A Mechanization of Phylogenetic Trees

(1)

MamounFilali

laliirit.fr

IRITCNRS

UniversitéPaulSabatier

118RoutedeNarnonne

F-31062Toulouse Frane

Abstrat. We study the mehanization of phylogeneti trees inhigher order logi.

After haraterizing trees within suh a logi, we state how to reason and to om-

puteaboutthem.Weintroduethesoalledgenerativepartitionsandrelationswhose

purposeis toallowthereonstrution ofatreefromitsleaves. Afterintroduingtree

transformations, we dene the graft operation. and onsider suient onditions for

thepreservationofthegenerativepartitionsorrelationsafteragraft.Itfollowsthatwe

anreonstrutatreegivenitssetofleavesanditsgenerativerelationwhihhasbeen

preservedalongthegrowthofthetree.Weapplythis resulttothereonstrutionofa

distributedomputation.

keywords:HOL,treestruture,veriation,ISAR.

1 Introdution

This paper gives a denitional formalization, in higher order logi (HOL), of

phylogenetitrees. Wealsoformalizehowtoreasonand ompute onsuhtrees.

We dene the notion of a generative relation, that aims at haraterizing in-

formation whihenables to rebuild atree. Finally,we propose a reonstrution

algorithm based on the set of leaves and a generative relation. The orretness

ofthealgorithmisestablished.Weintrodueanoperation,the graft,thatallows

to represent the growth of atree. a graftare stated.Weillustrate suh areon-

strution throughtheso-alledleafvetorsandaonretegenerativerelation.It

should bestressed that our study isnot onlyonerned withthe proposalof an

original algorithmbut alsoby theformal denitionsand proofs withina logial

framework.

The rest of this paper is organized as follows: Setion 2 gives the repre-

sentationand thebasi operations.Setion3introduesthegraftoperationand

studiesitsreonstrutionproperties.Setion4presentsaonreteexamplewhere

we apply the reonstrution algorithm. Setion 5 ontains the onlusions and

related works.

2 A phylogeneti tree representation and basi operations

In this setion, we introdue the formal representation of phylogeneti trees;

For suh a representation, we onsider how toreason about itand how toom-

(2)

with set theory only, we use type theoreti reasoning also. We have done the

mehanizationwithintheIsabellelogialframework[13℄. Atually,wehaveused

the Isabelle/Isar 1

[19℄ environment whih goal is to assist in the development

of human-readable proof douments omposed by the user and heked by the

mahine.

2.1 Notations and basi denitions

In this setion, we reall the basi set theory and order notions, wewill use.We

hope that the name of the denitions and their formal expression are self ex-

planatory.We have used thedenitions given in [6℄.Moreover, we expressthem

in the Isabellesyntax [13℄. For eah denition, rst, we give its signature, then

its formalexpression. For instane, we haveused the following denitions:

"Maximal

, λ

^S^. ^{m

∈

^S^.

∀

^m'

∈

^S^. ^m

⊆

^m'

⇒

^m ⁼ ^m'^}^"

"Down

, λ

⁽^S^,^e⁾^. ^{^s

∈

^S^. ^s

⊆

^e^}"

" PDown

, λ

⁽^S^,^e⁾^. ^{^s

∈

^S^. ^s

⊂

^e^}"

−−

^{

∗

^p^rô^pe^r ^pâ^r^tⁱ^tⁱôⁿ

∗

^}

"PPartition

, λ

⁽ⁿ^,^S⁾^.^P^a^r^tⁱ^tⁱ^oⁿ⁽ⁿ^,^S⁾

∧

⁽

∀

^e

∈

^S^. ^e

⊂

ⁿ⁾^"

"A//r

, S

x

∈

^A^. ^{^r^`^`^{^x^}}"

−−

^{

∗

^sê^t ô^f ê^qûⁱ^v ^lâ^s^sê^s

∗

^}

In Isabelle, the reexive transitive losure of relation r, denoted r

ˆ⋆

^, ^is ^in-

troduedas anindutivedata type[2℄. Itsintrodutionrules are rtranl_refl

whih speies that every ouple (a,a) belongs to the transitive losure, and

rtranl_into_rtranl whihspeiesthat if (a,b) belongstor

ˆ⋆

^and^(b,)

belongsto r, then (a,) belongs alsotor

ˆ⋆

^.

indutive "r^

∗

^"

intros

rtranl_refl : "(a, a)

∈

^r^{^}

∗

^"

rtranl_into_rtranl :

"(a, b)

∈

^r^{^}

∗ = ⇒

⁽^b^, ⁾

∈

^r

= ⇒

⁽^a^, ⁾

∈

^r^{^}

∗

^"

Withrespet tothe proofs,wehave usedthe Isabelle/Isarformat.Aproof is

established by asequene of intermediate results whih has to beproved reur-

sively or already established. Eventually, results are justied either as axioms

of the logi or by rules of the logi. With respet to proofs, Isar promotes the

1

(3)

so alled delarative style [18℄ whih is loser to the usual mathematial rea-

soningthan theproeduralformat.Letusmention that,basially,Isarsupports

natural dedution but alsosupports alulationalreasoning [7℄.

