UNIVERSITY OF ILLINOIS LIBRARY At urbana-champaign Digitized by the Internet Archive in 2013 http://archive.org/details/programmanualfor887hukc L-&( jr uiUCDCS-R-77-887 "TfldM UILU-ENG 77 17^3 PROGRAM MANUAL: FOR THE NETTRA SYSTEM % August, 1977 BY K. C. Hu DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS UIUCDCS-R-77-887 PROGRAM MANUAL: FOR THE NETTRA SYSTEM by K.C. Hu August 1977 Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 61801 This work was supported in part by the National Science Foundation under Grant No. NSF DCR 73-03421. Ill ABSTRACT This is the program manual for the NETTRA system which can design near-optimal, multiple-output, multi-level and loop- free NOR(NAND) networks under fan-in/fan-out restrictions and/or level restriction. Given function(s) may be completely or incom- pletely specified and both complemented and uncomplemented external variables are permitted as inputs. The user can specify the control sequence (the types of the initial network methods and the types and the order of the transduction procedures to be applied) to solve his problem. Besides, four control sequences are provided for the users who are not interested in the details of how to specify the control sequence. Facilities for treating unfinished jobs due to the ex- piration of computation time are also provided by the system. XV ACKNOWLEDGMENT The author is grateful for valuable discussions with Professor S. Muroga, H.C. Lai, J. Culliney and, in particular, Professor Muroga 's reading of the manuscript, The nice typing job done by Mrs. R. Taylor is also acknowledged. TABLE OF CONTENTS Page 1. GENERAL DESCRIPTION OF THE NETTRA SYSTEM 1 1.1 Outline of the system 1 1.2 Initial network methods 8 1.3 Transformation and transduction procedures 11 1.4 Control sequences 14 2. SET-UP OF INPUT DATA 16 3. RESTRICTIONS ON PROBLEM SIZE 30 4. EXAMPLES OF INPUT DATA SET-UP 31 5. OUTPUT FORMATS 39 6. INTERMEDIATE RESULTS FOR THE UNFINISHED JOBS 95 REFERENCES 102 1. GENERAL DESCRIPTION OF THE NETTRA-SYSTEM The NETTRA system(NETwork TRAnsduction system) is an effective tool for designing near-optimal, multiple-output, multiple-level and loop- free NOR (NAND) -gates networks under fan- in/ fan-out restrictions and level restriction. "Transduction" means "transformation" and "reduction". The pur- pose of network transductions is to reduce the cost of a network designed for a certain function (or functions) or to reconfigure (or transform) the network in such a way as to allow another transduction to eventually ac- complish the cost reduction. This cost, C, of a network is formally de- fined asAxR+BxI, where R is the number of gates, and I is the number of connections in the network; A and B are arbitrary non-negative weights. Different combinations of A and B give different cost criterion for the transduction. 1-1 Outline of the NETTRA system The NETTRA system can treat the following four types of problems: (1) Find near-optimal networks for the given function (s) under no fan-in/fan-out restrictions or level restriction. (2) Find near-optimal networks for the given function(s) under only fan-in/fan-out restrictions. (3) Find near-optimal networks for the given function(s) under only level restriction. (4) Find near-optimal networks for the given function (s) under both fan- in/fan-out restrictions and level restriction. Here by "under fan-in/fan-out restriction" we mean that the maximum fan-in of each gate and/or the maximum fan-out of each external vari- able or each gate in the network are restricted; by "under level re- striction" we mean that the maximum number of levels in the network is restricted. For each type of problem, the NETTRA system first produces the initial networks for the given functions by using some conventinal design methods [1] . The initial networks usually have many redundant gates and/or connections, although they can be produced in a very short computation time. So appropriate transformation and/or transduction procedures are applied by the NETTRA system to simplify the initial networks. For type (1) problems, the NETTRA system processes networks ac- cording to the flowchart shown in Fig. 1.1-1. In Fig. 1.1-1, after the initial network has been derived, the transduction procedures are applied, without considering fan-in/fan-out restrictions, to simplify the initial networks as much as possible. For type (2) problems, the corresponding flowchart is shown in Fig. 1.1-2. After obtaining the initial networks, the transduction procedures are applied to simplify the initial network first without considering the fan- in/fan-out restrictions. The simplified network has lower cost, but it may not satisfy the given fan- in/fan-out re- strictions. Therefore the transformation procedure is applied to transform the network into fan- in/fan-out restricted form. Then the transduction pro- cedures are applied again, considering the fan- in/fan-out restriction, to simplify the network as much as possible. The resulting network obtained at this point is a feasible solution for the given problem, i.e., a network which satisfies the given restrictions. But we can apply the non-f an-in/non- Start Find an initial network Apply a transduction procedure under no fan- in/ fan-out restriction ► 1 r Apply a transduction procedure under no fan-in/ fan-out restriction \ i C Stop J Fig. 1.1-1 Flowchart for solving type (1) problems. These loops will be executed repeatedly until there is no further improvement in the network cost. ( Start J Find an initial network H« Apply a transduction procedure under no fan-in/fan-out restriction 1 Apply a transduction procedure under no fan-in/ fan-out restriction Apply the fan-in/fan-out restricted transformation procedure 1 Apply a transduction procedure under fan-in/fan-out restrictions i Apply a transduction procedure under fan- in/fan-out restrictions ( Stop J Fig. 1.1-2 Flowchart for solving type (2) problems These loops can be executed repeatedly until there is no further improve- ment in the cost. fan-out restricted transductions, fan- in/fan-out restricted transformation, and fan- in/fan-out restricted transductions repeatedly in order to get a more simplified network. The flowchart for type (3) problems is shown in Fig. 1.1-3. First the NETTRA system finds an initial network whose number of levels is less than or equal to the specified limit, and then the trans- duction procedures are applied, considering the level restriction, to sim- plify the initial network as much as possible. The flowchart for type (4) problems is shown in Fig. 1.1-4. The NETTRA system tries to find an ini- tial network which satisifies the level restriction and has as less fan-in/ fan-out problems as possible. The transduction procedures will, then, be applied to simplify the initial network, considering the fan-in/fan-out re- strictions and level restriction. During the transductions, the number of levels of the network will never exceed the limit, and some fan-in/fan-out problems may be solved. If all fan-in/fan-out problems can be solved, then a feasible network has been obtained. If the transductions cannot solve all fan- in/fan-out problems, then no feasible network is obtained by the ap- proach shown in Fig. 1.1-4 even if there do exist feasible solutions for the given problem . In Fig. 1.1-1 through Fig. 1. 1-4, those loops marked with aster- isks can be executed as many times as the users wish. Besides, the users can specify the types of initial network methods and the types and the order of the transduction procedures to be applied to solve the problems according to If the fan- in of a gate exceeds the limit, then we say there is a fan-in problem at that gate. The fan-out problem of a gate or an external vari- able is similarly defined. t There may not exist feasible networks for a given problem if both fan-in/ fan-out and level restrictions both are imposed. ( start ) Find a level-restricted initial network Apply a transduction procedure under level restriction only Apply a transduction procedure under level restriction only ( st ° P ) Fig. 1.1-3 Flowchart for solving type (3) problems. These loops can be executed repeatedly until there is no further improvement in the cost. Start 1 f _J_ Find a level-restricted initial network Apply a transduction procedure under fan- in/fan-out restrictions and level restriction Apply a transduction procedure under fan-in/fan-out restrictions and level restriction f Stop J Fig. 1.1-4 Flowchart for solving type (4) problems. These loops can be executed repeatedly until there is no further improvement in the network cost. 8 their specific purposes. The general tendency of effectiveness and efficiei; of different transduction procedures and the characteristics of different ±\[ tial network methods will be given in the following sections. 1-2 Initial network methods Six different methods are implemented in the NETTRA system to proc initial networks. They are: 1. Universal NOR network method. 2. THREE-level network method. 3. Branch-and-bound method. 4. Tison's method. 5. Gimpel's algorithm. 6. Level-restricted initial network method. Among them, methods 1, 2 and 5 can produce networks with uncomple- mented external variables as inputs. Methods 2 and 5 produce only three-levl networks, and method 4 produces two-level or three-level networks depending n whether both uncomplemented and complemented external variables or only uncom- plemented external variables are available. Method 6 is for designing the lm restricted initial network when both fan-in/fan-out restrictions and level r- striction are imposed. Table 1.2-1 gives the mnemonic symbols for six initial network methods; it also shows what initial network methods can be used for what typ of problems (mentioned in section 1.1). The mnemonic symbols will be conve- nient for setting up data cards. In [1] many experiments have been conducted to find out the influ- ence of initial network methods upon the final results and to compare the computation time that different initial network methods spend. Table 1.2-2 CO B 1 0) C rH O X ■H O 4J M CO ex CU 14-1 U O CU CO x: o> 4-J (X >> Xl 4-1 C co ^ (1) H C O & ^H 4J co a) • •H £3 ^■N 4-> H •H rH • C CO H •H -H 4-1 C S-4 -H o o c •H 14-4 -H 4-J CJ CO CU -• H rH X X X X X U c/3 H hJ S3 O O PQ 2 2 > H 5 PQ J3 CO M H > w rJ |Z> H PQ H H H 2j C/3 X) o XI 4-1 CU E ^ M o o H c/1 3 O < w H 2 H l_4 XI O XI 4-1 CU CI 4-1 CU B O X! 4-1 CU XJ o rH CO •H u o B r-l X ■u CU B B 4-1 •H e •rH ■u O X 53 i — i CU S T3 4-1 XI C 4-J CI •H CU X) r4 4J pej CI O o o O O X X 60 •H 53 rH 1 4-1 rH S-l CU 13 CI E CO CO CU CI o a CU •H S-4 CO CO B > CI X u •H •H CU £> H PQ H o rJ CO B •H cu rH 4-1 CO CO rH •H U cu CU > 4-1 c CU •H CU rH E 60 •H rH O 1 rH 4-J B •H CU -o x > CU cO CU rH CO 3 E CU a; CU x. X X 4-1 4-1 o 60 £ CO CI CU r-i •H X cfl 4-1 £ 4-1 CI CU >-, CO CO rH CJ a ^ o X) o x xi X a: CU 4-1 r-l CO CU o 3 E 5 4-1 0) J* 01 X r-l o CI c s rH CO 4-1 CO CJ 0) •H c 4-1 CO •H Xi rH c o • CO •H X m •H 4-J 4-1 X3 OJ o •H CU E 4-1 c 4-1 •H CJ ^ r-{ •H h cO X) U o 3 cu 4-1 s cr 4-1 CO 4-J CU CJ CU CU •H r-l c !