### Table 1. Rewards (economic net returns), rd i, depending on state and action.

"... In PAGE 7: ... The economic net returns from the cow will of course depend on whether it is low yielding or high yielding and whether it is kept or replaced. In the model this is represented by a reward depending on state and action as appearing in Table1 . Those amounts are simply the annual net returns from a low, average and high yielding cow respectively.... In PAGE 16: ... However, a seasonal variation in rewards or physical outputs is easily modeled by including a state variable describing season (each state is usually defined by the value of a number of state variables describing the system). An advantage of the policy iteration method is that the equations in Table1 are general. Under any policy s we are able to calculate directly the economic consequences of following the policy by solution of the equations.... In PAGE 26: ... Thus the objective function (5) is applied, and no discounting is performed. In Table1 3, optimal policies under the three criteria are shown. It appears that the policies under the first two criteria are quite similar, but under the third criterion the optimal policy differs significantly.... ..."

### Table 1: Phylogenetic parsimony criteria.

"... In PAGE 31: ... The hypothesis encoded in this tree is preferred because it explains as much of the observed character distributions as possible by character-state transitions in a common ancestor, and invokes the fewest ad hoc hypotheses of subsequent character-state change [Far83]. There are several phylogenetic parsimony criteria, each of which encodes a di erent model of evolution by placing di erent restrictions on the types and numbers of character-state transitions allowable in a tree (see Table1 ). The Wagner Linear [KF69], Wagner General, and Fitch [Fit71] criteria assume the simplest model of evolution, in which character-state change is reversible.... In PAGE 33: ....2.1 Phylogenetic Parsimony Each of these problems is given as input a discrete character matrix for m taxa and d char- acters, and operates on an implicit graph G whose vertices are the set of all d-dimensional points de ned by the states of the given characters and whose edges are speci ed by the allow- able transitions between the states in these characters. Each phylogenetic parsimony problem seeks the evolutionary tree in G of minimum length that includes the given taxa, subject to the restrictions on character-state transitions that are particular to that problem apos;s criterion (see Table1 ). The given characters can be restricted in various ways to generate a family of... In PAGE 43: ... Question: Does the collection of characters C have a polarization such that there is a com- patible collection C0 C such that jC0j B? Unconstrained Qualitative Compatibility (UQC) Instance: Collection C of d qualitative characters de ned on a set of m objects; a positive integer B d. Question: Does the collection of characters C have a polarization and an ordering such that there is a compatible collection C0 C such that jC0j B? Table1 0: Character compatibility decision problems (adapted from [DS86]).... In PAGE 44: ... B0 = B BQC p m BCC [DS86] d0 = d m0 = m X0 = [x0i;j]; 1 i d0; 1 j m0 where a character apos;s most frequently occurring state becomes that charac- ter apos;s ancestral state in X0. B0 = B Table1 1: Reductions for character compatibility decision problems.... In PAGE 45: ... Question: Does there exist an additive tree T 2 Ad n such that X(D; A(T)) B? Fitting Unconstrained Matrices to Graph-Based Dominant Additive Trees (FUGT[ ]) Instance: Complete graph G = (V; E), jV j = n; semimetric D 2 Mn de ned on all pairs of vertices in G; set of taxa S V ; and a positive integer B. Question: Is there a subtree T of G that includes S such that Pfx;yg2T D(x; y) B and [ A(T)]S DS? Table1 2: Distance matrix tting decision problems (adapted from [Day83, KM86, Day87, Kri88]).... In PAGE 