Logical Limit Laws and Almost Sure Theories

103 Pages·2014·1.35 MB·English

U.U.D.M. Report 2014:4
To Infinity and Back: Logical Limit Laws
and Almost Sure Theories
Ove Ahlman
Filosofie licentiatavhandling
i matematik
som framläggs för offentlig granskning
den 19 maj 2014, kl 13.15, sal 2005,
Ångströmlaboratoriet, Uppsala
Department of Mathematics
Uppsala University

TO INFINITY AND BACK: LOGICAL LIMIT LAWS AND
ALMOST SURE THEORIES
OVE AHLMAN
This thesis consists of the following papers referred to by their Roman nu-
merals:
I O. Ahlman, V. Koponen, Random l-colourable structures with a prege-
ometry
II O. Ahlman, V. Koponen, Limit laws and automorphism groups of ran-
dom non-rigid structures
III O. Ahlman, Countably categorical almost sure theories
Introduction
History and preliminaries. A cornerstone of mathematical logic is the
abstract vocabulary of mathematics. A vocabulary V consists of constant,
function and relation symbols where each relation symbol and function sym-
bolhasanassociatedpositiveintegertoit,calleditsarity. Fromavocabulary
we may create formulas by adding variables, connectives (e.g. ∧ and ∨) and
quanti(cid:28)ers (e.g. ∃ and ∀) together with the symbols from the vocabulary. If
wewanttoevaluatesomekindoftruthofaformulaitneedstobeasentence
i.e. have no free variables. A quick example from graph theory is the sen-
tence ∀x∀y(E(x,y) ↔ E(y,x)) which states that the edge relation E (part
of the vocabulary in graph theory) is symmetric.
Di(cid:27)erent parts of mathematical logic do di(cid:27)erent things with languages.
The (cid:28)eld of proof theory studies what and how deductions from formulas are
possible using certain proof rules. Recursion theory describes what and how
fast we may compute functions in models of Peano arithmetic (and things
similar to Peano arithmetic). In this thesis we will focus on something in
between those (cid:28)elds called model theory, where we discover which sentences
are true or false in structures of some (often general) vocabulary and how
these structures relate to each other. We will consider vocabularies which
only contain relation symbols (called relational vocabularies) making the
structures a bit easier to study and avoids certain complications, especially
we have the property that any subset of the universe of a structure induces
a substructure.
A structure with vocabulary V consists of a universe set and, for each
ρ ∈ N, interpretations of each relation symbol of arity ρ as a set of or-
dered ρ−tuples of elements in the universe. We will denote the structures
with calligraphic letters M,N,A,B,... and their universes with the corre-
sponding non-calligraphic letter M,N,A,B,.... In order to relate structures
1
2 OVEAHLMAN
to each other we use embeddings and isomorphisms. An embedding (of
relational structures) between V−structures A,B is an injective function
f : A → B such that for each relation symbol R and a1,...,ar ∈ A we have
that R is true for the ordered tuple (a1,...,ar), noted A |= R(a1,...,ar), i(cid:27)
B |= R(f(a1),...,f(ar)). That a function is an embedding between struc-
tures will be written f : A → B. If f is also a bijection then we say that f
∼
is an isomorphism and then A is isomorphic to B, written A = B.
For each n ∈ N let Kn be some set of (cid:28)nite ((cid:28)nite universe) structures
over a vocabulary V and let µn : Kn → [0,1] ⊆ R be a probability measure
on Kn. The most natural example is to let Kn be all V−structures with
universe {1,...,n} and let µn be the uniform measure i.e. µn(N) = |K1n| for
each N ∈ Kn. In order to study the properties of the structures, we will
extend the probability measure µn so that for a property P (often a sentence
in the language) we have that
µn(P) = µn({N ∈ Kn : N satis(cid:28)es P}).
We may study what happens to the ’really big’ structures by regarding
limn→∞µn(P). This probability does not, for a general Kn, even converge.
However if a property P is such that limn→∞µn(P) = 1 we say that it is an
almost sure property. If each sentence ϕ i(cid:83)n the language is almost sure or
limn→∞µn(ϕ) = 0 then we say that K = ∞n=1Kn has a 0−1 law (under
the measure µn). That a set K has a 0−1 law is hence a proof that it is
very well behaved.
