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Countable universal relational structures

Petrus H Potgieter

www.potgieter.org

Department of Decision SciencesUniversity of South Africa

Siegen, 7 June 2011



Introductiona

The talk is an overview of certain results about universal objects for

countable relational structures, some of it joint work of the speakerand Willem Fouche.

There will be no proofs and only imprecise definitions.

aThis work has been partly supported by DFG/NRF grant GZ: 445 SUA-1 13/20/0 and

is also based in part upon research supported by the National Research Foundation grantIFR2011041500051.



Photograph used without the permission of Dr Konrad Swanepoel of the LSE.



Preliminaries

The Fraısse limit

L denotes a class of relational structures with given finite signature. Acountable L-structure is called universal in its class if every countableobject in the class can be embedded into it.

In 1954 Roland Fraısse showed that if the class of finite objects in Lhas the amalgamation property ,

if A, B1, B2 are finite objects in L and A can be embed-ded into B1 and into B2 then there exists a finite C suchthat B1 and B2 can both be embedded into C and the twoembeddings agree only on the respective images of A,

then there exists a countable and homogeneous L-structure X whichis universal for L. X is determined uniquely (up to isomorphism).

A structure is homogeneous if any isomorphism between finite sub-structure can be extended to an automorphism.



Rado’s random graph (early 1960s)

Rado’s random graph GR has the property that every countable graphis a subgraph of GR. It is therefore universal in the class L of countablegraphs.

GR is characterised by the extension property:

given any disjoint finite subgraphs U and V of GR, therealways exists a vertex x in GR such that x is adjacent to

every vertex of U and adjacent to no vertex of V .

There are many ways of constructing GR.



The random graph from primes

Let p1, p2, p3, . . . be any enumeration of the prime numbers. Constructa graph on the natural numbers by letting (n, m) be an edge whenever pn|m or pm|n.

47

_ _ _ 12

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35

511

_ _ _

613

_ _ _ 23

(m, n) iff pm|n ∨ pn|m



The graph from primes is the Rado graph

The graph constructed by (m, n) iff pm|n∨ pn|m has the expansionproperty which characterises the random graph.

Let disjoint sets U = {u1, u2, . . . , uk} and V ={v1, v2, . . . , v} be given. We will choose and x which is

adjacent to all of U and adjacent to none of V in the graphdefined above. Set

x = ( pu1 pu2 · · · puk)n

where n has been chosen large enough to guarantee that px

does not divide any of v1, v2, . . . , v.

Now x is adjacent to all of U and not adjacent to anyof V , as required (P, 1996).



A universal connected graph

Since GR is connected, it is actually also the universal countable objectfor the class L of connected graphs.

A universal tree

There is a universal countable tree but it is different from GR.

No universal planar graph

Pach (1981) showed that there is no universal planargraph (Komjath and Pach, 1991; Diestel and Kuhn,1999).



Universal objects by coin toss

The random graphErdos and Renyi showed in 1963 that if a countable graph is gener-ated in some canonical way from coin-tossing then it will be GR withprobability one.

The binary sequences that generate the random graph (by some fixedcanonical enumeration of the edges) are ubiquitous in category, i.e.the complement of the set is of first Baire category.

For each pair of finite vertex sets U and V , consider the setAU,V consisting of all sequences α for which the graph gen-

erated by the graph generated by α contains no x which isadjacent to all of U and adjacent to none of V . We ob-serve that each AU,V is closed and nowhere dense in the usualtopology on [0,1).

This is reported to have been pointed out by Peter Cameron.



A universal ranked diagram

Fouche and P (1998) gave a probabilistic as well as a deterministicconstruction of a universal ranked diagram.

A ranked diagram is the Hasse diagram of a poset of finite height nand there exists a universal object for each n.



A universal object for any L

Droste and Kuske (2003) describe the probabilistic construction of auniversal homogeneous object for any L that admits such an object.With respect to their so-called natural enumeration of the relations,the object constructed in this manner is universal for the class.

For instance, the natural enumeration for a single binary relation onthe natural numbers is

(1, 1); (1, 2);(2, 1); (2, 2); (1, 3);(2, 3); (3, 3);(1, 4);(2, 4); (3, 4);(4, 4); . . .

and similar enumerations are defined for more complex structures.Note: in the probabilistic construction of the random graph or of a universal ranked diagram, the enumeration scheme of the relationplayed no role.



