On the Structure of the Capacity Region of Asynchronous Memoryless Multiple-Access Channels

On the Structure of the Capacit y Region of Async hronous Memoryless Multiple-Access Channels Ninoslav Marina ∗ and Bixio Rimoldi † Abstract The asyn c hronous capacit y region of m emoryless m ultiple-ac cess c hannels is the union of certa in p olytop es. It is w ell-kno wn that ve rtices of such polytop es may be ap- proac hed via a tec hnique called su ccessiv e decod ing. I t is also kno wn that an extension of successiv e deco ding applies to the dominan t face of suc h p olytop es. The extension consists of forming groups of users in su c h a wa y that users within a group are deco ded join tly whereas groups are decod ed successiv ely . This pap er goes one step fur ther. It is sho wn that successiv e deco ding extends to every face of the abov e m en tioned p oly- top es. The group comp osition as w ell as the decodin g ord er for all rates on a face of in terest are obtained from a lab el assigned to that face. F rom the lab el one can extract a n umber of structural prop erties, suc h as the dimension of the corresp onding f ace and whether or not tw o faces inte rsect. Expr essions for the the num b er of faces of any giv en dimension are also deriv ed fr om the lab els. Index T erms − Multiple-access c hannel, p olytop es, faces, group succe ssive decod ing. 1 In tro duction The async hronous capacit y region of an M -user memoryless m ultiple-access channel (MAC) is the union of certain M -dimensional p olytop es. It is well know n that if a desired rate tuple lies on the vertex of the so-called dominant fac e of suc h a polytop e, one can deco de one user at a time suc c es sively , using t he co dew ords of already deco ded users as side inf o rmation [8, Section 14.3.2]. F or example, for a 2-user co de of blo c klength n and rat es R 1 and R 2 , resp ectiv ely , dec o ding user 1 and 2 success iv ely requires finding t he first co dew ord within a ∗ N. Marina was with the Schoo l o f Computer and Co mmu nication Sciences of the ´ Ecole Polytechnique F ´ ed´ erale de Lausanne, CH-101 5 Lausanne, Switzerland. He is now with the Depar tmen t of E lectrical Engi- neering, Univ ersity of Haw ai‘i at M¯ anoa, Honolulu, HI 9682 2, USA. This work was par tially supp orted b y the Swiss National Scie nce F oundation Gra nt Nr. 2 1-055 699.98 . The mater ia l in this paper was pres ent ed in part a t the IEEE International Symp osium o n Info r mation Theory , Chicago , USA, June/July 200 4 [2]. † B. Rimoldi is with the Schoo l of Computer and Communication Sciences of the ´ Ecole P olytechnique F ´ ed´ erale de Lausanne, CH-101 5 Lausanne, Switzerland. 1 co deb o ok of s ize 2 nR 1 and subse quen tly finding the second co dew ord in a co deb o ok of size 2 nR 2 . A decoder that mak es a join t searc h do es so in the bigger s pace of 2 nR 1 2 nR 2 pairs of alternativ es. Hence the attractivene ss of successiv e deco ding. In [15] it is shown that succe ssiv e deco ding for dominan t-f a ce v ertices extends to gr oup success iv e deco ding f o r rate t uples that are in the b o undar y of the do minant face. More sp ecifically , eac h p oin t on the b oundary of the dominan t face is on a fa ce of some dimension k ∈ { 0 , 1 , . . . , M − 2 } . F or a rate tuple on suc h a f ace of dimen sion k , succes siv e decoding requires forming M − k g r oups. F or instance , for a v ertex (a face of dimension 0) w e need M g r o ups, whic h means that eac h “group” con tains a single user, implying, as it should, single user deco ding of v ertices. Alternativ ely , if the ra te of intere st is o n a face of dimension 1, the n umber of groups is M − 1, i.e., all except tw o users can b e deco ded one at a time success iv ely , and the group of t wo is deco ded join tly . In [15] it is also sho wn that if the rate tuple of in terest is on the dominant face but not on its b oundary , then one can split a user and a c hannel input and make sure that the new ra t e tuple, whic h has an additional comp onen t, lies on the boundary of the do minant face of the newly created c hannel. By iterating this pro cedure one obtains rate splitting multiple-acce ss [9, 10]. In this pap er w e fo cus on some structural and op erational prop erties of the M -dimensional p olytop es that form the capacity region. W e extend the labeling tec hnique of [11, 15] so as to ha v e a lab el for ev ery face. The lab el is unique if the p olytop e is non-degenerated. A degenerated p olytop e (to b e prop erly defined la t er) is one for whic h certain faces collapse. T o a void complications due to the collapsing of faces w e consider only non-degenerated cases. F rom the lab el, w e can deduce structural prop erties suc h a s whic h faces in tersect and the dimensionalit y of a f a ce. The la b el also sp ecifies ho w to do succes siv e deco ding of groups, whic h is an op eratio