-4 r4 O 4J 4J rH CO 3 CO C CU O •H CO u 1 4-) X 1 CI •H 4-1 rH CO c CU 14H •H U > 1 — -. • (U CU c o CU 4-1 <-\ •H O CO CO 1 rH CU 1 mm t :• cs = centisecond min = minute Usually the initial networks obtained by Tison's method have lower costs tha; those obtained by other methods. But it has been seen that starting from the initial networks obtained by the branch-and-bound method, better final result, can usually be obtained. This is because the initial networks obtained by th> branch-and-bound method are usually multiple-level networks, and hence are more suitable for the transduction procedures to reconfigure. The initial networks obtained by Tison's method are restricted to be two-level or three- level, and hence are more difficult to reconfigure even though they have lowei 11 costs. The initial networks obtained by three-level network method are al- so restricted to be two-level or three-level; since they usually have higher costs than the initial networks obtained by Tison's method, the corres- ponding final results are worse. The initial networks obtained by the universal network method are multiple-level, but they usually contain too many gates and connections and hence are more difficult to simplify. 1-3 Transformation and transduction procedures In the NETTRA system, there is only one transformation procedure which can transform the network from non-fan-in/non-fan-out restricted form into fan-in/ fan-out restricted form. This transformation procedure is used for solving problems under only fan-in/fan-out restrictions, i.e., for solv- ing type (2) problems. The mnemonic symbol for this transformation proce- dure is JEFF. The transduction procedures can be classified into the following five groups according to their characteristics and capabilities: 1. Pruning procedures. 2. Procedures based on gate merging. 3. Procedures based on gate substitution. 4. Procedures based on connectable and disconnectable functions. 5. Procedures based on error-compensation. Table 1.3-1 gives the names and the mnemonic symbols of the cen- tral FORTRAN subroutines for realizing each group of procedures. The mnemonic symbols are originated from the transduction programs NETTRA-PG1, -Pg2, -PI, -P2, -Gl, -G2, -G3, -G4, and -El [1]. They are convenient for specifying input data cards. 12 Table 1.3-1 Central subroutines and their mnemonic symbols of the trans- duction procedures. TYPE OF TRANSDUCTION PROCEDURES NAMES OF CORRESPONDING CENTRAL SUBROUTINES MNEMONIC SYMBOLS PRUNING PROCEDURES MINI 2 NPGM RDTCNT NTP1 PROCI NTP2 PROCEDURES BASED ON GATE SUBSTITUTION SUBSTI NPGS PROCV NTG4 PROCEDURES BASED ON GATE MERGING GTMERG NTG3 PROCEDURES BASED ON CONNECTABLE AND DISCONNECTABLE FUNCTIONS PRIIFF NTG1 PROCIV NTG2 PROCEDURES BASED ON ERROR-COMPENSATION PROCCE NTEl Many experiments have been made to find out the effectiveness and the efficiency of the transduction procedures. The result is shown in Fig. 1.3-1; the detailed comparisons are given in [1]. Notice that all transduc- tion procedures except NTG4 can do transductions without violating the fan- in/fan-out restrictions and/or level restriction if the network before apply ing transductions is already fan-in/fan-out restricted and/or level-restrict i NTG4 is not included in Fig. 1.3-1, but usually it is less effective but mor time-consuming than NPGS in doing transductions under no fan- in/fan-out re- strictions and no level restriction [1]. 13 NTP1 NPGM NTP2 NPGS NTG3 NTG1 NTG2 NTE1 less effective more effective NPGM NTP1 NPGS NTP2 NTG3 NTG1 NTG2 NTE1 less time-consuming more time-consuming Fig. 1.3-1 General tendency of the effectiveness and the efficiency of the transduction procedures. (NTG4 is not included.) 14 1-4 Control sequences A control sequence consists of one or more initial network meth- ods and a TT-sequence (Transformation and Transduction sequence) , where each TT-sequence consists of the types, the order and the numbers of execu- tion times of the transduction procedures to be applied. Starting from the initial network obtained by one of the initial network methods, the TT- sequence will be applied to simplify the network as many times as users specified. The users can design control sequences by choosing appropriate initial network methods and the transduction procedures according to Table 1.2-1, Fig. 1.3-1 and the statistics in [1]. Besides, for users' conve- nience four control sequences, OPTION 1 through OPTION 4, are provided in- ternally for solving type (2) and type (4) problems. Since type (1) and type (3) problems are special cases of type (2) and type (4) problems, re- spectively, these control sequences can be used to solve type (1) and type (3) problems too. The contents of these built-in control sequences are shown in Table 1.4-1. Control sequences OPTION 1 through OPTION 3 are for designing fan-in/ fan-out restricted networks. The difference among them are that OPTION 1 aims at finding networks with very good costs, usually taking more computation time, OPTION 2 aims at finding networks with reasonably good costs in a reasonably short time, and OPTION 3 tries to find networks in a very short time, though their costs may not be good. Control sequence OPTION 4 is for designing fan-in/ fan-out restricted and level-restricted networks. It does not guarantee that feasible networks can always be obtained. If the users are not interested in setting up their own control sequences, they can simply select one of four built-in control sequences to solve their problems. 15 CO CD CJ a CD 3 o" CD CO I 3 X> U 3 O y-i •4-1 o CO 4-1 3 CD 4-> 3 o 43 CO H X! CD 4J O •h a #* A S-J CD 8 8 8 4-1 4-1 CO CO ■H CO CN CD W O O u a o H S3 8 8 S S3 8 4-1 >H 3 4-) r-l -H rH o o ft fa- o - « W 1 3 3 XJ 8 in S3 g 8 8 H S3 cO CO CN W iH M-l 3 o o o "-« CO C M H S3 ££ •H 4J I 3 cO fa 4J 3 3 O O X> -H 1 CD 4-1 C 4J eg co a b a fa fa fa iw i-l ft CD fa fa fa *■«* U O 4J H fa w 3 4-J 4-I CO 1-3 ■-> *-> •H CO CO 1 CD 3 3 H co CO U fa 4-1 XI CD 4-1 CJ •H h 4-1 CO cd a. i-i CD 8 4-1 8 8 4-) CO i-H 3 o M H O 3 H O O 1 o S3 H H 3 -H S3 S3 CO 4-1 M-l CJ A 1 3 * A 01 3 X> 8 8 8 O CO S3 3 co ro S ^ cO o O Q 3 l-i H H Cut •H 4-1 1 S3 S3 § 1 3 cO 14-1 1 3 O S3 • 3^2 MOO > hJ M PQ > > H 3 W M H H H w •J S3 W W H S3 § DH « PQ H h4 W H CN ro < H H (H M O O H H H H H O W fa Cm fa fa C/3 O O O O I CD > O U a- S •H O 3 CD CD x: 4-1 ■H 4-1 3 3 x> CD 4J CO CD Du CD l-l •3 CD •H t-\ a a cO CD X> O 4-» 00 3 •H O t>0 CO •H CD l-i 3 x> CD CJ O l-i a 3 o •H 4-1 CJ 3 X) co 3 cO l-i 4-1 X) • CD 4-) 4-1 CO a o CD O rH cd M CO 4-1 CO CD ,3 J3 4J 4J 16 2. SET-UP OF INPUT DATA For each separate run, a set of problems may be submitted. Input data cards consists of six types of cards: (i) < specification card > (ii) < heading card > (iii) < problem parameter card > (iv) < control sequence card > s (v) < output function card > s (vi) < connection description card > s The user must prepare the input cards (i) , (ii) , (iii) , (iv) and (v) for the first problem. Type (vi) cards must be prepared if the initial network is to be read in explicitly. For later problems, type (i) cards must not be prepared and other types of cards must or must not be prepared depending on the information of the < specification card >. The following is the formats of these cards. (i) < specification card > : This is the first card of the first problem. It contains several parameters which tell whether the following problems need types. (ii), (iii), (iv) , and (v) cards. There are 6 fields on this card. cols. K4: An integer, HEAD, which is right justified. This in- teger may be zero or nonzero. If it is zero, then the following problems do not have heading cards. This is the case when the following problems have the same heading information as the first problem. If HEAD is nonzero, then a heading card must be prepared for each of the following problems, 17 cols. 5^8: An integer, FUNC, which is right justified. It may be zero or nonzero. When it is zero, the following problems will have the same output function(s) as the first problem. This may happen when we want to realize the same function(s) but under different constraints or applying different control sequences. If FUNC is nonzero, then we must pre- pare < output function card >s. cols. 9^12: An integer, PARM, which is right justified. When it is zero, no < problem parameter card >s are required for the following problems, i.e., the following problems use the parameter values specified for the first problem. When PARM is nonzero, a < problem parameter card > must appear in each of the following problems. cols. 13^16: An integer, SEQC, which is right justified. When it is zero, the < control sequence card >s specified for the first problem will be used for the following problems. When SEQC is nonzero, the < con- trol sequence card >s must be specified for each of the following problems. cols. 17^20: An integer, PUNC, which is right justified. It may be zero or nonzero. If it is nonzero, then the best results for each pro- blem will be punched on the cards . The format of the punched cards is the same as that of the < connection-description card >s. If PUNC is zero, then no cards will be punched. cols. 21^24: An integer, TLIM, which is also right justified. It specifies the limit of computation time that the user likes to spend to solve each problem. If any problem cannot be finished in the specified time limit, then intermediate results will be punched so that the user can con- tinue running this problem next time. After punching the intermediate * These cards may be used as input data for the CALCOMP program for drawing the network. 