A Theory of Non-Deterministic Networks

Transcription

1 A Theory of Non-Deternstc Networs Alan Mshcheno and Robert K rayton Departent of EECS, Unversty of Calforna at ereley {alan, brayton}@eecsbereleyedu Abstract oth non-deterns and ult-level networs copactly characterze the flexblty allowed n pleentng a crcut A theory for representng and anpulatng non-deternstc (ND) ult-level networs s developed The theory supports all the networ anpulatons coonly appled to deternstc bnary networs, such as node nzaton, elnaton, and decoposton It s shown that an ND networ s behavor can be nterpreted n three ways, all of whch concde when the networ s deternstc Operatons perfored on an ND networ are analyzed under each nterpretaton for changes n a networ s behavor Modfcatons of a few operatons are gven whch ust be used to guarantee that a networ s behavor does not volate ts external fcaton These odfcatons depend on whch behavor s beng used and the locaton of related non-deterns Ths theory has been pleented n a syste, MVSIS We provde coparsons aong the uses of the varous behavors 1 Introducton A non-deternstc (ND) networ s slar to a oolean networ, except that, n general, each node has a ult-valued (MV) output and s represented by a non-deternstc relaton The falar don t cares used n logc synthess are a al for of non-deterns For exaple, don t cares fy for soe nput nters, the output can tae any of the values n ts range, whle ore generally, non-deterns occurs when, for an nput nter, the output can tae values fro a subset of values n the range of the output Non-deterns arses naturally n a synthess settng For exaple, a syste s fcaton ay be gven by an ND networ or autoaton Part of the syste ay be gven also To be syntheszed s an unnown sub-coponent The set of all possble behavors for the unnown can be derved as an ND relaton or ND autoaton, usng copleentaton and coposton operatons 0 In logc synthess, an ntal networ representaton can be gven wth copatble don t cares at the prary outputs Soe RTLs allow ncoplete behavor to be fed at nternal nodes Ths s nterpreted as don t care, e for the unfed nputs, the output can tae any value allowed for the varable Don t cares can also be derved fro a networ s functonalty n ters of observablty (ODC) and satsfablty (SDC) don t cares Generalzaton of these concepts to MV networs leads to the ore general noton of non-deterns Startng fro the ntal fcaton, synthess conssts of operatng on a oolean networ to obtan a saller, faster, ore effcent one, whch fnally s apped nto a set of logc gates for pleentaton n hardware When these operatons are generalzed to account for non-deterns, an analogous networ and set of operatons s desred The use of such networs can lead to fnal ore effcent deternstc bnary pleentatons, snce the generalzaton to MV ND networs allows a larger space for optzaton algorths to explore [3] We defne three networ sulaton odels (SS, NS, ) for ND networs, whch lead to three types of networ behavors A behavor s defned to be the set of all prary-nput praryoutput pars of vectors that can be sulated for the networ All three newly defned behavors reduce to the sae unque behavor f the networ s deternstc In the bnary case, one of these sulaton odels (SS) s analogous to ternary sulaton wth the three values, {0,1,} [1] We analyze how the correspondng ND networ behavors can change under varous coon networ operatons, such as decopose, substtute, elnate, collapse, node nze, and erge [8] It was found that soe of the classcal operatons need to be odfed to account for the effects of non-deterns We also study the lts (flexblty), wthn whch the functonalty of a node n an ND networ can be changed wthout volatng the external fcaton For all behavors, we derve a forula for coputng the coplete (axu) flexblty (CF) allowed at the node The paper s organzed as follows In Secton 2, we defne an ND networ and gve soe notaton Secton 3 dscusses the three ethods for nterpretng the behavor of an ND networ In Secton 4, we gve for each behavor type, ethods for coputng the coplete flexbltes (CFs) at a node and show that these cannot ncrease any of the retve networ behavors when any well-defned sub-relaton of the CF s used to replace the old relaton at the node Secton 5 dscusses the node elnaton process, Secton 6 extracton and decoposton, and Secton 7 ergng In each case, we analyze how the retve operatons can change the three types of networ behavors Secton 8 dscusses the relatve erts of the two coputatonally ore vable behavors, and SS Secton 9 dscusses how each of these behavors can be ade to ft nto a herarchcal theory where a networ can be parttoned and sub-parts can be optzed separately Secton 10 dscusses odfcatons on the two operatons (one for and one for SS), whch could ncrease the correspondng behavor, to ensure that the networ always satsfes ts fcaton Secton 11 dscusses soe experental results whch copare the use of the dfferent behavors n ters of the relatve szes of the flexbltes allowed Secton 12 concludes, suarzng the contrbutons and lstng soe longer-