nal property . In particular, w e will see that group decoding applies to ev ery face (not only the faces of the dominan t face). The pap er is organized as follows. In Section 2 w e define the relev an t p o lytop es and ch arac- terize and lab el their faces. The main result of Section 2 is Propo sition 5. It sp ecifies whic h faces inters ect a nd which do not. In Section 3 w e mak e the link b etw een the lab el and group success iv e deco ding. In Section 4 w e give expres sions for the n umber of faces of an y given dimension. Section 5 concludes the pap er. 2 Lab e ling face s Recall that an M -user discrete memoryless m ultiple-a ccess c hannel is defined in terms of M discrete input-alphab ets 1 X i , i ∈ { 1 , · · · , M } , an output alphab et Y , and a sto c hastic matrix W : X 1 × X 2 × · · · × X M → Y with en tries W Y | X 1 ,X 2 , ··· ,X M ( y | x 1 , x 2 , · · · , x M ) describing the probabilit y that the c hannel output is y when the inputs ar e x 1 , x 2 , · · · , x M . F or an y input 1 All r esults presented in this pap er car ry ov er to the Gauss ia n m ultiple-access channel. 2 distribution in pro duct form 2 P X 1 , · · · , P X M , define R to b e R = { R ∈ R M + : R ( S ) ≤ I ( X S ; Y | X S c ) , ∀S ⊆ [ M ] } , where R ( S ) △ = P i ∈S R i , X S △ =( X i ) i ∈S , S c △ =[ M ] \ S , [ M ] = { 1 , 2 , . . . , M } , and I ( X S ; Y | X S c ) is the mutual informatio n b etw een X S and Y giv en X S c . R + denotes the nonnegativ e reals. The capacit y region dep ends on whether or not there is sy nc hronism. A discrete-time ch annel is synchr onous if the transmitters are able to index c hannel input sequences in suc h a w a y that a ll inputs with time index n enter the c hannel at the same time. If this is not the case, meaning that there is a n unkno wn shift b et we en time indices, then the c hannel is said to b e asynchr onous. The capacit y region for either the sync hronous or async hro no us channel ma y b e described in terms of the region C D M C = [ P X 1 P X 2 ··· P X M R [ W ; P X 1 P X 2 · · · P X M ] , where the union is o v er all pr o duct input distributions. The capacit y region of the asyn- c hr o nous multiple-acces s c hannel with arbitrarily large shifts b et w een time indices is C D M C [3, 4], whereas if shifts are b ounded or the multiple -access c hannel is sync hronous then its capacit y region is the conv ex h ull of C D M C [5, 6, 7]. Definition 1 A region R is called non-degenerated if the following tw o conditions hold (a) I ( X S ; Y ) > 0 for all non-empt y sets S ⊆ [ M ], (b) I ( X S ; Y | X A ) < I ( X S ; Y | X B ) for all ∅ ⊂ S ⊂ [ M ], A ⊂ B ⊂ [ M ], and S ∩ B = ∅ . The abov e definition is natural. Essen tia lly it say s that eac h input carries information and all inputs inte rfere with one another. Not ice that for a non-degenerated channel it is also true that for all A ⊂ [ M ], ∅ ⊂ S ⊂ T ⊆ [ M ], and A ∩ T = ∅ , I ( X S ; Y | X A ) < I ( X T ; Y | X A ) . (1) T o see this, w e first observ e that the indep endence of the input random v ariables implies that I ( X S ; Y | X A ) ≥ I ( X S ; Y ) whenev er S and A do no t in tersect. Th us, condition (a) implies I ( X S ; Y | X A ) > 0 for eve ry non-empt y subset S of [ M ] and ev ery subset A o f [ M ] that do es not in tersect with S . No w w e can use the chain rule of m utual information to obtain I ( X T ; Y | X A ) = I ( X S ; Y | X A ) + I ( X T \S ; Y | X A∪S ) > I ( X S ; Y | X A ), where the inequalit y holds since the second term on its left m ust be p o sitive. An example of a channel that do es not fulfill condition (a) ab o v e is the tw o-user binary adder c ha nnel when the sum is mo dulo 2 and the inputs are assigned uniform probability . In this 2 Random v a riables and their sample v alues will be r epresented by capital and low ercas e letters, respe c - tively . 3 case condition (a) is violat ed since I ( X i ; Y ) = 0 for i = 1 , 2. The n R is a tr iangle as o pp osed to a pentagon. An example fo r whic h condition (b) is not fulfilled is when we hav e tw o parallel c hannels. In this case condition (b) is violated since I ( X 1 ; Y | X 2 ) = I ( X 1 ; Y ) = 1. The same is true if w e sw ap X 1 and X 2 . In this case R is a rectangle. Fig. 1 sho ws an example of a non-degenerated R (first subfigure) for M = 2 and all p ossible degenerated v ariations. Fig. 2 sho ws examples of degenerated cases for M = 3. All examples of F ig. 2 are for binary input c hannels and mo dulo 2 sums (when applicable). The first ro w depicts regions for the c hannel Y = X 1 + X 2 + X 3 . If we denote b y p i the probability that X i = 1, i = 1 , 2 , 3, then the first regio n (non degenerated) is obtained with p i ∈ [0 , 1] \ { 0 , 1 / 2 , 1 } , i = 1 , 2 , 3, the second r egio n in the same ro w may b e o btained with p 1 = 0 . 5, p 2 , p 3 ∈ [0 , 1] \ { 0 , 1 / 2 , 1 } , t he third with p 1 = p 2 = 0 . 