18 results, the next problem will be read in and processed. TLIM is implicitly expressed in seconds, and it may have the maxi- mum value 9999 seconds (or 2 hours, 46 minutes and 39 seconds). Recommended TLIM for a typical 3-, 4- or 5-variable problem is 2 minutes, 5 minutes or 10 minutes, respectively. Most of the 3-, 4- or 5-variable problems can be finished within the time limit recommended above. The < specification card > is helpful for saving the preparation of some data card& when a set of problems having the same information are submitted in one run. (ii) < heading card > : This card may contain any alphanumeric information in column 1-80. It is for the identification of each problem and no information on this card will be used in the actual computation. (iii) < problem parameter card > : This card specifies the nature of each problem the user wants to solve. There are fourteen fields in this card. Each field is composed of characters or numerals. cols. 1^4: An integer, N, which is right justified. It specifies the number n of external variables. Be sure to punch n, rather than 2 x n, for N in the case that both complemented and uncomplemented external vari- ables are available. cols. 5^8: An integer, M, which is right justified. This pa- rameter specifies the number m of output functions for the current problem. cols. 9vL2: An integer, A, which is right justified. The number A is the value of the non-negative weight for the number of gates in the cost function. Table 2-1 gives the typical combinations of values A and B for different network reduction problems. cols. 13^16: An integer, B, which is right justified. The number B is the value of the non-negative weight for the number of connections in 19 Network Reduction Problem Values of A and B reducing the number of gates only. A = 1 and B = reducing the number of gates primarily, then reducing the number of . connections secondarily . A = 1000 and B = 1 reducing the number of connections only. A = and B = 1 reducing the number of connections primarily, then reducing the number of gates secondarily. A = 1 and B = 1000 reducing the sum of the number of gates and the number of connections. A = B = 1 Table 2-1 Typical combinations of values A and B for different network reduction problems. Most of the programs in the NETTRA system are oriented toward this reduction problem, so the user will probably find this combination of A and B the most useful. 20 the cost function (see Table 2-1) . cols. 17^19: blank. col. 20: A blank or one of the characters "X" and "Y". The blank or the character X means that only uncomplemented external vari- ables are available as inputs. The character Y means that both comple- mented and uncomplemented external variables are available. cols. 21^24: An integer, TFI, which is right justified. This parameter specifies the maximum number of fan- in a gate may have. The default value is 100 when this field is zero or blank. cols. 25^28: An integer, TFO, which is right justified. It specifies the maximum number of fan-out for gates (not output gates) . The default value is 100 when this field is zero or blank . cols. 29^32: It specifies the maximum number of fan-out for ex- ternal variables. The default value is 100 when this field is zero or blank . cols. 33^36: An integer, TFOO, which is right justified. It specifies the maximum number of fan-out for output gates. The default t value is 100 when this field is zero or blank . If the user does not want to have some output gates having connections to other gates, he has to specify a negative number, e.g. -1. cols. 37^40: An integer, LREST, which is right justified. It specifies the maximum number of levels that the network may have. The default value is 100 when this field is zero or blank . * This is equivalent to solving the problem under no fan-in restrictions. t These are the cases of no fan-out restrictions for external variables or gates. This is the case when there exists no level restriction. 21 cols. 41^44: An integer, P0PT1, which is right justified. This parameter may have value 1 or 0. When it is 1, the detailed processes for initial network subroutine TISLEV will be printed. Usually this field is specified as zero or blank so that no intermediate result about TISLEV is printed. cols. 45^48: An integer, P0PT2, which is right justified. It has the value 1 or 0. When it is 1, the detailed information about the transduction procedures will be printed. Usually, it is left as blank. cols. 49^52: An integer, RERUN, which is right justified. This parameter indicates whether the current problem is run for the first time or not. If the current problem is run for the first time, then RERUN has to be set the value zero. For the problem which is not finished last time, the RERUN field on the parameter card in the punched deck is specified as 1. cols 53^56: An integer, NEPMAX, which is right justified. This parameter specifies the maximum number of error positions permitted in the transduction procedures based on error-compensation. Usually this field is left as blank even if error-compensation transduction is involved in the problem. The default value is 2 ^ n ~ ' , where n is the number of external variables. (iv) < control sequence card >s: The control sequence cards specify the types of initial network methods and the types, the order and the number of execution times of the transduction procedures to be applied for solving the current problem. The following four keywords denote four built-in control sequences They are 0PT1, 0PT2, 0PT3 and 0PT4, which correspond to control sequences 22 OPTION 1, OPTION 2, OPTION 3 and OPTION 4, respectively. The user need only specify one of these keywords on the control sequence card. Since free format is used to specify the control sequence cards, this keyword can appear anywhere on the card as the first four nonblank characters. For the user who intends to specify his own control sequence, the following descriptions must be followed. Twenty-two keywords are available for the specification of con- trol sequences. Each keyword consists of four characters. Table 2-2 shows these keywords and their corresponding meanings. Since free format is used, one or more blanks must be used to separate two keywords, a key- word and a numeral, or a keyword and a character (this will be clarified soon) . INTP is the first four nonblank characters that have to appear on the control sequence cards. It works as the start of a control sequence. The keywords for the selected initial network methods must be specified one by one after INTP. In Table 2-2, seven keywords are used for specifying initial network methods. Among them, EXNT is used for reading in an ini- tial network prepared by the user. The meanings for the six keywords ex- cept EXNT are introduced in Table 1.2-1. One of the keywords TDTP, JEFF, FLTP and NOIT must appear fol- lowing the keyword specifying one of the initial network methods. When- ever TDTP appears, the transduction procedures without fan-in/fan-out re- strictions and level restriction will be applied, and the keywords for the selected transduction procedures and the number of execution times must * It can be a positive integer or the character #. When # is used, the selected transduction procedure will be applied repeatedly until there is no further improvement in the network cost. 23 Table 2-2 Keywords for specifying control sequence cards KEYWORD MEANING INTP Types of initial network methods UNIV INITIAL NETWORK METHODS THRL BANB TISN TANT TLEV * EXNT TDTP No fan-in/fan-out restricted transduction step JEFF Fan- in/ fan-out restricted transformation step FLTP Fan- in/ fan-out restricted and/or level-restricted transduction step NOIT Number of iterations in the application of a TT- sequence STOP End of the control sequence NPGM TRANSDUCTION PROCEDURES NPGS NTP1 NTP2 NTG1 NTG2 NTG3 NTG4 NTE1 This keyword is for reading in an initial network prepared by the user, 24 appear following TDTP. Whenever FLTP appears, the transduction procedures will be applied under fan-in/fan-out restrictions and/or level restriction, and the keywords for the selected transduction procedures and the number of execution times must appear following FLTP. It should be noticed that NTG4 cannot be used for fan-in/fan-out restricted or level-restricted trans- duction. Whenever JEFF appears, the fan- in/fan-out restricted transforma- tion procedure will be applied. Whenever NOIT appears, the contents of the TT-sequence have been specified; so the number of iterations of the TT-sequence must be given after NOIT. This number can be a positive in- teger or the character //. When // is used, the TT-sequence will be applied repeatedly until there is no further improvement in the network cost. Notice that keywords JEFF and FLTP can never appear before the keyword TDTP if all of them appear on the card, and keyword FLTP cannot ap- pear before the keyword JEFF if they both appear on the card. Table 2-3 gives the order of keywords INTP, TDTP, JEFF, FLTP and NOIT for different types of problems. Table 2-3 Order of keywords for different types of problems Problems Type Keywords INTP TDTP JEFF FLTP NOIT (1) 1 2 3 (2) 1 2 3 4 5 (3) 1 2 3 (4) 1 2 3 It can be a positive integer or the character //. When // is used, the selected transduction procedure will be applied repeatedly until there is no further improvement in the network cost. 25 The following is an example for a type (2) problem. Example Suppose three initial network methods UNIV, THRL and BANB will be used to generate initial networks and the TT-sequence contains the following information: 1. Apply NPGS 3 times, NTG3 2 times, under no fan-in/ fan-out restrictions. 2. Apply the transformation procedure JEFF. 3. Apply NTG1 repeatedly, and then apply NTE1 2 times under fan-in/fan-out restrictions. 4. The TT-sequence is to be executed repeatedly. The specification of the control sequence for this problem is INTP UNIV THRL BANB TDTP NPGS 3 NTG3 2 JEFF FLTP NTG1 // NTE1 2 NOIT it STOP The shortest length of this sequence is 74 characters, so one card is enough. In any control sequence, the keyword STOP is used to indicate the end of the specification. (v) < output function card >s : The m < output function card >s specify the set of m output functions to be realized simultaneously. Each function is expressed in the truth table form. Each card contains the values of one output function, starting from column 1 of the card (i.e., left justified). The maximum number of binary digits for each function is limited to 32. (vi) < connection-description card >s: The < connection-descrip- tion card >s are used for inputting the configuration of a network prepared by the user. Each of these cards is divided into 40 fields of 2 columns each (i.e., columns 1^2, 3^4, ... 77^78, 79^80). Beginning with the first 26 field of the first card, continuing through the succeeding fields of that card and through the fields of as many additional cards as necessary, specify the network configuration right justified in their respective fields. The network configuration is expressed according to the follow- ing description. 1. Label each gate of the network uniquely by assigning to it one of the integers 1, 2, ..., R, such that the output gates receive the labels 1, 2, ..., m, where R is the number of gates in the network. 2. Assign the names XI, X2, ..., XN to the uncomplemented ex- ternal variables x. , x„, ..., x and the names Yl, Y2, . . . , YN to the 1 z n complemented external variables x,, x_, ..., x , respectively. 1 Z n 3. For each gate I, ln \ i ] 3C P So x 2 L 1 \ X„ A \ 3 > . Y» 1 ■y 1 > F: 1 f> 1 X l I L network s The given X l i -Ka) X ' h. "1 x 2 ■- x 3 Lg. 2 Fig. 2-1 (a) The given network col. 1 col. 56 + _1_2_3___2_4_5_6___3X1_6___4X1X2___5Y3X1___6Y1X2X3__** Fig. 2-1 (b) The card to specify the connection configuration for the network in (a) . 