the aim of this article is two - fold : first , to provide an analytical description of random dynamical decoupling because analytical expressions are often more manageable than combinatoric - numerical ones ; second , to use this description to propose a partial method of detecting intrinsic decoherence of quantum systems .dynamical decoupling is a method applied to stabilise states of quantum registers against undesired time - evolution . originally invented in nmr technology, it has been generalised to a wider context , in particular in quantum information theory .it works by application of repeated instantaneous unitary correction pulses on the quantum register , perturbing the original time - evolution .the procedure is particularly interesting and effective when performed in a random way .  while several general estimates and specialisations of the procedure have been proposed in the past ( cf . and the references therein ) , our focus here is on finding handy analytical descriptions of the time - evolution of expectation and distribution of physically interesting quantities like the density matrix process or the gate fidelity process arising from this random time - evolution .we believe that such descriptions are a valuable tool in future computations and enable predictions in experiments .we would like to provide now a rough overview of the content of this article .let stand for a generic finite - dimensional quantum system , with a finite - dimensional complex hilbert space and a possibly time - dependent lindblad generator ( cf . for general background information ) .we start in section [ sec : two ] by introducing dynamical decoupling and show that the decoupling condition can be satisfied only if  ] , for with unitary and selfadjoint .  by a _ decoupling set _ in we mean a finite group of unitaries such that and notice that in this case we automatically have .[ ex:1 ] the standard illustrative example of a finite - dimensional quantum system to keep in mind throughout this paper is an -qubit quantum system , so and ; there typically the decoupling set consists of the different combinations of pauli matrices on the tensor factors . here . , for all .given the cpt semigroup of time evolution maps of our system and a ( short " ) time , consider the externally modified time evolution where , and forms a certain sequence in with , meaning we apply _ instantaneous decoupling _ or _correction pulses _ at time ; set whenever . the sequence can be fixed or random , leading to _ deterministic _ or _ random dynamical decoupling_. it turns out that random decoupling has many advantages and moreover is mathematically more interesting , and that is why we want to investigate it here .our first goal is to find an analytical description of the externally modified time evolution . in the randomsetting , becomes a stochastic process ( a random walk with steps lasting time ) induced by the process with independent identically distributed ( iid ) and equidistributed increments in , and we are interested in the limit , which would enable nice analytical expressions .since , we find so the increment during the time interval is given by .we say that satisfies the _ decoupling condition _ for  the idea behind this condition is that it ensures cancellation of interaction at first order in , i.e. , for short time , and thus higher order terms contribute .we say ( the time evolution of ) our quantum system is _ purely unitary _ if , with selfadjoint , because in this case is induced by a one - parameter family of unitary matrices ; in this case .the `` opposite case '' , namely where we call _ purely dephasing_.  [ lem : deccond - unitary ] a decoupling set for satisfies the decoupling condition iff is purely unitary .we write out the generator in christensen - evans form  with a certain and completely positive which is not a multiple of ( w.l.o.g . , because adding to has the same result on as adding to ) .suppose first that is purely unitary ; then and . from , with indexing the decoupling set as above ,we obtain so satisfies the decoupling condition . if instead is not purely unitary , we have and hence is completely positive and nonzero .suppose equals then , for every rank - one projection , we have .but for every with , we have with each single term , due to the positivity of and the scalar product , and thus actually .in particular , since , we have , so .let us write in ( minimal ) kraus form with its kraus rank and certain  : this entails then , for any two mutually orthogonal vectors , hence , for every , we see that , or in other words , must be an eigenvector of .this holds for every , so , and thus is a multiple of , which contradicts our initial assumptions .therefore , , and i.e. ,  on does not satisfy the decoupling condition .  despite this result, it will turn out in the course of this paper that dynamical decoupling is still interesting beyond the unitary case .we continue with the notation and concepts introduced in the previous section .let us , in particular , assume all increments , with , in our random walk of decoupling pulses to be iid and equidistributed in as in the preceding section .the induced random walk lies in the completely positive maps on according to and .moreover , since completely positive maps are linear maps of the hilbert space and since all the increments are invertible , the random walk actually lies in the group of invertible linear maps of , and , the lie algebra of .this induced random walk has again iid increments and is described by the measure we would like to investigate it in the limit .however , since is an actual physical quantity in our set - up , we keep it and instead consider a fictitious limit , which should be good for small , as explained below .considering simply and the limit of , we would obtain a drift - like expression without fluctuations , which is not a really physical result but a good first approximation , cf .  and remark [ rem : drift - limit ] .in fact , the well - known donsker invariance principle basically says that the limit of a classical random walk is described suitably well by brownian motion , i.e. , by scaling length increments with the square - root of time increments , supposed that the expectation of every increment is .the following kind of central limit theorem helps us to treat these dissipation - fluctuation terms in the present noncommutative setting , and will thus become a building stone in our construction ; it has first been stated in but can also be found in the textbook ( * ? ? ? *th.4.4.2 ) . the necessary notation and concepts in lie groups and stochastic processes on lie groupsare lined out in appendix [ sec : app ] , and we suggest the reader to go through it before continuing here .[ theowehn ] let be an -dimensional lie group , with -chart , lie algebra basis and coordinate mappings extended to functions in and hence to the one - point compactification .let be a family of probability measures on converging to .suppose there are numbers such that is positive semi - definite and , for all and :  * * * for every -neighbourhood .then the sequence converges * -weakly to a measure on which belongs to the convolution semigroup whose corresponding operator semigroup on has infinitesimal generator with .we would like to apply this theorem to our setting , namely where and regarded as ( real ! ) linear lie group and algebra , respectively , and subspaces of .the plan is as follows : in a first step we shall construct a continuous - time stochastic process in , and in a second step use this to obtain a description of the induced behaviour of the density matrix process . in oursetting this means we first have to define suitable and physically realistic measures to which to apply our limit procedure of theorem [ theowehn ] .the drift part should correspond to the original drift part resulting from .putting and let us define the measures which conceptually imitate a diffusion part for the variation around the mean and a drift part for the mean movement .apart from being mathematically clear and plausible from the classical donsker invariance principle , the meaningfulness of this limit shall moreover be confirmed by numerical analysis carried out partially in the final section and mainly in .let us drop a quick side remark : as a rough first approximation for we might also study the purely drift - like similar to but with scaling variable instead of as we now would like to keep fixed . in analogy to the law of large numbersthis would lead to even nicer expressions and cpt dynamics but less faithful modelling ; sometimes we will consider it briefly for comparison reasons , cf .remark [ rem : drift - limit ] . it can be shown that all other types of scaling ( i.e. , others than or ) essential lead to either trivial or singular not well - defined expressions .we shall therefore stick to henceforth if not explicitly mentioned otherwise .moreover , it is clear that ( as is a group ) and hence .we have iff satisfies the decoupling condition for .using now the defining property of the coordinate maps in the limit , ( i ) is obtained with a series expansion of , namely : analogously , for ( ii ) we have for every . finally , it is easy to see that condition ( iii ) is satisfied as has discrete support in points only which converge to as .thus we get for the generator of the limit convolution semigroup on , which can be interpreted as a combination of drift and diffusion on .this has been the first big step in our construction , namely the construction of the convolution semigroup of measures on ; it implicitly describes a stochastic process on ( according to theorem [ theowehn ] ) with .our second step shall be to calculate the time evolution of the density matrix and related physically significant quantities out of the stochastic process .this is slightly involved , but can be done using some tools which we are now going to derive .a general fact is that , for every , we have = \int_{g_c } f(g ) { \operatorname{d}}\nu_t ( g )  = t_t f ( { \mathbf{1}}).\ ] ] we define the subsemigroup }:=\{g\in g : \|g\|\le 1\}\subset g\subset g_c ] .this implies that the convolutions of those measures are supported in } ] , meaning the gaussian process stays almost surely in } ] of bounded continuous functions on the complement } ] vanishing at the boundary : } } f(hg ) { \operatorname{d}}\nu_t ( h ) = 0,\ ] ] for every } ] , i.e. ,  has support in } ] can be identified with }) ] and v.v ., two extensions of a function }) ] , thus a unique element in }) ] , for every , so that }(g)= f(g) ]. then we get the quotient semigroup })_{t\in{\mathbb{r}}_+} ] , the twice differentiable functions on } ] for an arbitrary but fixed function in ( and hence ) coinciding with on } ] ( easy exercise ) ; moreover , following the notation of the preceding paragraph we write }:= q(f_{kl}^{[\infty]}) ] , we have , for every } ] , which is bounded by uniformly in } ] , so }\in{c^\infty}(k) ] ( cf .appendix [ sec : app ] ) .  