Pávó, I.

Vágó, I.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

ANALYSIS OF COMBINATIONAL NETWORKS BY TOPOLOGICAL METHOD By I. VAGO and I. PAVO Department of Theoretical Electricity. Technical University. Budapest Received: June 16. 1981 Introduction In 1958 at the Namur International Congress of Cybernetics. Professor Kalmar submitted a new construction for logical networks [1]. In his system the logical values are represented by special conductivity states between three poles while the two-argument Boolean operations can be realized by pure wire-boxes. The logical values and the general wire-box are illustrated in Fig. 1. Remark that in the construction by Kalmar the negation is represented by reversing the three-poles corresponding to the logical variable and the general wire-box has two isolated three-terminal plugs for the right-argument operation. Generally. one of the two-terminal plugs can be omitted, except in cases of equivalence and anti valence. Fig. 2 shows ten various wire-boxes associated with the two-argument logical operations. x, X2 x 2 o "false:' r ~ l p.~ ( 0 11 * 4 v 0 I ~ ~ . ( 0 it; i o "true", • logical values o ~ X ,*x2 o • i general operation (wire -box) Fig. 1 Xi x 2 1* 1 .) I according ? t ! to the ~ • .. operations , I 172 "---.---x < x c;> 0 ,--- --0----+--: I I r-'" x :!~i 666 ~ x x G. X > ~I I I ' i I ! i 90 (, lli c o GJiS ~u c;= >0.. cE c o o U 0.. E c o u C :J c o u c o u C :J .~ "Cl I. ~"AG6-1. p,.fl'6 x 1< 999 1-, , 1 ~ , ! : i i ~66 d66 '--..-' \~ X X X' I> X r--o" 0 I I ,----J I ' c o .0 .s:::. c c 0 .-.0 L c c , .:;. "-ASALYSIS OF COMBlc\ATlOSAL SETWORKS 173 Consider a logical formula of 11 variables constructed by negation and any of the .two-argument Boolean operations indicated in Fig. 2, and realize this forml}la by Kalmar boxes. Let this circuit be called Kalmar's network. Among others, the end value of a logical formula realized by Kalmar's network can be displayed on the Szeged Logical Machine [2J after choosing the input logical values of the variables. So this machine suits to establish the truth table of a logical formula. In this paper a topological method is presented for the analysis of the Kalmar's network giving a disjunctive normal form of a logical formula. This method can be used for digital analysis as well. Since an arbitrary Boolean function can be realized by Kalmar's network, the present method can be regarded as a topological procedure for the analysis of combinational networks. A property of the Kalmar network Let us consider a Boolean function F(x l' ... , XII) of 11 variables, negation and ten two-argument Boolean operations (Fig. 2). with a suitable Kalmar's network assigned to it. Terminate the input three-poles of the network by short-circuits in accordance with the actual truth values of the variables. Such , termination of the network is called an allowed termination. The following statement is valid: A Kalmar's network completed by an allo\ved termination is always a 2-tree. The middle and exactly one of the outer points of the output three-poles belong to the same component of the 2-tree. This property can be proved by inducation on the number of operations in formula F. If the formula contains only one operation the statement is trivially valid. Let us suppose that the number of the operations in the formula is n ~ 2. and the statement is valid to formulas containing less than n operations. Without violating generality it can be supposed that the first symbol of the formula is not a negation and F contains at least one two-argument operation. Hence: (1 ) where Kalmar's networks concerning formulas.1; and.1~ fulfil the induction hypothesis and * denotes a two-argument operation. Let us consider the Kalmar's netw'ork associated with the operation denoted by *. Completing the original Kalmar's network of F by an allowed termination the network associated with the operation * is completed by an allowed termination as well. This termination can be simulated by suitable short-circuits on the inputs of the network *, the graph of which is in fact a 2-tree. After this, exchange these short-circuits with the suitable parts of the 174 I. ~AG6-1. P.4V6 terminated Kalmar's network in that order corresponding to the simulation. So partly the original terminated Kalmar's network is obtained, and partly, the former graph contains neither circuits nor new components, which completes the proof. This property is a trivial proof that Kalmar's networks are always a k-trees. Derivation of the output function This property of Kalmar's network offers a direct possibility to derive the output function of the combinational network. Let us consider all possible allowed terminations of the Kalmar's network, and fix a termination.lfthe output function has a logical value "true" then there is a short-circuit between the middle and the lower poles of the output three-pole, otherwise there is an open-circuit. Since in the case of an output logical value ·'true", the upper point of the output three-pole belongs to one of the tree components. this point can be omitted from the network together \vith its tree component. Now there is a single path connecting the remaining output points. A conjunction can be constructed from the input logical variables (or their negations) corresponding to the series of short-circuits in the termination. Taking all the allowed terminations into consideration the disjunction the conjunctions formed along all the possible paths is really a disjunctive normal form of the output function. Repeating these considerations for the middle and top points of the output. to the sense. the negation 'of the output function is constructed. In order to take all the allowed terminations into account. terminate the input three-pole ofthc jth input variable by admittances ,Xj above and Xj belo\v so that the possible values of the admittances are 0 or 1. and .Xj * x j . Consider the graph of the network terminated in this manner. Omit from this graph the upper point of the output three-pole together with the connected input points and the admittances fitting them. Let us define a graph. called reduced graph. the edges of which are merely admittances. and the vertices are points of the abow network short-circuited by \vires. Topological formula for the output function: (2) - -'Yi . x,i = x j' where Pi(.X!, ...