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Constructive Combinatorics of Dickson’s Lemma

Iosif Petrakis

Ludwig-Maximilian Universitat Munich

Constructive Mathematics, Foundations and Practice

Nis, 25th June 2013

Iosif Petrakis Constructive Combinatorics of Dickson’s Lemma

The finite and the infinite versions of Dickson’s lemma

Theorem

1 DLp1, 2q: If α : NÑ N, there is a good index for α, or α is good on i i.e.,

αpiq ď αpi ` 1q.

2 DLp1, lq: If l ą 1 and α : NÑ N, there is good l-tuple of indices i1 ă i2 ă ... ă ili.e.,

αpi1q ď αpi2q ď ... ď αpil q.

3 DLp2, 2q: If α, β : NÑ N, there is a common good pair for α, β i.e.,

i ă j ^ αpiq ď αpjq ^ βpiq ď βpjq.

4 DLpk, lq: If k ě 1 and l ě 2 and α1, α2, ..., αk : N Ñ N, there exist indicesi1 ă i2 ă ... ă il on which αj ’s are simultaneously good.

5 DLp1,8q: If α : NÑ N, there is a weakly increasing subsequence of α.

6 DLp1, Sq: If α : S Ñ N, where S Ďunb N, there exists M Ďunb S such that α isgood on M.

Theorem

αpiq ď αpi ` 1q.

Theorem

αpiq ď αpi ` 1q.

Theorem

αpiq ď αpi ` 1q.

Theorem

αpiq ď αpi ` 1q.

Theorem

αpiq ď αpi ` 1q.

Two basic results connected to DL

1 @kDLpk, 2q implies the existence of Grobner bases for any finitely generated idealin K rX1, . . . ,Xns (termination of Buchberger’s algorithm).

2 (Simpson 1988): The following are equivalent over RCA0:

(i) ωω is well ordered.(ii) @kDLpk, 2q.(iii) HBT: If K is a countable field and n P N, then any ideal in K rX1, . . . ,Xns isfinitely generated.

Note: ordpRCA0q “ ordpWKL0q “ ωω , therefore the well orderedness of ωω cannotbe proved in RCA0.

Some initial remarks

Remark

DLp8, 2q doesn’t hold.

Proof.

Consider the sequence of sequences pαnq8n“1 defined for each n by

αn “ pn, n ´ 1, ..., 1, n ` 1, n ` 2, ...q.

We can not find a common good pair of indices for all αn: If n “ αnp1q, then αnpnq “ 1,for each n. Thus, if i ă j , then

αj piq ą 1 and αj pjq “ 1

i.e., pi , jq cannot be a good pair for αj .

Remark

DLp1,8q Ñ DLpk,8q Ñ DLpk, lq, for each k ą 2 and l ą 1.

DLp1,8q Ñ LPO

Proof.

If α : N Ñ 2, then we show Dnpαpnq “ 0q _ @npαpnq “ 1q. We define a sequenceβ : NÑ 2 by

βpnq “

1 , if @mďnpαpmq “ 1q0 , if Dmďnpαpmq “ 0q.

Applying DLp1,8q to β, we get a weakly increasing subsequence pβpinqqn of β.

1 If βpi1q “ 0, then Dmďi1 pαpmq “ 0q.

2 If βpi1q “ 1, then βpinq “ 1, for each n P N. In this case @npαpnq “ 1q:Fix n P N; since there is ik ą n we get

βpik q “ 1 Ñ @mďik pαpmq “ 1q Ñ αpnq “ 1.

Some connections to the pigeonhole principle

Proposition

(i) @lě2pDLp1, lq Ñ PHp2,8, lqq.

(ii) @lě2pDLp2, lq Ñ PHX1,X2p2,8, lqq.

(iii) DLp1,8q Ñ PHp2,8,8q.

Proof.

(i) By DLp1, lq on X ˝ α : NÑ 2 there are i1 ă i2 ă ... ă il s.t

X pαi1 q ď X pαi2 q ď ... ď X pαil q.

If X pαil q “ 0, αi1 , αi2 , ..., αil is monochromatic.If X pαil q “ 1, then we repeat the previous step on the tail αil`1, αil`2, ... of α. Afterat most l steps we find a monochromatic subsequence of α of length l .(iii) By DLp1,8q on X ˝ α : NÑ 2 there are i1 ă i1 ă i3 ă ..., s.t

X pαi1 q ď X pαi2 q ď X pαi3 q ď ... .