28 Notice that in specifying the input connection configuration for each gate, there is no restriction that external variables should appear earlier than other gates, and also there is no restriction that the gates should appear in order according to their numeric labels. These 6 groups of cards, (i) , (ii) , (iii) , (iv), (v) and (vi) , in this order constitute the necessary description for a single problem. In order to solve more than one problem in one computer run, the user can arrange serially the descriptions for other problems. Fig. 2-2 shows the input card sequence for the execution of more than one problem using typi- cal JCL statements for the IBM 360/75. 29 /*ID ... } JCL cards for problem identification /*ID SETUP UNIT=2314,ID=0K0043 // EXEC GOFORT.PROG=SYSTEM1,PARM=/OVLY,LIST,XCAL , ,REGION=390K //GO.STEPLIB DD DSN=USER.P1189.NETTRA,DISP=SHR //GO.SYSIN DD * /< specification card > /< heading card > < problem parameter card > [/< control sequence card >s /< output function card >s ,z < connection - description cards >s ) 1st problem Input Data Cards >l 2nd problem ! I *\ \ last problem /* Fig- 2-2 Input card sequence for the execution of problems by the NETTRA system 30 3. RESTRICTIONS ON PROBLEM SIZE In order to fit the program into a limited memory size (currently 400k bytes with IBM 360/75J), some restrictions on an acceptable size of a problem were made: 1. The number of external variables may not exceed 5. 2. The number m of output functions may not exceed 10 . 3. The number of gates, R, may not exceed 60-n in the case of uncomplemented external variables available (a blank or 'x' parameter). In the case of both complemented and uncomplemented external variables available, the limit is lowered to 60-2n. All of these limitations are essentially imposed by the array sizes in the program as presently written. *For initial network method TISN or TLEV, this is lowered to 4, 31 4. EXAMPLES OF INPUT DATA SETUP The following examples will illustrate, for using the NETTRA system, various possible input data card setups complying with the directions given in Chapter 2. Example 4-1 A one-bit full adder. One-bit full adder is an important arithmetic unit in most digital computers. The output functions are SUM = x © x © x and CARRY = x..x v In this example, we want to get results using the control sequence 0PT1 but under different fan- in/fan-out restrictions (TFI=TF0=TF0X=TF00=2 or 3 or 4) . Assume that the goal of cost reduction is to reduce the number of gates primarily and the number of connections secondarily. Also assume that only uncomplemented external variables are available. The input data cards for this example is shown below. (i) The < specification card > contains the following information: cols. l'H 0, no heading card for the following problems, cols. 5^8 0, using the same functions as the first problem for the following problems, cols. 9^12 1, using different parameters for each problem. cols. 13^16 0, using the same control sequence for each problem, cols. 17^20 0, no result will be punched. 32 cols . 1^4 3, cols. 5%8 2, cols. 9vL2 100, cols. 13VL6 1, cols. 17^20 x, cols. 21 24 600, for each problem the processing time is 600 seconds (10 minutes) . (ii) The < heading card > contains: *** ONE-BIT FULL ADDER : S=EX0R(X1 ,X2 ,X3) ,C=X1X2 Xl v X3v/X2vX3 *** (iii) Three < parameter card >s are required for three sets of fan-in/fan-out restrictions. But the following parameters have the same values. the value of N. the value of m. the value of A. the value of B. uncomplemented external variables available only, cols. 37^40 "blank" use the default value for LREST. cols. 41^44 and cols. 45^48 "blank" no printout for intermediate steps cols. 49^52 0, the problems are run for the first time, cols. 53^56 0, use the default for NGPMAX. cols. 21o,24, 25^28, 29^32, 33%36 are the parameters TFI, TF0, TF0X and TFOO. For our problem, they are set to the values 2's, 3's and 4's on each of the parameter cards, (iv) The < control sequence card > in this example contains the keyword 0PT1 only, (v) The < output function card >s consist of two cards, one for the sum and another one for the carry. Their binary repre- sentatives are: 01101001 — SUM 00010111 — CARRY 33 Since we do not read in the network explicitly no < connection- description card >s are necessary. The whole set of input cards for this problem is shown in Fig. 4-1. First col. 1 -1- Card ■+( 1 600 /*** ONE-BIT FULL ADDER : S=EX0R(X1,X2,X3) , C=XlX2vXLX3vX2X3 *** / 3 21000 1X2 2 2 2 /0PT1 /01101001 /00010111 / 3 21000 1X3 3 3 3 / 3 21000 i 1X4 4 4 4 Fig. 4.1 Input data setup for Example 4-1. Example 4-2 Parity functions In this example, we want to find near-optimal networks for 3-variable, 4-variable and 5-variable odd parity functions. For these parity functions, same fan- in, fan-out restrictions but different control sequences are used. Assume that both complemented and uncomplemented external variables are avail- able. The cost criterion is to reduce the number of gates primarily and the number of connections secondarily. The details of each card is explained be- low. (i) The < specification card > has the following contents: cols. 1^4 1, read a heading card for each problem, cols. 5^8 1, read a function card for each problem, cols. 9^12 1, read the parameter card for each pro- blem, cols. 13^16 1, read a control sequence card for each problem. 34 cols. 9^12 1000, cols. 13vL6 1, cols. 17^20 Y, cols. 17^20 0, no punch for the best results obtained. cols. 21^24 600, the maximum execution time is 10 min- utes for each problem, (ii) The contents for each < heading card > will be shown later, (iii) The following fields of the < problem parameter card >s have the same information for the three problems. cols. 5^ 8 1, the value of M the value of A the value of B complemented and uncomplemented vari- ables are available, the value of TFI the value of TFO. the value of TFOX. the value of TFOO. cols. 37^40 "blank", use the default value for LREST cols. 41^44 and cols. 45^48 "blank", no printout for intermediate steps. cols. 49V52 "blank", run the problem for the first time. cols. 53^56 "blank", use the default value for NEPMAX. Only the first field has different values for N for different problems, (iv) The following three different < control sequence card >s are for solving the 3-variable, 4-variable and 5-variable parity functions, respectively. 1. INTP BANB TDTP NTG1 // JEFF FLTP NTG1 // NOIT # STOP 2. INTP BANB TDTP NTG2 // JEFF FLTP NTG2 # NOIT // STOP 3. INTP BANB TDTP NTE1 // JEFF FLTP NTE1 // NOIT # STOP cols. 21^24 4, cols. 25^28 4, cols. 29%32 4, cols. 33^36 4, 35 (v) The < output function card > is different for each problem. 1. 01101001 2. 0110100110010110 x x * x 2 * x 3 x w x w x **» x 1 2 3 4 3. 01101001100101101001011001101001 x ® x ® x ® x, © x r 12 3 4 5 No < connection-description card >s are necessary. Fig. 4.2 shows the input data setup for this example. col. 1 First Card + [~ 600 ^*** THREE-VARIABLE PARITY FUNCTION ***" '01101001 11000 INTP BANB TDTP NTG1 # JEFF FLTP NTG1 # NOIT # STOP a** FOUR-VARIABLE PARITY FUNCTION *** 11000 f INTP BANB TDTP NTG2 // JEFF FLIP NTG2 # NOIT // STOP '0110100110010110 ^*** FIVE-VARIABLE PARITY FUNCTION *** r 11000 I INTP BANB TDTP NTE1 # JEFF FLIP NTE1 // NOIT # STOP K01101001100101101001011001101001 Fig. 4.2 Input data setup for Example 4-2. Example 4-3 ; Su-Nam's example functions There are 4-variable 4-output incompletely specified functions used by Su and Nam in [3]. We want to solve this example by the NETTRA system and compare the result with that obtained in [3] which has 25 gates, 42 connections and 6 levels. The restrictions used in [3] are: FI = FO = FOX = 2 and F00 = and both complemented and uncomplemented external vari- ables are available. Again the cost reduction criterion is to reduce the number of gates primarily and the number of connections secondarily. Fig. 4-3 shows the input data cards. 36 col. 1 4- First Card ■* / 600 !*** SU-NAM'S EXAMPLE *** /0PT4~ 41000 -1 ^000*1*1101*11111 /011*1*1100110000 /001*1*00001**000 output functions /001*1*0101111111 Fig. 4.3 Input data cards for Example 4-3. Notice that the < specification card > for this example contains 0's in the first five fields. This is because only a single problem is involved in this run. Example 4-4 : In this example, we want to compare the effective- ness and the efficiency of the transduction procedures under no fan-in/fan- out restrictions. The 5-variable function f (x. ,x~,x„ ,x, ,x,.) = E (0,4,5,6, 7,8,10,14,16,19,21,23,24,25,26,27,29,30) is used, and the control sequences composed of the initial network method UNIV and different transduction pro- cedures are applied. Assume only uncomplemented external variables are available and the cost criterion is to reduce the number of connections pri- marily and the number of gates secondarily. The input data deck is shown in Fig. 4-4. 37 First col. 1 4- Card ■* f ° 1 300 /*** FIVE-VARIABLE FUNCTION: HEX=8FA295F6 *** I 5 1 11000 X / INTP UNIV TDTP NPGM I NOIT 1 STOP /10001111101000101001010111110110 1 INTP UNIV TDTP NPGS // NOIT 1 STOP / INTP UNIV TDTP NTP1 // NOIT 1 STOP / INTP UNIV TDTP NTP2 // NOIT 1 STOP f INTP UNIV TDTP NTG1 # NOIT 1 STOP / INTP UNIV TDTP NTG2 // NOIT 1 STOP i 1 INTP UNIV TDTP NTG3 # NOIT 1 STOP 1 / INTP UNIV TDTP NTG4 // NOIT 1 STOP i / INTP UNIV TDTP NTE1 I NOIT 1 STOP Fig. 4-4 Input data deck for Example 4-4. In Fig. 4.4, the execution time for each problem is 300 seconds. No fan- in/ van-out restrictions and level restriction are imposed. Example 4-5 : In this example we want to get initial networks by different initial network methods. Since the initial network method TLEV is for the level-restricted case, it is not included in the control sequence. The same output function used in Example 4-4 is selected. Assume only un- complemented external variables are available. The input data deck is shown in Fig. 4-5. On the < problem parameter card >, the values of A and B are 1000 and 1, respectively. The values of A and B are not useful since no transduction or transformation is involved. But they are specified for printing the network costs. Another thing which should be noticed is that NOIT still has to be specified although no TT- sequence is included. 38 First col. 1 Card -»•/ 600 {*** FIND DIFFERENT INITIAL NETWORKS ***" 4 11000 1 X INTP UNIV THRL TISN BANB TANT NOIT 1 STOP /10001111101000101001010111110110 Fig. 4-5 Input data deck for example 4-5. 