The study of 0 − 1 laws for di(cid:27)erent sets K and measures µn started
with Glebskii, Kogan, Liogonkii and Talanov [7] and Fagin [5] who inde-
pendently proved that for each relational vocabulary V, if Kn is the set
of all V−structures with universe {1,...,n} then K has a 0−1 law under
the uniform measure (each structure has the same probability in Kn). It has
sincethenbeendiscoveredthatmanydi(cid:27)erentsetshave0−1laws, spanning
from Comptons [3] result about Kn being partial orders under the uniform
measure to Koponen [11] who proved that an l−coloured structure with a
pregeometry has a 0−1 law under the dimension conditional measure. In
these articles, the method Fagin used, when proving the (cid:28)rst 0−1 law, is
adapted to the context and since this method is vital for all three papers we
shall take a brief look at Fagins proof. Further examples of 0−1 laws may
be found in [1, 2, 4, 8, 9, 10, 13, 14, 15, 16, 17, 18].
The (cid:28)rst thing which is done by Fagin [5] is to (cid:28)nd certain sentences from
the language called extension axioms. Each extension axiom speaks about
two structures A ⊆ B such that |A| + 1 = |B| and the extension axiom
says that for each instance of A we may extend this into the structure B.
The proof then continues by proving that these extension axioms are almost
surely true in K, which consist of all (cid:28)nite structures of the speci(cid:28)ed rela-
tional vocabulary. A theory (collection of sentences) T is then created by
collectingallextensionaxiomsandthistheoryisshowntobeℵ0−categorical.
TO INFINITY AND BACK: LOGICAL LIMIT LAWS AND ALMOST SURE THEORIES3
The categoricity follows from a back and forth building of isomorphisms, us-
ing that any model of T satis(cid:28)es the extension axioms. An ℵ0−categorical
theory, without (cid:28)nite models, is complete and hence any formula is either
implied by the theory (thus must have probability 1) or its negation is (thus
the formula has probability 0). This theory T, which hence consist of all
formulas with probability 1, is called the almost sure theory. For more in-
formation about (cid:28)nite model theory the reader is suggested to look at [4].
A pregeometry G = (G,cl) consists of a set G and a closure operator
cl : P(G) → P(G) acting as a function on the power set of G. The closure
operator should satisfy the following axioms for each X,Y ⊆ G:
Re(cid:29)exivity: X ⊆ cl(X).
Monotonicity: X ⊆ cl(Y) (cid:83)⇒ cl(X) ⊆ cl(Y).
Finite character: cl(X) = {cl(X0) : X0 ⊆ X and |X0| < ℵ0}.
Exchange: Fora,b ∈ Gifa ∈ cl(X∪{b})−cl(X)thenb ∈ cl(X∪{a}).
The most trivial example, called the trivial pregeometry, is the pregeometry
where for each X ⊆ G we have that cl(X) = X. Pregeometries may be
viewed as generalizations of vector spaces and we get a direct example by
letting G be the vectors and cl be the span operator. In the spirit of vector
spaces we say that a set X ⊆ G is independent if for each a ∈ X we have a ∈/
cl(X−{a}). The dimension of a set X is the size of the largest independent
subset Y ⊆ X.
We will however not consider structures with an explicit closure operator,
but we may be able to de(cid:28)ne it. For a structure M and X ⊆ M we say
that the pregeometry G = (M,cl) is X−de(cid:28)nable in M if there are formulas
θ0(x0),...,θn(x0,...,xn),... (possibly with parameters from X) such that for
b,a1,...,an ∈ M we have
b ∈ cl(a ,...,a ) ⇔ M |= θ (b,a ,...,a ).
1 n n 1 n
If the pregeometry is ∅−de(cid:28)nable we may say that M is a pregeometry.
About paper I. An old famous problem in mathematics is the study of the
colouring of a map or, in mathematical terms, a planar graph. The basic
concept is that if we have two vertices adjacent to each other then they need
to have di(cid:27)erent colours. For Kn being l−coloured graphs with universe
{1,...,n} Kolaitis et al[10] showed that there is a 0−1 law under the uni-
form measure. However the concept of colouring is not trivially generalized
to vocabularies with higher arities. The key problem is that if we have a
relation R(a1,...,am), in some structure, then should all a1,...,am have dif-
ferent colours or should only at least two of them have di(cid:27)erent colours? In
[11] Koponen discussed a set Kn consisting of structures all of which have
the same ∅−de(cid:28)nable pregeometry with dimension n. But in order to give a
more general take on l−coloured structures, for some (cid:28)xed number l ∈ Z+,
Koponen studies coloured structures where the underlying pregeometry af-
fects how the colouring is done in the following way:
(1) Each element, except those in cl(∅), has one of the l colours.