recalling and noticing that } ] .since this holds for every basis vector , it holds for all elements in . applying it to the -valued density matrix stochastic process " , we find = { \operatorname{e}}^{t \hat{l}}(\rho_0),\ ] ] concluding our second step , too .we summarize this all in  [ th : cont - limit ] the continuous - time limit of the above random walk determined by a quantum system and random dynamical decoupling with decoupling set and leads to a contraction semigroup with generator .the density matrix is then a stochastic process in with expectation = { \operatorname{e}}^{t \hat{l}}(\rho_0 ) , \quad \forall t\ge 0,\ ] ] where   [ rem : non - const - evo ] if the intrinsic time evolution is not constant ( but still continuously differentiable ) , then the continuous - time limit can be carried out in the same way , resulting in a time - dependent generator and just a time - ordered integral  = \mathcal{t } { \operatorname{e}}^{\int_0^t \hat{l}(t'){\operatorname{d}}t'}(\rho_0 ) = \sum_{n=0}^\infty \int_0^{t}\int_0^{t'_{n } } \ldots \int_0^{t'_2 } \hat{l}(t'_n)\ldots \hat{l}(t'_1 ) { \operatorname{d}}t'_1 \ldots { \operatorname{d}}t'_n\ ] ] instead of the semigroup .however , this analytic expression will be a good approximation of the original random walk usually only if is sufficiently small such that .\ ] ] for simplicity we shall only deal with the time - independent version here below .  [ rem : drift - limit ] let us write for the generator and for the convolution semigroup of measures corresponding to the drift - like continuous - time limit of the random walk with as in instead of , and accordingly and for expectation and variances with respect to . then going through the construction of theorem [ th : cont - limit ], we see that the generator of becomes .hence , which vanishes iff the decoupling condition is fulfilled iff the original time evolution was unitary , according to theorem [ lem : deccond - unitary ] . in this case , for all , hence =\rho_0 ] and .\ ] ] now a variety of special cases may be investigated .if e.g.  acts only on the first qubit , i.e. , it can be written as , then so does .if moreover splits as a product state on the tensor factors , then so does  ] , for any given and initial density matrix .the result should be a good approximation for the actual random walk if .important related quantities like gate fidelity shall be computed in the following section . before concluding the present section let us derive here a tool that shall allow us to compute higher moments ( including variance ) of random variables , beyond the present linear ones ( expectation value ) .  [ prop : higher - mom ] in the setting of theorem [ th : cont - limit ] , for all , let and define the linear operator on by and linear extension .then  = \langle(y_1{\otimes} ... {\otimes}y_n ) , { \operatorname{e}}^{t\hat{l}^{(n ) } } ( x_1{\otimes} ... {\otimes}x_n)\rangle.\ ] ]  following the notation and the truncation and quotient space procedure exactly as in the case of , we can define ( not uniquely ) a smooth function } \in { c^\infty}(g) ] , which is analytic for , i.e. , in ; we can and do choose it such that }_{x_1 \ldots x_n , y_1\ldots y_n}=f^{[\infty]}_{x_1,y_1 } \cdots f^{[\infty]}_{x_n , y_n} ] and  ] .some readers might find the vanishing variance intuitively expected , given that the limiting procedure corresponds somehow to the classical law of large numbers where convergence is almost surely to the ( non - constant but time - dependent ) expectation value .the expression for follows immediately from that of in the preceding proof , specialising to : since the last line in becomes simply   for , we analogously compute : = & { \mathbb{e}}^{({\operatorname{drift}})}[f_t^2]- { \mathbb{e}}^{({\operatorname{drift } } ) } [ f_t]^2 = { \mathbb{e}}^{({\operatorname{drift}})}[(1-f_t)^2]- { \mathbb{e}}^{({\operatorname{drift } } ) } [ 1-f_t]^2\\ = & \frac{1}{d^2}\sum_{i , j , k , l=1}^d { \mathbb{e}}^{({\operatorname{drift } } ) } [ |\langle e_i , ( { \operatorname{id}}- \alpha'_t)(e_j ) \rangle|^2|\langle e_l , ( { \operatorname{id}}- \alpha'_t)(e_k ) \rangle|^2 ] \\ & - \big ( \frac{1}{d}\sum_{k , l=1}^d { \mathbb{e}}^{({\operatorname{drift } } ) } [ |\langle e_l , ( { \operatorname{id}}- \alpha'_t)(e_k ) \rangle|^2]\big)^2\\ = & \frac{1}{d^2}\sum_{i , j , k , l=1}^d |\langle e_i , ( { \operatorname{id}}- { \operatorname{e}}^{t\hat{l}^{({\operatorname{drift}})}})(e_j ) \rangle|^2|\langle e_l , ( { \operatorname{id}}- { \operatorname{e}}^{t\hat{l}^{({\operatorname{drift}})}})(e_k ) \rangle|^2 \\ & - \big(\frac{1}{d}\sum_{k , l=1}^d |\langle e_l , ( { \operatorname{id}}- { \operatorname{e}}^{t\hat{l}^{({\operatorname{drift}})}})(e_k ) \rangle|^2\big)^2\\ = & 0.\end{aligned}\ ] ]  we return to our former illustrative example [ ex:1 ] . since , we find that and hence as follows immediately from the respective definition in propositions [ prop : higher - mom ] and [ prop : modulus ] . for short times the results of proposition [ prop : eftvarft ] become : \approx & -\frac{1}{4^n}\sum_{k}^{4^n } 2 t \langle e_k , \hat{l}(e_k)\rangle  + \frac{1}{4^n}\sum_{k , l}^{4^n } t \langle e_k{\otimes}e_l , \hat{l}^{(2)}(e_l{\otimes}e_k)\rangle \\ = & \frac{2t\tau}{4^n|j| } \sum_{j\in j } \sum_{k , l}^{4^n } |\langle e_k , [ v_j h v_j , e_l]\rangle|^2\\ = & \frac{2t\tau}{4^n|j| } \sum_{j\in j } \sum_{k}^{4^n } |[v_j h v_j , e_k]|^2,\end{aligned}\ ] ] which for special cases of can be further simplified , but in general will be used in this form for a computer and is of order .a similar procedure may be applied to variance .in contrast , in the case of the drift - like limit , we would simply get =1 , \quad { \operatorname{var}}^{({\operatorname{drift}})}[f_t ] = 0 , \quad t\in{\mathbb{r}}_+,\ ] ] which is obviously less realistic than the diffusion - like limit , but on the other hand confirms that for unitary time - evolution dynamical decoupling works ( i.e. , decouples ) optimally , in contrast to other types of , cf .also theorem [ lem : deccond - unitary ] !theorem [ th : cont - limit ] gives us the expectation of our quantities in the continuum limit , but we must ask two questions:-  * how big is the difference between continuum limit and original discrete random paths ? * what is the distribution of the actual ( continuum limit ) paths around the expectation value ?these two errors add up to give the total maximal error , which we have to estimate now .concerning ( 1 ) , one has to work with a kind of berry - esseen theorem on the approximation of random walks by brownian motion .this is quite complicated , but we satisfy ourselves here with the fact that this error tends to as .  concerning ( 2 ), the deviation around the expectation value is expressed in the quantiles , which can be efficiently estimated using chebychev s inequality together with the variance expression .suppose a given quantum system undergoes decoherence caused by interaction with an external quantum heat bath described by another quantum system .then according to standard axioms of quantum mechanics , time evolution of the total ( closed ) system is unitary , thus described by a one - parameter automorphism family on the operators of the total hilbert space , namely and is the ( possibly time - dependent ) hamiltonian of the total system on and the time - ordered integral is defined in analogy to .the heat bath may be infinite - dimensional separable , but the involved hamiltonian is henceforth supposed to be uniformly bounded on compact intervals .it is unclear whether dynamical decoupling works without this assumption , and maybe alternative requirements would have to be made in case of unboundedness , cf .* app . ) for further discussion .the actual dynamics perceived on the subsystem is given by where is the partial trace ( conditional expectation ) onto the subsystem and the initial state of the heat bath .the resulting perceived dynamics then becomes a family of cpt maps . under special assumptions on , it actually produces the cpt semigroup with infinitesimal generator , the lindblad operator , but usually is no longer an automorphism .we call this phenomenon , where a cpt semigroup time evolution arises from interaction with an external quantum heat bath and unitary time evolution on the total system , _ extrinsic decoherence _ because the non - unitarity of time evolution of the original system is caused by interaction with the external heat bath .in contrast to this , _ intrinsic decoherence _we call the situation where time evolution of a _ closed _system is no longer unitary and the non - unitarity is intrinsic to the system , i.e. , does not arise from ( unitary ) interaction with a heat bath .it is a fundamental question whether this actually occurs in nature or whether the axiom of unitarity is always fulfilled on a sufficiently large total system .mathematically the two cases are described in the same way ( by cpt semigroups with unitary dilations ) , and also physically with usual observations they seem to be indistinguishable .  however , applying dynamical decoupling in the case of the above type of extrinsic decoherence , the time evolution of the total system is unitary , and so the perceived evolution on the subsystem is given by the discrete stochastic process now we notice that , if is satisfied for all , then it is also satisfied for all modulo .in fact , can be written as a finite sum , with certain traceless and with , and then   consider now for the ( possibly time - dependent ) hamiltonian .the heat bath is by definition in a thermal equilibrium state independent of time , i.e. ,  .let be the ( purely unitary ) lindbladian of the total system and thus , so , for all .then we obtain and hence = { \mathcal{e}}\circ { \mathcal{t}}{\operatorname{e}}^{\int_0^t\hat{l}'(t'){\operatorname{d}}t'}(\rho_0\otimes \rho^\theta).\ ] ] the main dynamics comes from , which leaves invariant , but changes it , so that higher - order terms disturb the invariance of the state.  we can conclude : if the system dynamics is determined by extrinsic decoherence then the decoupling condition is satisfied in first - order approximation and the total time evolution under decoupling in first - order approximation in and is described as in the unitary case ; will in general not be invariant under , but those effects are of higher order in .  we would like to have an estimate of  ] , average ( solid orange line ) , and the upper bound for  ] .this can be done by considering higher moments  ] as in proposition [ prop : eftvarft ] for an arbitrarily large number of runs , however , this is not necessary .figure [ fig1 ] illustrates the bounds with an average of concrete sample paths .although the precise relation between etc . is not clear and it is therefore difficult to compare the above bounds quantitatively , these values depend somehow monotonically on one another , i.e. , increase or decrease synchronically . in any case , when then the higher - order terms can be neglected and the difference between discrete random walk and continuum - limit tends to . at this point boundedness of needed ; otherwise alternative assumptions would have to be made that lead to future work , cf .also .moreover , it follows from the above bounds then that in the extrinsic case , whereas converges to some function in the intrinsic case .if extrinsic and intrinsic decoherence appear together : since the respective coupling strengths will not be known , it is impossible to compute the above bounds ; yet , for fixed , letting , the experimenter can check whether or not goes to , meaning pure extrinsic or intrinsic / mixed decoherence , respectively .the aim of this appendix section is to sketch the necessary definitions and facts about lie groups and convolution semigroups necessary to understand the third and fourth section . for a comprehensive study and the notation used in section [ sec : three ]we refer to any suitable textbook : e.g.  for ( linear ) lie groups and algebras , for one - parameter semigroups , ( * ? ? ?* ; * ? ? ?