5, p 3 ∈ [0 , 1] \ { 0 , 1 / 2 , 1 } , a nd the fourth with p 1 = p 2 = p 3 = 0 . 5. The first three subfigures o f the second ro w corresp ond to the channe l Y = ( Y 1 , Y 2 ) = ( X 1 + X 2 , X 2 + X 3 ). The fir st regio n ma y b e obtained with p i ∈ [0 , 1] \ { 0 , 1 / 2 , 1 } , i = 1 , 2 , 3, the second with p 1 = p 2 = 0 . 5, p 3 ∈ [0 , 1] \ { 0 , 1 / 2 , 1 } , a nd the third with p 1 = p 2 = p 3 = 0 . 5. The last region in the second row may b e obtained from the MA C Y = ( Y 1 , Y 2 , Y 3 ) = ( X 1 , X 2 , X 3 ) with p i ∈ [0 , 1] \ { 0 , 1 / 2 , 1 } , i = 1 , 2 , 3. The first three subfigures in the third r o w correspond t o the c hannel Y = ( Y 1 , Y 2 ) = ( X 1 + X 2 , X 3 ), with the input distributions of t he first one b eing p 1 , p 2 ∈ [0 , 1] \ { 0 , 1 / 2 , 1 } , p 3 ∈ (0 , 1), of the second b eing p 1 ∈ [0 , 1] \ { 0 , 1 / 2 , 1 } , p 2 = 0 . 5 , p 3 ∈ (0 , 1), and of the third b eing p 1 = p 2 = 0 . 5, p 3 ∈ (0 , 1). The last figure in the third ro w may b e obtained with p 1 ∈ { 0 , 1 } and p i ∈ (0 , 1), i = 2 , 3. R 1 R 2 Figure 1: Shapes of R for a 2- user c hannel. The first sub-figure is that of a non-degenerated case. In all pictures, the abscissa represen ts R 1 and the ordinate R 2 . An ob j ect of the form { R ∈ R M + : R ( S ) = c } , for some constan t c , is a n h yp erplane of R M + of dimens ion M − 1. The s et { R ∈ R M + : R ( S ) ≤ c } is one of the tw o half-spaces b ounded b y suc h an h yp erplane. R is a finite inte rsection of suc h half-spaces. A linear inequalit y Ra ≤ a 0 , where R is a ro w v ector, a a column v ector and a 0 is a scalar, is v alid for R if it is 4 R 1 R 2 R 3 Figure 2: Some versions of R for a 3-user channel. T he first sub-figure is that of a non- degenerated case. satisfied for all p oints R ∈ R . A fac e of R is defined as an y set of the form F = R ∩ { R ∈ R M + : Ra = a 0 } , where Ra ≤ a 0 is a v alid inequalit y for R . The dimension of a face is the dimension of its affine h ull, namely dim( F ) :=dim(aff( F )). In w ords, a face of R is the inte rsection of R with an ( M − 1) dimensional h yp erplane that k eeps R o n one side. Since the inequalit y R 0 ≤ 0 ( 0 b eing all zero v ector) is v alid for R , we observ e that R itself is a face. All the other faces F , called pr op er faces, satisfy F ⊂ R . Not e that the nu mber of faces of any dimension is maximal in the non-degenerated. In the follo wing text w e consider only channels with non-degenerated regions. F aces of dimension 0 , 1, M − 2 , and M − 1 are called v er tices , edg es , r idg es , and f acets , resp ectiv ely . In the non-degenerated cas e, f o r a single user c hannel, R ha s t w o v ertices and one edge and for a 2-user c hannel it has fiv e vertice s and fiv e edges. In Fig. 3 w e see that there are 16 v ertices, 24 edges and 10 facets for a non-degenerated region R of a 3-user c ha nnel. F or ev ery i ∈ [ M ], there is a b ack fac et of the form B i = R ∩ { R ∈ R M + : R i = 0 } , 5 and for ev ery S ⊆ [ M ], S 6 = ∅ , there is a fr ont fac et F S = R ∩ { R ∈ R M + : R ( S ) = I ( X S ; Y | X S c ) } . There are M bac k facets and 2 M − 1 fro nt facets, one for eac h non-empt y subset of [ M ]. It is con venie nt to extend the notation B i and F S as follo ws B A = \ i ∈A B i , with B ∅ = R b y con v en tion , F S 1 , S 2 ,..., S m = m \ j =1 F S j , with F ∅ = R b y con v en tion , F S 1 , S 2 ,..., S m |A = F S 1 , S 2 ,..., S m ∩ B A . Note that F S |∅ = F S , F ∅|A = B A , and F ∅|∅ = R . F ig. 3 sho ws a non-degenerated R for a 3-user c hannel and some of the lab els. R 3 R 1 F { 1 , 3 } F { 3 } F { 1 } F { 1 , 2 } F { 2 } F { 2 , 3 } F { 1 , 2 , 3 } , { 2 , 3 } , { 3 } R 2 F { 2 , 3 } , { 2 }|{ 1 } F { 1 , 2 , 3 } , { 1 , 3 } F { 1 }|{ 2 } Dominant F ace t F { 1 , 2 , 3 } Figure 3: Region R with lab els for a 3-user MA C. Next w e sho w that tw o fro n t facets in tersect if and only if the index set of one is a subs et of the index set of the other (Lemma 2) and that a front and a back f a cet in tersect if and only if the index of the bac k facet is not an elemen t of the set that defines the fron t facet (Lemma 4). Lemma 2 F S 1 ∩ F S 2 is not empt y iff S 1 ⊆ S 2 or S 2 ⊆ S 1 . 6 Pr o of: The “if ” direction is c learly true if S 1 = S 2 . Assume without lo ss of generality that S 1 ⊂ S 2 . W e w an t to sho w the existence of an R ∈ R suc h tha t R ( S 1 ) = I ( X S 1 ; Y | X S c 1 ) , and R ( S 2 ) = I ( X S 2 ; Y | X S c 2 ) . Without loss of generalit y , w e re- index users so that S 1 = [ k ] and S 2 = [ ℓ ], where ℓ > k . Consider R = ( R 1 , . . . , R M ) defined as follo ws R i = ( I ( X i ; Y | X i +1 , . . . , X M ) , i = 1 , . . . , M − 1 , I ( X M ; Y ) i = M . Observ e that R is a v ertex of the do minant face. Hence R ∈ R . F urthermore, from the c ha in rule for m utual informat io n R ([ i ]) = i X j =1 R j = i X j =1 I ( X j ; Y | X j +1 , . . . , X M ) = I ( X [ i ] ; Y | X [ i ] c ) . Th