47: ... Question: Does there exist an ultrametric tree U 2 Un;2 such that X(D; U(U)) B? Fitting Binary Matrices to Dominant Ultrametric Trees of Height 2 VIA STATISTIC X (FBUT2[X, ]) [X 2 fF1; F2g] Instance: Set S of n taxa; semimetric D 2 Bn; and a positive integer B. Question: Does there exist an ultrametric tree U 2 Un;2 such that X(D; U(U)) B and U(U) D? Table1 3: Auxiliary decision problems for NP-hardness proofs of distance matrix tting decision problems (adapted from [KM86, Day87, Kri88]). .... In PAGE 48: ... S0 = S + yi; 1 i apos; D0 = [d0 i;j] = quot; D M M0 1 #, where M = [mi;j], mi;j = for all 1 i n and 1 j apos;, M0 is the transpose of M, and 1 is a square matrix with zeros on, but ones o , the main diagonal. B0 = B VC p m FUGT[ ] V = f g S f vi j 1 i jVVCjg S f ej j 1 j jEV Cjg D = [di;j], where d( ; vi) = 1 d( ; ej) = 2 d(vi; vj) = 4 d(vi; ej) = 1 if ej = fvi; xg 2 EV C, d(vi; ej) = 3 otherwise d(ei; ej) = 2 S = f g S f ej j 1 j jEVCjg B = K + jEV Cj Table1 4: Reductions for distance matrix tting decision problems.... In PAGE 52: ...Thesis Literature Phylogenetic UBfC,QgCS fC,QgCS [DJS86] Parsimony UBfC,QgDo fC,QgDO [DJS86] UBfC,QgCI fC,QgCI [DJS86] UBW SPQ [GF82, Day83] UUW SPP [GF82, Day83] WUOWL WTP [Day83] Character BfQ,CgC BfQ,CgC [DS86] Compatibility UfQ,CgC UfQ,CgC [DS86] Distance Matrix FBUT[F1] bHICy [KM86] Fitting FBUT2[F1] bHIC3y [KM86], 2 1y [Kri86], FUT[1] [Day87] FBUT2[F2] 2 2y [Kri86], FUT[2] [Day87] FUUT[F1] 1y [Kri86], HICy [KM86] FUUT[F2] 2y [Kri86] FUUT[F1; ] P4 [Kri88] FUDT[ ], 2 fF1; F2g FAT[ ], 2 f1; 2g [Day87] FUGT[ ] AET [Day83] Table1 5: Correspondence between phylogenetic inference problems in this thesis and problems in the literature. All solution problems are marked with daggers (y); all other problems are decision problems.... In PAGE 74: ...Unweighted Weighted Given-Cost Given-Limit Decision - NP-complete Evaluation FPNP[O(logn)]-C FPNP jj -hard y - Solution FPNP jj -hard, properly FPNP[O(logn)]-hard, 2 NPMVg FPNP 2 NPMVg Spanning 2 Span(NPMVg FPNP ) 2 SpanP Enumeration 2 FP#P Random 2 FRP p 2 Generation Table1 6: Computational complexities of phylogenetic inference functions. y Most weighted distance matrix tting evaluation problems are only known to be properly FPNP[O(logn)]-hard (see Corollary 26).... In PAGE 78: ... Formula: maxT jfcj9x[(P(c; x) ^ x 2 T) _ (N(c; x) ^ :(x 2 T))]gj where P, N, and T are as de ned for SAT. Table1 7: Formulations of SAT in rst-order logic (adapted from [KT90, PY91]). MAX NP.... In PAGE 79: ... Formula:maxT jf(x1; x2; x3)j [ ((x1; x2; x3) 2 C0 ! x1 2 T _ x2 2 T _ x3 2 T) ^ ((x1; x2; x3) 2 C1 ! x1 62 T _ x2 2 T _ x3 2 T) ^ ((x1; x2; x3) 2 C2 ! x1 62 T _ x2 62 T _ x3 2 T) ^ ((x1; x2; x3) 2 C3 ! x1 62 T _ x2 62 T _ x3 62 T) ] gj; where C0, C1, C2, C3, and T are as de ned for 3SAT. Table1 8: Formulations of SAT in rst-order logic (cont apos;d from Table 17).... In PAGE 82: ... 4. Given a solution W of cost c to an instance of SOL-MIN-FBUT2[F1] derived by the reduction from X3C given in [KM86] (See Table1 4), in polynomial time we can nd a canonical solution W0 with cost c0 c. 5.... In PAGE 82: ... 5. Given a solution W of cost c to an instance of SOL-MIN-FUDT[F ] ( 2 f1; 2g) derived by the reductions from FBUT2[ ] given in [Day87] (see Table1 4), in polynomial time we can nd a canonical solution W0 of cost c0 c.... In PAGE 84: ... As the Generalized parsimony criterion can simulate any ordered phylogenetic parsimony problem, (5) can be proved by a variant on any of the proofs for (1 - 4). Proofs of (6 { 7): By the reductions given in Table1 1, solutions to SOL-MAX-BCC (SOL- MAX-BQC) yield solutions to SOL-MAX-CLIQUE (SOL-MAX-BCC) of the same cost. Hence, these reductions yield L-reductions with = = 1.... In PAGE 84: ... Hence, this reduction is an L-reduction. Proof of (10): Consider the reduction from FBUT2[F ] to FUDT[F ] 2 f1; 2g given in [Day87] (see Table1 4). As OPTFBUT2[F ] = OPTFUDT[F ], condition (L1) is satis ed with = 1.... ..."