* ch.4 ) for probability and convolution measures on lie groups .an _ -dimensional ( real ) lie group _ is an -dimensional smooth manifold which has a group structure with neutral element such that multiplication and inversion are smooth maps . in this paper always a _ linear algebraic lie group _ ,i.e. , a group of linear mappings on a finite - dimensional real vector space .the tangent space of in forms a lie algebra and is called the _ lie algebra of _ , denoted with scalar product ( nondegenerate bilinear form ) .there is a canonical diffeomorphism from a -neighbourhood in to some -neighbourhood mapping to and called the exponential map , which in the present case can be identified with the standard exponential function of matrices . given an orthonormal basis of , the corresponding _ ( coordinate ) -chart _ is the smooth function such that and is called the _-th coordinate map_. one may extend the functions to functions in denoted again by ; write for the one - point compactification of if is noncompact , otherwise we take , and every function is extended by to ; this is needed in section [ sec : three ] for technical reasons .the _ convolution _ of two probability measures on ( with usual borel -algebra ) is defined by suppose that and are supported in a subsemigroup .then for every , we have so is supported in , too .measures and convolution on can be trivially extended to by setting , for all .the set of probability measures on , equipped with the * -weak topology and convolution as multiplication , constitutes a topological monoid , where the dirac measure serves as the neutral element . herethe _ -weak topology _ on is defined as follows : a net of measures converges to a limit measure if for all ( the continuous -valued functions on ) the condition holds .a _ continuous convolution semigroup of probability measures on _ is a set of probability measures on ( trivially extended to ) such that for every and * -weakly .let be a continuous convolution semigroup of probability measures on .for define the operator then forms a _ strongly continuous one - parameter contraction semigroup on . to there corresponds an _ infinitesimal generator _ ( in the strong operator topology ) on a suitable -invariant dense domain .given a strongly continuous one - parameter semigroup with generator on a banach space , we write .a vector is called _ entire analytic for _ if is analytic , in which case it extends to an entire analytic function .nonzero analytic vectors need not exist for one - parameter semigroups , whereas for one - parameter groups they do .if is a strongly continuous one - parameter semigroup on a banach space with generator which leaves invariant a closed subspace , then it induces a strongly continuous one - parameter semigroup on the quotient banach space with infinitesimal generator as follows : denote the quotient map by , then and with dense .given a continuous convolution semigroup of probability measures on , there exists a probability space and a -valued markov process on this space such that its transition probabilities from to are given by .the most interesting processes on are the ones we encounter in section [ sec : three ] , the so - called _ gaussian processes _ , whose contraction semigroups have generators of the form with and forms a positive - definite matrix , and with .s. l. adler : _ quantum theory as an emergent phenomenon _ , cambridge university press ( 2004 ) c. arenz , r. hillier , d. burgarth : distinguishing decoherence from alternative quantum theories by dynamical decoupling , _ arxiv_:1405.7644v3 [ quant - ph ] ( 2014 ) h. breuer , f. petruccione : _ the theory of open quantum systems _ , oxford university press ( 2002 ) e. christensen , d. e. evans : cohomology of operator algebras and quantum dynamical semigroups , _ j. london math . soc . _ * 2 * , 2 , 358 - 368 ( 1979 ) e. b. davies : _ one - parameter semigroups _ , academic press ( 1980 ) k. j. engel , r. nagel : _ one - parameter semigroups for linear evolution equations _ , springer ( 2000 ) w. fulton , j. harris : _ representation theory . a first course _ , springer ( 1991 ) u. grenander : _ probabilities on algebraic structures _ , courier dover publications ( 2008 ) h. heyer: _ probability measures on locally compact groups _ , springer ( 1979 ) o. kern , g. alber , d. l. shepelyansky : quantum error correction of coherent errors by randomization , _ eur .j. d. _ * 22 * , 153 ( 2005 ) d. a. lidar , t. a. brun : _ quantum error correction _ , cambridge university press ( 2013 ) l. santos , l. viola : dynamical control of qubit coherence : random versus deterministic schemes , _ phys . rev . a _ * 72 * , 062303 ( 2005 ) a. n. shiryaev : _ probability _ , springer ( 1996 ) l. viola , e. knill and s. lloyd : dynamical decoupling of open quantum systems , _ phys .lett . _ * 82 * , 2417 ( 1999 ) l. viola , e. knill : random decoupling schemes for quantum dynamical control and error suppression , _ phys ._ , * 94 * , 060502 ( 2005 ) d. wehn : probability on lie groups , _ proc .usa _ * 48 * , 791 - 795 ( 1962 ) m. m. wolf : quantum channels and operations . guided tour .( lecture notes on http://www-m5.ma.tum.de/allgemeines/michaelwolfen ) ( 2011 )

Abstract: we discuss a few mathematical aspects of random dynamical decoupling , a key tool procedure in quantum information theory . in particular , we place it in the context of discrete stochastic processes , limit theorems and cpt semigroups on matrix algebras . we obtain precise analytical expressions for expectation and variance of the density matrix and fidelity over time in the continuum - time limit depending on the system lindbladian , which then lead to rough short - time estimates depending only on certain coupling strengths . we prove that dynamical decoupling does not work in the case of intrinsic ( i.e. , not environment - induced ) decoherence , and together with the above - mentioned estimates this yields a novel method of partially identifying intrinsic decoherence .


in this paper , we consider a single - source multicast discrete memoryless relay network in which the source wants to send the same message reliably to multiple destinations with the help of one or more relays . a model of relay networkswas introduced by van der meulen in  .however , the single - letter capacity characterization has been open even for three - node relay networks , i.e. , relay networks having a source , a relay , and a destination . in their seminal paper , cover and el gamal developed two fundamental coding strategies for three - node relay networks .one of them is decode - and - forward ( df ) , where the relay decodes the message and forwards it to the destination , which was shown to be optimal for physically degraded channels .df was generalized for multiple relays in  . in another strategy ,compress - and - forward ( cf ) , the relay compresses its received block and sends the compressed information to the destination .cf was shown to achieve the capacity for some classes of relay networks  .recently , cf was generalized to noisy network coding in for multiple relays , which includes many previous results on relay networks as special cases .a potentially better strategy is to decode as much as possible and compress the residual information , i.e. , a combination of df and cf  .indeed such a strategy was shown to be optimal by kang and ulukus for a certain class of diamond networks in  , which consists of a source , a noisy relay , a noiseless relay that receives exactly what the source sends , and a destination that has orthogonal finite - capacity links from relays .for this class of diamond networks , it was shown that a combination of df and cf at the noisy relay is optimal and the cut - set bound is in general loose  .  in this paper , we show the optimality of a combination of df and cf for a new class of single - source multicast relay networks with an arbitrary number of nodes , which includes the class of diamond networks in as a special case . in this class , which we call multicast tree networks ,a network has a tree topology in which the root node is the source and each parent node in the graph has at most one noisy child node and any number of noiseless child nodes .we note that the achievability and converse for diamond networks in can not be directly generalized to those for our multicast tree networks .first , the codebook constructions and relay operations of the coding scheme in for diamond networks , which has a single destination , vary according to the link capacities from relays to the destination .this can not be used for multicast tree networks since they have arbitrarily many destinations .next , it would not be easy to generalize the converse proof technique in for diamond networks , which have only four nodes in three levels , for our multicast tree networks , which have arbitrarily many nodes in arbitrarily high levels .therefore , for these two reasons , we need new techniques .the key technical contributions in the achievability and converse in this paper are as follows :  * achievability : for the generalization to multicast tree networks , we construct a _ robust _ coding scheme where codebook constructions and relay operations are independent for each node and do not depend on the network topology .such a robustness of the coding scheme makes the generalization from a single destination to multiple destinations possible . *converse : to get a very simple min - cut expression , we use a novel technique of iteratively manipulating inequalities , i.e. , we recursively reduce a number of inequalities into one using the tree topology .the organization of this paper is as follows .the model of a class of multicast tree networks is presented in section [ sec : model ] . in section [ sec : main ] , we present lower and upper bounds on the capacity of the class of multicast tree networks and show a condition for these two bounds to coincide . in section [ sec : achievability ] , we derive the lower bound by presenting a coding scheme where each noisy relay employs a combination of df and cf and each noiseless relay performs a random binning . in section [ sec : ub ] , the upper bound is shown using a recursion exploiting the tree topology . in section [ sec : diamond ] , we present an equivalent capacity expression for diamond networks that shows that without loss of optimality we can construct the coding scheme such that what is compressed after decoding at a noisy relay is a noisy observation of almost uncoded information . the conclusion of this paper is given in section [ sec : conclusion ] .the following notations will be used in the paper . fortwo integers and ,  ] and a collection of conditional probability mass functions where and for  ] .we note that for  ] .  a code for a single - source multicast discrete memoryless relay network of nodesconsists of a message set  ] assigns a symbol to every received sequence for  ] assigns an estimate to each received sequence .the message is chosen uniformly from the set .the average probability of error for a code is given as \right\}.\end{aligned}\ ] ] a rate is said to be _ achievable _ if there exists a sequence of codes such that as .the capacity is the supremum of all achievable rates .a single - source multicast discrete memoryless relay network is called a multicast tree network if the probability distribution has the form of }p\left(y_k|x_{p_k}\right ) % \label{eqn : joint_tree}\end{aligned}\ ] ] where is called the _ parent node _ of node and is called a _ child node _ of node .a child node is considered to be one level lower than its parent node .a node without a parent node is called the _ root node _ and a node that has no child node is called a _leaf node_. let for  ] denote the _ subtree _ of that consists of node and all of its descendants in . in this paper ,our goal is to present lower and upper bounds on the capacity of a class of multicast tree networks and to find some tightness conditions of those two bounds . in this class of multicast tree networks ,the source node is the root node , for  ] .let and for  ] denote the set of child nodes of node , i.e. , .from now on , we only consider this class of multicast tree networks .see fig .