39 5. OUTPUT FORMATS There are two types of output formats for the NETTRA system. One is for problems using the built-in control sequences 0PT1, 0PT2 and 0PT3, and the other is for problems using the built-in control sequence 0PT4 and the control sequences designed by the users themself. Example 4-1 uses the control sequence 0PT1 to find near-optimal networks for the one-bit full adder under different fan- in/ fan-out restric- tions. The printout for this example is shown in Fig. S-lCa^Ce). On the first page of the print-out for the first problem, the pro- blem number is shown first. Then the contents of the < heading card >, the number of external variables (n) t the number of output functions (m) , the cost coefficients (A and B) , the status of external variables (whether uncomplemented and complemented external variables are available) , the fan- in, fan-out and level restrictions the computation time specified for each problem in this run, the contents of the control sequence and the output functions (in row vec- tor form) are printed one by one. The cost, the computation time, and the network configuration for the best network are shown following the above printout. The network con- figuration is described in a way similar to the specification for the < connection-description card >s: describing the input connections for each gate in the network. The gate number, the gate level and the immedi- ate predecessors of a gate are shown in the columns entitled with "GATE", "LEVEL" and "FED BY", respectively. The gates, which exist in the initial network or in the network just after the application of the transformation procedure JEFF but then become redundant after the applications of the 40 * * * * * * » #o* # SB* # ♦ * W* # ^* ♦ CG» ♦ Q# ♦ 03* ♦ CU* * * * * * * * * * M CO x r- > (N X M II U X o X w II to G3 w Q o .-4 ►-J E> U* M CQ I W o * CO u o o o © &-. M b p- fr. o o o cc PC u w w cu CQ H 3C. X CO n jn O 2". ss U I I I XI to w t-i CQ M 03 «: > Q W e-i SB W an w .-» cu je o u z D fN e H h4 II CQ II O CO 03 v WIO CU ^W CQH W «C«< X HO F-" « O «s!EHO03 >DrO cu t'-. •EH II HD O (Ntaotoco rsi jh CCCCWPm 00>H II tupuipgcj II ^w Bt* CQ Q W H 0-. M O M CU CO u, u SB o w to .-> o 05 H Pu T~ (N o*-o«- O r- S5*~2 , .0 UOUO SB SB D !=> o u CU o 41 THIS IS THE BEST RESULT FOUND BY CONTROL SEQUENCE OPTION 1 TOTAL TIPE SPENT = 10742 CENTISECDNDS BEST COST = 12CP9 TRUTH TABLE 1=01101001 2 = 0001011 1 1 2 J 4 5 6 7 8 4 10 11 12 13 14 15 16 17 18 1i 20 11 23 24 25 26 27 28 29 30 i^VEL FED BY / 1/ 8 19 / 1/ 11 / 5/ 4 6 / §/ X1 X2 / 7/ X2 / 6/ 5 7 / 7/ X1 / 3/ 9 12 / 4/ •3 / 1/ / 2/ 6 8 / 4/ X3 / 1/ / 1/ / 1/ / 1/ / V / 1/ / 2/ X3 3 / 1 / / 1 / / / / 1 / / 1 / / ' / / 1 / / ' / / V / 1/ / 1 / Fig. 5-l(b) ********************** * PROBLEM NO. 2 * ********************** 42 NUMBER OF VARIABLES = 3 NUMBER OF FUNCTIONS = 2 COST COEFFICIENT A = 1000 B = 1 UNCOMPLEMENTED VARIABLES X - — FAN-IN = 3 FAN-OUT = 3 FAN-CUT FOR EX. VARIABLES = FAN-OUT FOR OUTPUT GATES = NO. OF LEVELS = 100 THIS IS THE BEST RESULT FOUND BY CONTROL SEQUENCE OPTION 1 TOTAL TIME SPENT = BEST COST = 9018 1600 CENTISECONDS TRUTH TABLE 1 = 1 1 1 1 2 = 1 1 1 1 iA'^E -. LEVEL FED BY 1 / 1/ ■j «; 9 / 1/ 6 7 3 / 2/ X2 6 4 / 1/ 5 / 2/ 6 7 8 o / 3/ X2 i 8 I / 4/ / 4/ X* 9 \l 9 / 5/ X3 10 / 5/ X1 1 1 / 1/ 12 / 1/ Fig. 5-l(c) 13 / V ********************** * PROBLEM NO- 3 * ********************** 43 NUMBER OF VARIABLES = 3 NUMBER OF FUNCTIONS = 2 COST COEFFICIENT A = 1000 B = 1 UNCOMPLEMENTED VARIABLES X FAN-IN = U FAN-OUT = 4 FAN-OUT FOR EX. VARIABLES FAN-OUT FOR OUTPUT GATES ■« NO. CF LEVELS = 100 THIS IS THE BEST RESULT FOUND BY CONTFOL SEQUENCE OPTION 1 TOTAL TIMF SPENT = PEST COST = 9018 661 CENTISECONES IRUTH ^ARLE 1=01101001 2 = 00010111 1 3 ji 5 6 7 i J 10 11 \i LEVEL FED BY / 1/ / 1/ / 2/ / 1/ / 2/ / ?/ / u/ / a/ / 5/ / 5/ / 1/ / V, / 3 5 6 7 X2 6 6 7 X2 7 X1 X3 9 10 X3 ▼ 1 Fig. 5-l(d) 44 ^» n O r- I/, 3 O \L> ~-- O r W r- r: H > r-J ('-4 O to o O 00 CD o ^ ^ f— o o O c< V) r- w a ♦ » » » # # * # -t# * * » * » * * t» » o» * z* * * • sr.» * w* # ■-)♦ # CQ* # o# » «« • &.« * * * * * * * o a w * * « * # * * * »♦ * * * co» * M» » 33« * C-«# * * * 01 » * O* * u.* * # * £« * J'# * r=* » to* * # * * * * * * * H o o o I/) o o o W r> «- r- w O r-I r»l » O X C-\ cri & o Cm 60 •H M (N CO » O O O o o o O O O (N (N IN ^, rn c ro a 'J 45 el 1 transduction procedures, are assigned as the first-level gates and they have no immediate predecessors. The best network for the first problem is shown in Fig. 5-2. x. 19 X 3~ 11 ig. 5-2 The best network obtained for the first problem in Example 5-1. Each gate has a gate number corresponding to the network configuration. For problems 2 and 3, only the information on the is shown, since the heading information, the control sequence and the output functions are the same as those of the first problem. The best net- works for problems 2 and 3 are found identical, and the network configura- tion is shown in Fig. 5-3. It is interesting to notice that the networks obtained in Fig. 5-3 has the minimal number of gates under given restric- tions. The optimality is proved by the integer programming logic design method based on branch-and-bound method. 46 level x 10 Fig. 5-3 The best network for problems _> in Example 4-1. The problem number for the fourth problem is printed after the results for the first three problems have been printed. Since there is no more input data card, "END OF FILE" is printed to indicate that all problems have been processed. A table, shown in Fig. 5-1 (e) is then printed to give a clear summary for this computer run. This summary contains the information about the problem size, fan- in, fan-out and level restrictions, the costs of the best networks, the numbers of levels of the networks and the computa- tion time spent. For the second type of printouts which correspond to the use of the built-in control sequence 0PT4 or the control sequences designed by the users, the results of the intermediate steps will be printed as well as the printout just described for the first type printout. In other words, the costs, the computation times, the network configurations of the ini- tial networks, the networks after applying the transformation and the 47 transductions will be printed. Fig. 5-4 (a)^(z) shows the printout for the problem which uses the following control sequence: INTP BANB THRL TDTP NPGM // NTG3 # JEFF FLIP NTG1 # NTG2 # NOIT # STOP The output function is the 5-variable single-output function f = Z (0,4,5,6,7,8,10,14,16,19,21,23,24,25,26,27,29,30). In Fig 5-4(a), the information about the problem is shown. The initial network obtained by the branch-and-bound algorithm is printed in Fig. 5-4(b). In Fig. 5-4(b) and (c) , the results after applying NPGM two iterations and then NTG3 two itera- tions are shown. Then the results after applying JEFF and the transduction procedures are shown one by one. In Fig. 5-4(0), the network with cost 12030 after applying the TT-sequence twice is obtained. Since the network cannot be simplified further, we go back to try the initial network method THRL and apply the same TT-sequence. Finally, in Fig. 5-4(y), a network with cost 13032 is obtained. The summary is shown in Fig. 5-4(z). For the problems using the built-in control sequence 0PT4 , the out- put formats are the same as the previous example, i.e. , the results of the intermediate steps are printed. Besides, the status (whether the network is fan-in, fan-out restricted and level-restricted or not) of the initial net- work, and the network after each application of the transduction is also shown. Fig. 5-5(a)^(t) gives the results of the one-bit full adder under both fan-in/fan-out and level restrictions using the control sequence 0PT4. The first problem, Fig. 5-5(a)^(g), is under the restrictions TFI = TFO = TFOX = TFOO = 3 and LREST = 4. The initial network derived by TISLEV is fan-in/ fan-out restricted and level-restricted. So after application of the trans- duction procedures, a feasible network with cost 12024 is obtained in Fig. u o © o o 48 ♦ ♦ # # # ♦ »-# * * » ♦ « .* # o# ♦ z» ♦ * * an# * FJ* * i-J# * C0# * a* ♦ 05* # # # * * * * Cm o> (N «=* 04 00 II w EC z o M E-< U z cm w •-) CQ M 05 > I W > * * in r» O t- O o II II II II CQ LO to W Z «c M O co H R •< H Z M U w 03 2 M << O O > Cm M Cm Cm Cm Cm o o W o 05. IM u U-) w CQ 00 H JC an lO D » o *£ » u X in W ►J «: M 05 > Q w E-t Z W ae W »-J ac o u Z sr an w -1 II CQ II o en 05 WW 04 ->w C0E-' M «<««! 33 HC Fh 05 O i «Dr O & Cm • H II XD Q awomia =r ^h QSOSWCm 00>H II PmPliHcj II JW HHH &• ZOOOCmW MOOOC i 1 1 1 1 W zzaa < >ac rt«««io MO E<0 Ut- z 3 Cm 49 THIS INITIAL NETWORK IS FOUND BY BRANCH AND BOUND ALGORITHM INITIAL NETWORK COST = 1U050 TIME ELAPSED = 59 CENTISECONDS [RUTH T.»BLE 1 = 10 1 11 1101000101001010111110110 [£ -. LEVEL FED BY 1 / 1/ I / 2/ 3 / 3/ 4 / 3/ 5 / 5/ / 5/ 7 / 8/ 3 / 7/ i / 6/ 10 / 6/ 11 / a/ 12 / a/ 13 / 7/ U / 9/ 2 3 u X5 9 xu 7 X3 7 X2 7 >'1 X3 X2 >:3 X2 X4 X2 X5 X4 X5 X1 6 X1 7 XU X5 10 8 8 8 14 XU X5 8 6 11 9 10 9 8 13 6 11 12 10 13 NETWORK DERIVED EY MINI2-NC FANIN/FANOUT LIMITS IN 2 ITERATIONS COST = 13038 TIME ELAPSED = 17 CENTISECONDS [RUTH TABLE 1 = 10001111101000101001010111110110 Fig. 5-4 (b) 50 It, . . LKVKL 1 / 1/ \ / 2/ / 3/ 4 / 3/ 5 / 3/ 7° / 3/ / 6/ 3 / 5/ 1 / 1/ 10 / 4/ 11 / 4/ / 4/ 13 / 5/ 14 / 7/ rtu bx 2 3 X5 X4 X3 X2 X1 X2 it 10 8 7 7 73 X3 X2 X5 X4 X5 X1 XI 7 X4 X5 11 11 8 8 14 X4 12 10 10 8 13 8 NETWOPK DERIVED BY GATE MEPGIKG-NO FANI N/FAKOOT LIMITS IN 2 ITERATIONS TIME ELAPSED = COST - 12036 40 CENTISECONDS TRUTH TABLE 1 = ^00011111010001010010101111101 GATii .- LEVEL FED BY 1 / 1/ 2 / 2/ 3 / V / 3/ 4 5 / 3/ 6 / 3/ 7 / 6/ 8 / 8/ 9 / 1/ 10 / 4/ n / 7/ 12 / 4/ *»3 / 5/ 14 / 1/ 2 3 4 5 6 X5 10 N X4 8 12 X3 7 8 10 X2 7 8 10 X1 X3 11 X2 X3 X4 X2 X5 8 13 X4 X5 8 X1 X1 7 Fig. 5-4 (c) 51 A NETWORK DERIVED BY JEFF COST - 13039 TIME ELAPSED = 9 CENTISECONDS T3TITH TABLE 1=10001111101000101001010111110110 liATE .. LEVEL FED BY 1 / 1/ 2 2 / 2/ 3 U 5 6 3 / 3/ X5 10 11 4 / 3/ XU 9 11 12 5 / 3/ X3 7 8 10 6 / 3/ X2 "7 8 10 7 / 6/ X1 X3 11 3 / 8/ X2 i3 XU 9 / U/ X2 X3 XU 10 / 4/ X2 X5 8 13 11 / 7/ XU XS 8 12 / U/ X1 13 / 5/ X1 7 14 / 1/ A NETWORK DERIVED BY 1-TH APPLICATION OF PF.IFF COST = 13037 TIME ELAPSED = 30 CENTISECONDS TROTH TABLE 1=10001111101000101001010111110110 GATE .. LEVEL FED BY 1 / 1/ 2 10 2 / 2/ 3 4 5 6 3 / 3/ X5 11 4 / 3/ XU 9 11 12 5 / 3/ X3 7 9 o / 3/ X2 7 9 7 / U/ X1 X3 11 8 / 6/ X2 X3 XU 9 / U/ X2 X3 XU 10 / 2/ X2 X5 8 13 11 / 5/ XU X5 8 12 / U/ X1 13 / 3/ X1 7 14 / 1/ Fig. 5-4 (d) 52 A NETWORK DERIVED BY 2-TH APPLICATION OF PRIFF COST = 13036 TIME ELAPSED = 22 CENTISECONDS T3FJTH TABLE 1=10001111101000101001010111110110 GATE .. LEVEL FED BY 2 10 3 4 5 6 X5 11 X4 8 11 12 X3 7 8 X? 7 8 X1 X3 11 X2 X3 X4 X4 11 X2 X5 9 13 XU X5 8 X1 X1 7 1 / 1/ 2 / 2/ 3 / 3/ 4 / 3/ 5 / 3/ 6 / 3/ 7 / 4/ 6 / 6/ 9 / 3/ 10 / 2/ 11 / 5/ 12 / 4/ 13 / 3/ 14 / 1/ A NETWOPK DERIVED BY 3-TH APPLICATION OF PRIFF COST = 130 33 TIME ELAPSED = 27 CENTISECONDS TRUTH '"ABLE 1 = 100011111010001010010 10111110110 Fig. 5-4(e) 53 TE . . LEVEL FFD BY 1 / 1/ 2 10 2 / 2/ 3 4 5 3 / 3/ X5 11 4 / 3/ xa 8 11 5 / 3/ X3 7 9 6 / 3/ X? 8 7 / 4/ X1 11 8 / 6/ X2 X3 9 / 4/ X4 11 10 / 2/ X2 X5 9 11 / 5/ X4 X5 8 12 / 4/ X1 13 / 3/ X1 8 14 / 1/ 12 13 A NETWORK DERIVED 3Y 4-TH APPLICATION OF PPIFF COST = 12033 TIME ELAPSED 17 CENTISECONDS TRUTH TABLE 1=10001111101000101001010111110110 GATE .. LEVEL 1 / 1/ i ' 2/ / 3/ 4 ' V 5 / 3/ 6 / 3/ 7 / 4/ 8 / 6/ 9 / 4/ 10 / 2/ 11 / 5/ 12 / 1/ 13 / 3/ 14 / 1/ FED BY 2 10 X5 1^f 5 6 X4 7 8 11 X3 7 9 X2 8 X1 11 ' X2 X3 X2 X3 X4 X2 X5 9 13 X4 X^ 3 X1 8 Fig. 5-4(f) 54 A NETWORK CEPIVED BY 5-TH APPLICATION OP PRIFF COST = 12033 TIME ELAPSED = 27 CENTISECCNDS I RUTH TABLE 1 = 1000111110100010100101011111011 i SATS .. LEVEL FED BY 2 10 3 4 5 6 X5 11 XU 9 11 13 X3 7 9 X2 8 X1 11 X2 X3 X2 X3 x4 X2 X5 9 13 xa X5 8 X1 8 1 / V 2 / 2/ 3 / 3/ 4 / 3/ 5 / 3/ 6 / 3/ 7 / 4/ 8 / 6/ 9 / 4/ 10 / 2/ 11 / 5/ 12 / 1/ 13 / 4/ 14 / 1/ ** NO PEDUNDANCY ?CUND ** A NETWORK DERIVED BY 1-TH APPLICATION OF PROCIV COST = 12033 TIME ELAPSED = 132 CEMTISECONDS TRUTH TABLE 1 = 1000111110100010100101011111011 Fig. 5-4 (g) 55 GATE .. LEVEL FED BY 1/ 2 10 2/ 3 4 5 6 3/ X5 11 3/ XU 7 9 11 3/ X3 7 9 3/ X2 8 4/ xi 11 6/ X2 X3 U/ X2 X3 X4 2/ X2 X5 9 13 5/ XU X5 8 1/ 3/ X1 8 1/ 1 / 2 / 3 / 4 / 5 / 6 • / 7 / 8 / l* / / 11 / 12 / 13 / 14 / ** NO PEDONDANCY FOUND ** NETWOPK EEEIVED EX MINI2-N0 FANIN/FANOUT LIKITS IN 1 ITERATIONS COST = 12033 TIME ELAPSED = 5 CENTISECONDS TROTH TABLE 1 = 10001111101000101001010111110110 GATi .