2 ter goals for the applcaton of ths theory and ts pleentaton ecause of restrcted space, no proofs are gven n ths paper However, all results have been proved and the proofs tested aganst a nuber of readers For the proofs, please refer to a ore extensve report [5] In addton, the theory has been pleented n a syste, MVSIS, ( and experental results are consstent wth the theory Our pleentaton ndcates that runtes penaltes ncurred for the generalzaton to MV and non-deterns are nal ecause ost algorths had to be copletely re-pleented, we used ths opportunty to prove the effcency of the algorths and data-structures Experents ndcate that runtes are about 5 tes faster than SIS, even though all algorths have been generalzed The qualty of results (when run on bnary deternstc exaples) s equal to or better than for SIS 2 ND Networs An ND networ s an acyclc drected graph A node represents an ND relaton between the node s nputs and ts one output An edge s drected fro node to f the relaton at node depends syntactcally on the varable y, assocated wth the output of node The output of node s ult-valued and taes values fro the doan D = {0, L, n 1} Prary nputs (PI) are nodes wth no nputs Prary output nodes (PO) delver the functonalty of the networ to ts envronent Sngle nput and output storage eleent nodes have the next state (NS) varables as nputs, and the present state (PS) varables as outputs Snce ths paper s concerned only wth the cobnatonal porton of the networ, the set (PI, PS) s called the cobnatonal nputs (CI) and represented by the vector, and the set (PO, NS) s called the cobnatonal outputs (CO) and represented by the vector An external fcaton of a networ, R (, ), s the set of all acceptable (CI, CO) nter pars, (, ), such that R (, ) = 1 f and only f the par (, ) s allowed Defnton: A relaton R (, ) s well-defned f for each nput nter, there exsts at least one allowed output nter n the relaton: ( R (, ) = 1) Defnton: Let R (, ) s output-syetrc f for any S1( ) L S( ) (, ) R (, ), where S( ) { v z, R (, z1, L, z 1, v, L, z N) = 1} Exaple Consder a networ wth two bnary outputs, z 1 and z 2 Suppose, for soe nter, the values that the outputs can tae are {00, 01} The relaton R(, ) s output-syetrc for ths nter, because S 1 ={0}, S 2 ={0,1}, and every cobnaton fro the product set {0} {0,1}={00, 01} belongs to the relaton If the sae outputs were to tae values {00, 01, 11} for ths nter,, t would not be output-syetrc because S 1 ={0,1}, S 2 ={0,1}, and there exsts a cobnaton {10} n product set, {0,1} {0,1}={00, 01, 10, 11}, whch does not belong to the relaton Output syetry has been used to defne copatble external don t cares n bnary networs The prarly reason ts use s that the choce of value ade at one output s ndependent of the choce ade at any other output For a general relaton, a choce ade at one output, can restrct the choces allowed at another output, and ths aes t uch harder to deal wth An ND relaton gvng the functonalty of a node n a networ can be fed by the characterstc functon relatng the nputs and output of the node The relaton at a node n the networ s denoted R( Y, y ) where Y s the set of fann varables, and y s the sngle output varable of the node For ease of notaton, soetes the arguents of a relaton wll be used to dentfy t, eg RY (, ) and RY (, y ) denote dfferent relatons even though each s naed R A relaton wth a sngle output s often stored as a set of deternstc ult-valued nput, bnary output functons, the th of whch s 1 for those fann nters that can produce value at the output These are called the -sets of the relaton and each can be represented n SOP (MV) for or as a MDD nary-output, MV-nput functons can be nzed usng a progra le Espresso-MV [2][7], resultng n a nzed MV su-ofproducts (MVSOP) expresson A product ter n an MVSOP s the conuncton of MV-lterals An MV-lteral of a varable y, for S exaple, y, s the bnary functon, whch s 1 f and only f y has a value n the set of values S A saller MVSOP representaton of the relaton can be obtaned by desgnatng one of the -sets as a default, whch s defned as the copleent of the other -sets 1 For bnary relatons, any overlap between the 0-set and the 1-set s called a don t care set, whch s typcally represented as a separate bnary functon In a oolean networ, the 0-set s usually taen as the default and don t cares are derved fro the networ structure (SDCs and ODCs) The notaton R (, ) wll be used to represent the fcaton of the networ An output syetrc fcaton has the advantage that t can be represented by a set of ndvdual sngle-output relatons, one for each CO, e R (, z ), = 1, L, N In the next secton, we defne three types of sulatons for an ND networ, {NS,, SS}, all of whch are the sae as the usual noton of sulaton when the networ s deternstc Defnton: The -behavor of an ND networ s the set of all CI/CO pars that can be sulated usng the sulaton of type 1 Note that a general ND relaton cannot be fully represented ths way because there ay be no -set that s dsont fro the unon of the others

