./is 7 EE PROEEDNGS, Vol., Pt. E, No., JULY 987

4 determined by the PL user. s with the omplement, the user has a hoie, as indiated by the dotted arrow, of onneting the PL olumn diretly to the PL-output metal summer or to the sense input of the inhibit. of stairase funtions used to obtain the representations listed just below the triangle. Note that this value is the maximum of the differene between any two adjaent onstant generator from unary funtion generators from unary funtion generators onstant generator PL output Fig. 5 (ZZ a omplement h nhibit omplement y -y Two PL output onfigurations PL output". Unary -valued funtions realised at the output of one PL olumn We now onsider the merits of the above onfigurations by analysing the funtions produed by eah. s a basis of omparison, we onsider the set of all unary -valued funtions and determine the minimum number of input onfigurations needed to realise eah. Our riteria for omparison is the total number of onfigurations needed for all unary funtions. n all but one of the PL onfigurations presented, we assume that the user an apply any input to as many input onfigurations as needed. Thus, unlike onventional PLs, an input variable may our in more than one row. The one exeption is the literal generator, where we allow only one ourrene of any variable, a restrition whih has applied in past studies. n the analysis whih follows, it will be onvenient to represent a unary -valued funtion/(x) as a four-tuple <^a o a l a a y, where a { =/(/), for ^ i ^... Stairase-funtion generator with a omplement or diret onnetion to the olumn output (i): unary funtion <a a l a a^) is onstant if a = a l = a = a. unary funtion (a o a l a a y is inreasing (dereasing) if a.-i^a, («,-_! ^fl,-) for ^ / ^. unary funtion (a o a a a ) is a K-funtion if there exists an integer m suh that fl,_! ^ a h for all ^ i ^ m and a } ^a j+i, for all m ^ j ^, where ^ m ^. For example, <> is a K-funtion, and < > is a onstant, inreasing, dereasing, and K-funtion. There are six stairase funtions, <>, <>, <>, <>, <>, and <>. The sum of two or more stairase funtions or onstant funtions, is a V- funtion. The sum of two stairase funtions an be a K-funtion whih is neither inreasing nor dereasing, e.g. <> = <> + <>. omplete haraterisation of the funtions realised is given in Referene 7. For ompleteness, we desribe this briefly below. Fig. 6 below shows the whole or partial K-funtions obtained by summing whole or partial stairase or onstant funtions. The three triangles along the top represent omplete inreasing funtions. For example, <> is the sum of <> and <>. The seond and third row represent partial sums of inreasing funtions where the values assoiated with x = and < x <, respetively, are missing. The integer along the side represents the number EE PROEEDNGS, Vol., Pt. E, No., JULY 987 Fig. 6 Whole or partial sums of stairase and onstant funtions logi values. This observation will be useful later in ounting the number of stairase funtions used in speifi realisations. y symmetry, there is a set of partial and whole dereasing funtions whih have the same representations as in Fig. 6 exept for the rotation about the vertial axis. ny funtion whih is the sum of stairase or onstant funtions is a ombination of the funtions in Fig. 6 or their reverses. Table shows all possible ombinations of stairase and onstant funtions plus omplements of suh ombinations. There are five olumns orresponding to the values of x for whih /(x) is minimum. Shown in the heading of the olumn is the form of the realised funtions using representations appearing in Fig. 6. The seond olumn from the left shows the onstant funtions, the next three show inreasing and dereasing funtions whih are not onstant funtions, and the next two olumns show K-funtions whih are neither inreasing nor dereasing. The entries show how many funtions are realised as the sum of s stairase funtions and possibly one onstant funtion, with the value of s shown in the extreme left-hand olumn. For example, <>, whih is the sum of two stairase funtions is ounted in the entry '' of the olumn orresponding to a funtionvalue minimum loated at m = (th olumn from the right) and in the row orresponding to s =. The other two funtions ounted are <> = < > + <> + <> and <> = <> + <> + <>. The '()' beside the '' orresponds to the funtions whih are the reverse of the funtions ounted in the '', i.e. <>, <>, and <>. For eah olumn, the funtion-value minima are listed at the heading, with paranthesised funtion values orresponding to parenthesised entries in the table. The seond olumn from the right in Table shows the total number of funtions realised as the sum of s stairase funtions and possibly one onstant funtion. There are funtions for whih s =, the onstant funtions. There are 8 funtions whih are the sum of stairase funtions exlusively. These are either inreasing or dereasing funtions. t the other extreme, there are 7