4 OVEAHLMAN
(2) Any two elements in the same 1−dimensional closure have the same
colour.
(3) ForeachrelationR(a1,...,an)thereareelementsb,c ∈ cl(a1,...,an)−
cl(∅) such that b and c have di(cid:27)erent colours.
Notice that if the underlying pregeometry is the trivial pregeometry then (3)
gives us exactly the colouring condition noted above i.e. that at least two
elements in a relation needs to have di(cid:27)erent colours. Following this we also
want to speak about strongly coloured structures who satisfy the following
condition (in addition to (1)-(3)):
(4) For each relation R(a1,...,an) and any independent elements b,c ∈
cl(a1,...,an) we have di(cid:27)erent colours of b and c.
Notice that one important part of being a coloured structure is that there
are unary relations P1,...,Pl which represent the colours in the language.
Koponen showed that for l−coloured structures Kn with pregeometries (as
described above) and random relations (not involved in the pregeometry) K
has a 0−1 law. This generalized the result of Kolaitis et al, which becomes
especially clear when the pregeometries are trivial.
However, the study of coloured structures becomes much harder when
removing the colours from the vocabulary, and just looking at colourable
structures. An(strongly)l−colourablestructureisastructurewherewemay
addlunaryrelationsymbolssuchthatthisnewstructurebecomes(strongly)
l−coloured. Koponen [11] studied the case of Kn being just l−colourable
structures, however she only presented a 0 − 1 law in the case where the
underlyingpregeometryistrivial. Thebigproblemwhennothavingatrivial
pregeometry is that we have no obvious knowledge about the possible colour
of an element, and hence even realizing if a structure is l−colourable or not
becomes a bit tricky.
In paper I we extend the 0−1 law of l−colourable structures, to those
structures who have a vector space pregeometry (and some closely related
variants). This is done by proving that there exists a binary formula ξ(x,y),
de(cid:28)nable without having explicit colours in the vocabulary, such that almost
surely in K, if some structure M |= ξ(a,b) then a and b must be assigned
the same colour, when colouring the structure M. We then use the formula
ξ(x,y) to create extension axioms which suit the class Kn and then prove
the 0 − 1 law in the spirit of Fagin’s [5] proof. Most of paper I is spent
actually creating the formula ξ which is done in surprisingly di(cid:27)erent ways,
depending on if we assume the colouring to be strong or not. In the strong
case, this is proven through a method inspired by Koponen [12] where we
de(cid:28)ne the formula by brute force. ξ(x,y) then states that there are so many
elements in relation to x and y such that any two elements satisfying ξ(x,y)
has to be of the same colour, since we know that (independent) elements in
relation to each other have di(cid:27)erent colour, and only l−colours exists. The
opposite direction, that any two elements with the same colour satisfy ξ, is
proven by using the extension properties developed for coloured structures
TO INFINITY AND BACK: LOGICAL LIMIT LAWS AND ALMOST SURE THEORIES5
byKoponen[11]whichcaninducethestructurewhichξ(x,y)claimstoexist.
The normal (non-strong) l−colourings on the other hand are proven using a
Ramseytheoretictheoremwhichsaysthatifwehavealargeenoughcoloured
vector space, then we need to have a large mono-coloured subspace. Using
this we create a structure which has as many relations as possible, and yet is
still l−colourable, in order to (cid:28)x the colouring of the structure. The formula
ξ(x,y) then says that x and y are part of the same mono-coloured subspace
of this structure, and hence must have the same colour. To show that any
elementswiththesamecolourmustsatisfyξ(x,y)weagainusetheextension
axioms for coloured structures to (cid:28)nd this large mono-coloured structure in
the right place.