3 , { NS,, SS} The -behavor of a networ s denoted as R (, ) Defnton: An ND networ -confors to, or -coples wth, ts external fcaton f R (, ) R (, ) 3 ehavors of ND Networs Each nterpretaton of the behavors to be defned for an ND networ s assocated wth a partcular type of sulaton odel We defne three, all of whch yeld the sae behavor f the networ s deternstc The nterpretatons to be defned are lsted n the order of ncreasng aount of behavor: 1 Noral sulaton (NS-behavor) 2 Noral sulaton ade copatble (output-syetrc) for all outputs (-behavor) 3 Set sulaton (SS-behavor) We defne each of the sulaton odels and dscuss ther relatve erts In ost applcatons, t s usually approprate to vew NS as the real behavor, and the others as easer-to-copute overapproxatons In anpulatng a networ, t s portant to use only one nterpretaton of a networ s behavor consstently Ths s because n soe operatons, a networ s behavor s perodcally copared wth ts external fcaton Changes are allowed provded they do not cause an ncrease beyond the fcaton Snce an ND networ can satsfy ts fcaton under one nterpretaton but not another, swtchng between dfferent nterpretatons could lead to a fnal networ that does not confor to ts external fcaton 31 ehavor by Noral Sulaton (NS) NS s the ost ntutve type of sulaton of an ND networ Proceedng n topologcal order, each ND node nondeternstcally selects one output value allowed by the current fann nter, and transts ths value to all of ts fanouts For ths type of sulaton, t s easy to obtan sngle pars (, ) of (CI, CO) nters of the behavor However, t s dffcult to obtan all pars, whch s ost often requred; n fact, of the three ethods, NS s the ost coputatonally coplex The coplete NS-behavor can be obtaned by the followng coputaton, NS R (, ) = R ( Y, y ) (31) y nternal nodes NS A par (, ) s n the MV ult-output relaton R (, ) precsely f s gven at the CI, and at each node there exsts a choce that s propagated to ts fanouts, such that fnally the vector appears at the COs NS A ore effcent ethod for coputng R (, ) s to use early quantfcaton of a conunctve relaton as t s done n soe foral verfcaton applcatons Even so, ths coputaton s stll probleatc snce, n general, there s one fnal relaton, whch ust relate all CIs wth all COs In contrast, the other two types of behavors to be dscussed can be represented by N ndependent relatons, each relatng CI vectors,, wth one CO, z, {1, L, N } In these cases, the set of CO vectors related to s the cross product of the sets of values at the ndvdual COs related to Thus, these two behavors produce outputsyetrc relatons 32 ehavor by NS ade Copatble () In, each CO s sulated ndependently wth NS, obtanng a set of relatons: Thus R (, z ), = 1, K, N N = 1 R (, ) R (, z ) s output-syetrc (the set { R (, z )} s copatble) Ths ncreases the behavor over NS snce each node that has ore than one CO n ts TFO s treated ndependently n each of the sulatons for the dfferent COs Ths s called -behavor, snce t represents the operaton of ang the NS-behavor copatble If each ND node has only one CO n ts TFO, then NS and are the sae Collapsng denotes the process of elnatng 2 all the nternal nodes n a networ, one by one, n soe unfed order After a networ s collapsed, only the output nodes rean and ther relatons wll depend only on the CI varables, Theore 31: The -behavor s equvalent to collapsng the networ n reverse topologcal order It s easy to show that collapsng n reverse topologcal order yelds the sallest set of output-syetrc relatons whch contans the NS behavor of the networ In ths sense, t s the sallest easy-to-copute output-syetrc over-approxaton of the NS behavor The followng s a useful observaton Theore 32: The NS and behavors of a networ are not changed by elnatng any deternstc node Thus for a deternstc networ, the order of elnaton durng collapsng s not portant 33 ehavor by Set Sulaton (SS) Set sulaton s perfored as follows Gven a nter, each CI has a sngle value (sngleton set) However, n general, an nternal node can have a subset of the allowed values for that node The sulaton proceeds n a topologcal order When a node s to be sulated, each of ts fanns has been assgned a set of values The node s output s the set of all values possble for that node gven ts fann sets; each fann nter can be taen fro the 2 A ore detaled dscusson of the elnaton operaton can be found n Secton 5

4 product set of the fann sets For exaple, suppose each nput has a set of values, S The output of a node s evaluated as the followng set: 3 S = { v R ( V, v) = 1, V S S L S } 1 2 Y Each fanout edge then receves the set S When all CO nodes have been coputed, the cross product of the CO output sets fors the set of nters { } allowed for Any such a par (, ) s n the SS-behavor of the networ 4, e SS (, ) R (, ) It s easy to observe that the SS-behavor s an outputsyetrc relaton and, hence, can be represented by a set of ndependent relatons, one for each output Slar to, a ey advantage of SS s that the networ can be anpulated as a networ of sngle-output MV nodes In contrast, the use of NSbehavor would lead to ult-output nodes and MV ult-output relatons at these nodes (see [5]) SS-behavor can be shown to be the sae as consderng the ND