Since DLp1,8q Ñ LPO, then either all terms of rX pαin qsn are 0, or there is some αins.t X pαin q “ 1. In the first case pαin qn itself is monochromatic, while in the second thetail αin , αin`1

, αin`2..., of pαin qn is monochromatic.

LPOÑ DLp1,8q

Remark

If Ppnq is a decidable predicate, then LPOÑ @nPpnq _ Dn Ppnq.

Definition

If i P N and α : NÑ N, the index i is a peak for α, Pαpiq, if @nąi pαpiq ą αpnqq.

Remark

If α : NÑ N, LPOÑ @i pPαpiq _ Dnąi pαpiq ď αpnqq.

Proof.

We fix i P N. If Nąi “ tn P N : n ą iu, we define the bijection e : NÑ Nąi

epnq “ pn ` 1q ` i , and the decidable predicate

Qi pnq :“ αpepnqq ă αpiq Ø αppn ` 1q ` iq ă αpiq.

By the previous remark we get

@npαppn ` 1q ` iq ă αpiq _ Dnpαpiq ď αpn ` 1` iqq.

LPOÑ DLp1,8q

Proof.

We fix a sequence α : NÑ N and by the decidability of Pαpiq we define

βpnq “

0 , if Dmąnpαpnq ď αpmqq1 , if Pαpnq.

By LPO we get the following two cases:

1 @npβpnq “ 0q Ø @nDmpm ą n ^ αpnq ď αpmqq. By ACd00

Dγ@npγpnq ą n ^ αpnq ď αpγpnqq.

0 ă γp0q ă γpγp0qq ă . . . , and αp0q ď αpγp0qq ď αpγpγp0qqq ď . . . .

2 Dnpβpnq “ 1q Ø DnpPαpnqq: if we consider the tail of α

αpn ` 1q, αpn ` 2q, αpn ` 3q, ..., then

αpjq P t0, 1, ..., αpnq ´ 1u,

for each j ě n ` 1. Since this tail of γ is a new sequence, then either it has positivelyno picks, and the previous case is applied, or there is some index n`m` 1 whichis a peak for it. Obviously, after at most αpnq ´ 1 steps we will have found a tailof α with no peaks and we can apply the argument of the first case.

Levels of constructive combinatorics

1 Constructive combinatorics of DL: the constructive study of the relations betweenthe various versions of DL and their connection to the rest of combinatorics.

2 Constructive combinatorics of well quasi-orderings: A w.q.o is a is a q.o W “

pW ,ĺq satisfying@αPWNDiăj pαpiq ĺ αpjqq.

The pair pi , jq is called a good pair of indices. Important theorems on specificq.o that are w.q.o: Higman’s lemma, Kruskal’s theorem, Graph minor theorem(Gupta-Fraser)

3 Constructive combinatorics: the constructive study of combinatoric theorems likeRamsey theorem, van der Waerden’s theorem, Hindman’s theorem, etc. The basicpropositions of (finite or infinite) combinatorics

[assert, crudely speaking, that every system of a certain class possesses a largesubsystem with a higher degree of organization than the original system.]

Versions of combinatoric theorems

Σ is a class of systems S .RpSq is a hereditary property on S i.e., t ĺ S Ñ RpSq Ñ Rptq, where t is an infinite(t P Σ) or a finite subsystem of S.Th: For each S there exists a t ĺ S such that Rptq (while in generally RpSq is not thecase).

1 The l-finite version of Th: t is of finite cardinality l , for some specific natural l forwhich |S| ą l .

2 The unbounded-finite version of Th: t is of any finite cardinality l , if |S | ě ℵ0.

3 The unbounded-infinite version of Th: t is an unbounded subsystem of S , if|S | ě ℵ0.

4 The infinite version of Th: t is of infinite cardinality ℵ0, where |S| ě ℵ0.

5 The higher-infinite version of Th: t is proved to be of infinite cardinality κ, forsome κ ą ℵ0, where |S| ě κ.

Definition

(i) A strong version of Th is one in which a bound is provided to the size of S neededto construct t such that Rptq.(ii) An optimal version of Th is one in which the smallest bound is provided to the sizeof S needed to construct t such that Rptq.

Ramsey and non-Ramsey properties

R is a Ramsey property, if

1 The (optimal) finite version of Th is constructively provable.2 The (optimal version of the) unbounded-finite version of Th is constructively prov-

able.3 The (unbounded-) infinite version of Th is classically provable.

