Our simulation has been implemented using CCC, as it takes advantage of low level optimization. Initially, it generates all possible hierarchies based on a given set of n taxa. Then it exhaustively traverses through all possible profiles of k hierarchies, together with all possible consensus trees, creating what we called a configuration. A configuration .H; H / is a pair of input trees (a profile) together with a consensus tree. Each configuration was compared against axiomatic properties and consensus functions in order to create the formal context.G; M; I /, withG the set of configurations,M the set of axiomatic properties and consensus
Implications of Axiomatic Consensus Properties 63
functions, and.g; m/ 2I if a configurationghas the (axiomatic) propertym. For example,..H; H /; .PO//2IifHcontains all common clusters ofH; likewise, ..H; H /; .Str//2IifH DStr.H/.
A first issue arose from the number of possible configurations. Given the number ofn-trees (Felsenstein 1978), and the NP-hard nature of some consensus functions (Phillips and Warnow 1996), we were able to run our application only forn 5.
FornD 4(resp.n D5,n D 6) andk D 3, we have 73,125 configurations (resp.
514,807,450,9:571012), and, for each, 22 properties (9 axioms and 13 functions) were tested on a laptop intel-core i5, 2.3 Gh. In order to significantly improve the running time, the consensus trees set was reduced to a more compact set of structure-based trees. All the trees were divided into equivalent classes such that all trees in a class are isomorphic up to a permutation of their labels. Consider two consensus trees H and H0 in the same class and the permutation between their labels, the configuration.H; H /has the same properties satisfied as the configuration . .H/; H0/.
Since the running time of the simulation increases exponentially with slight addition tonork, in order to have partial results from otherwise computationally impossible simulations, randomly selected profiles were chosen for every unique representative consensus tree in order to have a more accurate context and so a more precise set of implications.
Figure 1 shows the overall concept lattice, having 2,821 concepts. Although such a huge lattice is hard to read, it is strongly well-structured. There are only 82 implications on the canonical basis (Table 1). The lower (in the lattice) a property is, the less specific it is: the atoms define four big (overlapping) families of functions: (USP), (NlP), (cPO) and (PO), setting Nelson-Page function apart.
Under (PO) and (cPO), we can find the family of consensus functions satisfying both: (LM), (FD), (Med), (MajC), (Ol).
A few (well known) implications arise from the lattice. The meet of (NP) and (QSP) is the Adams’ consensus rule, thus uniquely defining it (Adams 1986). (USP) is a weakening of (QSP), which is a weakening of (SP). Relationships amongst axioms (Fig.2, left) are becoming clearer too: (PO) is satisfied if we have (Btw) (Neumann 1983), which is satisfied if we have (Dct). While considering the lattice of consensus functions (Fig.2, right), it is similarly well-structured. Apart from obvious special cases ((Str) and (Prj) implying (Ol), (Lo) implying (LM)) and previously known implications ((Maj) implying (Med)), all consensus functions are clearly independent and well-defined.
Our main result is a negative one: there are few unknown implications, and the consensus functions studied are independent. Unfortunately, a drawback of our approach is that we cannot implement fundamental (and desirable) axioms like Independence or Neutrality by construction as these properties are on two different profiles. We are planning to code more consensus functions (such as MRP, local, . . . ) in order to reach some exhaustive,or as close as it can be, study
64 F. Domenach and A. Tayari
Fig. 1 Concept lattice on 9 axioms and 13 consensus functions (Drawn with ConExpYevtushenko (2000))
of consensus functions on hierarchies. Similar work is scheduled to extend the simulation software to more general structures, such as weak hierarchies (Bandelt and Dress 1989), 2–3 hierarchies (Bertrand 2000), pyramids (Bertrand and Diday 1985), and different classes of lattices.
Implications of Axiomatic Consensus Properties 65
Table 1 Implications associated with the lattice
1. Btw!PO 42. PO cPO SP!Str
2. TPO!PO 43. Dct TPO USP Med!Prj
3. Dct!Btw 44. Dct USP NlP MajC !Ol
4. NP!TPO Btw 45. TPO Dur MajC !QSP
5. QSP!USP 46. Dct USP NlP FD!Ol
6. SP!cPO QSP 47. MajCAmed!Btw FD
7. Med!PO cPO 48. TPO USP NlP MajC !QSP
8. Ol!PO cPO 49. Btw Med LM Amed!Maj
9. MajC !PO cPO 50. Str Med!LM
10. FD!PO cPO 51. TPO USP NlP FD!QSP
11. Prj!Dct cPO Ol 52. TPO USP Med FD!QSP
12. Lo!LM 53. Dct TPO USP NlP!cPO Prj Ad Dur
13. Maj!Med LM; 54. TPO QSP Ol Dur MajC !Ad 14. Dur!USP Btw 55. Maj Ol Lo NlP MajC !FD
15. Amed!PO 56. Dct TPO Lo Med!NP
16. Ad!NP QSP Btw 57. TPO Dur FD!QSP
17. LM!PO cPO 58. Btw NlP LM Amed!Dct
18. Str!SP Ol 59. Str TPO Dur!Ad
19. Lo FD!MajC 60. Str MajC !Lo Med
20. NP USP Btw!QSP Ad 61. Dct TPO Lo NlP!NP 21. Dct USP Dur!cPO Prj 62. TPO QSP Ol Dur FD!Ad 22. Dct QSP!Prj Dur 63. USP Amed!Btw
23. Dct TPO USP Ol!Prj 64. Btw NlP MajCFD Amed!Dct 24. Dct NlP Med!Ol 65. Str FD!Lo Med MajC 25. FD Amed!Btw 66. TPO QSP Maj FD!Ol 26. Maj NlP FD!Ol 67. Btw NlP Med Amed!Prj
27. TPO Dur Med!Ad 68. PO cPO Btw Lo LM Amed!Dct MajCFD 28. Dct USP NlP LM!Ol 69. Btw Med MajCFD Amed!Maj
29. TPO Ol Dur LM!Ad 70. Str NlP!Med LM
30. USP Maj NlP!Ol 71. TPO Btw MajCFD Amed!Dct 31. LM Amed!Btw 72. Str Lo NlP Med!MajCFD 32. TPO USP Ol NlP!QSP 73. USP Btw LM Amed!Dct 33. TPO USP NlP LM!QSP 74. USP Btw FD Amed!Dct 34. Dct TPO Ol NlP LM!Prj 75. Str Btw NlP Med LM!TPO 35. Ol Amed!Prj 76. USP Btw NlP Amed!Prj FD 36. TPO NlP Dur!Ad 77. USP Btw Med Amed!Maj Prj 37. TPO USP NlP Med!QSP 78. Dct TPO Lo MajCFD Amed!NP 38. Dct TPO Ol NlP Med!Prj 79. Str SP Prj Lo Dur Med MajC !Maj 39. TPO USP Lo Med!QSP 80. Str TPO SP Prj NlP Ad Dur Med LM!Maj 40. Dct TPO Ol Lo!Prj 81. QSP Btw Amed!Str Maj Prj Lo Dur MajCFD 41. NlP Amed!Btw 82. USP Btw Dur Amed!Str QSP SP Maj Prj Lo
MajCFD
66 F. Domenach and A. Tayari
Fig. 2 Concept lattice associated with axioms (left) and consensus functions (right) (Drawn with ConExpYevtushenko(2000))
Acknowledgements The authors would like to thank the referees for their useful comments and references.
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