networ as a set of deternstc bnary nodes, one for each -set of each MV node For exaple, consder a ternary node The -sets of ths node (0-set, 1-set, and 2-set) are represented by MV-nput bnary-output functons In ths bnary nterpretaton, each nternal MV sgnal and each CO s replaced by a set of bnary sgnals and each correspondng lteral n any MVSOP s converted to a su of bnary lterals, eg {1,3,5} y y y y = b1 + b3 + b 5, where y b s the bnary output of the th -set of y The resultng networ s deternstc and can be anpulated le any such networ 5 ascally, ths converson s le representng MV sgnals usng postonal notaton whch allows for the representaton of sets The only sgnals that are ult-valued are the CIs, whch do not have to be converted snce they only carry sngleton sets Theore 33: The SS-behavor of an ND networ can be obtaned by treatng each -set as a separate bnary functon, collapsng the networ (n any order), and ergng each set of bnary outputs assocated wth a CO to for the -sets of that MV output Another ethod for coputng the SS-behavor s the followng Theore 34: The SS-behavor of an ND networ s exactly that obtaned by elnatng all nternal nodes n topologcal order The sae effect can be obtaned by unfoldng the networ nto a tree (usng duplcaton), resultng n a networ where each node has exactly one fanout It turns out that the SS-behavor s unchanged An ND node n ths tree has a unque path to one CO, so the effect 3 Note that even f the node relaton s deternstc, the output set can have ore that one eleent f soe of the nputs are sets wth ore than one value 4 Set sulaton s slar to what s done n ternary sulaton when values 0,1, are propagated stands for the set {0,1} 5 In fact, the networ s unate that an ND node n the orgnal networ can have on the SSbehavor s drectly related to the set of all paths fro the node to a CO Each te the output of an ND node branches to several fanouts, the effect s as f ndependent copes are ade of the ND node are ade As dscussed n subsequent sectons, any networ operaton that ncreases (decreases) the nuber of paths fro an ND node to a CO can ncrease (decrease) the SS-behavor of the networ 34 Coparson and Representaton of ehavors A networ s external fcaton gves the upper bound on the allowed networ behavor The fcaton can be outputsyetrc (ndependent relaton for each output) or a oolean relaton relatng all outputs An output-syetrc fcaton s analogous to gvng copatble external don t cares for a bnary networ Operatons on an ND networ can change ts -behavor, { NS,, SS} An ncrease n behavor s allowed only f t s stll contaned n the fcaton Output-syetrc fcatons have the advantage that they can be stored ndvdually for each output, eg as a set of bnaryoutput -set functons Other fcatons ay requre a sngle global ult-output relaton, relatng all nputs and outputs, whch can easly becoe too large If the fcaton s not outputsyetrc, one opton s to under-approxate t wth an output syetrc one; ths leads to a correct but conservatve approach The node nzaton operaton (as dscussed n Secton 4) uses the external fcaton drectly to test how uch a node s behavor (any of the -behavors) can be ncreased wthout volatng the fcaton In Secton 4, an ND relaton at a node s coputed to descrbe the axu flexblty (coplete flexblty CF) allowed n pleentng the node Dfferent nterpretatons of a networ s behavor wll lead to dfferent flexbltes (-CFs) allowed Node nzaton s the process of solvng for a welldefned sub-relaton of the -CF, whch gves the sallest representaton of the node [4] Usng an ND sub-relaton of the - CF allows for saller representatons 6 Another at s the ease of perforng networ anpulatons usng the dfferent behavors SS-behavor s the ost effcent because t s related to collapsng the networ n topologcal order Ths allows buldng global MDDs of each node, where only CI varables are needed at any stage n the collapsng process s also relatvely easy because collapsng n reverse topologcal order can be used, but buldng global MDDs s slghtly ore dffcult snce nternal varables (but only those representng the outputs of ND nodes) ust be used teporarly n the MDDs NS-behavor requres ether the use of ult-output relatons or nput deternzaton usng pseudo-nputs 6 A nu deternstc sub-relaton can never be saller that a nu ND sub-relaton

5 It s obvous that NS behavor s contaned n -behavor Also, s a subset of the SS-behavor One way to see ths s that n soe copes of ND relatons, whch lead to the sae CO, are ept synchronzed (nterdependent) durng the collapsng process In contrast, wth SS, all correlatons between dfferent fanouts of an ND node are lost when the node s elnated (snce elnaton s done each fanout at a te) As a result, we have, R NS (, ) R (, ) R SS (, ) (33) In Secton 4, t s shown that ths orderng has the reverse effect on the optzaton potentals (flexbltes) coputed usng these behavors, because the coputaton s based on coparng (by contanent) aganst the external fcaton For exaple, f SSbehavor s used, contanent s ore restrctve snce SS-behavor s the largest Thus the use of SS behavor wll lead to less flexblty allowed n