us, for i = k we get R ([ k ]) = I ( X S 1 ; Y | X S c 1 ) and for i = ℓ , R ([ ℓ ]) = I ( X S 2 ; Y | X S c 2 ). Hence R ∈ F S 1 ∩ F S 2 . T o pro v e the “only if ” direction, let R ∈ F S 1 ∩ F S 2 . Then I ( X S 1 ∪S 2 ; Y | X ( S 1 ∪S 2 ) c ) ( a ) ≥ R ( S 1 ∪ S 2 ) = R ( S 1 ) + R ( S 2 ) − R ( S 1 ∩ S 2 ) ( b ) = I ( X S 1 ; Y | X S c 1 ) + I ( X S 2 ; Y | X S c 2 ) − R ( S 1 ∩ S 2 ) ( c ) ≥ I ( X S 1 ; Y | X S c 1 ) + I ( X S 2 ; Y | X S c 2 ) − I ( X S 1 ∩S 2 ; Y | X ( S 1 ∩S 2 ) c ) ( d ) = I ( X S 1 \S 2 ; Y | X S c 1 ) + I ( X S 2 ; Y | X S c 2 ) ( e ) ≥ I ( X S 1 \S 2 ; Y | X ( S 1 ∪S 2 ) c ) + I ( X S 2 ; Y | X S c 2 ) = I ( X S 1 ∪S 2 ; Y | X ( S 1 ∪S 2 ) c ) where ( a ) and ( c ) follo w fro m the fact that R ∈ R , ( b ) from the definition of F S i , i = 1 , 2, ( d ) from the c hain rule fo r m utual information, and ( e ) holds since the inputs are inde p endent and conditioning on indep enden t inputs can not decreas e mutual informatio n. By comparing the first and the last term of the ab o v e c hain, w e see that (a), (c), and (e) mus t b e equalities. Equalit y in ( e ) means I ( X S 1 \S 2 ; Y | X S c 1 ) = I ( X S 1 \S 2 ; Y | X ( S 1 ∪S 2 ) c ) . Since R is non-degenerated (by a ssumption), the ab o v e equality implies that either S 1 \ S 2 = 0, i.e., S 1 ⊆ S 2 or S 1 = S 1 ∪ S 2 , i.e., S 2 ⊆ S 1 . This completes the pro of . ✷ The follow ing Lemma is from [15]. 7 Lemma 3 Assume R ∈ F S . Then for ev ery L ⊆ S I ( X L ; Y | X S c ) ≤ R ( L ) ≤ I ( X L ; Y | X L c ) . (2) Pr o of: The second inequalit y is true for ev ery R ∈ R . T o pro ve the firs t ineq uality observ e that R ( L ) = R ( S ) − R ( S \ L ) ( a ) = I ( X S ; Y | X S c ) − R ( S \ L ) ( b ) ≥ I ( X S ; Y | X S c ) − I ( X S \L ; Y | X ( S \L ) c ) ( c ) = I ( X L ; Y | X S c ) , (3) where ( a ) is true since R ∈ F S , ( b ) since R ∈ R and ( c ) follo ws from the chain rule for m utual information. ✷ Lemma 4 F S ∩ B A 6 = ∅ iff A ∩ S = ∅ . Pr o of: If A = ∅ then the Lemma is clearly true. Assume A 6 = ∅ . T o prov e o ne direction, let and R ∈ F S ∩ B A . Then 0 = R ( A ) = R ( S ∩ A ) ≥ I ( X S ∩A ; Y | X S c ), where the inequalit y follo ws from Lemma 3. This implies t ha t I ( X S ∩A ; Y | X S c ) = 0. Since R is non-degenerated, it follows that A ∩ S = ∅ . T o pro ve the other direction, assum e A ∩ S = ∅ and pic k a rate ˜ R suc h that ˜ R ∈ F S . Let R be obtained from ˜ R b y setting t o 0 all co ordinates with index in A . Clearly R ∈ B A but also R ∈ F S since R ( S ) = ˜ R ( S ). ✷ Prop osition 5 The in tersection F S 1 , S 2 ,..., S m |A is not empt y , if and o nly if the following tw o conditions are satisfied (i) The set sequence S 1 , S 2 , . . . , S m is telescopic, i.e., there is a perm utation π on the index set [ m ] suc h t hat S π (1) ⊃ S π (2) ⊃ . . . ⊃ S π ( m ) , and (ii) A ∩ S π (1) = ∅ . Pr o of: Assume that, after re-indexing if neces sary , S 1 ⊃ S 2 ⊃ . . . ⊃ S m and A ∩ S 1 = ∅ . The construction in the “if” part of the pro of of Lemma 2 leads to an ˜ R in F S 1 , S 2 ,..., S m . Let R b e obtained from ˜ R b y setting to 0 all co ordinates with index in A . This do es not affect co ordinates with index in S i . Hence R ∈ F S i , i = 1 , . . . , m and R ∈ B A . T o pro v e the con v erse, w e observ e t hat if S i is not con ta ined in S j or vice vers a, then b y Lemma 2, F S i ∩ F S j = ∅ . Similarly , if S 1 ∩ A 6 = ∅ , then according to Lemma 4, F S 1 ∩ B A = ∅ . This concludes the pro of. ✷ Note that prop osition 5 allow s us to define a unique lab el fo r each face in R . 8 There is o ne facet of R that stands out from the others. It is the d ominant fac et (commonly called dominant face) F [ M ] . It is sp ecial since p oin ts in the dominan t facet hav e maximal sum-rate. Observ e that, from Lemma 4, the dominan t facet is the only facet that do es not in tersect with any back f acet. The structure of the dominan t facet w as presen ted in [11 ]. In the one-user case, the dominan t facet is a v ertex, in the t w o-user case it is an edge that has t wo v ertices, in the three-user case a hexagon (Fig. 3). In Fig. 2, w e see that there are 24 v ertices, 3 6 edges and 14 t w o-dimensional faces in the dominan t fa cet of a 4- user c ha nnel. In general, the dominant facet is a g eometrical ob ject called p ermutahe dr on [18]. The notation for a v ertex in Fig . 2, has b een simplified. Instead o f writing the telescopic sequence F { 1 , 2 , 3 , 4 } , { 1 , 2 , 3 } , { 1 , 3 } , { 3 } , w e ha ve written the se quence of “decremen ts,” i.e., 4 , 2 , 1 , 3 (commas are not sho wn in Fig. 2). Besides b eing more compact, the sequence of decremen ts giv es the order in whic h users are deco ded. It is also a conv enien t no tation to coun t v ertices. Since eac h p erm uta tion on the set [ M ] is a v ertex in the dominant facet, it is clear that there are M ! suc h v ertices. 