### Table 1: Phylogenetic parsimony criteria.

"... In PAGE 26: ... The hypothesis encoded in this tree is preferred because it explains as much of the observed character distributions as possible by character-state transitions in a common ancestor, and invokes the fewest ad hoc hypotheses of subsequent character-state change [Far83]. There are several phylogenetic parsimony criteria, each of which encodes a di erent model of evolution by placing di erent restrictions on the types and numbers of character-state transitions allowable in a tree (see Table1 ). The Wagner Linear [KF69], Wagner General, and Fitch [Fit71] criteria assume the simplest model of evolution, in which character-state change is reversible.... In PAGE 28: ....2.1 Phylogenetic Parsimony Each of these problems is given as input a discrete character matrix for m taxa and d char- acters, and operates on an implicit graph G whose vertices are the set of all d-dimensional points de ned by the states of the given characters and whose edges are speci ed by the allow- able transitions between the states in these characters. Each phylogenetic parsimony problem seeks the evolutionary tree in G of minimum length that includes the given taxa, subject to the restrictions on character-state transitions that are particular to that problem apos;s criterion (see Table1 ). The given characters can be restricted in various ways to generate a family of... In PAGE 38: ... Question: Does the collection of characters C have a polarization such that there is a com- patible collection C0 C such that jC0j B? Unconstrained Qualitative Compatibility (UQC) Instance: Collection C of d qualitative characters de ned on a set of m objects; a positive integer B d. Question: Does the collection of characters C have a polarization and an ordering such that there is a compatible collection C0 C such that jC0j B? Table1 0: Character compatibility decision problems (adapted from [DS86]).... In PAGE 39: ... B0 = B BQC p m BCC [DS86] d0 = d m0 = m X0 = [x0i;j]; 1 i d0; 1 j m0 where a character apos;s most frequently occurring state becomes that charac- ter apos;s ancestral state in X0. B0 = B Table1 1: Reductions for character compatibility decision problems.... In PAGE 40: ... Question: Does there exist an additive tree T 2 Ad n such that X(D; A(T)) B? Fitting Unconstrained Matrices to Graph-Based Dominant Additive Trees (FUGT[ ]) Instance: Complete graph G = (V; E), jV j = n; semimetric D 2 Mn de ned on all pairs of vertices in G; set of taxa S V ; and a positive integer B. Question: Is there a subtree T of G that includes S such that Pfx;yg2T D(x; y) B and [ A(T)]S DS? Table1 2: Distance matrix tting decision problems (adapted from [Day83, KM86, Day87, Kri88]).... In PAGE 42: ... Question: Does there exist an ultrametric tree U 2 Un;2 such that X(D; U(U)) B? Fitting Binary Matrices to Dominant Ultrametric Trees of Height 2 VIA STATISTIC X (FBUT2[X, ]) [X 2 fF1; F2g] Instance: Set S of n taxa; semimetric D 2 Bn; and a positive integer B. Question: Does there exist an ultrametric tree U 2 Un;2 such that X(D; U(U)) B and U(U) D? Table1 3: Auxiliary decision problems for NP-hardness proofs of distance matrix tting decision problems (adapted from [KM86, Day87, Kri88]). .... In PAGE 43: ... S0 = S + yi; 1 i apos; D0 = [d0 i;j] = quot; D M M0 1 #, where M = [mi;j], mi;j = for all 1 i