pleentng a node On the other hand, SSbehavor s easer to copute wth 4 Node Flexbltes The coputaton of the coplete flexblty, CF, at a node y n an ND networ can be descrbed soewhat genercally for the dfferent behavors { NS,, SS} However, for each behavor, certan odfcatons need to be done Cut the networ at the output of node and consder the new networ (the cut networ), whch has an addtonal ndependent prary nput y Requre that the -behavor of the cut networ, R (, y, ), coples wth the networ fcaton R (, ): R (, y ) ( R (, y, ) R (, )), (41) whch sply says that for all outputs, the cut-networ behavor should be contaned n the fcaton Note that for both and SS, the behavor and the fcaton can be stated n ters of the ndvdual outputs, z, n whch case the coputaton becoes R (, y ) ( R (, y, z ) R (, z )) (42) z whch aes the coputaton for {, SS} uch ore effcent It turns out that f the flexblty for SS-behavor were coputed by Equaton 42, then Theore 42 below, about how t can be used, would not hold We need to odfy the coputaton of SS SS R (, y ) as follows When R (, y, z ) s coputed for the cut networ, t needs to be changed such that t can have set nputs at the y nput node n the cut networ, snce that s what can happen at the output of the y node when SS-sulaton s done on the uncut networ Ths can be done by ntroducng new bnary varables b, whch encode subsets of D, the doan of y For exaple, f D = {0,1,2}, there would be 3 bnary sgnals { b, b, b } as nputs to a odfed cut networ (for exaple, (0,1,1) would encode the subset {1,2} {0,1,2} ) A new node, η s ntroduced n place of node Its nputs are { b0, b1, b 2} and ts output y fans out to the sae nodes as n the orgnal networ The node relaton at set η s denoted R ( b, y ) and serves to translate between the bnary nputs and the MV set outputs Thus, n the exaple, (0,1,1,1) and (0,1,1,2) are n the relaton R set ( b, b, b, y ), and (0,1,1,0) s not SS Then, R ( b,, z ) s coputed for the odfed cut networ SS and ths s used n Equaton (42) to obtan R ( b, ), whch relates to allowed subsets of D Relatons R (, y ) express the Observablty Partal Care (OPC) for the node at output, whch s related to observablty don t cares coputed for a node n a bnary networ Note that the relatons depend on the CIs, Next we brng n what s analogous to satsfablty don t cares (SDC) to derve a local coplete flexblty (CF) Defne M ( Y, ) as the relaton between CI nters and vectors of values that the fann varables, Y, of node can tae durng - sulaton of the entre networ 7 The -CF s coputed by the forula N R ( Y, y) = ( M ( Y, ) R (, y )) 8 (43) = 1 Ths sply says that for all nput nters, those fann nters,, that can be produced by -sulaton ust be Y, related to the correspondng output values of the global flexblty It can be shown that SS NS R ( Y, y ) R ( Y, y ) R ( Y, y ) (44) We cla that each of these s axal, e no addtonal par of nters can be ncluded n any of the relatons whle antanng a vald flexblty relaton 7 A subtle pont s that n general for, ths s not the sae as the -behavor R ( Y, ) of the cut sub-networ whose COs are NS Y, but n fact M (, Y ) = R ( Y, ) Roughly, ths s because nternal nodes and output nodes are treated dfferently n ND networs Ths aes t ore dffcult to copute 8 For NS, there s no product over all outputs snce R NS (, y ) SS taes all outputs nto account For SS, we obtan R ( Y, b ) whch s a (ultple output) oolean relaton In general, to convert ths to an ND ult-valued relaton (sngle output), we need to choose, for each Y, one of the allowed sets as ndcated by there ay be several such sets b For a gven Y,

6 In general, the CFs, R ( Y, y ) for { NS,, SS}, are nondeternstc relatons Snce the current relaton at node, RY (, y ), s well defned and R ( Y, y ) R ( Y, y ), then also R ( Y, y ) s well-defned (assung that the current networ confors to the fcatons) Theore 41: The -CF for node s well-defned f and only f there exsts a relaton for node such that the resultng networ - confors, z TFO( ) z y -confors, z TFO( ) we ean that the contanent of z relatons (behavor s n the ) holds for those outputs n the TFO of node However, conflcts are possble at the outputs not n TFO() The portance of Theore 41 s that the CF can tell us f t s possble to correct the networ to eet ts fcatons at the TFO() by changng the relaton at node only The an port of the CF s the followng Theore 42: If any well-defned ND sub-relaton contaned n R ( Y, y ) s nserted at node, then the new networ, N, s - coplant, e R (, ) R (, ), at least for those outputs n the TFO() It could be that the ntal networ s not coplant Then the use of a well-defned sub-relaton can only correct those outputs that t can nfluence If the ntal networ s coplant, then t reans so after usng any well-defned sub-relaton contaned n ts CF In practce, one wants to fnd the well-defned sub-relaton wth the sallest representaton Ths s norally easured n ters of the total nuber of cubes n the non-default -sets In [4], a Qune-McClusey type algorth s gven for fndng a subrelaton wth the exact nu nuber of cubes Generally, the soluton s ND The correspondng proble