- LEVEL FEE BY 1/ 2 10 2/ 3 4 5 6- 3/ X5 11 3/ X4 7 9 11 3/ X3 7 9 3/ X2 8 a/ X1 11 6/ X2 X3 4/ X2 X3 X4 2/ X2 X5 9 13 5/ X4 X5 8 1/ 3/ X1 8 1 / 2 / 3 / i / D / O / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 1/ Fig. 5-4 (h) 56 NETWORK DERIVEE EY GATE MERGING-NO FANIH/FANOOT LIMITS IN 1 ITERATIONS TIME ELAPSED = COST ■ 12033 15 CENTISECONDS TRUTH TABLE 1 = 10001111101000 10100101011111011C ATri LEVEL 1 / 1/ 2 / 2/ 3 / 3/ 4 / 3/ 5 / 3/ 6 / 3/ 7 / u/ 6 / 6/ 9 / a/ 10 / 2/ 11 / 5/ 12 / 1/ %3 / 3/ 14 / 1/ FED BY 2 10 3 4 5 X5 11 X4 7 9 X3 7 9 X? 8 X1 11 X2 X3 X2 X3 X4 X2 XS 3 X4 X5 3 X1 11 13 A NETWORK DERIVED BY JEFF COST = 12033 TIME ELAPSED = 4 CENTISECONDS IRUTH TABLE 1=100011111010001010 0101011111011 Fig. 5-4(i) 57 GATE LEVEL FED BY 1 / V 2 / 2/ 3 / 3/ 4 / 3/ 5 / 3/ 5 / 3/ 7 / 4/ 3 / 6/ 9 / 4/ 10 / 2/ 11 / 5/ 12 / 1/ 13 / 3/ 14 / V 2 3 X5 XU x x x2 X1 72 X2 X2 X4 10 4 11 7 7 8 11 X3 X3 X5 X5 X1 8 X4 9 8 11 13 A NETWOPK DERIVED BY 1-TH APPLICATION 0? PRIPF COST = 12032 TIME PLAPSED = 25 CEKTISECOKDS 53UTH TABLE 1 = 10001111101000101001010111110110 GATE LEVEL FED PY 1 / 1/ 2 / 2/ 3 / 3/ 4 / 3/ 5 / 3/ 5 / 3/ 7 / 4/ 8 / 6/ 9 / 4/ 10 / 2/ 11 / 5/ 12 / 1/ 13 / 4/ 14 / 1/ 2 10 3 4 5 6 X5 11 X4 8 11 13 X3 7 9 X2 8 X1 11 X2 X3 X4 11 X2 X5 9 13 X4 X5 8 xi e Fig. 5-4 (j) 58 A NETWORK DERIVED BY 2-TH APPLICATION OF PRIPF COST ~ 12032 TIME EIAPFED = 25 CENTISECONDS TRUTH TABLE 1=10001111101000101001010111110110 IE . . LEVEL FED BY 1 / V 2 10 2 / 2/ 3 4 5 6 3 / 3/ X5 11 4 / 3/ X4 8 11 13 5 / 3/ X3 7 9 6 / 3/ X2 8 7 / 4/ X1 11 a / 6/ X2 X3 3 / 4/ X4 11 10 / 2/ X2 X5 9 13 11 / 5/ X4 X5 8 12 / 1/ 13 / 4/ X1 8 14 / 1/ ** NO REDUNDANCY FOUND ** A NETWORK DERIVED BY 1-TH APPLICATION OF PROCIV COST = 12032 TIME ELAPSED = 142 CENTISECONDS TRUTH TABLE 1 = 10001 1111 C1000101001010111110110 Fig. 5-4 (k) 59 GATii .. LEVEL 1 .2 3 4 / 3/ XU 7 8 11 5 6 7 6 9 10 / 2/ X2 X5 9 13 11 12 13 / 3/ X1 8 14 / 1/ / 2/ / 3/ / 3/ / 3/ / 3/ / 4/ / 6/ / 4/ / 2/ / 5/ / 1/ / 3/ / 1/ FED BY 2 10 3 4 5 X5 11 X4 7 8 X3 7 9 X2 8 X1 11 X2 X3 X4 11 X2 X5 9 X4 X5 8 ** NO REDUNDANCY FOUND ** NETWORK EIFIVED BY MINI2-N0 FANIN/FANOUT LIMITS IN 1 ITEPATICNS COST = 12032 TI!4E ELAPSED = 5 CENTISfiCONDS TfiUTH TUBLE 1=10001111101000101001010111110110 3&2£ . „ LEVEL FED FY 1 / 1/ 2 10 2 / 2/ 3 4 -J 3 / 3/ X5 11 4 / 3/ X4 7 8 5 / 3/ X3 7 9 o / 3/ X2 8 7 / 4/ X1 11 8 / 6/ X2 X3 9 / 4/ X4 11 10 / 2/ X2 X5 9 11 / 5/ X4 X5 8 12 / 1/ 13 / 3/ X1 8 14 / 1/ 11 13 Fig. 5-4U) 60 NETWORK DERIVEE BY GATE MERGING-NO FANIN/FANOUT LIMITS IN 1 ITERATIONS COST = 12032 TIME ELAPSED = 17 CENTISECONDS 1HOTH TABLE 1=1000111110 100010100101011111011 GATE LEVEL FED BY I / 1/ 2 10 2 / 2/ ■* 4 5 3 / 3/ X? 11 4 / 3/ xu 7 8 5 / 3/ X3 7 9 6 / 3/ X2 8 7 / 4/ XI 11- 8 / 6/ X2 9 / 4/ X4 11 10 / 2/ X2 X5 9 11 / 5/ XU X5 8 12 / 1/ 13 ' 2' X1 8 14 / v 11 13 A NETWORK EERIVEE BY JEFF COST = 12032 TIME ELAPSED = 2 CENTISECONDS TRUTH TABLE 1 *1 0001 1 111 01 00 01 01 00 10 10 1.1 1 1 1'0 1 1 Fig. 5-4 (m) 61 GATE .. LEVEL 1 / i / / 4 / 5 / 6 / 7 / a / 9 / 1 j / n / u / 13 / 14 / EL FED BY 1/ 2 10 2/ 3 U 5 3/ XF 11 3/ X4 7 8 3/ X3 7 9 3/ X2 3 6/ x2 xi 4/ xu 11 2/ X2 X5 9 5/ X4 X5 8 1/ 3/ X1 8 1/ 11 13 A NETWORK DEFIVJE BY 1-TH APPLICATION OF PEIFF COST = 12032 TIME ELAPSED = 32 CENTISECONDS TRUTH TABLE 1=10001111101000101001010111110110 GATE .. LEVEL FED BY 1/ 2 10 2/ 3 4 5 6 3/ X5 11 3/ X4 7 8 11 3/ X3 7 9 3/ X2 8 4/ XI 11 6/ X2 X3 4/ X4 11 2/ X2 X5 9 13 5/ X4 X5 8 1 / 2 / J / 4 / 5 / 6 / 7 / 8 / 3 / 10 / 11 / 12 / J 3 14 / / 1/ 3/ X1 V Fig. 5-4 (n) 62 ** NO REDUNDANCY FOUND ** A NETWCEK DERIVED BY 1-TH APPLICATION OP PPOCIV COS? = 12032 TIME ELAPSED = 135 CENTISECONDS TRUTH TABLE I =1 0001 1 1 1 1 010001 0100 10 1 OH 1 1 1 1 1 ( GATE -. LEVEL 1 / 1/ 2 / 2/ 3 / V '4 / 3/ 5 , / 3/ 6 / 3/ 7 / 4/ 3 / 6/ J- / 4/ 10 / 2/ 11 / 5/ 12 / V 13 / 4/ 14 / 1/ FED BY 2 10 3 4 5 6 X5 11 X4 8 11 13 X3 7 9 X2 8 X1 11 X2 X3 X4 11 X2 X5 a 13 X4 X5 8 X1 8 ** NO PEDUND2NCY FOUND ** Fig. 5-4(o) 63 *** TRY ANOTHER INITIAL NETWORK *** THIS INITIAL NETWORK IS FOUND BY THREE-LEVEL-NETWORK ALGORITHM INITIAL NETWORK COST = 26103 TIME ELAPSED = 10 CENTISECONDS TRfTTH TABLE 1 = 10001111101000101001010111110110 GATE .. LEVEL 1/1/ 4 11 13 15 16 17 19 21 26 2 3 4 5 6 7 8 j 10 M 13 14 15 lb 17 / 2/ X2 X4 X5 2 10 14 18 19 20 21 22 2i 44 25 26 / 2/ / 1/ / 3/ / 2/ / 2/ / 2/ / 3/ / 2/ / 3/ / 2/ / 3/ / fc / / 2/ / 3/ / £/ / 2/ / 2/ / 3/ / 2/ / 3/ / 2/ / 3/ / 3/ / 3/ / 3/ FED BY ■5 4 t; 7 9 XI X2 X3 X4 X5 XI X2 X3 X4 2 X1 X2 X3 X5 2 X1 X2 X3 2 XI X? X4 X5 X1 X3 34 6 xl X3 x5 X5 X1 X2 X4 X5 11 X4 X5 5 10 X2 X4 XI X4 6 12 X2 X3 X4 X5 X2 y^ X4 14 X2 X3 X5 2 U X2 X4 X5 2 X1 X2 X5 X2 X5 14 18 Y3 X4 X5 X4 X5 10 20 X2 X3 X4 X5 22 23 24 25 Fig. 5-4 (p) 64 NETWORK UEhlVKL tii HINl^-NU tTA N1N/EAN JUT LlfllTS IN 2 ITERATIONS COST = 190U7 TlfF EUPSED = 28 CENTISECONDS TRUTH TABLE 1 = 100011111010001010010101111101 1G CiAIE LEVEL 1 / 1/ / 1/ 2 3 / 1/ 4 / 1/ 5 / 1/ 6 / 3/ 7 / 3/ 3 9 ' 2' 10 / 3/ 1 1 / 1/ 12 / 3/ 13 / 2/ 14 / 3/ 15 / 2/ 16 / 2/ 17 / 1/ 18 / 3/ 19 / 2/ ^0 / 3/ 21 / 2/ 22 / 3/ 23 / 3/ 24 / 3/ 25 / 3/ 26 / 2/ FED BY 9 13 15 16 19 21 26 X3 X5 X X X3 X2 X1 X4 6 12 X3 X4 X5 X2 X3 X4 14 X2 X3 X5 14 X1 X2 X5 14 18 X3 X4 X5 10 20 X2 X3 X4 X5 22 23 24 25 NETWORK DERIVEE BY GATE MEPGINGJ-NO FANIN/FANOUT LIMITS IN 7 ITERATICNS COST = 14042 TIME ELAPSfD = 173 CENTISECONDS TRUTH TABLE 1 = 100011111010001010010101111101 Fig 5-4 (q) 65 GATE -. LEVEL FED BY 9 13 15 16 19 21 26 X1 X5 X1 X3 6 X2 X1 6 10 X3 X4 X5 X2 x3 X4 14 X2 X3 X5 14 1 / 1/ 2 / 1/ 3 / 1/ 4 / 1/ 5 / 1/ 6 / 3/ 7 / 1/ a n; 9 10 / 3/ 11 / 1/ 12 / 1/ 13 / 2/ 14 / 3/ 15 / 2/ 16 / 2/ 17 / 1/ 18 / 1/ 19 / 2/ 20 / 1/ 21 / 2/ 22 / 1/ 23 / 3/ 24 / 3/ Ab / 3/ 26 / 2/ X2 X5 6 14 74 X5 10 14 X3 X4 X5 10 23 24 25 A NETWORK DFRIVIC BY JEFF COST = 19046 TIME ELAPSED = 22 CENTISECONDS TRUTH TABLE 1=10001111101000101001010111110110 GATE .. LEVEL FFD BY 7 19 21 26 X3 X5 2 25 9 13 15 16 X1 X5 5 X1 X3 6 X2 1 / 1/ 2 / 7/ 3 / 6/ 4 / 3/ 5 / 3/ 6 / 5/ 7 / 2/ / 1/ 8 3 / 4/ 10 / 5/ 11 / 1/ Fig. 5-4(r) 66 U / 1/ / 4/ 14 / 5/ 15 / 4/ 16 / u/ 17 / 1/ 18 / 1/ 19 / 2/ 20 / 1/ 21 / 2/ 22 / 1/ 23 / 3/ 24 / 3/ 25 / u/ 26 / 2/ X1 6 10 X4 3 X2 X3 X4 14 X2 3 14 X2 4 6 14 X4 X5 10 14 X3 X4 X5 10 23 24 25 P NETWOFK DEFIVED BY 1-TH APPLICATION OF PEIFF COST = 19046 TIME ELAPSED = 54 CEN7ISEC0NDS TROTH TABLE 1=10001111101000101001010111110110 GATfi 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 1J / 14 / 15 / 16 / 17 / 18 / 19 / fl / / 22 / ^j / 24 / 25 / 26 / EL FED BY 1/ 7 19 21 26 6/ X3 X5 5/ X1 2 3/ 25 3/ Q 13 15 16 3/ XI 2/ 5 1/ 4/ ▼1 X? 25 5/ X2 1/ I* X1 10 25 5/ X4 4/ X2 X3 X4 2 4/ X2 X5 3 14 1/ 1/ 2/ Y, X2 4 6 23 X4 4 10 23 1/ 3/ X3 3/ X4 5/ X5 2/ 10 23 24 Fig 25 ;• 5-4(s) 67 ** NO REDUNDANCY FOUND ** A NETWOFK DERIVED BY 1-TH APPLICATION OF PROCIV COST = 13032 TIME ELAPSED = 629 CENTISECONDS TRUTH TJBLE 1 = 1000111110100010100101011111011 GATjS .. LEVEL FED BY 7 15 19 26 X5 X1 X2 23 9 16 X1 X2 XU X2 X3 XU X2 X5 14 XU X5 3 23 X3 2 10 14 23 1 / 1/ 2 / 5/ i / 3/ 4 / 1/ 3 / 3/ 6 / 1/ 7 ',V, 8 9 / 4/ 10 / 3/ 11 / 1/ 1 14 / 5/ 15 / 2/ 16 / 4/ 17 / 1/ 18 / 1/ 19 / 2/ 20 / V 21 / 1/ 22 / 1/ 23 / 4/ 24 / 1/ 25 / 1/ 25 / 2/ Fig. 5-4 (t) 68 A NETWORK DERIVED BY 2-TH APPLICATION OF PROCIV COST = 13032 TIME ELAPSED = 195 CENTISECONDS TRUTH TABLE 1 = 10001 1111 CI 00 01 01 00 1 10 111110 110 GAIE LEVEL 1 / 1/ 2 / 5/ 3 / 3/ 4 / V 5 / 3/ 6 / V 7 / 2/ 8 / 1/ 9 / 4/ H / 3/ / 1/ 12 / V 13 / 1/ 14 / 5/ 15 / 2/ 16 / 4/ 17 / 1/ 18 / 1/ 19 / 2/ 20 / 1/ 21 / 1/ 22 / 1/ z3 / 4/ 24 / V 25 / 1/ 26 / 2/ FED BY 7 15 19 26 X5 X1 X2 23 9 1* 3 5 2 X1 X2 X4 X2 X? X4 2 X2 X5 14 X4 X5 3 23 X3 2 10 14 2 3 ** NO REDUNDANCY FCUND ** Fig. 5-4 (u) 69 NETWORK DEPIVED EY MINI2-NO FANIN/FANOOT LIMITS IN 1 ITERATIONS COST = 13032 TIME ELAPSED = 7 CENTISECONDS 1RUTH TABLE 1=10001 1111 C10001 0100101011 11 1011 ATE .. LEVEL FED BY 7 15 19 26 X5 X1 X2 23 9 16 X1 X2 X4 X2 X3 X4 X2 X5 14 X4 X^ 3 23 X3 2 10 14 23 1 / 1/ 2 / 5/ 3 / 3/ 4 / 1/ 5 / ?/ 6 / 1/ 7 / 2/ 8 / V 9 / 4/ 10 / 3/ 11 / V u / 1/ 13 / 1/ 14 / 5/ 15 / 2/ 16 / 4/ 17 / 1/ 13 / 1/ 19 / 2/ 20 / 3/ 21 / 1/ 22 23 / 4/ 24 / 1/ 25 / 1/ NETWORK DEFIVEE BY GATE MEFGING-NO FANIN/FANOUT LIMITS IN 1 ITERATIONS COST = 13032 TIME ELAPSED = 15 CENTISECONDS TRUTH TPBLE 1 = 10001111101000101001010111110110 Fig. 5-4(v) 70 GATE .. LEVEL 1 / 1/ 2 / 5/ 3 / 1/ 4 5 / 3/ 6 / 1/ 7 / 2/ 8 / 1/ J / 4/ 10 / 3/ 11 / 1/ 12 / 1/ 13 / 1/ 14 / 5/ 15 / 2/ 16 / 4/ 17 / 1/ 18 / 1/ 1 J / 2/ 20 / 1/ 21 / 1/ 22 / 1/ 23 / 4/ 24 / 1/ 25 / 1/ zo / 2/ FED BY 7 15 19 X5 X1 X2 23 9 16 3 5 2 26 X1 X2 X4 X2 X3 X4 2 X? X5 14 X4 X5 3 23 X3 2 10 14 23 A METWOEK DEPIVED BY JEFF COS m = 13032 TIME ELAPSED = 7 CENTISECONDS TRUTH TABLE 1=1000111110100010100101011111011 JS . . LEVEL FED BY 1 2 3 4 5 ' 3/ / 5/ / 3/ / V / 3/ / 1/ 7 15 19 26 X5 XI X2 23 1 9 16 Fig. 5-4 (w) 71 7 / 2/ 8 / 1/ 9 / 4/ 10 / 3/ 11 / 1/ 12 / 1/ 13 / 1/ 14 / 5/ 11 / 2/ / 4/ 17 / 1/ 18 13 / 2/ 20 / 1/ 21 / 1/ 4.1 / 1/ 23 / 4/ 24 / V ^5 / 1/ 26 / 2/ X1 X2 X4 X2 X3 X4 2 X2 X5 14 X4 X5 3 23 X3 2 10 14 23 A NETWORK DERIVED 3Y 1-TH APPLICATION OF PRIFF COST = 13032 TIME ELAPSED - 2? CENTISECONDS TRUTH TABLE 1 = 1000111110100010100101011111011 GATE .. LEVEL FED BY 7 15 19 26 X5 X1 X2 23 9 16 X1 X2 X4 X2 X3 X4 X2 X5 14 X4 X5 3 23 X3 2 10 14 23 1 / 1/ 2 / 5/ 3 / 3/ 4 / 1/ 5 / 3/ 6 / 1/ 7 / 2/ 8 / 1/ 9 / 4/ 10 / 3/ 11 12 / 1/ 13 / 1/ 14 / 5/ 15 / 2/ If / 4/ / 1/ 18 / 1/ 19 / 2/ 20 / 1/ 21 / 1/ 22 / 1/ 23 / 4/ 24 / 1/ 25 / 1/ 26 / 2/ Fig. 5-4 (x) 72 ** NO REDUNDANCY FOUND ** A SETWCPK DERIVED BY 1-TH APPLICATION OF PROCIV COST = 13032 TIME ELAPSED = 190 CENTISECONDS TRUTH TABLE 1=10001111101000101001010111110110 TE - • LEVEL FEE Bl r 1 / V 7 15 19 26 2 / 5/ X5 3 / 3/ X1 X2 23 4 / 1/ 5 / 3/ 9 16 6 / 1/ 7 / 2/ 3 5 3 / 1/ 9 / 4/ X1 2 10 / 3/ X2 11 / 1/ 12 / 1/ 13 / 1/ 14 / 5/ X4 15 / 2/ X2 X3 X4 2 16 / 5/ X2 X5 14 17 / 1/ 18 / 1/ 19 / 2/ xu X* 3 23 2 J ', 1* 22 / 1/ 23 / 4/ X3 24 / 1/ 25 / 1/ 26 / 2/ 2 10 14 23 ** NO REDUNDANCY FCUND ** Fig. 5-4 (y) tj 53 r-- <*■) CD ul r-- ro 73 > w O UJ IT O so EH CO O (N * * 03* * * * to* * M* * 33* * t<* * * * «* * C'# * fc,* * * * x* * 01* * •** * JG* * X3* * D* * to* * * * * * * * * * 10 w pi o o o o M O o o o I in PJ r- r~ O O O o o o U »- f S3 O u 74 5-5 (g). The second problem is under the restrictions TFI = TFO = TFOX = TFOO = 2 and LREST = 6. The initial network shown in Fig. 5-5 (h) is level- restricted but not fan-out restricted: the fan-outs of external variable x and X- exceed the limit. After application of the transduction procedures, the fan-out of external variable x is reduced to 2 but the fan-out problem of external variable x still cannot be solved. Therefore, another initial network with 7 levels is obtained in Fig. 5-5(n),(o). This initial network is fan- in/van-out restricted, although not level-restricted. The transduc- tion procedures are applied to try to reduce the number of levels of this network. The final results is shown in Fig. 5-5(s) and (t) , where the net- work cost is reduced, but the network is still not level-restricted. This means that no feasible network can be obtained by the control sequence 0PT4 under the given restrictions. In the summary table, six asterisks are printed for problem 2 under the column entitled with BEST COST to mean that no feasible solution can be obtained. 