1243 1423 2143 2413 4213 4123 2431 4231 2341 3241 3421 4321 3412 3142 1342 1432 4132 4312 1234 1324 2134 2314 3214 3124 R 3 R 2 R 1 Figure 4: Dominan t facet of a 4 - user MA C. The 4th dime nsion, not sho wn here, has co or- dinate R 4 = I ( X { 1 , 2 , 3 , 4 } ; Y ) − R 1 − R 2 − R 3 . Lab els desc rib e the dec o ding order us ed to approac h the corresp onding ve rtex via successiv e deco ding. 9 3 Structur e , dimensi onalit y , and group s uccess iv e de- co ding In this section w e sho w that the faces of R consist of the Cartesian pro duct of fundamen tal regions and of dominan t facets of channels tha t are “spin-offs” from t he original c hannel W . T o distinguish those c hannels, we use subscripts that indicate the channel inputs and outputs. The original c hannel W will b e denoted b y W Y | X [ M ] . R ecall that the region R is completely sp ecified by the c hannel W Y | X [ M ] and b y the input distribution P X [ M ] . F or an y tw o sets U , V ⊂ [ M ] suc h that U ∩ V = ∅ , there is a c hannel with inputs X U and outputs ( Y , X V ). Sp ecifically , W Y X V | X U ( y , x V | x U ) = P X V ( x V ) W Y | X U ,X V ( y | x U , x V )) = P X V ( x V ) X x [ M ] \U \ V W Y X [ M ] \U \ V | X U ,X V ( y , x [ M ] \U \V | x U , x V ) = P X V ( x V ) X x [ M ] \U \ V P X [ M ] \U \ V ( x [ M ] \U \V ) W Y | X [ M ] ( y | x [ M ] ) , where b y con v en tion P X ∅ ( x ∅ ) = 1. A rate t uple fo r W Y X V | X U is an expression of the form R U △ =( R i ) i ∈U . The corresp onding fundamen t a l region R Y X V | X U is defined b y R Y X V | X U ∆ = R [ W Y X V | X U ; P X U ] = { R ∈ R |U | + : R ( L ) ≤ I ( X L ; Y | X V ∪ ( U \L ) ) , ∀L ⊆ U } . ( 4 ) The dimensionalit y of R Y X V | X U is | U | . Its dominan t facet is the ( |U | − 1)-dimensional sub- region obtained b y adding the equalit y R ( U ) = I ( X U ; Y | X V ) i.e., D Y X V | X U ∆ = D [ W Y X V | X U ; P X U ] = { R ∈ R |U | + : R ( L ) ≤ I ( X L ; Y | X V ∪ ( U \L ) ) , ∀L ⊂ U , R ( U ) = I ( X U ; Y | X V ) } . The follow ing sp ecial cases will b e used frequen tly R Y X S c | X S ∆ = R [ W Y X S c | X S ; P X S ] = { R ∈ R |S | + : R ( L ) ≤ I ( X L ; Y | X L c ) , ∀L ⊆ S } , D Y X S c | X S ∆ = D [ W Y X S c | X S ; P X S ] = { R ∈ R |S | + : R ( L ) ≤ I ( X L ; Y | X L c ) , ∀L ⊂ S , R ( S ) = I ( X S ; Y | X S c ) } , R Y | X S ∆ = R [ W Y | X S ; P X S ] = { R ∈ R |S | + : R ( L ) ≤ I ( X L ; Y | X S \L ) , ∀L ⊆ S } , D Y | X S ∆ = D [ W Y | X S ; P X S ] = { R ∈ R |S | + : R ( L ) ≤ I ( X L ; Y | X S \L ) , ∀L ⊂ S , R ( S ) = I ( X S ; Y ) } . The next lemma sa ys that F S is the Cartesian pro duct of a fundamen tal region R and a dominan t facet D . One exp ects this to be the case by lo oking at the facets F { 1 } , F { 2 } , and 10 F { 3 } of Fig . 3. F or instance F { 1 } is the Cartesian pro duct of a singleton a nd a p en tag o n. The singleton, the v alue of R 1 , is the dominant facet D of a single-user ch annel. The pentagon, the r egio n that contains R { 2 , 3 } , is the region R of a t wo-user multiple-acce ss channel. A p erhaps less ev iden t example is F { 1 , 2 } . This is the C artesian product of the dominan t fa cet D of a t w o user m ultiple-access c hannel and the f undamen ta l region R of a single user c ha nnel. Lemma 6 R ∈ F S iff R S c ∈ R Y | X S c and R S ∈ D Y X S c | X S . P roof : Let R ∈ F S . F rom the definition of F S , ∀L ⊂ S ⊆ [ M ], R ( L ) ≤ I ( X L ; Y | X L c ) and R ( S ) = I ( X S ; Y | X S c ). Therefore R S ∈ D Y X S c | X S . Moreov er, ∀T ⊂ S c w e may write [ M ] = S ∪ T ∪ Q as the union of disjoin t se ts. Then R ( T ) + R ( S ) ≤ I ( X T ∪S ; Y | X Q ) = I ( X T ; Y | X Q ) + I ( X S ; Y | X Q∪T ) = I ( X T ; Y | X Q ) + I ( X S ; Y | X S c ) = I ( X T ; Y | X S c \T ) + R ( S ) . Hence R ( T ) ≤ I ( X T ; Y | X S c \T ) and from (4) it follo ws that R S c ∈ R Y | X S c . T o prov e the con v erse, let R S ∈ D Y X S c | X S and R S c ∈ R Y | X S c . W e hav e t o prov e tha t R ( S ) = I ( X S ; Y | X S c ) and that for a ll L ⊆ [ M ], R ( L ) ≤ I ( X L ; Y | X L c ). The former is true sin ce R S ∈ D Y X S c | X S . T o pro v e the latter, let T = L ∩ S and Q = L ∩ S c . Since R S ∈ D Y X S c | X S , R ( T ) ≤ I ( X T ; Y | X S c ∪ ( S \T ) ) = I ( X T ; Y | X T c ) for all T ⊆ S . F urthermore, since R S c ∈ R Y | X S c , R ( Q ) ≤ I ( X Q ; Y | X S c \Q ) for all Q ⊆ S c . Hence R ( L ) = R ( T ∪ Q ) = R ( T ) + R ( Q ) ≤ I ( X T ; Y | X T c ) + I ( X Q ; Y | X S c \Q ) ≤ I ( X T ; Y | X T c ) + I ( X Q ; Y | X ( S c \Q ) ∪ ( S \T ) ) = I ( X T ∪Q ; Y | X ( T ∪Q ) c ) = I ( X L ; Y | X L c ) for all L ⊆ [ M ] and this completes the pro of. ✷ F rom Lemma 6 w e obtain the dimension of F S dim( F S ) = dim( R Y | X S c ) + dim( D Y X S c | X S ) = | S c | + | S | − 1 = M − 1 , whic h is to b e exp ected for a fa cet. Finally , Lemma 6 tells us that a rate po in t in F S ma y b e approached via group successiv e deco ding where gr o ups are deco ded in t