for fndng an optu deternstc sub-relaton s not solved 5 Elnaton Elnaton s the process of substtutng the relaton of a node nto all the relatons of ts fanouts Substtuton of the relaton at node nto a fanout s defned as replacng relaton R( Y, y ) wth R( Y, y ) R ( Y, y ) 9 After R has been substtuted nto all y ts fanouts, t can be reoved (elnated) fro the networ, snce y s no longer used anywhere The pact of elnatng a node on the behavor of the resultng networ s suarzed below Theore 51: Elnatng a node can ncrease the NS and behavors of a networ only f the node beng elnated s ND and has ore than one fanout Theore 52: Elnatng a node can ncrease a networ s behavor f and only f the node s ND and has reconvergent fanout Theore 53: Elnatng a node cannot ncrease the SSbehavor of the networ The orgnal reason for consderng SS-behavor was that elnaton effectvely substtutes a copy of the elnated node nto each fanout Each copy acts ndependently of the other copes and effectvely broadcasts an ndependent set of values to ts fanout Snce SS effectvely does the sae thng, elnaton can not ncrease the SS-behavor of a networ However, elnaton can decrease the SS-behavor f the nodes are not elnated n topologcal order (the nuber of paths to an output can decrease n ths case) 6 Extracton and Decoposton Extracton and decoposton are slar; the latter operates on a sngle node at a te, whle the forer operates on a set of nodes Wth decoposton, a new node (dvsor) s created, whch has only a sngle fanout; wth extracton there are two or ore fanouts The obectve s the sae, to fnd a good dvsor There are two fors of extracton/decoposton, dsont and non-dsont It s dsont f the fanns of the new node are not fanns of ts fanouts Theore 61: Extracton and decoposton cannot ncrease the NS and behavors of an ND networ Theore 62 The SS-behavor of a networ s not changed f, n a node decoposton/extracton, the non-dsont varables have no ND nodes n ther TFIs The SS-behavor s related to the nuber of paths fro an ND node to the outputs A non-dsont decoposton can ncrease ths nuber As an exaple, consder the networ n Fgure 2 A Fgure 2 Non-dsont decoposton of The decoposton of s non-dsont because the nputs of C are not dsont fro the nputs of Thus the nuber of paths fro A to has ncreased If A s non-deternstc or there s an ND node n TFI(A), then the SS-behavor could ncrease 7 Mergng C Mergng s the process of cobnng two or ore nodes (the ergng set) nto a sngle node wth ore values [6] A constrant on the ergng set s that the networ should rean acyclc The - sets of the new node are coposed of ntersectons of the -sets of the set of nodes beng erged A 9 y Y snce s a fanout of

7 Exaple: Consder the ergng of two nodes wth value ranges 3 and 5, retvely Then, the 0-set of the new node s the ntersecton of the 0-sets of the two relatons, the 1-set s the ntersecton of the 0-set and the 1-set, the 2-set the ntersecton of the 0-set and 2-set, etc, and the 14-set the ntersecton of the 2-set and the 4-set The second step of ergng nvolves substtutng the new node nto the unon of the fanouts of the ergng set by replacng lterals of the ergng set of each cube n the -set covers of a fanout by a sngle lteral of the new varable Exaple: In the above exaple, f a fanout cube n soe - {0,2} {1,3,4} set nvolves the product x y (x and y for the ergng set and are three-valued and fve-valued MV varables, retvely), ths product s replaced by the sngle lteral of the new 15-valued varable, say z, {1,3,4,11,13,14} z A cube wth the lteral x {1} but no y s {5,6,7,8,9} replaced by z snce the absence of a lteral of y ples all (fve) values of y Thus, the nuber of -set cubes n the fanouts cannot ncrease, but ost lely wll decrease (whch s one pont n dong a erge) after the resultng -sets are ade pre and rredundant Exaple: An exaple of the reducton s gven by the {0} {1} {1} {0} bnary OR gate wth nputs x and y: x y + x y It has two cubes and four lterals If x and y are erged nto a sngle node z, the MV-SOP of the gate becoes {1} {2} z + z, whch, when ade pre and rredundant, becoes one cube and one lteral, z {1,2} Theore 71 Mergng of nodes cannot change the NS or behavors and cannot ncrease the SS behavor of the networ Mergng can decrease the SS-behavor snce the nuber of paths to an output ay decrease 8 Coparng and SS Snce and SS behavors are coputatonally easer than NS, they are the lely canddates for pleentaton Coparson leads to the followng stateents 1 oth lead to output-syetrc relatons at the COs Ths allows the handlng of each CO separately 2 The coputatonal process for SS s ade easer snce collapsng n topologcal order allows for buldng global MDDs usng only the CI varables However, t s ade slghtly ore dffcult by the need to handle sets at the cut varable Snce s coputed n reverse topologcal order, nteredate varables at the ND nodes ust be part of the coputaton We found that ths aes t uch ore dffcult to copute 3 oth lead to networ operatons, slar to those for bnary networs, allowng operatons on sngle nodes and creatng only sngle output nodes 4 One operaton n each case can ncrease behavor