[ fig : tree_ex ] .a practical example of our multicast tree networks is depicted in fig .[ fig : sensor ] , which represents a sensor network where a sensor node wants to send a message to the gateway nodes at the boundary connected with infinite - capacity wired links . in this example , each relay node has outgoing links to its neighbor relays such that one of the links is arbitrarily noisy and the others are noiseless . motivation for assuming noiseless links comes from a practical scenario where a transmitter is using a fixed modulation scheme tuned for the worst link and thus the transmission from the transmitter to the other receivers with better channel qualities looks almost noiselesslet us present lower and upper bounds on the capacity of multicast tree networks .[ eqn : cardinality ] latexmath:[\ ] ]  now , note that the right - hand side of ( [ eqn : multicast_cov ] ) is clearly upper - bounded by }c(1 , d) ] .we have where is from csiszr sum identity  , which proves ( [ eqn : cov_basic1 ] ) .we have note that combining ( [ eqn : basic_second_proof ] ) with ( [ eqn : cov_basic1 ] ) proves ( [ eqn : cov_basic2 ] ) .we have where is from the following markov chains : which proves ( [ eqn : cov_basic3 ] ) .for  ] .let and denote the triplets of random variables given as we will show the existence of that satisfies ( [ eqn : another ] ) separately for the cases of and .first , consider the case of .let , , and .note that is a continuous function of and becomes and when and , respectively . from the intermediate value theorem, there exists  ] such that .furthermore , ( [ eqn : another2 ] ) and ( [ eqn : another3 ] ) are satisfied from and respectively .

Abstract: in this paper , we characterize the capacity of a new class of single - source multicast discrete memoryless relay networks having a tree topology in which the root node is the source and each parent node in the graph has at most one noisy child node and any number of noiseless child nodes . this class of multicast tree networks includes the class of diamond networks studied by kang and ulukus as a special case , where they showed that the capacity can be strictly lower than the cut - set bound . for achievablity , a novel coding scheme is constructed where each noisy relay employs a combination of decode - and - forward ( df ) and compress - and - forward ( cf ) and each noiseless relay performs a random binning such that codebook constructions and relay operations are independent for each node and do not depend on the network topology . for converse , a new technique of iteratively manipulating inequalities exploiting the tree topology is used .  relay network , compress - and - forward , decode - and - forward , diamond network , multicast tree network


a binary star is a system of two stars and they orbit around their common centre of mass ( walter , 1940 ; kuiper , 1941 ) .in fact , at least of the stars are found to be in binary systems ( abt , 1983 ; pinfield et al . ,the orbital periods ( porb ) of binary stars range from 11 minutes to  years .( podsiadlowski , 2001 ) . in a binary starroche lobe plays a very major role .it actually coins the specific region about a component star in a binary system within which the orbiting material remains gravitationally bound with respect to that star ( herbig , 1957 ; plavec , 1966 ) .detached binaries are binary stars where each component is within its roche lobe and neither star fills its roche lobe .mass transfers are unlikely for this category of binaries . in case of semi - detached binaries ( kopal , 1955 )one of the components fills the roche lobe but the other does not . incase of a contact binary ( kopal , 1955 ) both the components fill their roche lobes .  in case of semi - detached and contact binaries ,mass transfer is quite expected .in fact , the majority of binaries are in fairly wide systems that do not interact strongly and both the stars evolve essentially as single star .but there is a large fraction of systems ( for porb at the left hand neighbourhood of 10 years ) that are close enough ( podsiadlowski , 2001 ) that mass is transferred from one star to another which changes the structures of both the stars and their subsequent evolution .they are termed as close binaries .although the exact numbers are somewhat uncertain , binary surveys suggest that the close binaries range between to ( duquennoy and mayor , 1991 ; kobulnicky and fryer , 2007 ) of the total binary population .  crawford ( 1955 ) and hoyle ( 1955 ) independently proposed that the observed secondaries are originally the more massive stars . at the primary level both the companions in a binary evolve independently .but as the donor expands beyond its roche lobe , then the material can escape the gravitational pull of the star and eventually the process of mass transfer starts from donor to gainer .the material will fall in through the inner lagrangian point .normally there are two possible mechanisms for mass transfer between stars in a close binary system .the first one is conservative mass transfer in which both the total mass and angular momentum of the binary are conserved and the second one is non - conservative mass transfer where both the total mass and angular momentum decays with time .observationally , there are evidences for both conservative and non conservative mass transfer in close binaries ( podsiadlowski , 2001 ; yakut , 2006 ; manzoori , 2011 ; pols , 2012 ) .the modelling of non - conservative mass transfer is a challenging one as in this case governing mechanism becomes very much complicated and uncertain due to the loss of the mass as well as the angular momentum in the binary .mass transfer in close binaries is often non - conservative and the ejected material moves slowly enough that it can remain available for subsequent star formation ( izzard et al . , 2013 ) .some communications are available which have tried to address some issues in the non - conservative mass transfer . packet ( 1981 ) proposed that accretion and spin up continues until the mass of the gainer increases by about at which point it rotates so fast that material at its equator is unbounded .rappaport et al .( 1983 ) and stepien ( 1995 ) calculated the relative angular momentum lost from the system for a non - conservative mass transfer with uniform accretion rate .soberman et al .( 1997 ) discussed on different types of modes of mass transfer like roche lobe overflow , jeans mode and produced detailed description of orbital evolution for non - conservative mass transfer for different modes as well as for a combination of several modes .podsiadlowski ( 2001 ) discussed briefly the magnetic braking for non - conservative mass transfer in a binary system . .interestingly , nelson and eggleton ( 2001 ) made a survey of case a ( with short initial orbital periods ) binary evolution with comparison to observed algol - type systems under the assumption of conservative mass transfer .they arrived at the conclusion that this simplified assumption has much less acceptability in view of the real observation and they pointed out the consideration of loss of mass and angular momentum in order to understand the dynamics more precisely .sepinsky et al .( 2006 ) discussed the mass loss in the non - conservative mass transfer and loss of angular momentum in eccentric binaries .van rensbergen et al . (2010 ) discussed on the mass loss out of close binaries .manzoori ( 2011 ) sketched the effects of magnetic fields on the mass loss and mass transfer for non - conservative mass exchange with uniform rate of accretion by the gainer .davis et al .( 2013 ) demonstrated mass transfer in eccentric binary systems using binary evolution code .interestingly , the above works mainly dealt with uniform mass accretion rate by the gainer .but real situations may not always play by this simple rule . in this regardwe must mention that stepien and kiraga ( 2013 ) while detailing on the evolutionary process under the non - conservative mass transfer in close binary system argued for non - uniform rate of mass accretion with respect to time .  in the present work we have tried to introduce a new mathematical model for the non - conservative mass transfer in the close binary which can address the issues like the mode of mass transfer and mass loss under the consideration of the critical mass limit ( for the gainer ) ( packet , 1981 ) of the transfer when the gainer is spun up through an accretion disk .the present model has be en prescribed taking in to account the gradually decreasing profile of the mass accretion rate by the gainer star with respect to time as well as with respect to the increase in mass of the gainer star and consequent time dependent mass profiles of the component stars and the orbital angular momentum of the close binary system have been demonstrated .we have also provided a numerical model in view of our present consideration .we first assume that is the mass of the gainer star and is the mass of the donor star at time t. initially i.e. greater mass is discharging mass and lower mass is in taking mass .we here consider that the process of this mass exchange continues within the range , where is supposed to be the critical mass of the gainer beyond which the process completely stops .this assumption is made following the argument made by packet ( 1981 ) that the process of mass transfer continues till the gainer gets increase in its weight when the gainer is spun up through an accretion disk as beyond this the accretor rotates so fast that material at the equator gets unbounded .we consider the following model of non - conservative mass transfer :      it is to be noted that there already exists a conventional model of non - conservative mass transfer in close binaries ( podsiadlowski et al . , ( 1992 ) ; sepinsky et al . , ( 2006 ) ; van rensbergen et al . , (2010 ) ; davis et al . , ( 2013 ) ) assuming the mass exchange rate to be uniform with respect to time and the proposed model in this context was where and carry the same meaning as in ( 1 ) .later stepien and kiraga ( 2013 ) showed that for non - conservative mass transfer this mass exchange rate decreases gradually with time and also according to izzard et al .( 2013 ) this mass exchange rate is very slow and decreases as the gainer captures mass . in view of thiswe introduced two factors in the above model given in ( 1 ) : the first factor is to trace the gradually decreasing profile of mass exchange rate with time and in this connection the parameter must be small enough so that this rate of exchange remains feeble altogether .the second factor is to capture the gradually decreasing profile of this exchange rate with the increase in within the limit as mentioned earlier . herethe parameter is dimensionless but the parameter has the unit per unit time.  now applying bernoulli s law to the gas flow through the inner lagrangian point , we get    where a is a numerical constant likely to be between 1 and 2 .since , ( pols , 2012 ) where is the total timescale of mass transfer and p is is the orbital time period we therefore have ,    for stars with convective envelops , i.e. for red - giants or low - mass main sequence stars , and this gives us from ( 3 ) ,    using ( 4 ) in ( 1 ) ,    we consider that at t=0 ( when this mass exchange started ) the initial masses of gainer and donor were and respectively .as is taken to be the total time scale of mass transfer we can believe that is the hypothetical time taken by the gainer to reach the mass starting from , provided there is no reverse mass transfer or any other issue preventing this mass exchange .  