About paper II. An automorphism on a structure M is an isomorphism
from M to M. It is an easy result to show that the set of all automorphisms
on M, Aut(M), form a group under compositions. This group acts with
a group action on M where one of the more important concepts are the
support of an automorphism, which are the elements which are not mapped
to themselves. For Kn being all V−structures with universe {1,...,n} under
the uniform measure, Fagin [6] proved that the automorphism group is al-
most surely the trivial group for structures in K. This has as a consequence
that the set K consisting of only (cid:28)nite structures with a trivial automor-
phism group, has a 0−1 law. One natural follow up to this is to discover
what happens if K consists only of structures without trivial automorphism
group. These kinds of sets are what we discuss in paper II, where we do a
general investigation in the area, discussing what happens when we (cid:28)x the
automorphism group or size of the support.
In the case of K consisting of all structures we see that the support is
almost surely empty, since the automorphism group is almost surely trivial.
This does in some way give a hint that the support is almost surely ’small’.
In paper II we prove that when comparing the sets of structures K and C
where all structures in C has ’a bit larger’ support on some automorphism,
than the support of any automorphism in K, then |Cn| → 0. This is the
|Kn|
basis of the paper since it means that whenever we look at structures with
large support, or large automorphism group, we may restrict the amount of
elements moved by the automorphism. After further work we deduce that
we may actually reduce this to having a bounded amount of elements which
are actually moved by some automorphism.
Using these results we now have a quite a clear picture of how these struc-
tures without trivial automorphism group look, one bounded part where the
automorphisms can move elements around, and one place which is (cid:28)xed un-
der any automorphisms. This knowledge leads to an estimate of the amount
of structures having a certain structure of the elements in the support. Hav-
ing an estimate of the supports it then becomes easier to compare sets K
and C with di(cid:27)erent automorphism group with each other and receive yet
another estimate of which of them are more common.
6 OVEAHLMAN
Thispaperleadsuptoa0−1law. Howeveritisonlypossibletodeduceit
for a very restricted set of structures having a (cid:28)xed structure on the support
and the automorphism group being somewhat (cid:28)xed. In the case where we
onlyrestrictthesupportofstructuresinKortheautomorphismgroupsthen
we just get a convergence law i.e. the asymptotic probability limn→∞µn(ϕ)
converges, but not necessarily to 0 or 1.
About paper III. In paper III we work in a very general setting using any
set of V−structures Kn under any kind of probability measure µn, with the
only restriction being that the structures in Kn grow in size (almost surely)
as n grow. Using this general K we study how the almost sure theory of K
a(cid:27)ects K in the case where K has a 0−1 law under the measure µn. Espe-
cially important for this study are the equivalence relations de(cid:28)nable (using
formulas from the vocabulary) in the models of the almost sure theories,
which we show induce a restriction of which sizes the structures in K may
attain. Especially this induces a restriction on the sizes of structures which
de(cid:28)ne a pregeometry since being independent is an equivalence relation.
As mentioned in the preliminaries, Fagin’s [5] method for proving a 0−1
law for K consisting of all (cid:28)nite V−structure, involves proving that the al-
most sure theoryis ℵ0−categorical and hencecomplete. In paper III we do a
generalization of this concept and study how an almost sure theory which is
ℵ0−categoricalinducespropertiesonthesetofstructuresfromwhichitcame
from. Theresultisthatthesetofstructuresalmostsurelysatis(cid:28)essomekind
ofextensionaxioms(notnecessarilyaseasilydescribedasthosein[5]),ifand
only if the almost sure theory is ℵ0−categorical. So Fagin’s method works
to prove the 0 − 1 law of any set K which has an ℵ0−categorical almost
sure theory (and a 0−1 law). Furthermore the sets K with ℵ0−categorical
almost sure theories may be extended in the same way as Meq, getting ele-
ments representing equivalence classes. The nice thing about this extension
isthatthesetofextendedstructureshasa0−1lawifandonlyiftheoriginal
set had one, giving us a new way to transform sets of structures in order to
(cid:28)nd out if they have a 0−1 law or not.
A theory T is strongly minimal if for each model M of T, tuple a¯ ∈ M
and formula ϕ(x,y¯) either ϕ(x,a¯) or ¬ϕ(x,a¯) is only satis(cid:28)ed by a (cid:28)nite
amount of elements. Paper III ends with a classi(cid:28)cation of sets K with a
0−1 law whose almost sure theory is both ℵ0−categorical and strongly min-
imal. These kinds of theories have quite many nice properties which help in
producing the result. The classi(cid:28)cation comes down to that the sets K who
induce these almost sure theories need to have structures where the auto-
morphism group permutes almost all elements. This condition is equivalent
with having an ω−categorical strongly minimal almost sure theory.