It has been characterzed when ths happens and, n both cases, the operatons can be easly odfed to ensure that the behavor can t ncrease The probleatc operaton s elnaton for and s extracton for SS 5 can provde ore flexblty at a node, but fro experents, the proveent s only about 1% The theory based on SS-behavor was the one pleented ntally n MVSIS, beng the ost coputatonally effcent pleentaton has ust been copleted Table 1 suarzes the coparson between behavor and SS-behavor n ters of possble changes of the networ behavor after the correspondng operaton An ncrease n behavor could cause non-coplance Operaton SS-behavor -behavor elnaton can t ncrease ay ncrease extracton node nzaton ay ncrease (see Theore 62) can t ncrease (see 52) node flexblty less ore can t ncrease can t ncrease ergng can t ncrease can t change Theore Table 1 Coparng two coputatonally vable theores 9 Herarchcal Theory An nterestng queston s what type of sulaton has been assued f the fcaton s gven by the ntal ND networ 10 Ths occurs n the followng stuatons: 1) Networ N has been cut out of a larger networ, N, and N acts as ts own fcaton Ths ght happen f N s so large that optzaton algorths cannot be appled to t Thus, sub-networs N are cut out; ther nputs and outputs are treated as PIs and POs No external don t cares are gven for N because these would have to be derved fro N The obectve s to re-synthesze N to obtan a saller sub-networ whose behavor s contaned n N The result s then sttched bac nto N It s portant to guarantee that the networ contanng the optzed subnetwor stll satsfes the fcatons for N, because t s te consung to chec contanent after each step 10 If the fed networ s deternstc, ths queston does not arse

8 2) A sub-networ N s cut out of a larger networ, but ts contents are gnored The fcaton N s derved only fro the surroundng envronent of N n N and the fcaton Ths s slar to coputng the CFs, except that, n general, the cut-out sub-networ ay have several outputs The second type of optzaton n a herarchcal theory s probleatc, snce a type of CF would need to be derved for ultple output nodes Ths s slar to what has been done for deternstc bnary networs [1] n ters of oolean relatons We leave ths type of optzaton for another paper For the frst type, we can state the followng results: 1 If the NS-behavor of N s not ncreased, then the NSbehavor N s not ncreased 2 For, a sall odfcaton needs to be done n treatng groups of outputs (those that have paths to the sae output n N ) of N as dependent 3 To guarantee coplance for SS-behavors, the nputs of the carved-out networ N should be consdered as general set nputs, rather than sngleton-set nputs as t s done n the case of NS and Ths odfcaton s slar to the change n the coputaton of flexblty usng SS-behavor, dscussed n Secton 4, where set nputs were requred Hence, under these odfcatons, f the orgnal networ N has ts {NS,, SS}-behavor s contaned n the fcaton, then the odfed networ contanng the odfed carved networ would also {NS,, SS}-confor Thus {NS,, SS}- behavors can be ade sutable for sub-networ optzaton Ths observaton s portant snce t allows for parttonng a large networ, optzng the sub-parts separately, and coposng the results, leadng to a vald odfed networ 10 Modfyng Two Networ Operatons In Table 1, two operatons are probleatc one for and one for SS These need to be odfed to ensure that the networ behavor s not ncreased to possbly ae t non-confor The only networ operaton that could cause the -behavor to ncrease s elnaton, and then only f an ND node wth reconvergent fanout s elnated Thus, the followng odfcaton to the elnaton operaton can be ade when behavor s used Chec a node to be elnated for beng both ND and havng reconvergent fanout; f both condtons hold, then the node relaton s deternzed (replaced by a well-defned deternstc sub-relaton) before elnaton Snce all other networ elnatons can t ncrease the behavor, the ND networ can never get out of -coplance f ths odfcaton s used For SS, the only operaton that could cause non-coplance s extracton/decoposton y Theore 62, ths can happen only f there s a non-dsont varable n the decoposton wth an ND node n ts TFI The followng odfcaton can be ade Durng extracton, chec the new dvsor for nputs that are not dsont If found, then ether deternze all ND nodes n the TFI of ths dvsor, or loo for another dvsor Addtonally, one could accept the new dvsor and then chec ts SS-CF If t s well defned, then accordng to Theore 41, the dvsor can be odfed to correct ths non-conforance 11 Experental Results We copared the aount of flexblty obtaned by usng versus SS Ten benchars were selected that were ult-level and had ult-valued varables Statstcs Nae I/O N Nnd Rec ND, % 4ac 9/ bpds 12/ cc 15/ cop 4/ ep 7/ sort 8/ sy 9/ clp 9/ cordc 23/ c432 36/ Aver 1097 Table 2: Statstcs of benchar exaples These benchars were odfed by nsertng at each node about 10% ore non-deterns than orgnally present The average aount of non-deterns per node s shown n Colun 6 of Table 2, where N s the nuber of nodes, Nnd the nuber of ND nodes, and Rec the average nuber of reconvergent paths per node ehavor Flexblty Nae SS, %, % NS, % SS, % NS, % 4ac bpds cc cop ep sort sy clp cordc c Aver Table 3: Coparson of and SS flexbltes The aount of global non-deterns was coputed for the three behavors for the resultng networ shown n Coluns 2, 3, and 4 of Table 3 The SS behavor was taen as the external