he ord( S c , S ). (F or a rigorous pro of of this fact w e 11 need to use co des that hav e indep endent and identically distributed comp onen ts. This ma y b e done using random co ding argumen ts a s in [15].) Then next result is a generalization of Lemma 6. It says that when F S 1 , S 2 ,..., S m is no t empt y it is the Cartesian pro duct of a fundamen ta l region and m dominan t fa cets. Theorem 7 Let S 1 ⊃ S 2 . . . ⊃ S m form a telescopic sequence . R ∈ F S 1 , S 2 ,..., S m iff R S c 1 ∈ R Y | X S c 1 and R S i \S i +1 ∈ D Y X S c i | X S i \S i +1 for i = 1 , . . . , m , w here b y w ay of conv en tion w e ha v e defined S m +1 = ∅ . P roof : Let R ∈ F S 1 ,..., S m and recall that F S 1 , S 2 ,..., S m = T m i =1 F S i . F rom Lemma 6 w e hav e R S i ∈ D Y X S c i | X S i and R S c i ∈ R Y | X S c i , i = 1 , . . . , m. (5) This prov es R S c 1 ∈ R Y | X S c 1 . In order to complete the pro of of the direct part, it is s ufficien t to sho w that (5) implies R S i \S i +1 ∈ D Y X S c i | X S i \S i +1 , ∀ i = 1 , . . . , m − 1 . F or this, it is enough to sho w that, R ( K ) ≤ I ( X K ; Y | X S c i ∪ ( S i \S i +1 \K ) ) = I ( X K ; Y | X ( S i +1 ∪K ) c ) ∀K ⊆ S i \ S i +1 , i = 1 , . . . , m , with equalit y if K = S i \ S i +1 . F rom (5), for any K ⊆ S i , R ( K ) ≤ I ( X K ; Y | X K c ) with equalit y if K = S i and for any L ⊆ S i +1 , R ( L ) ≤ I ( X L ; Y | X L c ) with equalit y if L = S i +1 . Then for K ⊆ S i \ S i +1 w e hav e R ( K ) = R ( S i ) − R ( S i +1 ) − R ( S i \ S i +1 \ K ) = I ( X S i ; Y | X S c i ) − I ( X S i +1 ; Y | X S c i +1 ) − R ( S i \ S i +1 \ K ) ( a ) ≤ I ( X S i ; Y | X S c i ) − I ( X S i +1 ; Y | X S c i +1 ) − I ( X S i \S i +1 \K ; Y | X S c i ) = I ( X K ; Y | X ( S i +1 ∪K ) c ) , where ( a ) follow s from the fact that ∀Q ⊂ S i , R ( Q ) ≥ I ( X Q ; Y | X S c i ). The equality in ( a ) holds if K = S i \ S i +1 . This pro v es the direct part. T o prov e the con v erse, let R S c 1 ∈ R Y | X S c 1 , and R S i \S i +1 ∈ D Y X S c i | X S i \S i +1 , for i = 1 , . . . , m . W e hav e to pro ve that R ( L ) ≤ I ( X L ; Y | X L c ) holds for all L ⊆ [ M ] w ith eq uality if L = S i , i = 1 , . . . , m . R ( S i ) = I ( X S i ; Y | X S c i ) is true since R S i \S i +1 ∈ D Y X S c i | X S i \S i +1 and R ( S i ) = m X j = i R ( S j \ S j +1 ) = m X j =1 I ( X S j \S j +1 ; Y | X S c j ) = I ( X S i ; Y | X S c i ) , i = 1 , . . . , m. No w let L ⊆ [ M ] and define L i = L ∩ S i \ S i +1 , i = 0 , 1 , . . . , m with S 0 = [ M ] b y con- v ention. Then L = S m i =0 L i is a disjoint pa r t ition. F r o m R S c 1 ∈ R Y | X S c 1 and L 0 ⊆ S c 1 it 12 follo ws R ( L 0 ) ≤ I ( X L 0 ; Y | X S c 1 \L 0 ). F urthermore, since R S i \S i +1 ∈ D Y X S c i | X S i \S i +1 , R ( L i ) ≤ I ( X L i ; Y | X S c i ∪ ( S i \S i +1 \L i ) ) = I ( X L i ; Y | X S c i +1 \L i ) for all L i ⊆ S i \ S i +1 . Therefore, I ( X L ; Y | X L c ) = m X i =0 I ( X L i ; Y | X S i − 1 j =0 L j ∪L c ) ( a ) ≥ m X i =0 I ( X L i ; Y | X S c i +1 \L i ) ≥ m X i =0 R ( L i ) = R ( L ) , where ( a ) holds since i − 1 [ j =0 L j ∪ L c = [ M ] \ m [ j = i L j ⊇ S c i +1 \ m [ j = i L j = S c i +1 \ L i , and (b) holds since for j ≥ i + 1, L j ⊆ S i +1 implies that L j do es not in tersect with S c i +1 . This completes the pro of. ✷ F rom the previous theorem, the dimension of F S 1 , S 2 ,..., S m is dim( F S 1 , S 2 ,..., S m ) = dim( R Y | X S c 1 ) + m X i =1 dim( D Y X S c i | X S i \S i +1 ) = M − |S 1 | + m X i =1 ( |S i | − |S i +1 | − 1) = M − m. The theorem also implies that all po ints in F S 1 , S 2 ,..., S m ma y b e approache d via g roup succes- siv e deco ding with groups of users deco ded according to the follo wing order: ( S c 1 , S 1 \ S 2 , S 2 \ S 3 , ..., S m − 1 \ S m , S m ). Corollary 8 Let S 1 ⊃ S 2 ⊃ . . . ⊃ S m b e a telesc opic sequence. R ∈ F S 1 ,..., S m |A iff R S c 1 ∈ R Y | X S c 1 , R S i \S i +1 ∈ D Y X S c i | X S i \S i +1 for i = 1 , . . . , m , and R A = 0 . P roof : Recall that F S 1 ,..., S m |A = F S 1 ,..., S m ∩ B A . Hence R ∈ F S 1 ,..., S m |A iff R ∈ F S 1 ,... S m and R A = 0 . The rest follo ws from Theorem 7. ✷ F rom the ab ov e Corollary w e conclude that dim( F S 1 , S 2 ,..., S m |A ) = dim( F S 1 , S 2 ,..., S m ) − |A | = M − |A| − m. F urthermore, R ∈ F S 1 ,..., S m |A ma y b e appro ac hed by decoding gro ups of users in the order ([ M ] \ A \ S 1 , S 1 \ S 2 , S 2 \ S 3 , ..., S m − 1 \ S m , S m ). 