5 Table : Funtions realised as the sum of stairase funtions and funtions whih are the omplement of suh sums ( and olumn PLs) P(s) Q(s) 5 6 () [] [()] () [()] () [()] () Number of funtions realised without omplement Number of extra funtions realised with omplement [5(5)] 5(5) [6(6)] 5(5) [6(6)] [] () [(D] [] [] [] [] [] m - value of x for whih f(x) is minimum s = number of stairase funtions used in the realisations P(s) = number of funtions requiring s stairase funtions (without omplement) Q(s) = number of funtions requiring s stairase funtions (with omplement) Underlined entries ontain funtions requiring a nonzero summer in its minimal realisation [] [] [Z] [6] [6] [8] [] [7] onstant applied to the output metal three funtions whih are the sum of 6 stairase funtions. These are <>, <OO>, and <O>. Summing more than 6 stairase funtions yields no additional funtions. t should be noted that only minimal realisations are ounted in Table. For example, while <> an be realised as the sum of stairase funtions, < >, <>, <>, and <>, a minimal realisation requires only stairase funtions and a onstant, <>, <>, <>, and <>. Thus, <> is ounted among the 5 funtions in the entry orresponding to the olumn m = and the row s =. n examination of Fig. 6 and Table shows that not all K-funtions are realised as the sum of stairase funtions. For example, <> is not inluded. The first two logi values,, an only be realised as the sum of two inreasing funtions <> and <>. However, the orresponding funtion would be <> not <>. n fat, any K-funtion with as a subpattern is not realisable. There are 5 other nonrealisable subpatterns. Observation: Let f(x) = ^a o a l a a } be a unary - valued funtion realised as the sum of stairase funtions and possibly a onstant funtion, where/(x) is in the set of K-funtions not ontaining forbidden subpatterns,,,,, and. The minimum number of stairase funtions used to realise/(x) is max (a,_ x a,) + max (a j+ i a j) ^ / < m m ^ j' ^ where m is a value of x for whih/(x) is minimum. The minimum number of stairase funtions required to form a given realisable K-funtion is the sum of the maximum differene between adjaent logi values along the dereasing part of the funtion and along the inreasing part. This follows from an earlier observation on partial stairase funtions. f we allow omplementation at the output of a metal summer, then additional funtions an be realised. For example, any K-funtion that is neither inreasing nor dereasing, produes an inverted K-funtion whih is neither inreasing nor dereasing when omplemented. Suh a funtion is not the sum of stairase funtions. n Table the number of funtions realised using the omplement is indiated by brakets []. Thus, in the two olumns orresponding to K-funtions whih are neither inreasing nor dereasing, some entries of the form i(i) or i are repeated in brakets, [i(iy\ or [i], respetively. omplements of inreasing (dereasing) funtions are deteasing (inreasing) funtions. The only additional funtions generated by omplementing suh funtions are those whih ontain a forbidden subpattern,,,,,, and. Table shows there are, in all, suh funtions. For example, <> ontains a forbidden subpattern, and thus is not the sum of stairase funtions and possibly a onstant. However, <> is the omplement of <>, whih is a dereasing funtion realisable as the sum of a single stairase funtion <> and the onstant <>. Thus, <> is an additional funtion only realisable beause of the omplement. Table shows that the total number of additional funtions obtained by using the omplement is 7. This is in addition to the 9 funtions whih are realised as the sum of stairase funtions and perhaps a onstant funtion. The omplement is neessary if all funtions are to be realised in a PL using a stairase-funtion generator. For example, <> is not realised as the sum of any 7 EE PROEEDNGS, Vol., Pt. E, No., JULY 987