1 The infinite Ramsey theorem: If X : rNs2 Ñ 2, there is an infinite sub-sequence Asuch that rAs2 is X -monochromatic.

2 The infinite pigeonhole principle: if X is an n-coloring of N, then there exists aninfinite X -monochromatic set

3 Hindman’s theorem: If N is finitely colored, there exists an infinite A Ď N s.t. ΣpAq(the set of all distinct sums of elements of A) is monochromatic (the unbounded-finite version of this is Folkman’s theorem).

R is a non-Ramsey property, if the unbounded-finite version of Th is classicallyprovable but its infinite version doesn’t hold. E.g.,

1 @kDLpk, 2q holds, but DLp8, 2q does not.2 There are arbitrary large finite bad sequences of naturals (n ą n ´ 1 ą . . . ą 0),

but not an infinite one).3 van der Waerden’s theorem: Given a 2-coloring of N, there exist arbitrary long

monochromatic arithmetic progressions. But there is a 2-coloring for which thereexists no infinite monochromatic arithmetic progression.

Some observations on combinatoric theorems

1 We know the optimal version of DLp1, 2q, but not of DLp2, 2q, although there areat least two strong versions of DLp2, 2q.

2 We do not know a strong version of Higman’s lemma (the embedding of words isa w.q.o) even for words on an alphabet of two letters.

3 There are propositions of finite combinatorics (Friedman’s Proposition B) which isprovable only with the use of large cardinals, or the proposition of Paris-Harringtonwhich is provable in second-order analysis but not in Peano arithmetic (also withno constructive proof).

4 There are non-trivial infinite combinatoric propositions with a constructive proof.E.g., Veldman’s intuitionistic proof of the infinite Ramsey theorem, or its 2012-generalization and simplification by Vytiniotis, Coquand and Wahlstedt (new in-ductive characterization of an almost full relation, avoidance of bar induction).

5 It is not uncommon that non-constructive proofs inspire or have a constructivecounterpart e.g., minimal-bad-sequence-proofs of Higman’s lemma, or of DL in-spired corresponding constructive proofs of them. Also, our proof of DLp2, 2q isthe constructive counterpart of the classical proof DLp1,8q Ñ DLp2, 2q.

6 An open question (maybe too difficult to answer): Given a class Σ can we findgeneral conditions such that whenever some property R satisfies them, then R isa Ramsey property? And if such conditions are found, can we prove that theycharacterize Ramsey properties?

7 Many combinatoric propositions have (classical or constructive) proofs which arebased on the use of repetitive arguments. It is not unusual in combinatoricsto reach non-trivial conclusions by repeating seemingly trivial thought-steps (nooptimal versions).

Constructive studies of DL

1 1913-1993 Classical proofs: e.g., DLp1,8q Ñ DLp2, 2q, through the classicallyproved HL.

2 Veldman, Bezem 1993: as a corollary of his intuitionistic Ramsey theorem.

3 Coquand, Persson 1999, and Persson 2001: proof within TT with inductive defini-tions, Agda and Coq-implementations.

4 Mateos et.al, 2003 (unbounded-infinite version) and Sustik 2003 (explicit ordinalmap): proofs and implementations in ACL2.

5 Veldman 2000/2004: intuitionistic proof independently from Ramsey theorem (alsointuitionistic proofs of Higman and Kruskal).

6 Herz 2004: providing bounds through Simpson’s theorem.

7 Munich logic group 2002-12: refined A-translation of the DLp1, Sq Ñ DLp2, 2q-proof of the Π0

2-formula DLp2, 2q (U. Berger, Bucholz, Schwichtenberg, later Ratiu)[related work of Seisenberger on Higman’s lemma] and minlog-implementations.

8 J. Berger, Schwichtenberg: direct constructive proof of DLp2, 2q, providing also abound, with a corresponding minlog-implementation.

The extraction of a bound from a proof: the example of DLp1, 2q

A minimal bad sequence-proof: Suppose there is a sequence α which is bad. Let k0

be the minimum natural for which there is a bad sequence β starting with k0. Itsexistence is justified by the the minimum principle of natural numbers. Since β ą pk0q,

β “ k0, β1, β2, ... .

Obviously, β1 ă k0, otherwise β would be good. But, if β is bad, then also

β`1 “ β1, β2, ...

is bad, which contradicts the minimum property of k0.