75 * * X to u o o o o SO m ** * »-* * * #* o* 55* * w* ►0* CD* o* re* * * ** m x r- X > CM X r— X II u X » X X ns O X w II CO o Q Cm En M CQ I CO * * * ro CN o o o II II 10 CO w ti *c -1 o CQ H EH <«* H 55 M U w CG a H «S o U > CM H Cm CM Cm Cm o O CO O cr; « o w w cu CQ H ac SH 10 p C5 O 55 55 U CO X to w CQ *u M > Q po EH ss CO s co ►0 ac o o a CO sc CO -i II CQ II O to a; wto a, »-iw CQH CO «u<: tn HU H en a* ««EH « >E> O Cm Cm tE-» II x» Q fflMOMW m hIh UlCri WCm 00>H || CmCkCOU II -JW HHH Cm 55 S3 ==> 3 Cm to HOOOO 1 1 1 1 W us 55 as so i£ •t-} o CG H SB a o m l •H Pm t * OrCr O r- oou»~ Mr-MO E-ir-fHO OOUO PS SB CD O Cm Cm Em O 76 THIS INITIAL NETWORK IS FOUND BY TISLEV METHOD INITIAL NETWORK COST = 15029 TIKE ELAPSED = 57 CENTISECONDS TRUTH TABLE 1 = 1 1 1 1 2 = 1 1 1 1 GATE .. LEVEL FED BY 1 / 1/ 6 7 8 2 / 1/ 10 12 13 3 / 4/ XI 4 / 4/ X2 5 / 4/ X3 o / 2/ 3 9 10 7 / 2/ 4 5 14 3 / 2/ 14 15 16 / 1/ / 1/ / 4/ / 4/ / 4/ / 2/ / 2/ / 2/ / 3/ / 3/ / 1/ / 2/ / 2/ / 3/ / 3/ / 3/ 9 / 3/ 4 5 10 / 3/ X2 X3 11 12 / 2/ 14 16 13 / 2/ X1 15 i4 / 3/ 3 15 / 3/ 4 16 / 3/ 5 ** LEVEL RESTRICTED ** A NETWORK DERIVED BY 1-TH APPLICATION OF SOBSTI & HINI2 CC3 T = 1^02° TIME ELAPSED = 12 CENTISECONDS TRUTH TABLE 1=01101001 2 = 00010111 Fig. 5-5 (b) 77 TJS . . LEVEL FED BY 1 / 1/ 6 7 8 2 / 1/ 10 12 13 3 / 4/ X1 4 / 4/ X2 5 / 4/ X3 6 / 2/ 3 9 10 7 / 2/ 4 5 14 6 / 2/ 14 15 16 9 / 3/ a e 10 / 3/ X2 X3 11 / 1/ 12 / 2/ 14 16 13 / 2/ X1 15 14 / 3/ 3 15 / 3/ 4 16 / 3/ c ** LEVEL RESTRICTED ** A NETWOPK DERIVEE BY 1-TH APPLICATION OF GTHERG COST = 13024 TIKE ELAPSED = 32 CENTISECONDS TRTTTH TABLE 1=01101001 I - 1 1 1 1 GATE 1 2 3 5 6 7 8 10 11 12 13 14 n ** LEVEL EVEL FED BY / 1/ 6 7 8 / V 10 12 / 4/ X1 / 4/ X2 / 4/ X3 / 2/ 3 9 10 / 2/ U 5 14 / 2/ 14 15 / 3/ u 5 / 4/ 72 X3 / 1/ / 2/ 14 / 1/ / 3/ t M* 10 L RESTRICTED ** Fig. 5-5(c) 78 A NETWOEK DERIVEE BY 2-TH APPLICATION OF GTMERG COST TIME = 13024 ELAPSED CENTISECONDS TRUTH TABLE 1=01101001 2 = 00010111 TE . - LEVEL FED BY 1 / 1/ 6 7 8 2 / 1/ 10 12 3 / 4/ XI 4 / I*/ X2 5 / 4/ X3 6 / 2/ 3 9 10 7 / 2/ a 5 14 d / 2/ 14 15 9 / 3/ 4 5 10 / 4/ X2 X3 11 / 1/ 12 / 2/ 9 14 13 / 1/ 14 / 3/ 3 15 / 3/ 10 1o / 1/ ** LEVEL RESTRICTED ** A NETWORK DERIVED BY 1-TH APPLICATION OF PFIFF COST = 12024 TIME ELAPSED = 15 CENTISECONDS TRTTTH TABLE 1=01101001 2 = 00010111 Fig. 5-5 (d) GATjS 79 LEVEL FID BY 1 / V 6 7 8 2 / 1/ 10 12 3 / 4/ X1 4 / 4/ X2 5 / 4/ X3 6 / 2/ •? 9 10 7 / 3/ XI a 5 8 / 2/ X1 X2 X3 9 / 3/ u 5 10 / 3/ X2 X3 11 / 1/ 12 / 2/ 7 14 13 / 1/ 14 / 3/ 3 15 / 1/ 16 / 1/ ** LEVEL RESTRICTED ** A NETWORK DERIVED BY 2-TH APPLICATION OF PEIFF COST = 120 2U TIFF ELAPSED = 22 CENTISECONDS T3UJTH TABLE 1=01101001 2 = 00010111 EVEL FEE BY GATE 1 2 3 4 5 D 7 & 3 10 11 \± 13 X\ 16 ** LEVEL RESTRICTED ** / 1/ f 7 8 / 1/ 10 12 / 4/ X1 / 4/ X2 / 4/ Y3 / 2/ 3 9 10 / 2/ a 5 14 / 2/ X2 X3 14 / 3/ a c / 3/ X2 X3 / 1/ / 2/ X1 9 / 1/ / 2/ 3 / 1/ / 1/ Fig. 5-5 (e) 80 A NETWORK DERIVED BY 1-TH APPLICATION OF PPOCIV COST = 12024 TIKE ELAPSED = 82 CENTISECONDS TRUTH TABLE 1=01101001 2 = 00010111 TE . . LEVEL FED BY 1 / 1/ 6 7 8 2 / 1/ 2? 12 3 / 4/ 4 / 4/ . X2 5 / 4/ X3 6 / 2/ 3 9 10 7 / 3/ X1 4 5 8 / 2/ X1 X2 X3 J / 3/ 4 5 10 / 3/ X2 X3 11 / 1/ 12 / 2/ 7 14 13 / 1/ 14 / 3/ 3 15 / 1/ 16 / 1/ ** LEVEL PESTRICTED ** A NETWORK DERIVED BY 1-TH APPLICATION OF PROCCE COST - 12024 TIME ELAPSED = 85 CENTISECONDS Fig. 5-5(f) 81 TRUTH TABLE 1=0-1101001 2 = 00010111 GATE .. LEVEL FED BY 1 /I/ 6 7 8 2 / 1/ 10 12 3 / 4/ X1 4 / 4/ X2 5 / a/ X3 6 / 2/ 3 9 10 7 / 3/ XI a 5 b / 2/ XI X2 X3 } / 3/ U 5 10 / 3/ X2 X3 U / 1/ 12 / 2, / 2/ 7 1U 13 / 1/ 14 / 3/ 3 16 / 1/ ** LEVEL RESTRICTED ** ** THIS 13 A FEASIBLE SOLUTION ** ********************** * PROBLEM NO. 2 * ********************** NUMBER OF VARIABLES = 3 NUMBER OF FUNCTIONS = 2 COST COEFFICIENT A = 1000 B = 1 Fig. 5-5 (g) 82 UNCOMPLEMENTED VARIABLES X FAN-IN = 2 FAN-OUT = 2 FAN-OUT FOR EX. VARIABLES FAN-OUT FOR OUTPUT GATES » NO, OF LEVELS = 6 THIS INITIAL NETWORK IS FOUND BY TISLEV METHOD INITIAL NETWORK COST = 18029 TIFE ELAPSJD = 117 CENTISECONDS TR'JTH TABLE 1 = 1 1 1 1 i 4. = 1 1 1 1 GAliS .. LEVEL FEE 3Y 1 2 3 4 5 6 7 8 10 11 12 13 14 15 16 17 13 19 20 21 22 / V / 1/ / 5/ / 1/ / 6/ / 2/ / 2/ / 3/ / 4/ / 3/ / 5/ / 4/ / 1/ / 4/ / 4/ / 1/ / 5/ / 1/ / 6/ / 5/ / 5/ / 6/ X1 X3 8 10 o 11 14 X2 5 20 3 9 11 12 n 22 17 21 21 XI X2 X2 19 c XI FaS-OUT PROBLEM IK EXTERNAL VARIABLE X1 = 3 FAN-OUT PROBLEM IN EXTERNAL VARIABLE X2 = 3 ** LEVEL RESTRICTED ** Fig. 5-5 (h) 83 A NETWORK DERIVED BY 1-TH APPLICATION OF SU3STI & MINI2 COST = 17028 TIME ELAPSED ■ 32 CENTISECONDS TRUTH TABLE J = 1 1 1 1 2 = 00010111 GATE LEVEL 1 / 1/ 2 / 1/ 3 / 5/ 4 / 1/ 5 / 6/ 6 / 2/ 7 / 2/ 8 lit 3 10 / 3/ U / 5/ / tt/ 13 / 1/ 14 / a/ 15 / u/ 16 / V 17 / 6/ 18 / 1/ 13 / 1/ 20 / 5/ 21 / 5/ 22 / 6/ FFD BY 6 7 8 X1 73 8 9 10 11 iU2 1a 15 x m 20 21 3 21 X1 X2 17 X1 FAN-OUT PROBLEM IN EXTERNAL VARIABLE X1 ** IEVEL FESTRICTED ** Fig. 5-5(i) 84 A NETWORK DERIVED BY 2-TH APPLICATION OP SUBSTI S MINI! COST = 17028 TIME ELAPSED 1U CENTISECONDS TRUTH TABLE 1=01101001 2=00010111 GATiS .- LEVEL 1 / 1/ 2 / 1/ 3 ' 5/ 4 / 1/ 5 / 6/ 6 / 2/ 7 / 2/ a / 3/ 9 / a/ 10 / 3/ 11 / 5/ 12 / 4/ M / 1/ / 4/ 15 / 4/ 16 / 1/ 17 / 6/ 18 ' 1/ 19 / 1/ 20 / 5/ 21 / 5/ *2 / 6/ FED BY 6 7 8 X1 X3 8 10 9 11 1ti X2 5 20 •3 9 11 12 22 15 Vi 21 21 X1 X2 17 X FAN-OUT PROBLEM IN EXTEPNAL VARIABLE X1 = ** LEVEL RESTRICTED ** A NETWORK DERIVED BY 1-TH APPLICATION OF GTMERG COST = 17C28 THE ELAPSED = 27 CENTISECONDS TRUTH TABLE 1=01101001 2=00010111 Fig. 5-5 (j) 85 TE . . LEVEL FED BY 1 / 1/ 6 7 2 / 1/ 8 3 / 5/ X1 «• / 1/ 5 / 6/ X3 6 / 2/ iSi? 7 / 2/ 8 / 3/ 9 12 9 / 4/ 11 22 10 / 3/ n is 11 / 5/ X2 22 u / u/ 5 17 / V 14 / 4/ 20 21 15 / 4/ 3 21 16 / 1/ 17 / 6/ T1 X2 16 / 1/ 13 / 1/ 20 / 5/ 17 21 / 5/ c 22 / 6/ X1 FAN-OUT PPOBLFM IN EXTERNAL VARIABLE X1 = ** LEVEL RESTRICTED ** A NETWORK DERIVED BY 1-TH APPLICATION OF PRIFF COST = 17C28 TlilE ELAPSED = 2U CENTISECONDS TRUTH TABLE 1=01101001 2 = 000101^1 GATL h . . LEVEL FED BY 1 / 1/ f 7 2 / 1/ e 3 / 5/ X1 ft / 1/ j / 6/ X3 6 / 2/ 8 9 7 / 2/ 10 11 9 12 8 / 3/ Fig. 5-5 (k) 86 1 10 J <4 5 6 7 I 20 21 22 / 4/ / 3/ / 5/ / 4/ / 1/ / 4/ / a/ / V / 6/ / V, / / 5/ / 5/ / 6/ --11 1U 15 x: v> X3 20 21 22 X1 X2 17 5 X1 FAS-OrjT PROBLEM IN EXTERNAL VARIABLE X1 ** LEVEL RESTRICTED ** A NETWORK DERIVED BY 1-TH APPLICATION OF PPOCIV COST = 16027 TIKE ELAPSED = 18 CENTTSECONDS TiTJTH TABLE 1=01101001 2=00010111 GATE . . LEVEL FFD BY I / 1/ f 7 / 1/ 8 3 / 5/ X1 4 / 1/ 5 / 6/ X"> 6 / 2/ "fa 9 7 / 2/ 10 11 a / 3/ 9 12 9 / 4/ 11 22 10 / 3/ 14 15 11 / 5/ X2 22 u / / fc 5 17 14 / 4/ X? 20 15 / 4/ 3 12 16 / 1/ 17 / 6/ X1 X2 18 / 1/ 1 9 / 1/ 20 / 5/ 17 21 / 1/ 22 / 6/ X1 FArf-OUT PROBLEM IN EXTERNAL VARIABLE X1 = ** LEVEL RESTRICTED ** Fig. 5-5(£) 87 A NETWOFK DERIVED BY 2-TH APPLICATION OF PPOCIV COST = 16027 TIME ELAPSED = 142 CENTISECONDS TRUTH TABLE 1=01101001 2=00010111 TE . . LEVEL FED BY 1 / 1/ 6 7 2 / 1/ 8 3 / 5/ X1 4 / 1/ 5 / 6/ X3 6 / 2/ 8 9 7 / 2/ 10 11 8 / 3/ 9 12 3 / a/ •j 11 10 / 3/ 1U 15 11 / 5/ X2 22 *i 2 / 5/ 5 17 13 / 1/ 14 / 4/ X3 20 15 / |/ 3 12 1o / 1/ 17 / 6/ X1 X2 18 / 1/ 19 / 1/ 20 / 5/ 1 -7 21 / 1/ 22 / 6/ X1 FAH-OOT PROBLEM IK EXTERNAL VARIABLE X1 = ** LEVEL RESTRICTED ** Fig. 5-5 (m) 88 ». NETrfOFK DERIVED BY 1-TH APPLICATION OF PBOCCE COST = 16077 TIME EIJPSED = 197 CENTISECONDS TRUTH TABLE 1 = 01101001 I = 1 1 1 1 I J . . LEVEL FFD BY 1 / 1/ 6 7 2 / 1/ e J / 5/ X1 4 / 1/ 5 / 6/ r < b / 2/ "8 9 7 / 2/ 10 11 3 / 3/ 9 12 9 / 4/ 3 11 10 / 3/ 14 15 11 / 5/ X2 22 12 / 5/ 5 17 13 / 1/ 14 / 4/ X3 20 15 / 4/ 3 12 1o / V 17 / 6/ X1 X2 13 / 1/ 19 / 1/ *0 / 5/ 17 21 / 1/ 22 / 6/ X1 FAS-OUT PROBLEM IN EXTERNAL VARIABLE X1 - ** LEVEL RESTRICTED ** *** TRY ANOTHER INITIAL NETWORK *** INITIAL NEIWORK COST = 19030 TIME ELAPSED = 75 CENTISECONDS Fig. 5-5 (n) 89 iHUTH TABLE 1=01101001 2 = 00010111 GkTd LEVEL FED BY 1 / 1/ 6 7 2 / «/ 16 17 3 / 7/ X1 4 / 7/ X2 5 / 1/ 6 / 2/ 8 9 7 / 2/ 10 11 / 3/ 2 9 / 6/ 4 2 2 10 / 3/ 14 ^5 11 / 3/ 19 22 12 / 1/ 13 / 1/ 14 / u/ 19 21 15 / 4/ « 21 16 / 5/ X3 9 17 / 5/ X2 23 Id / 1/ u / 5/ 4 20 / 6/ X3 ±\ / 5/ 20 22 / 7/ X1 li / 6/ 3 ** NOT LEVEL PESTEICTED ** A NFTWOFK DEPIVEE BY 1-TH APPLICATION OF SUBSTI 8 MINI2 COST = 19030 TIME ELAPSED = 19 CEMTISECONDS TRUTH TABLE 1=01101001 2 = 00010111 GATii 1 2 3 4 LEVEL / 1/ / 4/ / 7/ / 7/ FED BY 6 7 16 17 X1 X2 Fig. 5-5(o) 90 b / 1/ 6 / 2/ 8 9 7 / 2/ 10 11 8 / 3/ 2 i / 6/ 4 22 10 / 3/ 14 15 11 / 3/ 19 22 12 / 1/ 13 / 1/ 14 / u/ 19 21 15 / 4/ 3 21 16 / 5/ X3 9 M / / fc X2 23 19 / 5/ 4 20 / 6/ X3 l\ / / fc i? 23 / 6/ 3 ** NOT LEVEL RESTRICTED ** A NETWOPK CEFIVED BY 1-TH APPLICATION OF GTMERG COST = 19030 TIME ELAPSED = 34 CENTISECONDS TRUTH TABLE 2 = 00010111 GATE -. LEVEL FED BY 1 / 1/ 6 7 2 / 4/ 16 17 3 / 7/ X1 4. / 7/ X2 I / 1/ 6 / 2/ 8 9 7 / 2/ 10 11 d / 3/ 2 ^ 9 / 6/ 4 22 10 / 3/ 14 15 11 / 3/ 19 22 12 / 1/ 14 / 4/ 19 21 15 / 4/ 3 21 16 / 5/ X3 9 17 / 5/ X2 23 18 / 1/ Fig. 5-5(p) 91 iy 20 21 22 23 ** NOT / b/ 4 / 6/ X3 / 5/ 20 / 7/ X1 / 6/ 3 LEVEL RESTRICTED ** ft NETWORK DERIVED BY 1-TH APPLICATION OF PPIFF COST TIME = 1Q030 ELAPSED 22 CENTISECONDS TRUTH TABLE 1=011010 01 2 = 00010111 GATii 1 2 3 4 5 6 7 d ) 10 11 12 13 14 15 16 17 18 19 20 21 <:2 23 ** LEVEL FED FY NOT / 1/ 6 7 / 4/ 16 17 / 2/ X1 / 7/ X2 / 1/ / 2/ fi 9 / 2/ 10 11 / 3/ 2 / 6/ 3 4 / 3/ 14 15 / 3/ 2 17 / V / 1/ / 4/ 19 21 / 4/ 21 22 / 5/ X3 9 / 5/ 19 23 / 1/ / 6/ 4 / 6/ X3 / 5/ 20 / 7/ X1 / 6/ 22 LEVEL RESTRICTED ** Fig. 5-5 (q) 92 A NETWORK DERIVED BY 1-TH APPLICATION OF PROCIV COST = 13021 TIME ELAPSED = 238 CENTISECONDS TRUTH TABLE 1=01101001 2 = 00010111 Id - . LEVEL FED BY 