n and 1 j apos;, M0 is the transpose of M, and 1 is a square matrix with zeros on, but ones o , the main diagonal. B0 = B VC p m FUGT[ ] V = f g S f vi j 1 i jVVCjg S f ej j 1 j jEV Cjg D = [di;j], where d( ; vi) = 1 d( ; ej) = 2 d(vi; vj) = 4 d(vi; ej) = 1 if ej = fvi; xg 2 EV C, d(vi; ej) = 3 otherwise d(ei; ej) = 2 S = f g S f ej j 1 j jEVCjg B = K + jEV Cj Table1 4: Reductions for distance matrix tting decision problems.... In PAGE 47: ...Thesis Literature Phylogenetic UBfC,QgCS fC,QgCS [DJS86] Parsimony UBfC,QgDo fC,QgDO [DJS86] UBfC,QgCI fC,QgCI [DJS86] UBW SPQ [GF82, Day83] UUW SPP [GF82, Day83] WUOWL WTP [Day83] Character BfQ,CgC BfQ,CgC [DS86] Compatibility UfQ,CgC UfQ,CgC [DS86] Distance Matrix FBUT[F1] bHICy [KM86] Fitting FBUT2[F1] bHIC3y [KM86], 2 1y [Kri86], FUT[1] [Day87] FBUT2[F2] 2 2y [Kri86], FUT[2] [Day87] FUUT[F1] 1y [Kri86], HICy [KM86] FUUT[F2] 2y [Kri86] FUUT[F1; ] P4 [Kri88] FUDT[ ], 2 fF1; F2g FAT[ ], 2 f1; 2g [Day87] FUGT[ ] AET [Day83] Table1 5: Correspondence between phylogenetic inference problems in this thesis and problems in the literature. All solution problems are marked with daggers (y); all other problems are decision problems.... In PAGE 68: ...Unweighted Weighted Given-Cost Given-Limit Decision - NP-complete Evaluation FPNP[O(logn)]-C FPNP jj -hard y - Solution FPNP jj -hard, properly FPNP[O(logn)]-hard, 2 NPMVg FPNP 2 NPMVg Spanning 2 Span(NPMVg FPNP ) 2 SpanP Enumeration 2 FP#P Random 2 FRP p 2 Generation Table1 6: Computational complexities of phylogenetic inference functions. y Most weighted distance matrix tting evaluation problems are only known to be properly FPNP[O(logn)]-hard (see Corollary 26).... In PAGE 71: ... Formula: maxT jfcj9x[(P(c; x) ^ x 2 T) _ (N(c; x) ^ :(x 2 T))]gj where P, N, and T are as de ned for SAT. Table1 7: Formulations of SAT in rst-order logic (adapted from [KT90, PY91]). MAX NP.... In PAGE 72: ... Formula:maxT jf(x1; x2; x3)j [ ((x1; x2; x3) 2 C0 ! x1 2 T _ x2 2 T _ x3 2 T) ^ ((x1; x2; x3) 2 C1 ! x1 62 T _ x2 2 T _ x3 2 T) ^ ((x1; x2; x3) 2 C2 ! x1 62 T _ x2 62 T _ x3 2 T) ^ ((x1; x2; x3) 2 C3 ! x1 62 T _ x2 62 T _ x3 62 T) ] gj; where C0, C1, C2, C3, and T are as de ned for 3SAT. Table1 8: Formulations of SAT in rst-order logic (cont apos;d from Table 17).... In PAGE 75: ... 4. Given a solution W of cost c to an instance of SOL-MIN-FBUT2[F1] derived by the reduction from X3C given in [KM86] (See Table1 4), in polynomial time we can nd a canonical solution W0 with cost c0 c. 5.... In PAGE 75: ... 5. Given a solution W of cost c to an instance of SOL-MIN-FUDT[F ] ( 2 f1; 2g) derived by the reductions from FBUT2[ ] given in [Day87] (see Table1 4), in polynomial time we can nd a canonical solution W0 of cost c0 c.... In PAGE 77: ... As the Generalized parsimony criterion can simulate any ordered phylogenetic parsimony problem, (5) can be proved by a variant on any of the proofs for (1 - 4). Proofs of (6 { 7): By the reductions given in Table1 1, solutions to SOL-MAX-BCC (SOL- MAX-BQC) yield solutions to SOL-MAX-CLIQUE (SOL-MAX-BCC) of the same cost. Hence, these reductions yield L-reductions with = = 1.... In PAGE 77: ... Hence, this reduction is an L-reduction. Proof of (10): Consider the reduction from FBUT2[F ] to FUDT[F ] 2 f1; 2g given in [Day87] (see Table1 4). As OPTFBUT2[F ] = OPTFUDT[F ], condition (L1) is satis ed with = 1.... ..."