Asanexample,thefollowingstatementwhihonsistsinassumptions 2

(assumes),

a onlusion (shows) and a proof sript (proof) establishes that the union of

two hierarhies (see setion 2.2) is also a hierarhy. A basi statement of the

proof has the format:

from h

^fats

i have

^label

^′ : ^′ h

proposition

i by h

^method

i

whih aimisto establishproposition fromfats by applying method.

t h e o r e m Hierarhy_union:

assumes h1:"H1

∈

^Hⁱ^e^r^ar^h^y^"

assumes h2:"H2

∈

^Hⁱ^e^r^ar^h^y^"

assumes s: "

∀

ⁿ¹

∈

^H1^.

∀

ⁿ²

∈

^H2^. ^SDS(ⁿ¹^,ⁿ²⁾^"

s h o w s "(H1

∪

^H2⁾

∈

^Hⁱ^e^r^ar^h^y^"

proof

−

f r o m h1 h2 h a v e e: "

∅ 6∈

^H1

∪

^H2" ^by ⁽^unfold

Hierarhy_def , blast)

f r o m s h a v e "

∀

ⁿ¹

∈

^H1^.

∀

ⁿ²

∈

^H2^. ^SDS⁽ⁿ²^,ⁿ¹⁾^"

by (auto simp only: SDS_def)

f r o m this h a v e "

∀

ⁿ¹

∈

^H2^.

∀

ⁿ²

∈

^H1^. ^SDS⁽ⁿ¹^,ⁿ²⁾^" ^by

auto

f r o m s this h1 h2 h a v e sds: "

∀

ⁿ¹

∈

^H1

∪

^H2^.

∀

ⁿ²

∈

^H1

∪

H2. SDS(n1,n2)"

by (unfold Hierarhy_def , blast)

f r o m h1 h2 h a v e f : "finite (H1

∪

^H2)^" ^by ⁽^u^nfold

Hierarhy_def , auto)

f r o m h1 h2 h a v e "

∀

ⁿ

∈

^H1

∪

^H2^. ^fⁱⁿⁱ^tê ^n" ^by(ûnfôld

Hierarhy_def , auto)

f r o m e sds f this show ?thesis by (unfold Hierarhy_def ,

blast)

q e d

2.2 Hierarhies and trees

Our mehanization is based on the introdution of phylogeneti trees starting

from the basi notionsof set theory. Forsuh a purpose, we rst onsider hier-

arhies [3℄ and then introdue trees as restrited hierarhies. Along with these

hierarhies, wegivesome general denitions that willbe used later.

2

(4)

The basi idea of the following representations is to infer a struture from

the relationsbetween itselements;the strutureis not enoded diretly.Suh a

ontent based enoding is motivated by the fat that our basi onern is the

reonstrution startingfromsome oftheelements,namelytheleaves,ofthetree

struture.

Hierarhies. We rst introdue ageneri graphas aset ofnodes. Anode is a

set of generielements.

types

'e graph = "('e set) set"

−−

^{

∗

^gêⁿê^rⁱ ^grâph

∗

^}

'e node = "('e set)"

−−

^{

∗

^g^eⁿ^e^rⁱ ^node

∗

^}

Hierarhies are nite graphs whih elements are nite and non empty and

obey to the SDS: Subset Disjoint Subset relation:

"SDS

, λ

⁽^s1 ^,^s²⁾^. ^s1

⊆

^s²

∨

^s¹

∩

^s2 ⁼

∅ ∨

^s²

⊆

^s¹^"

"Hierarhy

,

^{H^. ^fⁱⁿⁱ^t^e^(H)

∧

⁽

∀

ⁿ

∈

^H^. ^fⁱⁿⁱ^t^e⁽ⁿ⁾⁾

∧ ∅ 6∈

^H

∧

⁽

∀

ⁿ¹

∈

^H^.

∀

ⁿ²

∈

^H^. ^SDS ⁽ⁿ¹^,ⁿ²⁾⁾

}"

In the following, we give the formaldenitions that willbe used.

"Leaves

, λ

^h^. ^{^l

∈

^h^. ^PDown⁽^h^,^l⁾ ⁼

∅

^}"

"ROOT

, λ

^h^.

S

h"

"Subtrees

, λ

^t^. ⁱ^mage ⁽

λ

^e^. ^Down(^t^,^e⁾⁾ ⁽^Maximal⁽^t⁾⁾^"

−−

^{

∗

^p^rôpê^r ^sû^b^t^rêê^s

∗

^}

"PSubtrees

, λ

^t^. ^Su^b^t^r^e^e^s ⁽^t

−

^Maximal⁽^t⁾⁾^"

−−

^{

∗

^rôô^t^s ô^f ^p^ro^pê^r ^sû^bt^rêê^s ^, ^hⁱ^l^d ^no^des

∗

^}

"R1

, λ

^t^. îmagê ^RÔÔ^T ⁽^P^Su^bt^reê^s⁽^t⁾⁾^"

"Sigma

, λ

^S^. ^{

S

(

S

S)}

∪

⁽

S

S)"

Dueto the lak of spae,we donot state allthe establishedresults. Wewill

givethem on the y whenneeded.