on integration ,( 4 ) gives ,    this gives ,    again integrating ( 5 ) we get ,    using ( 6 ) in ( 8) ,    this gives   for  + now as the process of mass transfer continues till the gainer gets increase in its weight when the gainer is spun up through accretion disk ( packet , 1981 ) we have ,    thus from ( 10 ) we get , ,\ ] ] for   we expect that at , appreciably nears to the critical value and this suggests us to take at the right hand neighbourhood of zero .the parameters and must assume their magnitudes in accordance with this .  as we know that in non - conservative mass transfer , less than of the mass ejected from the donor reaches the gainer i.e. as the accretion efficiency is less than about 0.25 ( de mink et al . , 2009 )we have , this gives ,  \ ] ] now , \ ] ] is an increasing function of t. hence minimum of this function occurs at t=0 .this gives , =\frac{11}{4 } ~~[~using~(11 ) ~]\ ] ]  we have , ( pols , 2012 ) where , r being the radius of the donor and being the roche lobe radius .we also have ( pols , 2012 ) .so we can take ,   the orbital period should be less than 100 days so that the star can fill its roche lobe during its expansion through main sequence phase till to a red giant ( monzoori , 2011 ) .this gives us ( days ) and from ( 14 ) we can consider days i.e. years . on the other hand , we have years ( postonov and yungelson , 2014 ) .this gives finally ,    as a is expected to lie in the range between 1 and 2 and further , the expression is expected to be in the range of order between 1 and depending on the ratio .thus in order to bring the expression at the vicinity of zero as argued previously we must take in the order of 10 which gives also , as a is expected to be in the range between 1 and 2 .this gives ,  \ ] ]  the orbital angular momentum ( oam ) of a circularized close binary system at time t is given by , ( demircan et al ., 2006 )    where , is the total mass of the system at time t , a is the separation between and , and g is the gravitational constant .( 17 ) can be written as    where  being the binary angular velocity .we have from ( 18 ) ,    where the orbital period is   from ( 20 ) combining the results of ( 6 ) and ( 12 ) , we can obtain the time dependent profile of the orbital angular momentum j. however rappaport et al .( 1983 ) worked in the framework of non - conservative mass transfer with uniform exchange rate to estimate the amount of orbital angular momentum that is lost by the binary system on the basis of the understanding that the angular momentum lost for the system is equivalent to the angular momentum of the lost mass at any instant .but that might not work consistently in the long run .so taking in to account the non - conservative mass transfer with non - uniform accretion rate we prefer to furnish the above time dependent model of orbital angular momentum directly as an explicit function of the changing masses of both gainer and donor star during the process of non - conservative mass transfer .we here produce a numerical example for the non - conservative mass transfer in close binary system taking the initial masses of the gainer and donor as (gm ) and ( gm ) . for the present calculation, we consider the values of the parameters as ( per year ) and .the profiles of the change in masses for both gainer and donor as well as the changing profile of the orbital angular velocity with time are observed within the pre - assigned time scale of mass exchange as ( years ) and orbital period p=90 ( days ) . in this regard the graphs for against t ( fig .1 ) , against t ( fig .2 ) and j against t ( fig .3 ) are demonstrated to visualize this changing profile in the binary .interestingly in the present calculation we can observe some increase in the magnitude of j at the very initial phase ( around up to 3000 years ) .but after that phase we can see a steady decrease in j ( see fig .this might have occurred due to the reason that at the initial level increases sharply while experiences steady decrease .this altogether may have generated this kind of convex shape in the graph of j at the initial phase .but afterwards as the mass accretion rate by the gainer varies slowly and consequently shows a very weak variation with time , j continues to show a steady decreasing profile with the decrease in [ in view of ( 20 ) ] .in this paper we have made an effort to establish a mathematical model of non - conservative mass transfer in close binary systems which can address the gradually reducing rate of the mass accretion by the gainer from the donor with respect to time ( stepien and kiraga , 2013 ) as well as with respect to the increase in mass of the gainer ( izzard et al . , 2013 ) .in the present model we have taken in to account a critical mass limit of the gainer for this mass exchange ( packet , 1981 ) beyond which this process may not be continued . in view of this we have presented the time dependent profiles of the masses of both the component stars in a close binary as well as of the orbital angular momentum of the corresponding binary system .we have presented a numerical example in this regard to demonstrate the changing profiles of masses of the component stars in a close binary as well as the orbital angular momentum of the system .however the formula of packet ( 1981 ) used in the present model is valid when the gainer is spun up through an accretion disk .when the gainer is spun up through direct impact this formula may not be appropriate . in those situationsthe spin up of the gainer can be modulated by tidal interactions .hence in the process the gainer can accrete much more than of its mass without rotating at critical velocity for a long period of time .we will work in future in this direction to understand the critical limit of mass accretion under the consideration that the gainer is spun up through direct impact .magnetic field can play a significant role in this mass transfer and mass loss for some close binary systems ( podsiadlowski , 2001 ; manzoori , 2011 ) .this can be an interesting future study to incorporate the effect of the magnetic field in the present model .on the other hand there have been claims on the possibility of reverse mass transfer in binary systems ( crawford , 1955 ; hoyle , 1955 ; nelson and eggleton , 2001 ; stepien and kiraga , 2013 ) .mass loss is also common in the process of single star evolution .so for a component star in a binary system mass loss may occur both for its single star evolutionary process as well as for binary star evolution . it can be critical study in this context to separate these two issues for better understanding .there is also a suggestion to include the effect of neutrino emission according to the photon - neutrino coupling theory for better understanding of the evolution of binary systems ( raychaudhuri , 2013 ) .future study in this context may focus also on these issues .fr gratefully acknowledges support from the inter - university centre for astronomy and astrophysics ( iucaa ) , pune , india , for providing research facility .all three authors express their sincere gratitude to prof .probhas raychaudhuri , professor ( retd . ) , department of applied mathematics , university of calcutta , kolkata , india ) for a careful reading of the manuscript and for his valuable suggestions in this context .[ 1 ] abt , h.a .( 1983)-normal and abnormal binary frequencies , ann .a a , 21 , pp.343 - 372 .+ [ 2 ] crawford , j.a .( 1955)-astrophys . j. , 121 , 71 .+ [ 3 ] davis , p.j . , siess , l. and deschamps , r , ( 2013)-mass transfer in eccentric binary systems using the binary evolution code binstar , a , 556 , 4 .+ [ 4 ] de mink , s.e . , pols , o.r . , langer , n. and izzard , r.g .( 2009)-a a , 507 , l1 . +[ 5 ] duquenennoy , a. and mayor , m. ( 1991)-a , 248 , 485 .+ [ 6 ] demircan , o. , eker , z. , karatas , y. and bilir , s. ( 2006)-mnras , 366 , 1511 . + [ 7 ] herbig , g.h .( 1957)-non stable stars , part v , cambridge university press , london .+ [ 8 ] hoyle , f. ( 1955)-the frontiers of astronomy ( london heinamann ) , pp .195 - 202 . +[ 9 ] izzard , r.g ., de mink , s.e . , pols o.r . ,langer , n. , sana , h. and de koter , a. ( 2013)-massive binary stars and enrichment of globular clusters , mem.s.a.it ., 1,pp . 1 - 4 .+ [ 10 ] kobulnicky , h.a . and fryer , c.l .( 2007)-astrophys . j. , 670 , 747 .+ [ 11 ] kopal , z. ( 1955 ) ann.dap , 18 , 379 . + [ 12 ] kuiper , g.p .( 1941 ) astrophys .j. , 93 , 133 .+ [ 13 ] manzoori , d. ( 2011 ) mass transfer and effects of magnetic fields on the mass transfer in close binary system in advanced topics in mass transfer by el - amin , m. ( ed . ) , in tech . ,+ [ 14 ] nelson , c.a . and eggleton , p.p .( 2001 ) a complete survey of case a binary evolution with comparison to observed algol - type systems , astrophys .j. , 552 , pp 664 - 678 .+ [ 15 ] packet , w. ( 1981 ) a , 102 , 17 . + [ 16 ] pinfield , d. j. , dobbie , p. d. , jameson , r. f. , steele , i. a. , jones , h. r. a. and katsiyannis , a. c. ( 2003)- brown dwarfs and low - mass stars in the pleiades and praesepe : membership and binarity , mnras , 342 ( 4 ) , pp .1241 - 1259 .+ [ 17 ] plavec , m. ( 1966 ) trans .b12 , 508 .+ [ 18 ] podsiadlowski , p. , joss , p.c . and hsu , j.j.l .( 1992)- presupernova evolution in massive interacting binaries .astrophys.j . ,391 , 246 .+ [ 19 ] podsiadlowski , p. ( 2001)-the evolution of binary systems , in accretion processes in astrophysics , ( ed . )matrtinez - pais , i.g . ,shahbaz , t. and velazquez , j.c . , cambridge university press , pp .+ [ 20 ] pols , o. r. ( 2012)-lecture notes on binary stars , department of astrophysics / imapp , radboud university nijmegen , the netherlands ( electronic source : / ) . + [ 21 ] postnov , k.a . and lev yungelson , r. ( 2014)-the evolution of compact binary star systems .living rev .relativity , 17 , 3 .+ [ 22 ] rappaport , s. , joss , p.c and verbunt , f. ( 1983)-a new technique for calculations of binary stellar evolution , with application to magnetic braking .astrophys.j . ,275 , 713 .+ [ 23 ] raychaudhuri , p. ( 2013 ) roche lobe and close binary stars , bull .soc . , 105 ( 5 ) , pp .369 - 378 .+ [ 24 ] sepinsky , j.f ., willems , b. , kalogera , v. and rasio , f.a .( 2006)-interacting binaries with eccentric orbits ii secular orbital evolution due to non - conservative mass transfer , arxiv:0903.0621v1 .+ [ 25 ] soberman , g.e , phinney , e.s , van den heuvel , e.p.j .( 1997)-stability criteria for mass transfer in binary steller evolution , a , 327,620 - 635 .+ [ 26 ] stepien , k. ( 1995)-mnras , 274 , 1019 . + [ 27 ] stepien , k. and kiraga , m. ( 2013)- evolution of cool close binaries : rapid mass transfer and near contact binaries , acta astronomica , 63 , pp:239 - 266 .+ [ 28 ] van rensbergen , w. , de greve , j.p ., mennekens , n. , jansen , k. and de loore , c. ( 2010 ) -mass loss out of close binaries , a , 510 , a13 .+ [ 29 ] walter , k. ( 1940)- z. astrophys . , 18 , 157 . + [ 30 ] yakut , k. ( 2006)-an observational study of unevolved close binary stars , phd thesis , university of leuven , belgium .