• xn) is a Boolean product of admittances formed along the ith path connecting vertices associated to the output points (output vertices) in the reduced graph. and the Boolean summation (2) affects all possible paths. ASALYSlS OF COMBlSATlOSAL SETWOR/(S 175 Examples 1. Let us construct the Kalmafs network shown in Fig. 3 from antivalence and conjunction boxes. For producing a disjunctive normal form of the output function let us omit output point 13 together with points 4. 7 and 12 which are connected to it (See Fig. 4). Terminate the remaining network inputs by the proper admittances. Fig. 5 shows the reduced graph. Now there are two paths betw'een the vertices associated with the output points 14 and 15. By the earlier convention for the admittances and this expression has but one disjunction term .\"5: logical the formula obtained is a conjunction. namely F(x . .1') = (.\"+t+y) /\.\" = .\} . r 7 o-t-+~ y < 8 l 9 . L, 0f---------j-...J X , 5 O~-----J • 6 0>----------' Fig. 3 x x 5 6 'i 9 Y Fig . .f 176 /. VAGO-I. PAVO Fig. 5 ; 1 I 2 ° i I. 3 L. b i 5o-J I I l6~ I i 7 I I lJ:16 b 1~~ LL;17 F (o,b,c,d) I 18 I I I C 11 I 12 ' 13~ i j d 1L.~ 15 Fi!!. () 2. Consider the Kalmar's network in Fig. 6. Now it is more practical to write the negated output function, for omitting point 18 yields a simpler reduced graph than omitting point 16. Fig. 7 shows the reduced graph for establishing the formula F(a. b. c, d). There are two paths between the vertices associated with 16 and 17, but both provide disjunctions, so we have: F(a, b. c, d)=dcba+dcba. ASALYSIS OF COMBISATIOAAL AETII'ORJ.;S 177 b a Fig. 7 Computer analysis A simple flo\v chart of the procedure necessary for the analysis is illustrated in Fig. 8. For cOJ)1puter implementation it is most essential to generate the edge admittance product Pi(."': 1- ... _ .\11) formed along the path connecting the output vertices. The generation is possible in several \vays. The results of applied graph theory can be used to search the paths. The problem can be reduced to tree Kalmar network I reduced graph i = 1 F.=O Fig. 8 termination x1' ,x n VI1 )=1 n: =1 i: =1 s~y ,. ! Vln)::s I. V.-iGO·-I. P.4J·O I I L, i etul ~ I ~--r------~~---. e Fig. 9 generation of a graph since any path between the output vertices can be included into a tree of the reduced graph. For such a generation the method presented in [3] may be applied. In this case it is very practical to take the vertex associated with the middle point of output as the negative root. So the cycle check starting from the outside vertex. part of the complete cycle check. gives exactly the searched path. This method is very elegant but from the point of view of computer technique it has the inconvenience that a path can occur more than once. so that finally the equivalent terms of the disjunctive normal form have to be contracted. There is a possibility to produce a single copy of each path wanted from the modified adjacency matrix of the reduced graph [3]. For this purpose let the outside output vertex be marked by I and the middle one by m in the reduced graph, where m is the number of vertices. Starting from the first row of the adjacency matrix, all paths leading from I to m can be obtained one by one. A,\'ALl"SIS OF COMBINATIONAL A'ETWORKS 179 A flow chart is seen in Fig. 9 for the construction of a computer program. The input data ofthe program are elements Clij of the modified adjacency matrix of size m x m while the output is the actual vector Vof size /1, the components of which are the vertices of the path (/1 is the number of the vertices in the path. wanted IJ ~m). For the conjunction of the admittances the pairs of the vertices may be produce<;l from the elements of V in suitable order. Summary In this paper combinational circuits are realized by Kalmilr"s network, The Kalmar"s network completed by an allowed termination is shown to be always a 2-tree. so a topological formula can be constructed for the output Boolean function of a combinational network. Simple examples are giwn on writing the output function by this method. and finally a flow chart for computer technique is presented, References I, KALI>!..\R. L: A new principle for construction of logical machines: Proc. of The 2nd International Congress of Cybernetics. Namur. 1958. pp 458-463. 2. NE~IES. T.: Cybernetic Machines (in Hungarian): PubL House of the HAS. Budapest. 1962. pp 93--95. 3. P . \vo. L: Generation of the k-trees of a graph: Acta Cybernetica. Szeged. Tom. I. Fasc. 2. 1971. pp. 57-68. Prof. Dr. Istvan VAGO. Budapest H-1521 Dr. Imre PA VO, Szeged, 6720, Somogyi ut 7. 

ANALYSIS OF COMBINATIONAL NETWORKS BY TOPOLOGICAL METHOD

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ANALYSIS OF COMBINATIONAL NETWORKS BY TOPOLOGICAL METHOD

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Periodica Polytechnica (Budapest University of Technology and Economics)