Trees and phylogeneti trees. Starting from hierarhies, we rst dene a

(5)

"Tree

,

^{h

∈

^Hⁱê^râr^h^y^. ^Maximal⁽^h⁾ ⁼ ^{RÔÔ^T(^h⁾^}}"

Then,we introduephylogeneti treesas treeswhihnodes are either leaves

or the union of allits subnodes:

"Phylo

,

{t

∈

^Tree^.

∀

ⁿ

∈

^t^. ⁿ

∈

^L^ea^v^es⁽^t⁾

∨

ⁿ ⁼

S

PDown(t,n)}"

With respet to phylogeneti trees, we just mention the following equality

that willallowus to say that the reonstrution an proeed starting from the

leaves, whilethestatementofthe reonstrutiontheorem isoverthe root.Atu-

ally, for a phylogeneti tree

t

^, ^we ^have: ^ROOT

(t) = S

Leaves

(t)

^. ^Moreover, ^we

will relyon the following result about the union of phylogenetitrees:

lemma phylo_union:

assumes t1: "t1

∈

^Phyl^o^"

assumes t2: "t2

∈

^Phyl^o^"

assumes u: "ROOT(t2)

∈

^Le^av^es⁽^t1⁾^"

s h o w s "t1

∪

^t²

∈

^Phy^lo^"

proof ... q e d

Examples. Thegure1illustratestherepresentationof phylogenetitrees.For

instane, with respet tothe previous denitions and the tree lett2, we have:

t0 {x}

{a,e,b}

{a} {e} {b}

t1

{x,y,a,e,b,u,v,w}

{x,y} {a,e,b} {u,v,w}

{x} {y} {a} {e} {b} {u} {v} {w}

t2

(6)

t2 = {{x, y, a, e, b, u, v, w}, {x, y}, {x}, {y}

, {a, e, b}, {a}, {e}, {b}, {u, v, w}, {u}, {v}, {w}}

Leaves

(t2) = {{x}, {y}, {a}, {e}, {b}, {u}, {v}, {w}}

ROOT

(t2) = {x, y, a, e, b, u, v, w}

R1

( t 2) = {{x, y }, {a, e, b}, {u, v, w}}

The deomposition and indution theorems. In order to reason about

phylogeneti trees, we rst introdue a deomposition theorem: a tree is either

a singletonontainingits ROOT, orthe sum (Sigma) of its proper subtrees.

t h e o r e m phylo_ases:

assumes t: "t

∈

^Phy^lo^"

s h o w s "t = {ROOT(t)}

∨

^t ⁼ ^Sigma ⁽^P^Su^bt^r^ee^s⁽^t⁾⁾^"

proof ... q e d

We state the indution theorem about phylogeneti trees as follows:

t h e o r e m phylo_indut:

assumes b: "

∀

^e^. ^P({^e^}⁾^"

assumes r: "

∀

^T

∈

^domSigma^. ⁽

∀

^t

∈

^T. ^t

∈

^Phy^lo

∧

^P(^t⁾⁾

⇒

^P(^Sigma^(T)⁾^"

s h o w s "

∀

^t

∈

^Phyl^o^. ^P(^t⁾^"

proof ... q e d

where domSigma speies the set of trees whih an be joined to form a phy-

logenetitree:

"domSigma

,

{S. S

6= ∅ ∧

^fⁱⁿⁱ^t^e⁽^S⁾

∧

^S

⊆

^Tr^ee

∧

⁽

∀

^t¹

∈

^S^.

∀

^t²

∈

^S^. ^t1

6=

^t²

⇒ S

t1

∩ S

t2 =

∅

⁾

∧

^P^P^a^r^tⁱ^tⁱ^oⁿ⁽

S S

S, image ROOT S)

}"

2.3 Transformations

The basi property of the studiedtransformations istopreserve the underlying

struture while transforming the nodes.

Hierarhy transformations and preservation theorem. First, we intro-

due generaltransformationswhihbasi property istopreserve theardinality

of a set of nodes.

"G _ t r

, λ

^g^. ^{^t^r^.

∀

ⁿ¹

∈

^g^.

∀

ⁿ²

∈

^g^.

(tr(n1) = tr(n2)) = (n1 = n2)}"

(7)

A hierarhy transformation is a generaltransformation whih preserves the

relations between the nodes of a hierarhy:

{

∗

^hⁱê^râ^r^h^y ^t^râⁿ^s^fô^r^mâ^tⁱôⁿ^s ^sê^t

∗

^}

"H_tr

, λ

^h^.

{tr

∈

^{G _ t r}⁽^h⁾^. ⁽

∀

ⁿ

∈

^h^. ^fⁱⁿⁱ^t^e⁽ⁿ⁾

⇒

^fⁱⁿⁱ^t^e⁽^t^r⁽ⁿ⁾⁾⁾

∧

⁽

∀

ⁿ

∈

^h^. ⁿ

6= ∅ ⇒

^t^r⁽ⁿ⁾

6= ∅

⁾

∧

⁽

∀

ⁿ¹

∈

^h^.

∀

ⁿ²

∈

^h^. ⁿ¹

⊆

ⁿ²

⇒

^t^r⁽ⁿ¹⁾

⊆

^t^r⁽ⁿ²⁾⁾

∧

⁽

∀

ⁿ¹

∈

^h^.