Abstract: mass transfer in close binaries is often non - conservative and the modeling of this kind of mass transfer is mathematically challenging as in this case due to the loss of mass as well as angular momentum the governing system gets complicated and uncertain . in the present work a new mathematical model has been prescribed for the non - conservative mass transfer in a close binary system taking in to account the gradually decreasing profile of the mass accretion rate by the gainer star with respect to time as well as with respect to the increase in mass of the gainer star . the process of mass transfer is understood to occur up to a critical mass limit of the gainer star beyond which this process may cease to work under the consideration that the gainer is spun up through an accretion disk . +  + * keywords : * close binary ; non - conservative mass transfer ; primary star ; secondary star


in the drawing on the left of figure [ fig : chart ] , we give an example of a simple context - free chart , where we have assumed the edges to be conventionally oriented from left to right .  seen in geometric terms , this chart is an oriented planar graph dividing the plane in five internal regions and one external one .each of the internal regions can be transformed into a disk by a topological transformation of the plane preserving oritentation .the five corresponding circular graphs , or s , are shown on the right .relative to one of these cells , an edge can be considered oriented positively if it goes clockwise , or negatively if it goes counter - clockwise .thus , the np edge of the s - np - vp cell is oriented negatively , while the np edge of the np - john cell is oriented positively .it is then an obvious observation that the chart on the left can be obtained by a process of `` stitching '' together the five cells along _ identically labelled but inversely oriented _ edges .in the course of this operation , each cell undergoes an orientation - preserving topological deformation .the cells of figure [ fig : chart ] provide a geometric presentation of context - free productions .each cell has the property that it has exactly one positively oriented edge , labelled with a nonterminal , and one or several negatively oriented edges , labelled with nonterminals or terminals .if , in the context - free chart , one chooses a point in each internal region and draws lines between two points belonging to adjacent regions , then one obtains a planar graph which is tree - like , and which corresponds to the usual notion of derivation tree in a context - free grammar .  if , instead of a context - free grammar , one considers a general type-0 grammar , charts can be generalized to graphs which do not have this `` tree - like '' property .an illustration of such a case is given in figure [ fig : qsys ] , which shows a derivation of the french prepositional phrase `` du livre '' , involving the contraction `` du '' of the preposition `` de '' with the determiner `` le '' .the graph on the left could be called a `` type-0 chart '' , or a `` q - system chart '' , in reference to .the cells associated with this chart are shown on the right .in contrast to the context - free case , one of the cells now contains two positively oriented edges .  in the computation systems we are proposing here , we will consider cells with an arbitrary number of positively or negatively oriented edges to be the building blocks from which computations are built .our formalization will rely on the mathematical concept of a .cancellation diagrams were introduced in 1933 by van kampen and have been playing since the seventies an increasing role in combinatorial group theory .* definition*. a , or simply , over the vocabulary is a finite graph which is : ( 1 ) planar , that is , embedded in the plane in such a way that two edges can only intersect at a vertex ; ( 2 ) connected ; ( 3 ) directed , that is , the edges carry an orientation ; ( 4 ) labelled , that is , each edge carries a label taken in . in the limit , a graph consisting of a single vertex is also considered to be a diagram .a diagram separates the plane in connected open sets : the exterior ( set of points that can be connected to a point at infinity without crossing an edge ) , and open internal regions , called s , each consisting of points which can be connected without crossing an edge , but which are separated from the exterior .the set of points in the plane which belong to an edge or to an internal region of the diagram is called the of the diagram .the locus of a diagram is always a connected and simply connected ( no holes ) closed region of the plane .an edge whose points are not in the closure of a cell region is called a edge of the diagram .an example of a diagram over the vocabulary is given in .this diagram is made up of three cells and contains one thin edge .the of a cell is the set of edges which constitute its topological boundary .the boundary of a diagram is the set of edges which are such that all their points are connected to the exterior .if one choses an arbitrary vertex ( such as o in the figure ) on the boundary of a diagram , and if one moves on the boundary in a conventional clockwise fashion , then one collects a list of edges which are either directed in the same way as the movement , or contrary to it . by producing a sequence of labels with exponent in the first case , in the second case , onecan then construct a word over the vocabulary ( that is , a word in the _ free group _ over , see below ) ; this word is said to be a of the diagram .we will say that a diagram is if there does not exist a pair of edges with a common vertex , with the same label , oriented oppositely relative to ( that is , both edges point towards or both point from ) , and such that at least one of the two `` angles '' formed by the two edges is `` free '' , that is , does not `` contain '' another diagram edge ( see ) .suppose that one is given a fixed set of cells over a vocabulary .let s consider the following generative process for producing diagrams ( only informally described here ) :  1 .initialize the diagram by choosing a point in the plane ; 2 .iterate an arbitrary number of times the following procedure any of the following steps : * add a new oriented labelled edge to the exterior of the current diagram by connecting it to one vertex on the boundary of the diagram ; * add a new cell taken from to the exterior of the current diagram , either by connecting one vertex of the cell to one vertex on the boundary of the diagram or by `` pasting '' the cell to the boundary of the diagram along consecutive edges having the same labels and arrow directions ; * reduce the current diagram by `` folding together '' two adjacent edges on its boundary which have the same label but opposite directions ( informally , the deformation involves gradually reducing the external `` angle '' between the two edges until they are identified , that is , the folding is towards the interior of the diagram ) ;  it is easy to prove that this process generates exactly the set of diagrams over such that all the cells of are elements of .the boundary words of elements of can be seen as _ coding _ results of the computations determined by the `` specification '' .  for instance , going back to the example of figure [ fig : chart ] , the boundary word is the result of a computation over the specification consisting of the five circular cells in the figure .any word thus obtained which is of the form can be seen as coding the fact that is a sentence relative to the `` grammar '' specified by .this informal notion of computation with diagrams will now be made more precise by turning to its algebraic counterpart , _ group computation structures_.we start by quickly reviewing some basic concepts of group theory before turning to group computation structures .a is a set together with a product , written , such that :  * this product is associative ; * there is an element ( the neutral element ) with for all .a is a monoid in which every element has an inverse such that .a of is a subset of containing and closed under the product of .a of is a submonoid of which is closed under inversion .two elements in a group are said to be if there exists such that .  a subset ( resp .a subgroup , a submonoid ) of a group is said to be a ( resp ., of iff when it contains , it contains all the conjugates of in .  if is a subset of , the intersection of all normal submonoids of containing ( resp . of all subgroups of containing )is a normal submonoid of ( resp . a normal subgroup of ) and is called the of in ( resp . the of in ) .the notion of normal subgroup is central in algebra . for our purposes here ,the less usual notion of normal submonoid will be the more important notion .let s consider an arbitrary set , called the , and let s form the so - called set of atoms on , which is notated and is obtained by taking elements in as well as the formal inverses of these elements .we now consider the set consisting of the empty string , notated , and of strings of the form , where is an atom on .it is assumed that such a string is , that is , never contains two consecutive atoms which are inverse of each other : no substring or is allowed to appear in a reduced string .  when and are two reduced strings , their concatenation can be reduced by eliminating all substrings of the form or .it can be proven that the reduced string obtained in this way is independent of the order of such eliminations . in this way , a product on is defined , and it is easily shown that becomes a ( non - commutative ) group , called the over  .we will say that an ordered pair is a if :  1 . is a set , called the , or the set of 2 . is a subset of , called the , or the set of . if it contains this additional assumption is made , becomes equal to . ]  the submonoid closure of in is called the of the group computation structure .the elements of will be called , or simply .if is a relator , and if is an arbitrary element of , then will be called a of the group computation structure .it is easily seen that the set of quasi - relators is equal to the normal subset closure of in , and that is equal to .a relative to is a finite sequence of quasi - relators .the product in is evidently a result , and is called the .it can be shown that the result monoid is entirely covered in this way : each result is the result of some computation .a computation can thus be seen as a `` witness '' , or as a `` proof '' , of the fact that a given element of is a result of the computation structure .  