Acknowledgments
I would like to thank my advisor Vera Koponen, who has put down a lot
of work discussing things I have written and who is always positive to my
TO INFINITY AND BACK: LOGICAL LIMIT LAWS AND ALMOST SURE THEORIES7
ideas. My graduate student colleagues at the Department of Mathematics
also deserve gratitude, for listening and being part of multiple conversations
inside and outside of the area of mathematics.
Finally I dedicate this thesis to the shade of my heart, my (cid:28)ancØe Karin,
for always being encouraging towards me, even when I am not.
References
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ties, advances in mathematics, Volume 65 (1987) 65-96.
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Inform. and comput. 78 (1988), 108-123.
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[6] R. Fagin, The number of (cid:28)nite relational structures, Discrete mathematics, Vol. 19
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(cid:28)abilityofformulasoverthelowerpredicatecalculus,KibernetykaVol.2(1969)17-27.
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structures, Annals of pure and applied logic, Vol 163 (2012) 391-438.
[12] V. Koponen, A limitlaw of almost l-partite graphs, The journal of symbolic logic,
Volume 78 (2013) 911-936.
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l
Random -colourable structures with a pregeometry
Ove Ahlman and Vera Koponen
July 20, 2012
Abstract
We study (cid:28)nite l-colourable structures with an underlying pregeometry. The prob-
ability measure that is used corresponds to a process of generating such structures
(withagivenunderlyingpregeometry)bywhichcoloursare(cid:28)rstrandomlyassigned
to all 1-dimensional subspaces and then relationships are assigned in such a way
thatthecolouringconditionsaresatis(cid:28)edbutapartfromthisinarandomway. We
can then ask what the probability is that the resulting structure, where we now
forget the speci(cid:28)c colouring of the generating process, has a given property. With
this measure we get the following results:
1. A zero-one law.
2. The set of sentences with asymptotic probability 1 has an explicit axiomati-
sation which is presented.
3. Thereisaformulaξ(x,y)(notdirectlyspeakingaboutcolours)suchthat,with
asymptoticprobability1,therelation(cid:16)thereisanl-colouringwhichassignsthe
same colour to x and y(cid:17) is de(cid:28)ned by ξ(x,y).
4. Withasymptoticprobability1,anl-colourablestructurehasauniquel-colouring
(up to permutation of the colours).
Keywords: model theory, (cid:28)nite structure, zero-one law, colouring, pregeometry.
1 Introduction
We begin with some background. Let l ≥ 2 be an integer. Random l-colourable (undi-
rected) graphs were studied by Kolaitis, Pr(cid:246)mel and Rothschild in [7] as part of proving
azero-onelawfor(l+1)-clique-freegraphs. Theyprovedthatrandoml-colorablegraphs
satis(cid:28)esa(labelled)zero-onelaw,whentheuniformprobabilitymeasureisused. Inother
words,ifCndenotesthesetofundirectedl-colourablegraphswithvertices1,...,n,then,
for every sentence ϕ in a language with only a binary relation symbol (besides the iden-
tity symbol), the proportion of graphs in Cn which satisfy ϕ approaches either 0 or 1.
They also showed that the proportion of graphs in Cn which have a unique l-colouring
(up to permuting the colours) approaches 1 as n → ∞. In [7] its authors also proved
the other statements labelled 1(cid:21)4 in this paper’s abstract, when using the uniform prob-
ability measure on Cn, although in case of 3 it is not made explicit. This work was
preceeded, and probably stimulated, by an article of Erd(cid:246)s, Kleitman and Rothschild
[4] in which it was proved that proportion of triangle-free graphs with vertices 1,...,n
which are bipartite (2-colourable) approaches 1 as n→∞.
One can generalise l-colourings from structures with only binary relations to struc-
tures with relations of any arity r ≥2 by saying that a structure M is l-coloured if the
elements of M can be assigned colours from the set of colours {1,...,l} in such that if
M |= R(a1,...,ar) for some relation symbol R, then {a1,...,ar} contains at least two
1