∀

ⁿ²

∈

^h^. ⁿ¹

∩

ⁿ² ⁼

∅ ⇒

^t^r⁽ⁿ¹⁾

∩

^t^r⁽ⁿ²⁾ ⁼

∅

⁾

}"

A hierarhy is preserved by a hierarhy transformation:

t h e o r e m hierarhy_trans:

assumes t: "t

∈

^Hⁱ^e^r^ar^h^y^"

assumes tr: "tr

∈

^H_tr⁽^t⁾^"

s h o w s "image tr t

∈

^Hⁱ^e^r^ar^h^y^"

proof ... q e d

A tree isalso preserved by a hierarhy transformation.

Phylogeneti transformations and preservation theorem. Intuitively,

when aphylogenetitransformationisappliedtoanon-leafnode, the deompo-

sition into its desendant nodes is preserved. The haraterizing property of a

phylogeneti transformationis expressed asfollows:

"P_tr

, λ

^h^. ^{ ^t^r

∈

^H_tr⁽^h⁾^.

∀

ⁿ

∈

^h^.

n

∈

^Le^av^es⁽^h⁾

∨

^t^r⁽ⁿ⁾ ⁼

S

(image tr (R1(Down(h,n))))}"

Then,we state the preservation theorem:

t h e o r e m phylo_trans:

assumes t: "t

∈

^Phyl^o^"

assumes tr: "tr

∈

^P_tr⁽^t⁾^"

s h o w s "image tr t

∈

^Phyl^o^"

proof ... q e d

Example. AMutation isa transformationthatonerns the nodesup a graph

node: gp, suh up nodes ontain gp, and a mutation isexpressed as follows:

"Mutation

, λ

(gp,R).

λ

n. if gp

⊆

ⁿ ^then ⁽ⁿ

−

^gp⁾

∪

^R ^el^se ^n"

WeshowthattheMutation transformationisaphylogenetitransformation:

theorem Mutation_P_tr:

a s s u m e s h: "h

∈

^Phylo^"

a s s u m e s g: "g

∈

^Phylo^"

(8)

a s s u m e s pre: "PreGraft(h,gp,g)"

shows "Mutation(gp,ROOT(g))

∈

^P_tr^(h⁾^"

proof ... qed

where PreGraft (We will use this prediate as the preondition of the Graft

operation.) isdened asfollows:

"PreGraft

, λ

(h,gp,g). h

∈

^Hiera^rh^y

∧

⁽⁽^(RO^O^T ^h⁾

∩

^(RO^O^T ^g⁾⁾ ⁼

∅

⁾

∧

gp

∈

^Leaves^(h⁾

∧

^g

∈

^Hiera^rh^y

∧

^g

6 = ∅

^"

2.4 Generative partitions and relations

One of our onerns is the reonstrution of a phylogeneti tree starting from

the set of its leaves. The basi idea of suh a reonstrution is to partition

the leaves aording to its diret proper subtrees and to apply reursively the

reonstrution to eah of the sets of the partition. These suessive partitions

dene the sets whih are generated by a generative partition.

Generative partitions. Given aphylogenetitree h,GP isalled agenerative

partition of h, if eah node is either a leaf or partitioned aording the diret

sub-rootsof n (Down(h,n) is the subtree of h whih root isn).

"GenerativePartition

, λ

⁽^h^,^G^P)^.

∀

ⁿ

∈

^h^.

GP(n) = (if n

∈

^Lêa^ve^s⁽^h⁾ ^then ^{n} ê^lsê ^R1(Down⁽^h^,ⁿ⁾⁾⁾^"

Generative relations. Semantially, the generative relation is a symmetri

relationof whihthe transitivelosureis agenerativepartition.The motivation

for introduinggenerativerelations is to make loalthe reasoning about the of

growth the tree and onsequently easier than a globalone. First, we dene R2P

whih onverts a relation to the partition funtion given by its reexive and

transitive losure: a node n is partitioned by the lasses of the orresponding

equivalene relation.

"R2P(r)

, λ

ⁿ^. ⁽ⁿ ^// ⁽⁽^r⁽ⁿ⁾⁾^{^}

∗

⁾⁾^"

"GenerativeRelation

, λ

⁽^h^,G^R)^.

(

∀

ⁿ

∈

^h^. ^G^R(ⁿ⁾

⊆

ⁿ

×

ⁿ

∧

^{s y m}^(G^R(ⁿ⁾⁾⁾

∧

^Gêⁿê^râ^tⁱ^vê^Pâ^r^tⁱ^tⁱôⁿ⁽^h^, ^R2P^(G^R)⁾^"

2.5 The reonstrution funtion and theorem

We introdue the auxiliary funtion Reonstrut. Its denition is set up in

order tobeaeptedasawelldened funtionby Isabelle: sineitisareursive

(9)

at eah all. The ondition of the if expression ensures it. The reonstrut

funtion is aurryed version of Reonstrut.

redef Reonstrut "m e a s u r e (

λ

^(GP,^s⁾^. ^ar^d ^s⁾^"

"Reonstrut(GP,s) =

(if (finite s)

∧

⁽

∀

^s^'

∈

^G^P(^s⁾^. ^s^'

⊂

^s⁾ ^then

Sigma (image (

λ

ê^. ^Rêôn^s^t^ru^t^(GP,ê⁾⁾ ^(GP ^s⁾⁾

else {s})"

(hints simp add: psubset_ard_mono)

"reonstrut(GP)

, λ

^s^. ^R^e^on^s^t^ru^t^(GP,^s⁾^"

The theorem haraterizing the reonstrution is statedas follows:

t h e o r e m generative_partition_reonstrution :

s h o w s

"

∀

^G^P.