Remark: There is no direct way to extract a bound for DLp1, 2q.

The extraction of a bound from a proof: the example of DLp1, 2q

If u “ pu0, u1, ..., un´1q P Năω , we define the deidable and monotone predicategoodpuq by

goodpuq :“ Diăjď|u|pui ď uj q.

Let good|u be the corresponding predicate “good inductively bars u”, defined by the ηand z rules:

goodpuq

good|u

@nPNpgood|u a nq

good|u.

By the corresponding induction principle we get

good|u Ñ @αąuDmpgoodpαpmqqq.

Theorem (DLp1, 2q within TT ` ID)

good| ăą.

Proof.

good|p0q is trivial. Supposing good|pmq, for each m ă n, we get good|pnq by showing

good|pn, 0q, good|pn, 1q, ..., good|pn, n ´ 1q.

good|p0q Ñ good|pnq ˚ p0q, ..., good|pn ´ 1q Ñ good|pnq ˚ pn ´ 1q.

A variation of the formal system BIM (Howard-Veldman)

1 Extensionality: @α,βpα “1 β Ø @npαpnq “0 βpnqq.

2 Constant S :pS1q @np Spnq “ 0q,pS2q @m,npSpmq “ Spnq Ñ m “ nq.

3 Constant 0: @np0pnq “ 0q.

4 Constants J,K , L:@m,n,l pKpJpm, nqq “ m ^ LpJpm, nqq “ n ^ JpKplq, Lplqq “ lq.

5 Composition: @α,βDγ@npγpnq “ αpβpnqqq.

6 Primitive Recursion:

@α,βDγ@m,npγpm, 0q “ αpmq ^ γpm,Spnqq “ βpm, n, γpm, nqqq.

7 Full Induction: If ϕ is any formula of L, then

ϕp0q ^ @npϕpnq Ñ ϕpSpnqq Ñ @nϕpnq.

8 Decidable Countable Choice (ACd00):

@nDmRpn,mq Ñ Dα@nRpn, αpnqq,

where Rpn,mq is a formula in which the variable α does not appear free, and it isdecidable, i.e., BIM $ @n,mpRpn,mq _ Rpn,mqq.

Some characteristics of BIM

1 BIM is a formalization of a very weak subsystem of BISH, like Kleene’s M or EL.

2 Proofs in BIM are translated into proofs in TCF, hence they are, in principle,minlog-implementable.

3 BIM proves the existence of the Picard iterations of α starting at n0.

4 (Howard-Kreisel 1966) BIM˚ $ ACd00 Ñ DCd

0 , and BIM˚ $ AC00 Ñ DC0.

5 BIM˚ $ ACd00 Ø ACd

6 BIM can be considered the common ground of classical and constructive mathe-matics. Thus we may describe formal varieties of constructivism by adding to BIMextra principles like CP,FAN, etc. (Veldman).

Some more constructive reverse mathematics over BIM

IRT: If X : rNs2 Ñ 2, then there is A “ ti1 ă i2 ă ...u Ď N, s.t rAs2 ismonochromatic under X .WIRT: If W is a q.o, then Wp1, 2q ÑWp1,8q.

Proposition

(i) IRTÑWIRTÑ DLp1,8q.

(ii) IRTÑ IPHÑ LPO.

(iii) LPOÑ IPHÑ IRT.

(iv) WIRT`DCÑ IRT.

DLp1, 2q and its relation to DLp1, 3q

Proposition

The following optimal version DLop1, 2q of DLp1, 2q is proved within BIM˚:

@αDMDi pi ă M ^ αpiq ď αpi ` 1qq,

where M “ αp0q ` 1. We also write M “ Mαp1, 2q.

Proposition

Ef :NÑN@α@iăj pf pαpiqq ď f pαpjqq Ñ αpiq ď αpjq ď αpj ` 1qq. I.e., DLp1, 2q doesn’timply DLp1, 3q in one only step.

Proof.

By DLop1, 2q there exists k ă f p0q ` 1 s.t f pkq ď f pk ` 1q. Take α and i ă j s.tαpiq “ k, αpjq “ k ` 1 and αpj ` 1q ă αpjq.