9 fcaton, and was used n calculatng the average, over all nodes, of the aount of non-deterns n the resultng and SS flexbltes These are shown n Coluns 5 and 6 of Table 3 12 Conclusons A theory of non-deternstc networs has been developed and pleented to erge the two concepts of ult-level networs and non-deterns Space ltatons allowed for only statng the results of ths nvestgaton For a ore coplete understandng, t s necessary to refer to [5] The legalty of varous classcal networ anpulatons was analyzed under three dfferent defntons of behavor, whch correspond to three ethods of sulaton: noral (NS), noral copatble (), and set sulaton (SS) SS s the sae as that obtaned by elnatng all nternal nodes n the networ n topologcal order and s equvalent to elnaton n reverse topologcal order It was shown that NS SS For { NS,, SS}, algorths were gven to copute correspondng coplete flexbltes (-CFs) at a node It s claed that these flexbltes are axu, and all have the property that any well-defned sub-relaton contaned n the antans coplance Two operatons could cause non-coplance when nondeterns s present; extracton s probleatc for SS, whle elnaton s probleatc for However, easy fc odfcatons were gven, whch alter the way extracton or elnaton s perfored n those cases where soe nondeterns s related to the node under operaton The use of NS behavor sees too coputatonally expensve for larger crcuts snce t s equvalent to coputng a oolean relaton for the entre networ In contrast, and SS behavors have reasonable coputatonal costs (as verfed by our current pleentatons), snce they can be forulated n ters of a relaton at each of the separate outputs All behavors can be ft nto a herarchcal theory, whch can handle very large networs The anpulaton of ND networs usng SS-behavor was pleented n the second generaton of the MVSIS syste [6] Recently, behavor has been pleented n MVSIS The experental results show that the flexblty coputed usng both types of behavor are sgnfcantly larger than the aount of nondeterns present at the node (73% vs 10%) The addtonal flexblty can prove the qualty of the optzaton algorths Surprsngly, experental results ndcate that yelds only about 1% ore flexblty than for SS Snce we also found that coputng flexblty was uch ore dffcult that we frst thought, we conclude that the use of the ore effcent coputaton (SS) does not loose uch n flexblty Intal experence wth odfcaton of those operatons that could cause the networ to becoe non-coplant, show that t s easy to odfy these on-the-fly Effcency of handlng ND networs s on a par wth the anpulaton of coparable deternstc bnary networs; thus the penalty for ncludng the generalzatons to ult-valued non-deternstc nodes s sall 11 A near-ter future goal s to experent wth enhancng the wellnown bnary operatons by allowng varous optzaton algorths to search n a larger (MV) space A longer-range goal s to open up, to any applcatons, the possblty of anpulatng non-deternstc probles For exaple, we can use ND networ anpulatons to operate on ND regular autoata by developng effcent operatons of copleentaton and coposton These could be appled drectly to ult-level networ representatons of the autoata yeldng possble proveents n effcency over exstng technques Many applcatons, such as protocol synthess, cryptography, dscrete control probles, solvng gaes etc, could beneft fro ths capablty References [1] D rand, Verfcaton of large syntheszed desgns, Proc ICCAD 93, pp [2] R K rayton, G D Hachtel, C T McMullen, A L Sangovann-Vncentell, Logc Mnzaton Algorths for VLSI Synthess Kluwer Acadec Publshers, Dordrecht, 1984 [3] A Mshcheno, and R rayton, A oolean paradg for ult-valued logc synthess, Proc IWLS 02, pp [4] A Mshcheno and R rayton, Splfcaton of nondeternstc ult-valued networs, Proc ICCAD 02, pp [5] A Mshcheno, and R rayton, A Theory of Non- Deternstc Networs, UC ereley Techncal Report, ERL, EECS Departent, Feb 2003 [6] MVSIS Group MVSIS UC ereley [7] R L Rudell and A Sangovann-Vncentell, Multple-valued nzaton for PLA optzaton IEEE Trans CAD, Vol 6(5), pp , Sep 1987 [8] E Sentovch, et al, SIS: A syste for sequental crcut synthess, Tech Rep UC/ERI, M92/41, ERL, Dept of EECS, Unv of Calforna, ereley, 1992 [9] Y Watanabe, L Guerra and R K rayton, Logc optzaton wth ult-output gates, Proc ICCD 93, pp [10] N Yevtusheno, T Vlla, R K rayton, A Petreno, and A L Sangovann-Vncentell, Soluton of parallel language equatons for logc synthess, Proc ICCAD 01, pp In fact, the new theory gave us the otvaton to re-pleent all the technology ndependent operatons of SIS n ths ore general settng The result s a logc synthess capablty whch s uch ore effcent there was lttle loss because of the generalzaton whle the re-thnng of the algorths and data-structures proved the effcency greatly