∀

^t

∈

^Phy^lo^. ^Gêⁿê^râ^tⁱ^vê^Pâ^r^tⁱ^tⁱôⁿ⁽^t ^,G^P)

⇒

⁽^rêôⁿ^s^t^rû^t^(G^{P )}^(RÔÔ^T(^t⁾⁾ ⁼ ^t⁾^"

proof ... q e d

This theorem is established thanks to the indution theorem over phyloge-

neti trees (2.2).

2.6 Disussion

In this setion, we disuss the denition of phylogeneti trees that has been

elaborated.Withrespettothestrutural pointofview:aphylogenetitreean

be dened as either a singleton node or as the Sigma of its subtrees. Suh a

set basedonstrution isnot admitted by most of the type theory based logial

frameworks [9,1,4,13℄. In fat, in suh frameworks a tree is usually reursively

dened through the list of its subtrees, orthrough a map of itssubtrees from a

givenindextype.Wehavetriedtoworkwitheahoftheserepresentations.Their

maindrawbakistobreak theunderlyingnaturalonuene.Forinstane,with

suhrepresentations,insertingasubtreeafteratuallyremovingit,doesnotyield

the originaltree.Suhaonueneisfundamentalforestablishingnaturallyour

reonstrution result. Otherwise, we would have to introdue modulo relations

inorder tonotdistinguish between trees ofwhihsubtrees are identialbut not

in the same order.

3 The graft operation

In our setting, the graft operation models the growth of a tree. As its name

suggests, the graft operation onsists ingrafting a tree ata given node. In this

(10)

3.1 Graft deomposition

Lethbeagraph,gpanode of hwherethe graftshouldourandgthegraphto

graft.Weexpress the graftthrough twobasi operations:rst, h istransformed

through aMutation, seond,g isgraftedthrough theunion(

∪

⁾^operation.^Suh

a deomposition isillustrated by gure2. The Graft is expressed as follows:

"Graft

, λ

⁽^h^,^gp^,^g⁾^.⁽ⁱ^mage ⁽^Mûtatiôn ⁽^gp^,RÔÔ^T(^g⁾⁾⁾ ^h⁾

∪

^g"

{x,y,a,e,b,u,v,w}

{x,y} {a,e,b} {u,v,w}

{x} {y} {a} {e} {b} {u} {v} {w}

Graft

{x,y,a,n1,n2,b,u,v,w}

{x,y} {a,n1,n2,b} {u,v,w}

{x} {y} {a} {n1,n2} {b} {u} {v} {w}

{n1} {n2}

{x,y,a,e,b,u,v,w}

{x,y} {a,e,b} {u,v,w}

{x} {y} {a} {e} {b} {u} {v} {w}

{x,y,a,n1,n2,b,u,v,w}

{x,y} {a,n1,n2,b} {u,v,w}

{x} {y} {a} {n1,n2} {b} {u} {v} {w}

{n1,n2}

{n1} {n2}

transformation

union

Fig.2. Deompositionofagraft

Weshowthatthe graftedtree isalsoaphylogenetitree.Theproofisestab-

lishedthanks tothedeompositionofthe graftoperation;werst establishthat

Mutation is a phylogeneti transformation, then thanks to the union theorem,

g being phylogeneti,it follows that the graftedtree isphylogeneti.

t h e o r e m graft_phylo:

assumes h: "h

∈

^Phyl^o^"

assumes g: "g

∈

^Phyl^o^"

assumes pre: "PreGraft(h,gp,g)"

s h o w s "Graft(gp,g)(h)

∈

^Phy^lo^"

proof ... q e d

(11)

3.2 Reonstruting a graft through a generative partition

This setion states a general result about the preservation of a generative par-

tition GP. Infat, wehavea preonditionabout the partitioningofthe mutated

nodes.

t h e o r e m generative_partition_graft_phylo:

assumes h: "h

∈

^Phy^lo^"

assumes g: "g

∈

^Phy^lo^"

assumes pre: "PreGraft(h,gp,g)"

assumes gp_h: "GenerativePartition(h,GP)"

assumes g p _ g: "GenerativePartition(g,GP)"

assumes gp_tr:

"

∀

ⁿ

∈

^h^. ^G^P(^Mutâtiôn⁽^gp^,RÔÔ^T(^g⁾⁾⁽ⁿ⁾⁾ ⁼

(if n

∈

^Lêavê^s⁽^h⁾ ^then ^{^Mûtatîon⁽^gp^,RÔÔ^T(^g⁾⁾⁽ⁿ⁾^}

else image (Mutation(gp,ROOT(g))) ( GP(n)))"

s h o w s "GenerativePartition(Graft(h,gp,g),GP)"

proof ... q e d

3.3 Reonstruting a graft through a generative relation

In the same way, a generative relationan be preserved while extending a tree

through a graft.Thanks tothis preservation: a tree, growing through graft op-

erations,willalways bereonstrutiblefromitsleavesthrough itsinvariantgen-

erative relation.