### Table 4 Comparisons of Feasibility Modeling Techniques Probabilistic

1999

"... In PAGE 9: ...1 and 3.2, the features of various existing methods for modeling feasibility robustness are summarized and compared in Table4 . We have considered various attributes in this comparison, such as whether the constraint function requires statistical evaluation, whether the description of uncertainty distribution has to be given, how the performance distributions are described, and whether the calculation of partial differential of the function is needed, etc.... ..."

Cited by 22

### Table 1 Summary of the four probabilistic models

1994

Cited by 13

### Table 4: Parameter Setting for OKAPI Probabilistic Model

"... In PAGE 5: ... In order to define an quot;optimal quot; parameter setting for the BM25 model, we have to conduct a set of experiments based on the CACM and CISI test- collections [Savoy 1995]. The results are depicted in Table4 . However, in our current context, we have set our retrieval scheme according to the parameter values given by [Robertson et al.... ..."

### Table 5. A decomposable probabilistic model is in-

in A Simple Approach to Building Ensembles of Naive Bayesian Classifiers for Word Sense Disambiguation

### Table 1: Results of the probabilistic model for two patients.

### Table 2: Deterministic vs. Probabilistic Models

### Table 1: Probabilistic ranking for the queries

2004

"... In PAGE 6: ...e., car, tank, and rocket) are shown in Table1 , as well as their rankings, computed by the mixture models. The first column in Table 1 indicates the query group and the model it comes from, the second column in- dicates the circular shift applied (i.... In PAGE 6: ...he query results for three of our models (i.e., car, tank, and rocket) are shown in Table 1, as well as their rankings, computed by the mixture models. The first column in Table1 indicates the query group and the model it comes from, the second column in- dicates the circular shift applied (i.e.... In PAGE 6: ... Fig. 3 shows the verification re- sults for the hypotheses listed in Table1 in the case of the rocket model. We received extremely small MSE errors in all of our experiments using artificial data sets.... In PAGE 6: ... We received extremely small MSE errors in all of our experiments using artificial data sets. Table1 shows that the hypotheses with the high- est probabilities were also the correct hypotheses in all cases except in one case (i.... ..."

Cited by 2