for specific computation tasks ,one focusses on results of a certain sort , for instance results which express a relationship of input - output , where input and output are assumed to belong to certain object types .for example , in computational linguistics , one is often interested in results which express a relationship between a fixed semantic input and a possible textual output ( generation mode ) , or conversely in results which express a relationship between a fixed textual input and a possible semantic output ( parsing mode ) .if is a group computation structure , and if is a given subset of , then we will call the pair a .we will say that is the set of acceptors , or the , of . a result of which belongs to the public interface will be called a of .the two views of computation provided on the one hand by diagrams and on the other hand by group computation structures are in fact equivalent to each other .this equivalence is the consequence of the `` the fundamental theorem of combinatorial group theory '' .* definition*. a word on is said to be iff every cyclic permutation of it is reduced .it is easy to see that :  * a reduced word is cyclically reduced iff it is not of the form with an atom ( positive or negative ) ; * if a word is cyclically reduced , then all its cyclic permutations are cyclically reduced ; * for any word , there is a conjugate of which is cyclically reduced ; * two conjugates of a word which are cyclically reduced are cyclic permutations of each other .it is often convenient to picture the set of all cyclic permutations of a cyclically reduced word as a circular diagram of labelled oriented edges such that no adjacent edges cancel each other . for any word , such a diagram provides a canonical representation of the cyclically reduced conjugates of .consider a group computation structure . without loss of generality, it can be assumed that the relators in are cyclically reduced , because the result monoid is invariant when one considers a new set of relators consisting of conjugates of the original ones . from now on , unless stated otherwise , this assumption will be made for all relators considered .take any such cyclically reduced relator , where and , and construct a labelled cell in the following way : take a circle and divide it in arcs ; label the clockwise- arc and orient it clockwise if , anti - clockwise otherwise .the labelled cell thus obtained is call the associated with .  rather than presenting the through a set of relator words as we have done before, it is now possible to present it through a set of relator cells ; if one gives such a set , a standard presentation of the can be derived by taking an arbitrary origin on each cell and `` reading '' the relator word clockwise from this origin ; the origin chosen does not matter : any other origin leads to a conjugate relator , and this does not affect the notion of result .we are now able to state what j. rotman calls `` the fundamental theorem of combinatorial group theory '' .we give the theorem in a slightly extended form , adapted to the case of a , that is , using normal _ sub - monoid _ closure rather than normal _ subgroup _ closure ; the subgroup case follows immediately by taking a set or relators containing along with .[ fundamental theorem of cgt ] let be a group computation structure such that all relators are cyclically reduced . if is a cyclically reduced word in , then if and only if there exists a reduced diagram having boundary word and whose regions are relator cells associated with the elements of .the proof is not provided ; it can easily be recovered from the property demonstrated in ( chapter 5 , section 1 ) .the proof involves the following remark .if one considers a product with and arbitrary elements of , this product can be read as the boundary word of the `` star '' diagram represented in , starting at and progressing clockwise .this star diagram is in general not in reduced form , but it can be reduced by a stepwise process of `` stitching together '' edges which do not respect the definition of a reduced diagram .  * example*. let s consider a with vocabulary and set of ( cyclically reduced ) relators .the cyclically reduced word is an element of , for it can be obtained by forming the product if we form the star diagram for this product , we obtain the first diagram shown in .this diagram is not reduced , for instance the two straight edges labelled are offending the reduction condition .if one `` stitches '' these two edges together , one obtains the second diagram in the figure .this stitching corresponds to a one - step reduction of the boundary word of the first diagram , into the boundary word of the second . by continuing in this way, one obtains the fifth diagram of the figure , which is reduced , and whose boundary is the desired result .we will now show how the formal concepts introduced above can be applied to the problems of grammatical description and computation .we start by introducing a simple grammar for a fragment of english . in figure[ fig : g - grammar ] , this grammar is presented algebraically in terms of relators , and in figure [ fig : linguistic - cells ] the same grammar is presented geometrically in terms of relator cells .formally , is a group computation structure with acceptors over a vocabulary consisting of a set of logical forms and a disjoint set of phonological elements ( in the example , words ) .thus , are phonological elements , j , s(j , l ) are logical forms .  the grammar lexicon , or set of relators , is given as a list of `` lexical schemes '' .thus , in figure [ fig : g - grammar ] , each line is a lexical scheme and represents a set of relators in .the first line is a ground scheme , which corresponds to the single relator j i , and so are the next four lines .the sixth line is a non - ground scheme , which corresponds to an infinite set of relators , obtained by instanciating the _ term meta - variable _ a ( notated in uppercase ) to a logical form .so are the remaining lines .the vocabulary and the set of relators that we have just specified define a group computation structure .let s now describe a set of acceptors for this computation structure .we take to be the set of elements of which are products of the following form : where is a logical form ( stands for `` semantics '' ) , and where each is a phonological element ( stands for `` word '' ) .the expression above is a way of encoding the ordered pair consisting of the logical form and the phonological string ( that is , the inverse of the product ) .` j i ` + ` l i ` + ` p i ` + ` m i ` + ` w i ` + ` r(a ) i ai ` + ` s(a , b ) bi i ai ` + ` i(e , a ) ai i ei ` + ` t(n ) ni i `   we now show how to compute a proof of the fact that i(s(j , l),p ) is a public result for , or , in other words , that the logical form and the phonological form are in correspondance relative to .we start by an informal geometric computation of this result and follow by an algebraic computation of it . the geometric computation if the more intuitive of the two , the algebraic one the one for which we have the more precise formal definition and which more directly displays group - theoretical characteristics .consider the diagram of figure [ fig : parsing - generation - diagram ] .the cells of this diagram are instances of the cell schemes of figure [ fig : linguistic - cells ] . from theorem [ fundamental theorem of cgt ], this means that i(s(j , l),p ) is a public result of .the diagram in figure [ fig : parsing - generation - diagram ] can be obtained by several computations , in the sense of section [ diagrams as computational devices ] .one such computation , corresponding to a generation mode , starts from the top edge ( i(s(j , l),p ) ) and progressively adds cells in a top - down way . a second computation, corresponding to a parsing mode , starts by building a diagram consisting of the sequence of the five phonological edges at the bottom and progressively adds cells in a bottom - up way .still another geometric computation , establishing the connection with the algebraic computation , will be shown below .consider the following relators , instanciations of relator schemes of figure [ fig : g - grammar ] :  ` r1 = i(s(j , l),p ) pi i s(j , l)i ` + ` r2 = s(j , l ) li i ji ` + ` r3 = j i ` + ` r4 = l i ` + ` r5= p i `  and the quasi - relators :  ` r1 = r1 ` + ` r2 = r2 ` + ` r3 = r3 ` + ` r4 = ( ) r4 ( ) i ` + ` r5 = ( ) r5 ( ) i `  then we have :  ` r1 r2 r3 r4 r5 = ` + ` i(s(j , l),p ) i i i i i `  and therefore i(s(j , l),p ) iii ii is the result of a computation ( r1,r2,r3,r4,r5 ) , as announced .what is the relationship of this computation with the geometric view of figure [ fig : parsing - generation - diagram ] ?the answer is given by the geometric computation indicated by the diagram in figure [ fig : john - saw - star ] .if we read the boundary of this diagram clockwise starting from o , we obtain the unreduced expression r1 r2 r3 r4 r5. but if we perform further geometric reduction steps on the diagram of figure [ fig : john - saw - star ] , we obtain the diagram of figure [ fig : parsing - generation - diagram ] .the linguistic examples that we have presented up to now have been rather simple .we will now introduce an extension of group computation structure which gives rise to forms of non - local grammar dependencies .let be a finite multiset ( unordered list ) of words in .we will call such an expression a over .we will say that a word is a of iff can be expressed as a product , where are elements of .let be the set of multi - conjugates of ._ remark : _ this notion is well - defined , for any ordering of the multiset leads to the same set of multi - conjugates ( by a simple property of conjugacy ) .it is also easy to check that replacing any by a cyclic permutation of does not change , and furthermore , that is a normal subset of .if we are given a ( finite or infinite ) collection of multi - relators , we can consider the group computation structure obtained by taking as set of relators the set :   we will call the gcs thus obtained the .when presenting a group computation structure with multi - relators , we will sometimes use the notation : for presenting a multi - relator .a multi - relator consisting of only one relator will be written simply ; specifying such a multi - relator has exactly the same effect on the computation structure as specifying the simple relator , because is normal in .we will now state a theorem which is an extension of the the fundamental theorem of combinatorial group theory for the case of gcs s with multi - relators .we first need the notion of multi - cell associated with a multi - relator .we first remark that we can always assume that a multi - relator is such that each is cyclically reduced , because taking the cyclically reduced conjugate of does not change the notion of multi - conjugate .we will assume this is always the case in the sequel .if is a multi - relator , and if is the cell associated with in the manner of [ relator - cells ] , then we will call the multiset of cells the associated with this multi - relator .consider a finite multiset of multi - relators and take the multi - set obtained by forming the multiset union of the multi - cells associated with these multi - relators .a diagram whose cells are exactly ( that is , taking account of the cell counts ) those of the multiset will be said to be a the gcs with multi - relators under consideration .one can easily prove the following extension of theorem [ fundamental theorem of cgt ] .