6 stairase funtions, but it is realised as the omplement of suh a sum (<> + <> + <». Thus, the funtion <> itself is not realisable unless the omplement is available. ny funtion on n variables is realisable provided that there are enough olumns. This follows from the fat that any produt of a onstant and literal funtions (a term in the summation of eqn. ) an be realised by an appropriate hoie of stairase funtions, at most 6 for eah variable. For example, i xl is realised by summing the six stairase funtions <>, <>, <>, <>, <>, and <> eah with input x k. p u of eqn. is realised by hoosing a onstant to the metal summer equal to p u. The underlined entries in Table orrespond to funtions whih require a nonzero value for the output of the onstant generator at the PL output metal summer. For example, one funtion ounted in the entry [, ()] of the fourth olumn ( ^ m ^, ( ^ m ^ )) and the third row (s = ) is <>, with the realisation, <> = <> + <> + <llll>. n all, there are suh funtions. The other 9 are = <> + <> = <> + <> + <> - = <) + <) + <)- <> = <> + <> + <> + <> <> = <> + <> + <> + <> = <> + <> + <> - = <> + <> + <> - = <> + <> + <> + <> + = <> + <> + <> + <> + The onstant is neessary in the realisation of these funtions. Without it, they would be unrealisable in a oneolumn PL... Stairase-funtion generator with an inhibit or diret onnetion to the olumn output (ii): f an inhibit is used instead of the omplement at the PL olumn output, a different lass of funtions is realised. The additional funtions, shown in Table, produe two Table : Funtions realised as the sum of stairase funtions applied to an inhibit Funtion ondition U(s) V(s) = < < = <> < << _ ^~ ^ ^ < Number of funtions improved with inhibit Number of extra funtions realised with inhibit 5 = number of stairase funtions used in minimal realisation U(s) = number of funtions whih are realised as a sum of stairase funtions without inhibit, but whih are also realised with fewer stairase funtions using an inhibit V{s)= number of funtions requiring s stairase funtions (with inhibit) 8 output logi levels only. The entries orrespond to the ^a o a l a a ) notation for unary funtions. For example, the first row of the olumn headed by 'Funtion',, orresponds to the unary funtion <a o a i a a > = ( }. The third olumn speifies the range of values for and. For this funtion = and ^ <. Thus, represents the two funtions <> and <>. n addition to the 9 funtions whih are sums of stairase funtions, the inhibit produes additional funtions, less than one half the number of additional funtions produed by the omplement. However, unlike the omplement, the inhibit improves on the funtions whih are the sum of stairase funtions. For example, <> an be realised as the sum of three stairase funtions «> = <> + <> + <», but requires only one when the inhibit is used «> = inhibit ). There are altogether funtions whih have a better realisation when the inhibit is used... Step-funtion generator with a omplement or diret onnetion to the olumn output {Hi): Table + () shows the funtions realised by summing step funtions and onstants. The funtion-entry notation is idential to that of Table. ompared to the 8 realised by the sum of one stairase funtion and possibly a onstant funtion, there are 6 funtions realised as the sum of one step funtion and possibly a onstant funtion. t is interesting to note that the sum of more than step funtions and possibly a onstant produes no more funtions than are realised by summing or fewer funtions. Reall that for stairase funtions, up to 6 may be summed to produe a distint funtion. n all, 56 funtions are realised as the sum of step funtions, approximately 5% more than the 9 realised as the sum of stairase funtions. However, the input onfigurations required to realise the step funtion is more omplex than for the stairase funtion. lso shown in Table are funtions whih are omplements of the sums of step funtions and possibly a onstant funtion. s with the nonomplemented funtions, the form of eah realisation is given. rakets [] are used to enlose the number of funtions realised when a omplement is used. n all, 6 additional funtions are produed. Thus, the total number of funtions realised by the omplement operation at the olumn output is 8 using step funtions or 6 using stairase funtions at the input. s with the stairase funtions, the omplement is neessary in order to realise all funtions. For example, neither <> nor <> is realised as a sum of step funtions, but both are the omplement of the sum of step funtions and a onstant (e.g. <> = <OO> + <> + <>). Given enough PL olumns, any -valued funtion an be realised... Step-funtion generator with an inhibit or diret onnetion to the olumn output (iv): Table also shows EE PROEEDNGS, Vol., Pt. E, No., JULY 987 7