A strong version of DLp1, 3q within BIM˚ requires the repetitive use ofDLop1, 2q

Proposition

@αDMDi,j,k pi ă j ă k ď M ^ αpiq ď αpjq ď αpkqq, where for a given α

M “ Mαp1, 3q “Nÿ

Mj , N “ αpip1qq ` 2, M1 “ αp0q ` 1, Mj`1 “ αpMj q ` 1,

for each j P t1, ...,N ´ 1u, and ip1q is determined by DLop1, 2q on α.

Proof.

By DLop1, 2q on α we get ip1q ď αp0q s.t αpip1qq ď αpip1q ` 1q, andM1 “ Mαp1, 2q “ αp0q ` 1. By DLop1, 2q on the tail of α from M1,αp1qpnq “ αpM1 ` nq, we get ip2q ď αp1qp0q “ αpM1q, s.t αp1qpip2qq ď αp1qpip2q ` 1q,and M2 “ αpM1q ` 1. Repeating these steps N “ αpip1qq ` 2 number of times we get

ip1q ă ip2q ă ... ă ipNq,

and by DLop1, 2q on αpip1qq, αpip2qq, ..., αpipNqq we get

αpipkqq ď αpipk`1qq ď αpipk`1q ` 1q.

Some remarks on the previous proof

1 Behind the repetitive use of DLop1, 2q is the formal fact:

@npDLp1, 2, nqq Ñ DLp1, 3q,

DLp1, 2, nq : @αDuPNăω p|u| “ n ^ smonpuq ^ @uiPupαpui q ď αpui ` 1qq.

2 Within this proof the rightmost pair of the indices on which a sequence α weaklyincreases are consecutive numbers. Generally, all indices are not consecutive.

Proposition

Df :NÑN@αDiăj pf pαpiqq ď f pαpjqq Ñ αpiq ď αpjq ď αpj ` 1qq.

Proof.

Take f “ idN and use the last remark.

Proposition

Ef :NÑN@α@i1ăi2ă...ăil pf pαpi1qq ď f pαpi2qq ď . . . ď f pαpil qq Ñ αpi1q ď αpi2q ď . . . ďαpil ` 1qq. I.e., DLp1, lq doesn’t imply DLp1, l ` 1q in one only step.

A strong version of DLp1, lq within BIM˚ requires the repetitive use ofDLsp1, l ´ 1q

Proposition

If l ě 3, then

@αDMDi1,i2,...,il pi1 ă i2 ă ... ă il ď M ^ αpi1q ď αpi2q ď . . . ď αpil qq,

where for a given α

M “ Mαp1, lq “Nÿ

Mj , N “ αpip1qq ` 2,

M1 “ Mαp1, l ´ 1q, Mj`1 “ Mαpjq p1, l ´ 1q,

for each j P t1, ...,N ´ 1u. Also

Mαp1, l ´ 1q is the bound for DLsp1, l ´ 1q on α,αpjq is the tail of α starting from the index Mj ,

Mαpjq p1, l ´ 1q is the bound for DLsp1, l ´ 1q on the sequence αpjq,

ip1q is the first index determined by the application of DLsp1, l ´ 1q on α.

DLop1, 2q doesn’t prove DLp2, 2q in one step

Proposition

(i) Ef :N2ÑN@n1,n2,m1,m2 pf pn1, n2q ď f pm1,m2q Ñ pn1, n2q ď pm1,m2qq.

(ii) DLop1, 2q doesn’t prove DLp2, 2q in one step.

(iii) @f :N2ÑNDn1,n2,m1,m2 pf pn1, n2q ď f pm1,m2q ^ pn1, n2q ę pm1,m2qq.

Proof.

(i) Suppose that exists such a function f . Then f p0, 1q ą f p1, 0q ą f p0, 1q.(ii) If there was such a function f and α, β given sequences applying DLop1, 2q onf pαpnq, βpnqqn we would get i s.t

f pαpiq, βpiqq ď f pαpi ` 1q, βpi ` 1qq Ñ pαpiq, βpiqq ď pαpi ` 1q, βpi ` 1qq.

(iii) Consider the sequence

αp0q “ f p1, 0q, αp1q “ f p0, 1q, αp2q “ f p2, 0q, αp3q “ f p0, 2q, . . . .

By DLop1, 2q on α we get i ă αp0q ` 1 s.t αpiq ď αpi ` 1q, and for each j ą 0p0, jq ę pj ` 1, 0q and pj , 0q ę p0, jq.