13 4 Num b er o f faces of dimension D No w w e ar e ready to derive the num b er of D - dimensional faces in R for any D = 0 , 1 , . . . , M . W e start b y describing the n um b er of D -dimensional faces of the dominan t facet. Prop osition 9 The n um b er of D - dimensional faces in the dominan t facet of R is N d ( M , D ) = M − D X j =1  M − D j  ( − 1) M − D − j j M . (6) Pr o of: Any D -dimensional face on the dominant facet is lab eled by F [ M ] , S 2 ,..., S M − D . The difference sets [ M ] \ S 2 , S 2 \ S 3 , . . . , S i \ S i +1 , . . . , S M − D \ ∅ fo rm an ( M − D ) partition of [ M ]. There is a one-t o -one correspo ndence b et w een a D -dimensional face and suc h a pa r tition. The n um b er of such ordered partitions is N d ( M , D ) = X m 1 ,m 2 ,...,m M − D m i ≥ 1 , ∀ i P i m i = M  M m 1 , m 2 , . . . , m M − D  = X m 1 ,m 2 ,...,m M − D m i ≥ 1 , ∀ i P i m i = M M ! Q i m i ! . (7) T o go further, w e expand the followin g p olynomial  x 1! + x 2 2! + · · · + x M M !  M − D = M ( M − D ) X k = M − D x k X m 1 ,...,m M − D m i ≥ 1 , ∀ i P i m i = k 1 m 1 ! m 2 ! . . . m M − D ! , and note that the co efficien t in fron t of x M m ultiplied by M ! give s (7). Therefore, N d ( M , D ) = M ! co eff   M X i =1 x i i ! ! M − D , x M   ( a ) = M ! co eff   ∞ X i =1 x i i ! ! M − D , x M   ( b ) = M ! co eff  ( e x − 1) M − D , x M  ( c ) = d M dx M ( e x − 1) M − D     x =0 , where co eff( f ( x ) , x i ) is the co efficien t of x i in the T aylor series expansion around zero of t he function f ( x ), ( a ) is true since taking all the terms up to M o r up to infinit y will not change 14 the co efficien t in front of x M , ( b ) follo ws from the T aylor expansion of e x , and ( c ) follo ws from the definition of the T a ylor expansion. T o pro v e (6), we use the Binomial f orm ula to expand ( e x − 1) M − D , namely ( e x − 1) M − D = M − D X j =0  M − D j  e j x ( − 1) M − D − j . T aking the M -t h de riv ative , d M dx M ( e x − 1) M − D = M − D X j =1  M − D j  e j x ( − 1) M − D − j j M , and setting x = 0 we obtain (6 ). ✷ Observ e that b y letting D = 0, using the fact that there a r e M ! v ertices in the dominan t facet, from (6) w e obtain an alt ernativ e expression for M ! that is M ! = M X j =1  M j  ( − 1) M − j j M . The n um b er o f faces in the dominan t facet is directly connected to t he Stirling n umb er of the se c ond kind [14, 1 7 ], denoted b y  M n  . This is kno wn as Kar amata n otation [13]. The Stirling n um b er of the second kind is the n um b er of w a ys w e can partition a set of M ele men ts in to n nonempt y subs ets. In calculating the num b er of D -dimensional faces, all p ermu tations of suc h partitions ha ve to be counted. That is, N d ( M , D ) = ( M − D )!  M M − D  . Lik e for facets, it is useful to distinguish b etw een fron t and back faces. Hence w e sa y that a face F S 1 , S 2 ,..., S m |A , is called a fr ont fac e if A = ∅ and a b ack fac e if A 6 = ∅ . Prop osition 10 The total n umber of front fa ces of dimension D , denoted b y N f ( M , D ) , equals N f ( M , D ) = N d ( M , D ) + N d ( M , D − 1) . (8) Pr o of: An y D - dimensional f ron t face has a la b el F S 1 , S 2 ,..., S M − D |∅ for some S 1 ⊆ [ M ]. If S 1 = [ M ] , the fro n t face is in the dominant facet and there are N d ( M , D ) suc h faces. If S 1 ⊂ [ M ] the f r o n t face is not in the dominan t facet. Since there is a one-to- one relationship b et w een the subscripts of F S 1 , S 2 ,..., S M − D |∅ and those o f F [ M ] , S 1 , S 2 ,..., S M − D |∅ when S 1 ⊂ [ M ], if 15 follo ws that the t o tal n um b er of front faces not in the dominan t f a cet is exactly N d ( M , D − 1). T o obta in the total num b er of fron t D -faces w e hav e t o add this n umber and the n umber N d ( M , D ) of D - faces in the dominant facet. ✷ W e no w hav e an expression for N d ( M , D ) (Prop osition 9) and an expression for N f ( M , D ) (Prop osition 10). Next w e derive an expression f o r the n umber of back faces N b ( M , D ) . Prop osition 11 The total num b er of D -dimensional bac k faces in R is giv en by N b ( M , D ) = M − 1 X i = D  M i  N f ( i, D ) . (9) Pr o of: T o deriv e ( 9 ), we observ e that all bac k faces are front faces for some other c ha nnel with few er users. This can b e seen from the lab el F S 1 , S 2 ,..., S m |A of a bac k face, where A 6 = ∅ . The dimension of this face is M − m − |A| . Recall tha t A ∩ S 1 = ∅ . If we remo v e a ll users with index in A , w e obtain the front face F S 1 , S 2 ,..., S m |∅ of an ( M − |A| )- user MA C. The dimensionalit y of this fa ce is also M − |A| − m . Running o ver all pertinent subsets A ⊂ [ M ] yields N b ( M , D ) = X A⊂ [ M ] 0 < |A|≤ M − D N f ( M − |A| , D ) . Since there are  M |A|  subsets of cardinalit y |A | , N b ( M , D ) = M − D X |A| =1  M |A|  N f ( M − |A| , D ) = M − D X i =1  M i  N f ( M − i, D ) = M − D X i =1  M M − i  N f ( M − i, D ) = M − 1 X i = D  M i  N f ( i, D ) . ✷ No w we ar e ready to deriv e an expres sion for the total num b er of D -dimensional faces in R . Theorem 12 The total n um b er of D - dimensional faces in R , 0 ≤ D ≤ M , is N ( M , D ) = M X i = D  M i  " ( i + 1 − D ) i − i − D X j =1  i − D j − 1  ( − 1) i − D − j j i # . (10) Pr o of: First w e observ e that N ( M , D ) = N f ( M , D ) + N b ( M , D ) = M X i = D  M i  N f ( i, D ) . 