7 Table : Funtions realised as the sum of step funtions and funtions whih are the omplement or inhibit of suh sums ( and olumn PLs) s Funtion ondition R(s) S(s) Us) R(s) S(s) T(s) ^< <^ 6 6 [] {} <^ < <, s$ 78 [] {} ~~ ool ool same, exept = :+ < = '\:-= [] = : <> ^ -./- {} /"" D O + + << same, exept i d ^+ [] 8 [] {} same, exept = == [] ~ ~ + ++ = =, = = [] Number of funtions realised without omplement or inhibit Number of extra funtions realised with omplement [6] [6] Number of extra funtions realised with inhibit {} {} s = number of step funtions used in the realisations R{s) = number of funtions requiring s step funtions S(s) = number of funtions requiring s step funtions (with omplement) T(s) = number of funtions requiring s step funtions (with inhibit) the additional funtions realised when an inhibit is used at the output of the PL olumn. The number of additional funtions is, as shown in braes {}. Thus, the inhibit produes about one fifth of the additional funtions produed by the omplement. gain, any -valued funtion an be realised using this ombination of input and output onfigurations...5 Literal generator with an inhibit at the olumn output and no onstant generator {v): literal generator onsists of two stairase-funtion generators, one with x f on the input well and the other with -x k. The funtion produed at the PL olumn is the arithmeti sum of a stairase-up and a stairase-down funtion. f the two funtions are for a range of x k, a k ^ x k ^ b k, the sum is for that range. Thus, if n opies of the literal generator are applied to the summer olumn, eah with x f applied at the input for ^ k < n, the olumn has harge if and only if eah x f is bounded as a k ^ x f ^ b k. With an inhibit at the olumn output, the inhibit output will be /, the logi apaity of the inhibit output well, if and only if a k ^ x f ^ b k for all ^ k ^ n. The output funtion is. ai v bi. ^... <»n Y bn / whih is one produt term in the sum-of-produts expression of eqn.. The metal summer to whih the inhibit output is applied, realises the sum of eqn.. We have hosen not to use the onstant generator at the PL output, sine the produt of literals formulation for a 7 EE PROEEDNGS, Vol., Pt. E, No., JULY 987

8 Table : Number of funtions realised using the fewest input onfigurations against the number of olumns and number of input onfigurations. Number of olumns Number of input onfigurations omplement or diret Output onfiguration inhibit or diret nput onfiguration inhibit only stairase funtion step funtion stairase funtion step funtion literal without onstants o o o o Total number of funtions Total number of input onfigurations Total ost of input onfigurations multiple-valued funtion without a onstant given in eqn. is ommon, and thus serves as a basis for omparison with previous formulations. For the same reason, no onstant generator is part of the olumn driven by input onfigurations. Sine eah olumn output is summed on the metal summer whih serves as the PL output, the PL realises a sum-of-produts involving literals with Sum being normal addition trunated to when the sum exeeds. s long as there are suffiiently many olumns (produt terms), any n-variable -valued funtions an be realised []. The number of funtions realised at the output of one olumn is, sine there are literal funtions for eah of nonzero logi levels and the onstant funtion <>. The advantage of the literal generator over the stairase-funtion generator with the inhibit at the olumn output is in the additional logi available to the user of the PL. With the literal generator, both halves of the pair of stairase-funtion generators are inluded whether both are needed or not. With the stairasefuntion generator, however, only one half of the pair will be used when only one half is needed. Thus, the latter PL makes more effiient use of hip area. This will be disussed in Setion..5 Funtions realised by PL s with more than one olumn n this setion, we analyse the -valued unary funtions realised by PLs with one, two, or more olumns, whih are in one of the five onfigurations disussed above. program was written whih enumerates the unary funtions aording to the number of input onfigurations and olumns required. Starting with funtions realised by a single olumn, all traditional unary funtions realised with olumns were generated, then olumns, et. t eah step, the realisation requiring the least number of input onfigurations is retained. The results are shown in Table. The right five olumns represent the number of unary funtions realized by the following PL onfigurations : stairase-funtion generator/omplement or diret step-funtion generator/omplement or diret stairase-funtion generator/inhibit or diret step-funtion generator/inhibit or diret literal generator/inhibit always The top two setions of rows orrespond to PLs with or olumn and the data here agrees, as it should, with Tables, and. t is interesting that the onstant funtions <>, <>, <>, and <> are ounted in the funtions whih require one olumn, sine with the literal generator, there is no onstant generator at the PL output. t should be noted that the data represent a minimisation of the total number of input units, not olumns. That is, the 8 funtions using olumns in the minimal realisations assoiated with stairase-funtion generator/ omplement or diret an all be realised in suh PLs using only olumns. However, with olumns, there is EE PROEEDNGS, Vol., Pt. E, No., JULY