DLp1, 3q doesn’t prove DLp2, 2q in one step

Proposition

(i) Ef :N2ÑN@n1,n2,m1,m2,k1,k2pf pn1, n2q ď f pm1,m2q ď f pk1, k2q Ñ pn1, n2q ď

pm1,m2q _ pn1, n2q ď pk1, k2q _ pm1,m2q ď pk1, k2qq.

(ii) DLp1, 3q doesn’t prove DLp2, 2q in one step.

Proof.

(ii) If there was such a function f then by DLp1, 3q on γpnq “ f pαpnq, βpnqq we wouldget a common good pair for α, β.

A strong version of DLp2, 2q within BIM requires @lě2DLsp1, lq

Proposition

@α,βDMDi,j pi ă j ă M ^ αpiq ď αpjq ^ βpiq ď βpjqq,

M “ Mα,βp2, 2q “ MpKq “Kÿ

1 ď K ď N, N “ αpip1q1 q ` 2, M1 “ Mαp1, 3q,

βpip1q1 q ď 1 Ñ K “ 1,

βpip1q1 q ě 2 Ñ M2 “ Mαp1q p1, βpi

p1q1 q ` 1q,

where ip1q1 is the first index determined by DLsp1, 3q on α and αp1q is the tail of α

starting from the index M1. Also, if 1 ď j ď N ´ 1, then

βpipj`1q1 q ď βpi

pjq1 q ´ 1 Ñ K “ j ` 1,

βpipj`1q1 q ě βpi

pjq1 q Ñ Mj`1 “ Mαpjq p1, βpi

pj`1q1 q ` 1q,

where ipj`1q1 is the first index of the application of DLsp1, βpi

pjq1 ` 1qq on αpjq, which is

the tail of α starting from the index Mj .

Proof of DLsp2, 2q within BIM

By DLsp1, 3q on α there are ip1q1 ă i

p1q2 ă i

p1q3 , for which

αpip1q1 q ď αpi

p1q2 q ď αpi

p1q3 q,

based on the initial segment of α of length M1 “ Mαp1, 3q. We also consider the terms

βpip1q1 q, βpi

p1q2 q, βpi

p1q3 q.

If βpip1q1 q ď 1, by DLop1, 2q there is a common good pair for α, β, and K “ 1.

If βpip1q1 q “ µ ě 2, we consider the tail αp1q of α which starts from M1. By

DLsp1, µ` 1q on αp1q there are indices ip2q1 ă i

p2q2 ă ... ă i

p2qµ`1, for which

ip1q1 ă i

p1q2 ă i

p1q3 ă i

p2q1 ă i

p2q2 ă ... ă i

p2qµ`1,

αp1qpip2q1 q ď αp1qpi

p2q2 q ď ... ď αp1qpi

p2qµ`1q.

Of course, α also weakly increases on them. Considering βpip2q1 q we work as follows:

If βpip2q1 q ď µ´ 1, then we can find the needed pair of indices as previously.

If βpip2q1 q ě µ “ βpi

p1q1 q, we repeat the previous step working with the tail αp2q of α

which starts from M1 `M2, where M2 “ Mαp1q p1, βpip1q1 q ` 1q.

Proof of DLsp2, 2q within BIM

If we are at step j , where 1 ď j ď N ´ 1, we find an index ipj`1q1 , which is the first

index of the application of DLsp1, βpipjq1 ` 1qq on αpjq, the tail of α starting from Mj .

If βpipj`1q1 q ď βpi

pjq1 q ´ 1, then, by DLsp1, 2q, a common good pair is found, and

K “ j ` 1.

If βpipj`1q1 q ě βpi

pjq1 q, we repeat the procedure at most N “ αpi

p1q1 q ` 2 number of

times. Then we haveip1q1 ă i

p2q1 ă ... ă i

pNq1 ,

βpip1q1 q ď βpi

p2q1 q ď ... ď βp

pNq1 q.

By DLop1, 2q on

αpip1q1 q, αpi

p2q1 q, ...αp

pNq1 q

we find a common good pair for α, β based on an initial segment of them of length atmost M “

řKj“1 Mj .

1 It is a characteristic example of a constructive proof from which a bound is ex-tracted.

2 It reveals the dependence of DLp2, 2q on the versions DLp1, lq.

3 It is the constructive analogue of the classical proof

WIRTÑ rDLp1, 2q Ñ DLp1,8qs,

DLp1,8q Ñ DLp2, 2q.