1 / 1/ 6 7 2 / 4/ 16 17 3 / 1/ 4 / 7/ X2 5 / 1/ 6 / 2/ 8 9 7 / 2/ 10 11 8 / 3/ 2 9 / $/ U 22 10 / 3/ 20 11 / 3/ 2 17 12 / 1/ 13 / 1/ 14 / 1/ 15 / 1/ 16 / 5/ X3 9 17 / 5/ X1 X2 18 / 1/ 1J / 1/ 20 / 4/ X3 21 / 1/ 22 / 7/ X1 2} / 1/ ** NOT LEVEL RESTRICTED ** A NETWORK DERIVED BY 2-TH APPLICATICN OF PROCIV COST = 13021 TIME ELAPSED = 120 CENTISECONDS IRUTH TABLE Fig. 5-5 (r) 1=011010 01 2 = 00010111 93 JATK •- LEVEL 1 / V 2 / a/ 3 ' 1< / 7/ 4 3 / 1/ 6 / 2/ 7 / 2/ 3 / 3/ i / 6/ 10 / 3/ 11 / 3/ -\l / 1/ w / 1/ 15 / 1/ 16 / 5/ 1 7 / 5/ Id / 1/ 1^ / 1/ 20 / 4/ 21 / V / 7/ 22 23 / 1/ FED BY 6 7 16 17 X2 6 9 10 11 2 4 22 20 2 17 X3 9 X1 X2 X3 X1 NOT LEVEL RESTRICTED ** A NETWOFK DERIVED BY 1-TH APPLICATION OF PROCCE COST = 13021 TIKE ELAPSED = 115 CENTISECONDS TRUTH TABLE 1=01101001 2 = 00010111 GATE £ . . LEVEL 1 2 3 4 / 1/ / 4/ ' V / 7/ 5 6 / 2/ FED BY e 7 16 17 X2 8 9 Fig. 5-5(s) 94 -» o m in «- «o (J rr\ fi * * * • * •• ♦ o« * SB* » * » W« #■-«• • cc* #o* • 0'» « Cu» • • » • #»* O a M • # »S5» • D« » o-» • • • in* »M# • X* *H» « • »«# • o» • lb* • • #»• #BS* • <<• » C* • »# • O* • tn» o <"> . When the specified time is to expire but the problem currently under execution is not finished, then the intermediate results are punched on the cards so that the user can use these cards to run this problem next time. During the execution of a problem, the computation time, TIME, spent so far will be compared with the specified limit, TLIM, each time after the initial network is obtained or after the transformation or the transduction is ap- plied on a network.' If the comparison satisfies one of the following three inequalities , then this problem will be executed continually. Otherwise the intermediate results will be prepared. TIME + 2 sec 126, or when TIME > 6 sec. 96 The problem is not finished because the specified time is not enough. So some necessary information is printed in Fig. 6-l(b), the intermediate results are punched and the system goes to read another problem. Since only one problem is included, the summary table is printed and then the execution is terminated. The information shown in Fig. 6-1 (b) is for rerunning the program and can be ignored by the user. (In Fig. 6-1 (b), six asterisks are printed under the column entitled with "BEST COST", because no result which satisifes the given restrictions is obtained yet) . The punched deck contains the following cards: (1) The first four fields of the < specification card > are specified as 1, since we prepare the < heading card >, the < problem parameter card >, the < output function card >s and the < control sequence card >s for each unfinished pro- blem. The last two fields contain the same values as the original problem; i.e., the same values for PUNC and TLIM. The user may change the value of TLIM before rerunning the program . (2) The < heading card > usually has the following format: OLD PROB. # X , FROM INITIAL NETWORK XXXX, INTERMEDIATE RESULTS | (3) The values of the parameters, except the last two fields, on the < problem parameter card > are the same as the original N-l problem. Parameter NEPMAX has the value 2 if this field was blank in the original problem; and parameter RERUN has the value 1 to indicate that this is not the first time that 97 to u o o VO I £ I *** # * * O {/) H pq * * H m r- O 1 r- 3 * ■* 5* S3 W •-) II CQ II O to Cfi WtO CI, ►-1W CQ6-I W *:■<: x MO H w o r>»-o CU Cu • Ft II XD Q ^twotnw •3- hlH KWW&) 00>H || fchWU II hIW F»FiH Oi ten cd ft, co HOOOO 1 1 I 1 w aazz ii; «<« -i a § ^ to to w 55 •oj »-) o CQ H H «: Ft 5n H U w « 25 H *< a U > fM H Cv, fe, Pn &< O o M O en CK u UJ w cu cu Ft S3 S3 to D » O ss *s U «C Cu Oj S3 u 2 ST. a => w > £ £ K , ft P o I to I O fc:.0 * 5 ° # O c>o O hr W F>*- to uc Sr. M CD O PL, F' O u (N cu o * » * * * * .» * CQ* * O* * r>* * * * w* * a:# * E-<* * » * e>* * as* * H* * 3* * 2* * H* * H* « 2* * o* * o* * * * cq* * o* * Pu* * * * *:* » U# * W* 98 _. m w fN u r- *— ' w *2 H H * Q* * * * Q* * W* * K* * U* * 25* * o* * c* * * * w* * a=* * H* * * * w* * w* * O* * ■# » « * t* * >h* * M* * hJ* * «!* * O* * H* * Eh* * «<* * £* * n* * H* *D* *«** * * #Q* * W* *Cn* * Cu* * O* * Eh# * w* * # * v>* * «n* * 3t» * » * t* * a* * o# » «* * a.* * * * w* * M* * DC* * H* » * * »* * a* * w* * ft.* * M# * cu* * x# » w* « * * W* * 03* * * * O* * H* * ♦ * W# * £« * M* * M* * * * ** w o asrs o M H cu o kJO MO > &H E-<0 WO UO E- w ►J o Pu O o •- EH W o * u * » H * W * u * m w EH W w H W Cm W Eh C/l CG«- W EH ;»0 P0 -c H W * * 05* * * * W* *M* * 35* * Eh* * * * 01* * O* * P-.* * * ft** * 03* * «e* * s* * sc* * D* * w* * * * * * * * * * £h O W O W 1- o o 6h o O I * Pu M 3- Pu CD »- «: o o o 5= IT O O u 99 the problem will be run. (4) The < control sequence card >s are divided into five sets of cards (usually five cards) . The first set of cards, consisting of one card, contains the names of the initial network methods which were specified for the orig- inal problem but have not been applied. It has the following format: /INTP EXNT XXXX • ■ • XXXX initial network methods which have not been ap- plied the network obtained so far will be an ini- tial network next time The second set of cards (usually one card is enough) corresponds to the specification of the transduction procedures under no fan-in/fan- out restrictions and no level restriction. The corresponding format is shown below. XXXX X number of iterations /TDTP XXXX X ' / v name of the transduction procedure The third set of cards have only one card. It has 4 characters, JEFF, only. If "JEFF" is not specified on the control sequence cards in the original problem, then this card is not punched. The fourth set of cards correspond to the fan-in/fan-out re- stricted and/or level-restricted transduction steps. The corresponding format is shown below. 100 FLTP XXXX X XXXX X ^ \ name of the number of interations transduction procedure The fifth set of cards consist of only one card. It contains the number of iterations that the TT-sequence to be applied. The corresponding format is shown below. NOIT X ^ STOP | ^^ number of iterations (5) The < output function card >s are the same as the original problem. (6) The < current status card > is a new type of card which con- tains the information about the status of the current pro- blem. This information will be used for rerunning the pro- blem. (7) The < connection-description card >s for the network which which was just processed when the problem stopped have the same format as the < connection-description card >s for the input data. The network described by these cards will be treated as the initial network for the next run. (8) The < connection-description card >s for the best network obtained so far are also punched (if there is no network satisfying the given restrictions, then these cards do not exist) . This network will also be read in and stored so that we can keo the "real" best network at the end of the processes. 101 Fig. 6-2 gives the punched deck for the example shown in Fig. 6-1. Notice that there are no < connection-description card >s for the best net- work since no network which satisfies the given restrictions has been ob- tained. Another thing which should be mentioned is that the character "//" which was specified in the original < control sequence card >s is now re- placed by 99, i.e., the selected transduction procedure or the TT-sequence will be executed at most 99 times. Apparently this is equivalent to exe- cuting it repeatedly until there is no further improvement in the cost. 126 'OLD PROB. # 1, FROM INITIAL NETWORK BANDB . INTERMEDIATE RESULTS r 5 11000 1 X 4 4 4 4 100 1 16 flNTP EXNT 'TDTP NTG3 99 NTG1 99 'JEFF 'FLTP NTE1 99 'N0IT 99 STOP '01101001100101101001011001101001 1110 2000000 626100 1 1 2 913 2 3 4 5 627 3 5 6192123 4X3 5X214 6X11525 9 3111615212426 '11X2X418 13X315161921 14X115 15X1X3202227 16X2X3232425 18X4X52526 19X4 18 20 1 ^26 20X1X3X41826 21X5182224 22X1X3X51824 23X2X4 24X2X3X526 25X1X224 26X2X3 'X427 27X1X2X4X5 ** 1. < specification card > 4. < control sequence card >s 2. < heading card > 5. < output function card >s 3. < problem parameter card > 6. < current status card > 7. < connection description card >s for the current network Fig. 6-2 Punched deck for the unfinished job in Fig. 6-1 102 REFERENCES [1] Hu, K.C. and S. Muroga, "NOR(NAND) Network Transduction System, (The Principle of NETTRA System)," Report No. UIUCDCS-R-77-885, Department of Computer Science, University of Illinois, Urbana, Illinois. [2] Hu, K.C, "Level-Restricted NOR Network Transduction Programs," Report No. UIUCDCS-R-77-849, Department of Computer Science, University of Illinois, Urbana, Illinois. [3] Su, Y.H. and C. W. Nam, "Computer-Aided Synthesis of Multiple Output Multi-Level NAND Networks with Fan-In and Fan-Out Constraints," IEEE Trans. Comput . , Vol. C-20, December, 1971. BIBLIOGRAPHIC DATA HEET . Tit le and Subtitle ROGRAM MANUAL: 1. Report No. UIUCDCS-R-77-887 3. Recipient's Accession No. FOR THE NETTRA SYSTEM 5. Report Date August, 1977 . Author(s) K. C. Hu 8- Performing Organization Rept. No. . Performing Organization Name and Address Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 61801 10. Project/Task/Work Unit No. 11. Contract/Grant No. sISF DCR 73-03421 2. Sponsoring Organization Name and Address National Science Foundation 1800 G Street, N.W. Washington, D.C. 20550 13. Type of Report & Period Covered Technical 14. 5. Supplementary Notes This is the program manual for the NETTRA system which can design near-opti- al, multi-level and loop-free NOR(NAND) networks under fan-in/fan-out restriction and/ r level restriction. Given function(s) may be completely or incompletely specified rid both complemented and uncomplemented external variables are permitted as inputs, he user can specify the control sequence (the types of the initial network methods and he types and the order of the transduction procedures to be applied) to solve his roblem. Besides, four control sequences are provided for the users who are not in- erested in the details of how to specify the control sequence. Facilities for treating nfinished jobs due to the expiration of computation time are also provided by the ystem. 1. Key Words and Document Analysis. 17a. Descriptors Logic design, logic circuits, logic elements, programs (computers). 'b. Identifiers/Open-Ended Terms Computer-aided design, transduction procedures, transformations, near-optimal net- works, optimal networks, fan-in, fan-out, level, NOR, NAM), NETTRA system, integer programming logic design, program manual. 'e. COSATl Field/Group • Availability Statement Release Unlimited "M NTIS-35 ( 10-70) 19. Security Class (This Report) _,, UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 22. Price USCOMM-DC 40329-P71 OCT 2 'MO v>unoj; UNIVERSITY OF ILL INOIS URBANA 510 84 II 6R no COO? no 886 893(1977 Generating binary treat lexicographical 3 0112 088403594 wm HI KB H ■ l ■'.V. I HI H Pi H mm @ ■ II