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Theorem 4128 Let U be a possibility distribution on a set U of (discrete) attributes and let G = (U, E) be an undirected graph over U If U is decomposable wrt G, then G is a conditional independence graph wrt U If G is a conditional independence graph wrt U and if it has hypertree structure, then U is decomposable wrt G Proof The full proof can be found in Section A5 in the appendix The first part of the theorem can be proven in direct analogy to the corresponding part of the proof of the probabilistic counterpart of the above theorem Again it is exploited that two attributes that are u-separated cannot be in the same clique and therefore the maximal cliques can be divided into two sets.

def CreateLeg( self , tag , flows , pay or receive , adjuvant table , payoff): try: adjuvants = None if adjuvant table: adjuvants = TradeServer.retrieve(adjuvant table, adjuvants ) leg = \ ppf.core.leg( TradeServer.retrieve(flows, flows ) , eval("ppf.core."+pay or receive) , adjuvants , eval("ppf.pricer.payoffs."+payoff)()) TradeServer. legs[tag] = leg return tag except RuntimeError, e: utils.raise com exception(e) def CreateTrade( self , tag , legs , exercise sched , exercise type): try: tl = [TradeServer.retrieve(l, legs ) for l in legs[1:]] if exercise sched: exercises = TradeServer.retrieve(exercise sched, exercises ) if not exercise type: raise RuntimeError, "missing exercise type" call cancel = eval("ppf.core.exercise type."+exercise type) trade = ppf.core.trade(tl, (exercises, call cancel)) else: trade = ppf.core.trade(tl, None) TradeServer. trades[tag] = trade return tag except RuntimeError, e: utils.raise com exception(e) if name == " main ": utils.register com class(TradeServer)

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Note that this part of the theorem does not require that the graph has hypertree structure and thus it is valid for arbitrary undirected graphs For the second part of the theorem the hypertree structure of G is essential, because it allows an induction on a construction sequence for the graph G that exploits the global Markov property of G The construction sequence is derived from the ordering of the maximal cliques that results from the ordering underlying the running intersection property Unfortunately, the above theorem cannot be generalized to arbitrary graphs as the following example demonstrates (this is a slightly modified version of an example given in [Gebhardt 1997]) Consider the undirected graph and the simple relation shown in Figure 412 (and recall that a relation is only a special possibility distribution).

During the late 1970s, a political factor became noticeable which would gradually turn out to be very important, namely the pressure in many countries (starting in the US) for the introduction of competition in the telecom eld Traditionally, the European telecom operators had always been state monopolies, but in 1982, the UK government decided to open the eld of mobile communications for competition The rst system to be used this way would be TACS, a modi ed version of the AMPS system used in the US Today, when there is competition in the telecom eld in most countries, it is a bit strange to read the discussions of the early 1980s with predictions of the dire consequences of competition in telecommunications.

1111 = {{A, B, C, D}}, 1122 = {{A, B}, {C, D}}, 1211 = {{A, C, D}, {B}}, 1222 = {{A}, {B, C, D}}, 1232 = {{A}, {B, D}, {C}}, 1112 = {{A, B, C}, {D}}, 1123 = {{A, B}, {C}, {D}}, 1212 = {{A, C}, {B, D}}, 1213 = {{A, C}, {B}, {D}}, 1221 = {{A, D}, {B, C}}, 1223 = {{A}, {B, C}, {D}}, 1231 = {{A, D}, {B}, {C}}, 1233 = {{A}, {B}, {C, D}}, 1234 = {{A}, {B}, {C}, {D}}. 1121 = {{A, B, D}, {C}},

It is easy to check that the graph is a conditional independence graph of the relation, since both conditional independence statements that can be read from it, namely A R C | {B, D} and B R D | {A, C}, hold in the relation However, it does not have hypertree structure, because the set of its four maximal cliques does not have the running intersection property, and, indeed, the relation is not decomposable wrt the graph If it were decomposable, then we would have a dom(A) : b dom(B) : C dom(C) : d dom(D) : R(a, b, c, d) = min{R(a, b), R(b, c), R(c, d), R(d, a)}, where R(a, b) is an abbreviation of R(A = a, B = b) etc.

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