A simplied mutation: the basi update upd. For the purpose of our

appliation, we onsider the following node transformation:

"upd

, λ

⁽^l ^,N)^.

λ

^S^. ⁱ^f ^l

∈

^S ^then ^S

−

^{^l^}

∪

^N ^e^ls^e ^S"

Sine we have: upd

(l, N ) =

^Mutation

({l}, N )

^, ûpd înherits ^the ^property ôf

Mutation; then itis a phylogenetitransformation.

Moreover, in order to simplify the proof obligations for establishing that

a generative relation is preserved after a graft, we have elaborated suient

onditions that should be established by the update funtion. Due to the lak

of spae, we donot detailthem.

We haveestablished the preservation of the generativerelation for the graft

of a so alledanonial tree whih onsistsof a rootand aset of leaves:

"Canoni

, λ

N. {N}

∪

⁽

S

e

∈

^N. ^{{^e^}}⁾^"

We have the following invariant theorem establishing the preservation of a

(12)

lemma generative_relation_graft_phylo:

assumes t: "t

∈

^Phyl^o^"

assumes up: "{l}

∈

^t^"

assumes gp1: "GenerativeRelation(t,GR)"

assumes gr_m: "

∀

ⁿ^. ^G^R(ⁿ⁾

⊆

ⁿ

×

ⁿ

∧

^{s y m}^(G^R(ⁿ⁾⁾^"

assumes terminal: "Terminal(t)"

assumes N: "

∀

ⁿ

∈

^t^. ^N

∩

ⁿ ⁼

∅

^"

assumes N_e: "N

6= ∅ ∧

^fⁱⁿⁱ^t^e^(N)^"

assumes gp2: "GenerativeRelation(Canoni(N),GR)"

assumes gr_tr:

"

∀

ⁿ

∈

^t^. ^l

∈

ⁿ

⇒

^r_upd⁽^l ^,^N)⁽ⁿ⁾^(G^R(ⁿ⁾ ^,G^R(^upd⁽^l ^,N)⁽ⁿ⁾⁾⁾^"

s h o w s "GenerativeRelation(Graft({l},Canoni(N))(t) ,GR)"

proof ... q e d

4 Appliation

As an appliation of phylogeneti trees, we onsider distributed diusing om-

putations. In the initial state, one site (or proess) multi-asts a message to a

subsetof othernodes.Then,allnodes sharethesame behavior:whenamessage

is reeived, the reeiver performs a omputation step and, possibly, multi-asts

a messageto asubset of other nodes.

We are interested in the following problem: how to reonstrut the global

history ofsuhaomputation,afteritstermination 3

,frominformationgathered

during the omputation.For suh adiusing omputation, the ontrolow is a

treeinwhihthenodesaretheomputationstepsandtheedgesare themessage

ommuniations.Ouralgorithmonsistsinolletinganenoded representation

oftheseleaves.Fromthisleavesset,weapplythereonstrutionalgorithmbased

upon agenerative relation dened onthe omputationas aphylogeneti tree.

4.1 Control tree enoding

We dene an enoding for the ontrol tree. The nodes generated during the

omputation (temporary leaves) are enoded as vetors. At eah site, a loal

ounter is inremented by

p − 1

êah ^time â ômputation ^step ^multi-asts

p

messages. Thus, one 4

plus the sum of the loalounters represents the number

of the ontrol tree leaves. Moreover, the value of the ounter of site

s

^is ^the

maximum of the vetors omponent atthe index

s

^.

3

Suhareonstrutionisusuallyusedfordebuggingpurposes.

4

Wehavetotakeintoaounttheinitialstateswheretheountersareallnullandthetreeonsists

(13)

collected visit tag (0,[0,0,0,0])

(1,[1,0,0,0]) S0

S1

S2

S3

(2,[1,1,0,0])

(3,[1,1,2,0])

(site,[x,y,z,t]) internal node tag (site,[x,y,z,t]) (2,[2,1,0,0])

(2,[1,1,2,2]) (1,[2,1,0,0])

(1,[1,1,2,2]) (0,[1,1,2,0]) (0,[1,1,2,2])

(0,[1,1,0,0])

(1,[1,1,2,0])

(2,[1,0,0,0])

Fig.3.Visittagsenoding

We assoiate a "visit tag" to eah node. This tag is omposed of the site of

thenodeandanaturalintegersvetor.Thisvetor

V

^has^a^size

N

^,orresponding to the number of sites, and is assigned the loal ounter values of the sites it

has visited. Figure3 shows the taggingof the nodes of a diusing omputation

with this enoding.

The state spae of all the appliation is modeled by a global type State.

It ontains the elds related to the network, the loal omputations and the

olletor. The omputation is onernedby the followingelds:

The eld lounter implements the loalounter of eah site;

The eld olleted reordsthe visittags of the omputationleaves.