[ extended fundamental theorem of cgt ] .let be a group computation structure with multi - relators .if is a cyclically reduced word in , then is in the result monoid of iff there exists a reduced diagram relative to having boundary .let s consider the extension of presented in figure [ fig : g - grammar with multi - relators ] .the first nine entries are the same as before , but three new multi - relator schemes have been added to the end .they correspond to definitions of the quantifiers `` every '' and `` some '' and to the definition of the relative pronoun `` that '' .the notation p [ ] is employed to express the fact that a logical form containing an _ argument identifier _  is equal to the application of the abstraction p to .the _ identifier meta - variable _ x in p[x ] ranges over such identifiers ( , , , ... ) , which are notated in lower - case italics ( and are always ground ) .the meta - variable p ranges over logical form abstractions missing one argument ( for instance .s(j , ) ) .when matching meta - variables in logical forms , use of higher - order unification will be allowed .for instance , one can match p[x ] to s(j , ) by taking and .  `j i ` + ` l i ` + ` p i ` + ` m i ` + ` w i ` + ` r(a ) i ai ` + ` s(a , b ) bi i ai ` + ` i(e , a ) ai i ei ` + ` t(n ) ni i ` + ` ev(n , x , p[x ] ) p[x]i ; x ni i ` + ` sm(n , x , p[x ] ) p[x]i ; x ni i ` + ` tt(n , x , p[x ] ) p[x]i i ni ; x `  the geometric presentation of is given in figure [ fig : multi - cells ] , for the three new multi - relators ( we have omitted reproducing the cells of figure [ fig : linguistic - cells ] , which also belong to the specification ) .the dotted lines indicate which cells belong to a given multi - cell .the cells belonging to a multi - cell always work in solidarity : they have to appear together in a diagram or not at all .   the figures [ fig : every - some ] , [ fig : some - every ] , and [ fig : man - that ] illustrate some computations with .the first diagram consists of seven cells : three `` mono''-cells , and two `` bi''-cells ( a ) and ( b ) , corresponding respectively to the entries for `` every '' and for `` some '' in the grammar .the boundary of the diagram can be read ev(m,,sm(w,,s ( , ) ) ) . because of theorem [ extended fundamental theorem of cgt ] ,this proves the correspondence relative to of the logical form ev(m,,sm(w,,s ( , ) ) ) with the sentence .the second diagram is similar to the first , but with a different layout between the multi - cells for `` every '' and `` some '' .it establishes a different scoping for the quantifiers of the same sentence .the third diagram establishes the correspondence between the sentence and the logical form r(t(tt(m , x , s(l , x ) ) ) ) .in section [ groups , monoids , normal subsets ] , we stated without justification that , for the purpose of presenting group computation structures , we would use normal submonoids of a free group rather than normal subgroups , which are much more central in algebra .we would like now to give some support to this claim .suppose that we were to define computation results as being elements of the normal subgroup closure of a set of relators , rather than of the normal submonoid closure .then would partition the elements of into equivalence classes : two elements being equivalent iff .for all purposes two such elements would become undistinguishable . for linguistic formalization , where we need to model the relationship between a logical form and a phonological form , this approach would neutralize important differences : if we had , say , the results , , and  meaning that is associated with , and is associated with both and , then would all be in the same equivalence class , meaning in particular that would be associated with , counter - intuitively . and , it would be possible to specify both relators and in the group computation structure .one might then perhaps even think of _ quotienting _ the group computation structure by the equivalence relation thus created . ]this problem does not occur with the choice of : in this case the relation between defined by is only a preorder , not an equivalence relation ( see ) .we started this paper in section [ grammar and geometry ] by taking a geometric viewpoint on conventional rewriting systems such as context - free grammars or type-0 grammars . before ending, we would like to consider the following question : what , if any , is the difference between group computation and these rewriting systems ?a detailed discussion of this question is outside the scope of this paper ( see ) , and only some brief indications will be given here .let s consider a type-0 rewriting system and a type-0 chart relative to this system .we have seen in section [ grammar and geometry ] that such a chart can be obtained by `` pasting '' together cells corresponding to the rules of the rewriting system .that is , the chart can be seen as a diagram relative to the defined by this collection of cells .but such a diagram has an interesting property among all diagrams that could be produced relative to the : the cells of are partially ordered by the relation which considers a cell of a diagram to immediately precede a cell of the same diagram iff there is a common edge between and which is negatively oriented relative to and positively oriented relative to . for a standard presentation of the chart ,this partial order is just the usual `` top - down '' order between the cells .the fact that the cells of the diagram are partially - ordered by the precedence relation just defined is a _ global _ property of the diagram which means that the precedence relation does not create cycles among cells .it is a property which is foreign to group computation , where cells are assembled relative to a purely _ local _ criterion .this means that , in general , the translation of a type-0 system into a produces diagrams that can not be interpreted as derivations of the original type-0 system .however , in the case where the relator cells resulting from the translation of the rewriting system can be _ statically _ partially - ordered by the precedence relation where iff has a negatively oriented edge labelled and has a positively oriented edge with the same label , _ then _ no cycles can ever appear in diagrams of the , and then the two notions of computation become identical .  the static condition just described is very restrictive for type-0 systems _ stricto sensu _ , that is , having a finite vocabulary of non - terminals , because it precludes recursivity .it is , however , much more interesting in the case of rewriting systems defined by rule - schemes over terms , such as dcgs , or their type-0 counterparts .an example of such a situation is provided by the cells of figure [ fig : linguistic - cells ] , which can be seen to correspond to an `` extended '' type-0 system of this kind , and where the instanciated relator cells can be partially ordered by the precedence relation just described ( see ) .let s first define the _ semantic size _ of a logical form term as the number of nodes in this term which are different from argument identifiers such as , , ... ( thus , ) , and the semantic size of a phonological word as 0 .let s also define the _ phonological size _ of a phonological word as 1 , and the phonological size of a logical form as 0 .if one looks at the multi - relator schemes of figure [ fig : g - grammar with multi - relators ] , it can be checked that any grounded multi - conjugate instance of each multi - relator scheme is such that .for example , taking the multi - relator scheme ev(n , x , p[x ] ) p[x]i ; x nii we see that any multi - conjugate instance of this scheme is such that ) } ) + h({\tt p[x]\i } ) + h({\tt x } ) + h({\tt n\i } ) + h(\w{every}\i),\ ] ] because taking conjugates does not change the value of a morphism like which takes its values in a commutative group .thus we have : } ) - \ss({\tt p[x ] } ) +0 - \ss({\tt n } ) , -1 ) = ( 1,-1),\ ] ] and so on for the other relator schemes .consider now a computation of a result involving multi - relators ( in geometric terms , a diagram with boundary involving multi - cells ) .the value of is then taken times , that is , . this has the following consequences ( stated informally here ) for the complexity of parsing and generation :  1 .  if a phonological string and a logical form are in correspondence relative to the grammar , then it is possible to bound the phonological size of the string as a function of the semantic size of the logical form ( in fact they are equal ) .if we have a string of phonological size to parse , then any computation will involve at most ( and in fact exactly ) multi - cells ; this implies that parsing is decidable and of bounded complexity .if we have a logical form of semantic size to generate , then any computation will involve at most ( and in fact exactly ) multi - cells ; this implies that generation is decidable and of bounded complexity .although the grammar is rather remarkable in admitting a morphism such as which has the same value on all the multi - relators , what is really needed for the complexity properties to hold is a less demanding requirement ( see ) of finding a morphism which realize some reasonable `` exchange '' of phonological material for semantic material on each multi - relator .alain colmerauer .les systmes - q ou un formalisme pour analyser et synthtiser des phrases sur ordinateur .publication interne 43 , dpartement dinformatique , universit de montral , montral , septembre 1970 .

Abstract: this paper , following , presents an approach to grammar description and processing based on the geometry of _ cancellation diagrams _ , a concept which plays a central role in combinatorial group theory . the focus here is on the geometric intuitions and on relating group - theoretical diagrams to the traditional charts associated with context - free grammars and type-0 rewriting systems . the paper is structured as follows . we begin in section 1 by analyzing charts in terms of constructs called _ cells _ , which are a geometrical counterpart to rules . then we move in section 2 to a presentation of cancellation diagrams and show how they can be used computationally . in section 3 we give a formal algebraic presentation of the concept of _ group computation structure _ , which is based on the standard notions of free group and conjugacy . we then relate in section 4 the geometric and the algebraic views of computation by using the fundamental theorem of combinatorial group theory . in section 5 we study in more detail the relationship between the two views on the basis of a simple grammar stated as a group computation structure . in section 6 we extend this grammar to handle non - local constructs such as relative pronouns and quantifiers . we conclude in section 7 with some brief notes on the differences between normal submonoids and normal subgroups , group computation versus rewriting systems , and the use of group morphisms to study the computational complexity of parsing and generation .