4 Using the bounds Mαp1, lq we avoid the use of some choice principle (Richman-Schuster).

5 Using the bounds Mαp1, lq we can reformulate our propositions in non (infinite)sequential terms (Richman 2001).

6 The whole proof-procedure and bound-extraction is generalizable to DLpk, lq, al-though the description of the bounds is more complex.

DLp2, 2q doesn’t prove DLp3, 2q in one step

Proposition

(i) Ef1,f2:N3ÑN@n1,n2,n3,m1,m2,m3 pf1pn1, n2, n3q ď f1pm1,m2,m3q ^ f2pn1, n2, n3q ď

f2pm1,m2,m3q Ñ pn1, n2, n3q ď pm1,m2,m3qq.

(ii) DLp2, 2q doesn’t prove DLp3, 2q in one step.

Proof.

(ii) Suppose that α1, α2, α3 : NÑ N. If there were such functions f1, f2, then applyingDLp2, 2q on

pf1pβpnqqqn, pf2pβpnqqqn,

where, for each n P N,βpnq “ pα1pnq, α2pnq, α3pnqq,

we get i ă j s.t

f1pα1piq, α2piq, α3piqq ď f1pα1pjq, α2pjq, α3pjqq ^

f2pα1piq, α2piq, α3piqq ď f2pα1pjq, α2pjq, α3pjqq Ñ

pα1piq, α2piq, α3piqq ď pα1pjq, α2pjq, α3pjqq.

On the relation of DL with Higman’s lemma, HL2p1, 2q, on words over analphabet with two letters

Theorem (Higman’s lemma, HL2p1, 2q)

@α:NÑ2ăωDi,j pi ă j ^ αi ĺ αj q, where ĺ is the embedding relation between 0, 1-words.

Remark

HL2p1, 2q Ñ @k,2DLpk, 2q.

Proposition (I.P 2011)

Suppose that α : N Ñ 2ăω and the sequence of lengths p|αpnq|qn is bounded orunbounded. Then,

@kDLpk, 2q Ñ HL2p1, 2q.

Proof.

For the unbounded case we use induction on the length |αp0q|.

J. Berger (2012) @kDLpk , 2q Ñ HL2p1, 2q

Proof.

It suffices to prove it for words starting with 0. Use DLpk, 2q, where

k “ 2|αp0q| ` 1

on the k components ofF : N` Ñ Nk`1

m ÞÑ pδpαpmq,Πk p∆npαpmqqq,

whereδpuq is a weight function counting the number of changes from 0, 1 into u.∆npuq outputs the shortest extension, or largest restriction of u of weight n.If δpuq “ n, Πn is the bijection of all these words onto Nn

u ÞÑ pm1, . . . ,mnq

u “ 0 . . . 0loomoon

1 . . . 1loomoon

. . . i . . . iloomoon

Some remarks on Berger’s result

1 This is a proof formalizable in BIM.

2 In principle we can derive from the extracted bounds of the corresponding versionsof DLpk, 2q a bound of HL2p1, 2q.

3 Generalization of Berger’s proof method in the case of HLmp1, 2q?

4 Study of the reverse combinatorics of this relation, namely to show that no specificversion DLpk, lq suffices to prove HL2p1, 2q. Already done for some simple cases.

Further topics of study extending the constructive combinatorics of DL

1 Search for the optimal bound for DLp2, 2q.

2 Constructive proofs with extracted bounds for w.q.o on specific ordinals pα,ďq.This is a bottom-up approach according to which the extracted bounds reveal theinitial segment of the sequence f : N Ñ α required to find a good pair of f(extension of BIM with ordinals). Note that Herz follows a top-down approachwithin which the bounds are not that explicit.

3 Extension of this approach to other w.q.o in order to derive e.g., Wp2, 2q from@lWp1, lq avoiding choice principles.

4 Study of the ideal on R of all its subsets X for which pX ,ď|XˆX q is a w.q.o andcomplete description of such an X .

5 Characterization of the subsets of N which satisfy DLp8, 2q. Clearly, a sequenceof sequences with arbitrary large bad initial segments violate DLp8, 2q.

A final comment

Bishop: meaningful distinctions need to be preserved Dco, Dcl, but also a distinctionbetween a constructive proof which does not provide a bound and one which providesit. Of course, in principle a constructive proof should provide one, even implicitly. It isimportant to make it as explicit as possible.

constructive combinatorics of dickson's lemma petrakis.… · 5 dlp1;8q: if : n Ñn, there is...

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