Twotransitionsareonsideredand eahofthemislaunhedwhenamessage

is reeived:

ReeiveAndEnd desribes a omputing step without further message multi-

ast. In this ase, the visit tag ontained in the reeived message is sent to

the olletor;

ReeiveAndSplitdesribesaomputing stepterminatedby amessagemul-

tiast.Inthis ase, anew tagisreated: itholds the destinationsite (

d

⁾^and

avetor whih isidential tothe tag vetor of the splittingnode (

m.V

^),^ex-

eptforthesplittingsite(self)omponent,whihgetsthenewloalounter

value

lcounter[self]

âssigned ^by ^this ômputation ^step. ^No ^message îs ^sent

(14)

With respet to phylogeneti trees, the diusing omputation is seen as a

tree. A ReeiveAndEnd assigns to a node the denitive leaf status. While a

ReeiveAndSplit extends a tree with new leaves. We interpret it as a Graft

operation.Then,forvalidationpurposes,wehaveanauxiliaryvariableauxTree

for reording the growth of suha superposed tree: we prove that attermina-

tion, this auxiliary tree and the reonstruted tree are the same.

4.2 Termination detetion and reonstrution

We introdue a olletor proess to gather vetors: a vetor is sent to the ol-

letorwhen aproess performsaomputationstep withoutmulti-astinganew

message. In suh a ase, this step generates a leaf with respet to the ontrol

ow ofthe omputation.Then, withrespet tophylogeneti trees,the olleted

tagged messages are in fat leaves of the phylogeneti tree superposed to the

diusing omputation (and reorded in the auxiliaryvariableauxTree).

The reonstrution of the ontrol tree an only start when the global om-

putationisterminated.Several distributedalgorithmsansolvethe termination

problem,espeially, thanks toaolletor proess[12℄. However, the enoding it-

self provides a simple riterion for termination detetion [8℄: a omputation is

terminated when the number of olleted leavesis equalto one plus the sum of

the elementsin the maximum of the olleted visit vetors 5

:

|

^olleted

|= 1 + X

s ∈

^Site

v ∈ max collected v.V [s]

The generative relationfor the diusing omputationas a phylogeneti tree

is dened asfollows:

"gr

, λ

ⁿ^. ^{⁽^v1^,^v2⁾^. ^v1

∈

ⁿ

∧

^v²

∈

ⁿ

∧

(if V(v1) = m i n _ o n(n)

∨

^V(^v2⁾ ⁼ m i n _ o n(n) then (v1 = v2)

else (

∃

^s^. ^V(^v¹⁾⁽^s⁾ ⁼ ^V(^v2⁾⁽^s⁾

∧

^s

6=

^w(^v1⁾

∧

s

6=

^w(^v2⁾

∧

^V(^v1⁾⁽^s⁾

6=

m i n _ o n(n)(s))) }"

Wederivethe orretness ofthe reonstrutionthrough the followinginvari-

ant:

"ReonstrutionInvariant

, λ

^s^t^. ^au^xTr^ee⁽^s^t⁾ ⁼

reonstrut(R2P(gr))(olleted(st)

∪

ⁿ^etwor^k⁽^s^t⁾⁾^"

Then, when termination is reahed, the network is empty, and the reon-

strution appliedtothe olleted messages givesthe omputationtree.

5

|

^_

|

^denotes^the^ardinality^of^_^.

(15)

5 Conlusion

In this paper, wehave proposeda mehanization ofphylogenetitrees. Starting

frombasiset theory,wehaveintroduedphylogenetitreesthrough hierarhies

and trees. Then, we have dened generi transformations. We note that sets

based representations, although already suggested in the literature[10℄,are not

widely used inomputer siene. To the best of our knowledge, the representa-

tion of a tree through the set of its leaves together with a generative partition

or relation, as well as the study of dediated transformations, are original. We

have given a onrete example, where suh notions have been applied to tree

reonstrution and shown how suh a reonstrution ould be validated. It is

interesting to remark that thanks to theorem proving tehniques, suh a vali-

dation was possible; atually we have onsidered an unknown number of nodes

and unboundednaturalvetors.Usualmodelhekingtehniquesannothandle

suh problems.

Most of our results have been proved formally within Isabelle. In fat, our

treesareunordered trees.Suhadatatypeouldbeonsideredasanindutive

data type where Sigma would play the role of a onstrutor; however, due to

the negativeourrene 6

most ofthe logialframeworks(HOL[9℄,Isabelle[13℄,

PVS[4℄,Coq[1℄) donotsupportsuhadenitionshema.VosandSwiestra[17℄

havestudiedrestritionsforaeptingindutivedatatypeswithnegativeour-

renes; sine our trees are nite, we ould have reused their work. This work

is not known tobe availablewithin the Isabelleframework. An alternative way

would have been to introdue unordered trees through an equivalene rela-

tion [14℄ over ordered trees where subtrees are onstrutedwith a list.It would

be interesting to ompare the subsequent developments of phylogeneti trees,

generativerelations and partitions.