16 Using (8) w e obtain N ( M , D ) = M X i = D  M i  [ N d ( i, D ) + N d ( i, D − 1)] , (11) where N d ( D , D ) = 0, N d ( D , D − 1) = 1 and, b y conv en tion, N d ( i, − 1) = 0 (t he latter is needed for the case D = 0). F urthermore, from (6) we obtain N d ( i, D ) + N d ( i, D − 1) = ( i − D + 1) i + i − D X j =0 j i ( − 1) i − D − j +1  i − D + 1 j  −  i − D j  = ( i − D + 1) i − i − D X j =0  i − D j  j i +1 ( − 1) i − D − j i − D + 1 − j = ( i − D + 1) i − i − D X j =1  i − D j − 1  ( − 1) i − D − j j i . (12) Inserting (12) in to (11) yields (10) and completes the pro of. ✷ Next we determine a closed f o rm expression for the total n um b er N ( M , 0) of v ertices and the total n um b er N ( M , 1) o f edges. Lemma 13 The total num b er o f v ertices in R is ⌊ eM ! ⌋ . Pr o of: F ro m (11), N ( M , 0) = M X i =0  M i  N d ( i, 0) ( a ) = M X i =0  M i  i ! = M X i =0 M ! ( M − i )! = M X i =0 M ! i ! = M ! ∞ X i =0 1 i ! − M ! ∞ X i = M +1 1 i ! ( b ) = eM ! − M ! ∞ X i = M +1 1 i ! , where in ( a ) w e ha ve used the w ell kno wn fact that the n um b er of ve rtices N d ( i, 0) o f the dominan t f acet of an i -user region is i ! and ( b ) follo ws fro m the T aylor series expansion of e . 17 Since eM ! − M ! P ∞ i = M +1 1 i ! is an in teger, and ∞ X i = M +1 M ! i ! = ∞ X i =1 M ! ( M + i )! = ∞ X i =1 1 Q i j =1 ( M + j ) < ∞ X i =1 i Y j =1 1 M + 1 = ∞ X i =1  1 M + 1  i = 1 / ( M + 1) 1 − 1 / ( M + 1) = 1 M ≤ 1 , it follow s that N ( M , 0) = M X i =0 M ! i ! = $ N ( M , 0) + ∞ X i = M +1 M ! i ! % = ⌊ eM ! ⌋ . (13) ✷ Lemma 14 The total num b er o f edges in R is M 2 ⌊ eM ! ⌋ . Pr o of: F ro m (11) w e hav e N ( M , 1) = M X i =1  M i  ( N d ( i, 1) + N d ( i, 0)) . F urthermore, since N d ( i, 0) = i !, fro m (7), N d ( i, 1) = X m 1 ,m 2 ,...,m i − 1 m j ≥ 1 , ∀ j P j m j = i  i m 1 , . . . , m i − 1  = ( i − 1)  i 2 , 1 , . . . , 1  = i !( i − 1 ) 2 . Therefore, N ( M , 1) = M X i =1  M i   i ! + i − 1 2 i !  = 1 2 M X i =1  M i  i ! ( i + 1 ) = 1 2 M X i =1 M ! ( M − i )! ( i + 1) = 1 2 M − 1 X j =0 M ! j ! ( M − j + 1) = M + 1 2 M − 1 X j =0 M ! j ! − 1 2 M − 2 X k =0 M ! k ! ( a ) = 1 2 [( M + 1)( ⌊ eM ! ⌋ − 1) − ( ⌊ eM ! ⌋ − M − 1)] = M 2 ⌊ eM ! ⌋ , where in ( a ) w e us e (13) to obta in P M − 1 j =0 M ! /j ! = ⌊ eM ! ⌋ − 1 and P M − 2 j =0 M ! /j ! = ⌊ eM ! ⌋ − M − 1. ✷ 18 5 Summary The capacit y region of an async hronous memoryless m ultiple-access channel is the union of certain p olytop es. The p oin ts in those p olytop es are exactly the rate tuples that can b e appro a c hed at an arbitrarily small error probabilit y . In this pap er w e ha v e dev elop ed op erational and structural prop erties that apply to those p olytop es. The cen terpiece of o ur dev elopmen ts are the lab els that we use to tag their faces. F or non-degenerated cases (the only kind considered in this pap er), the set of lab els is the set of expressions o f the form ( S 1 , S 2 , . . . , S m |A ), where A ⊆ [ M ] and [ M ] \ A ⊃ S 1 ⊃ S 2 , . . . , ⊃ S m . This extends the lab eling in tro duced in [11]. Eac h lab el o f the ab o v e form tags o ne face and eac h face has a unique suc h tag. W e hav e sho wn that the la b el S 1 , S 2 , . . . , S m |A tags a face of dimension M − m − |A| . By coun ting the n um b er of suc h expres sions for a fixed k , w e find the n um b er of faces of a giv en dimension. W e ha v e also sho wn that a rate tuple on t he fa ce with lab el S 1 , S 2 , . . . , S m |A ma y b e ap- proac hed via succe ssiv e deco ding, as follows: the users with index in ([ M ] \ A \ S 1 are deco ded first, follow ed by t he users with index in S 1 \ S 2 , f ollo w ed by those with in- dex in S 2 \ S 3 etc. The users with index in S m are decoded last. The use rs with in- dex in A do not need to b e deco ded since they ha ve v anishing rate. The deco ding order ([ M ] \ A \ S 1 , S 1 \ S 2 , S 2 \ S 3 , ..., S m − 1 \ S m , S m ) is an equiv alen t alternative w a y to lab el faces. T able 1 summarizes the expressions for the num b er of faces of a g iv en dimension, where f ( i ) n (0) = d dx ( e x − 1) n   x =0 . The logarithm of the total num b er of D -dimensional faces as a function of D , for M = 1 , 2 , . . . , 20 is sho wn in Fig . 5. T able 1: Num b er of v ertices, edges, fa cets and D -dimensional faces for an M -user MA C. Ob jects In R In the dominan t facet V ertices ⌊ eM ! ⌋ M ! Edges M 2 ⌊ eM ! ⌋ M !( M − 1) / 2 F acets M + 2 M − 1 2 M − 2 D -faces P M i = D  M i   f ( i ) i − D (0) + f ( i ) i − D +1 (0)  f ( M ) M − D (0) 19 0 5 10 15 20 0 5 10 15 20 25 dimension of the face − D log 10 N(M,D) Figure 5: T o tal num b er of D - dimensional faces (expressed in logarithmic form) a s a function of D . Eac h curv e corresponds to a v a lue of M . T he curv e that corresp onds to M = m , m = 1 , 2 , . . . , 2 0, is the o ne that hits the abscissa at D = m . . References [1] N. 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