9 a realisation whih requires fewer input units overall than with any -olumn realisation. With this one exeption, there are no funtions for whih there is an advantage to using more olumns than the minimum, in any of the three onfigurations. Observation: The number of olumns neessary to realise any unary -valued funtion in a PL using either the stairase- or step-funtion generator and possibly a omplement at the olumn output is at most, while in a PL using the literal generator and an inhibit at the olumn output is at most. onluding remarks The next to last row in Table shows the total number of input onfigurations needed to realise all 56 -valued unary funtions. t shows that stairase-funtion generator/omplement or diret requires signifiantly more input onfigurations, 896, than any of the other onfigurations, eah requiring about 6 onfigurations eah. However, onsidering a ost whih reflets hip area oupied, the stairase-funtion generator/ omplement or diret is very good, ranking nd out of 5. The main results of this paper are summarised in the last row of Table. t shows the total ost (hip area) of the input onfigurations required to realise all unary funtions. Eah entry is derived by multiplying the number of required input onfigurations in the row just above by the ost of eah onfiguration. n this omparison, the stairase-funtion generator/inhibit or diret requires the least hip area, with stairase-funtion generator/omplement or diret seond best, onsuming about 5% more hip area. The two ombinations using the step-funtion generator are extremely ostly by omparison. s disussed earlier, the step-funtion generator is the same as the stairase-funtion generator exept that an inhibit has been added. From this analysis, it is lear that the extra iruit omplexity of the inhibit does not inrease the number of realised funtions suffiiently to offset the extra hip area. s disussed earlier, the literal-funtion generator onsists of two opies of the stairase-funtion generator. We observed that for some literal funtions, one of the two opies is not needed. From Table, it an be seen that it uses almost twie the hip area of the stairase-funtion generator/inhibit or diret, suggesting that there is signifiant waste in the use of the added stairase-funtion generator. The main onlusion we reah is that the stairasefuntion generator together with the inhibit at the olumn output is signifiantly better than the other ombinations. Thus, an important problem is the development of an effiient synthesis tehnique for this PL design. 5 knowledgment The authors aknowledge valuable omments by three referees of this paper. This work was funded by NTO Grant /8. 6 Referenes KERKHOFF, H.G.: 'Theory, design, and appliations of digital harge-oupled devies'. Ph.D. Thesis, Twente University of Tehnology, Enshede, The Netherlands, pril 98 TRUML, P., and UTLER, J.T.: 'On the realization of multiple-valued logi funtions using D PL's'. Proeedings of the th nternational Symposium on Multiple-Values Logi, Winnipeg, Manitoba, anada, May 98, pp. - KUO, H.-L., and FNG, K.-Y., 'The multiple-valued programmable logi array and its appliation in modular design'. Proeedings of the 5th nternational Symposium on Multiple-Valued Logi, Kingston, Ontario, anada, May 985, pp. -8 MME, M., and PPHRSTOU,..: 'Simplifiation of MVL funtions and implementation via a VLS array struture', ibid., pp SSO, T.: 'n algorithm to derive the omplement of a binary funtion with multiple-valued inputs'. EEE Trans., 985, -, pp. - 6 ENDER, E., UTLER, J.T., and KERKHOFF, H.G.: 'omparing the SUM with the MX for use in four-valued PLY. Proeedings of the 5th nternational Symposium on Multiple-Valued Logi, Kingston, Ontario, anada, May 985, pp KERKHOFF, H.G., and UTLER, J.T.: 'Design of a high-radix programmable logi array using profiled peristalti harge-oupled devies'. Proeedings of the 6th nternational Symposium on Multiple-Valued Logi, laksburg, Virginia, US, May 986, pp LLEN, M., and GVONE, D.D.: ' minimization tehnique for multiple-valued logi systems', EEE Trans., 968, -7, pp EE PROEEDNGS, Vol., Pt. E, No., JULY 987