Withrespet totheformalizationoftreesandbiologyrelatedresults,numer-

ous works have been published. Among the more reent, we an ite [16℄ who

onsider the problem of tree inlusionin a ategorial setting. [11℄ reviews ba-

si network models for reasoning about biology; he noties that appliations to

biologyofexistingtoolsfromalgebraisjustbeginning.Tothebestofourknowl-

edge, the mehanization of these works has not been onsidered yet. We think

that our work ould be reused as a starting point for establishing algorithms

orretness but also forthe orretness of their proposed proofs 7

.

6

ThenegativeourreneisduetothefatthattheparameterofSigma,onsideredasaonstrutor,

isasetoftrees.

7

It is interesting to remarkthat the analysis of the algorithm of [5℄ is reportedto be inorret

(16)

Referenes

1. B. Barras, S.Boutin, C. Cornes, J. Courant, J. Filliatre, E. Giménez, H. Herbelin, G. Huet,

C. Muñoz, C. Murthy, C. Parent, C. Paulin, A. Saïbi, and B. Werner. The Coq Proof

Assistant Referene Manual Version V6.1. Tehnial Report 0203, INRIA, August 1997.

http://oq.inria.fr.

2. S.BerghoferandM. Wenzel. IndutivedatatypesinHOL-lessonslearnedinformal-logiengi-

neering.InSpringer-Verlag,editor,TheoremProvinginHigherOrderLogis,volume1690,pages

1936,1999.

3. S.BokerandA.W.Dress.Anoteonmaximalhierarhies.AdvanesinMathematis,(151):270

282,2000.

4. S. Crow, S. Owre, J. Rushby, N. Shankar, and S. Mandayam. A Tutorial Introdution

to PVS. In Workshop on Industrial-Strength Formal Speiation Tehniques, Boa Raton,

http://www.sl.sri.om/pvs,April1995.

5. J.Culbersonand P.Rudniki. A fastalgorithm for onstrutingtrees from distane matries.

InformationProessingLetters,30(4):215220,may1989.

6. B.DaveyandH.Priestley. Introdution toLattiesandOrder. CambridgeMathematialText-

books.CambridgeUniversityPress,1990.

7. E.W.DijkstraandC.S.Sholten. PrediateCalulusandProgramSemantis. Springer-Verlag,

1989.

8. M. Filali, P. Mauran, G.Padiou, P. Quéinne, and X. Thirioux. Renement based validation

ofadistributedterminationdetetionalgorithm . InFMPPTA'2000,Canun,volume 1800of

Leture NotesinComputerSiene,pages10271036.Springer-Verlag,may2000.

9. M.GordonandT.Melham. IntrodutiontoHOL. CambridgeUniversityPress, 1994.

10. D. E.Knuth. The art of omputer programming Fundamentalalgorithms, volume 1. Addison-

Wesley,1969.

11. R.Laubenbaher. Algebrai modelsinSystemsbiology. InH. AnaiandK.Horimoto, editors,

Algebrai Biology 2005 -Computer Algebrain Biology,pages3340. UniversalAademipress,

Tokyo,Japan,2005.

12. F.Mattern. Globalquiesenedetetionbasedonreditdistributionandreovery.Information

ProessingLetters,30(4):195200,Feb.1989.

13. T.Nipkow,L.Paulson,andM.Wenzel.Isabelle/HOL.AProofAssistantforHigher-OrderLogi.

Number2283inLetureNotesinComputerSiene.Springer-Verlag,2002.

14. L.C.Paulson. Deningfuntionsonequivalenelasses. ACMTransations onComputational

Logi,7(4):658675, 2006.

15. L.ReyzinandN.Srivastava.Onthelongestpathalgorithmforreonstrutingtreesfromdistane

matries. InformationProessingLetters,101(1):98100,january2007.

16. F.RoselloandG.Valiente. Analgebrai viewofthe relationbetweenlargestommonsubtrees

andsmallestommonsupertrees. TheoretialComputer Siene,(362):3353,2006.

17. T. E. Vos and S. D. Swierstra. Indutive data typeswith negative ourrenes in HOL. In

WorkshoponThirtyFiveyearsofAutomath,Edinburgh,UK,2002.

18. M.WenzelandF.Wiedijk. Aomparison ofthemathematialprooflanguages MizarandIsar.

JournalofAutomatedReasoning, 29:389411,2002.

19. M. M. Wenzel. Isar a generi interpretative approah toreadable proof douments. Number

1690inLetureNotesinComputerSiene.Springer-Verlag,1999.

A Mechanization of Phylogenetic Trees

, λ

∈

∀

∈

⊆

⇒

, λ

∈

⊆

, λ

∈

⊂

−−

∗

∗

, λ

∧

∀

∈

⊂

, S

∈

−−

∗

∗

ˆ⋆

ˆ⋆

ˆ⋆

∗

∈

∗

∈

∗ = ⇒

∈

= ⇒

∈

∗

from h

i have

′ : ′ h

i by h

i

∈

∈

∀

∈

∀

∈

∪

∈

−

∅ 6∈

∪

∀

∈

∀

∈

∀

∈

∀

∈

∀

∈

∪

∀

∈

∪

∪

∀

∈

∪

−−

∗

∗

−−

∗

∗

, λ

⊆

^′ : ^′ h