A random linear order

Using the natural enumeration for a single complete irreflexive binaryrelation on the natural numbers

(1, 2); (1, 3);(2, 3); (1, 4); (2, 4);(3, 4); (1, 5);(2, 5);(3, 5); (4, 5); . . .

we choose any infinite binary sequence. Proceeding from the left, fora pair (i, j) put

i j if either this is implied by (the transitive closure of)the ordering up to this point or if the next unused digit of thesequence is 1

or

j i if either this is implied by (the transitive closure of)the ordering up to this point or if the next unused digit of thesequence is 0.

With probability one this procedure will yield a dense chain withoutendpoints (Droste and Kuske, 2003).



Using specific random sequences

Algorithmic (KCSL/Martin-Lof) randomness

Fix a universal prefix Turing machine (UPTM) with alphabet {0, 1}.

Associate the UPTM with a partial function G : {0, 1}∗ → {0, 1}∗

where G(y) = x iff the Turing machine outputs x on input y.

Define K G(x) = min{|y| : G(y) = x}.

We shall call an infinite binary sequence α = α1α2α3 . . . algorith-mically random in the sense of Kolmogorov-Chaitin-Schnorr-Levin if

there exists a c (depending on α) such thatK (α1α2 . . . α3) > n − c for all n.

This definition is independent of the choice of UPTM and the KCSLrandom sequences are the same as the Martin-Lof random sequences.



Dense linear order from a random sequence

The preceding D-K procedure always generates a countable linear orderfrom a binary sequence and with probability one it is also dense andwithout endpoints.

Given a binary sequence, the property being dense with no endpoints for the linear order generated by the KD scheme can be described as

follows:

∀i∀ j∃k ((i j → i k j) ∧ (i = j → k j)) .

Furthermore, given (i,j,k), it is possible to write an effective proce-dure to verify (from the generating binary sequence) whether i k

j in the order or not.Being dense without endpoints is therefore—for a countable linearorder—an almost sure Π0

2property and each such property is known

to be satisfied by all algorithmically random (Martin-Lof) sequences(Fouche, 1994; Kurtz, 1981, cited by Downey et al. 2004).



Universal structures with comptable construction

Suppose L is a class of (finite or countable) relational structures withfinite signaturea. Via the natural enumeration of all the possible

Ri (x1, . . . , xki)

the procedure of Droste and Kuske (2003) generates a universal L-

structure.

At each step, one should check whether either Ri (x1, . . . , xki) or¬Ri (x1, . . . , xki) is strictly required by the preceding. If not, a cointoss is used to decide which relation to add to the structure.

If the check is computable at each stage in the preceding choices,then the resultant structure will be computable in the sequence of coin tosses.

aIt should be closed under the union of increasing chains of finite structures and taking of substructures.



Martin-Lof random sequences generate universal structures

For a binary sequence α denote by Ak the substructure on{1, 2, . . . , n} by the procedure above. Use of the natural enumer-ation makes Ak computable in k and α.

Suppose the finite structures in L can be enumerated, e.g. through

V m computable in m. Then universality is the Π02 property

∀m∃n (V m a substructure of An) .

From Droste and Kuske (2003) we know that with this procedure, the

probability of getting a universal structure is 1.It follows that every algorithmically random (Martin-Lof) sequence α generates a universal object for L,using Fouche (1994); Kurtz (1981).



A Ramsey property

The property of being universal is immune to the partition of copiesof some finite substructure by random sequences.

Let X be a recursive homogeneous structure with a denseage. For each β ∈ Age(X ) and each algorithmically random(Martin-Lof) ε, there exists an embedding ν : X → X suchthat

χε(β) = 1 for each β ∈ [ν (X ), β].

Further, ν can be so constructed that it is recursive relativeto ε (Fouche and P, 1998).

Here [Y, γ ] denotes the copies of γ in Y .



References

Diestel, R. and Kuhn, D. (1999). A universal planar graph under the minor relation. Journal of

Graph Theory , 32(2):191–206.

Downey, R. G., Griffiths, E. J., and Reid, S. (2004). On Kurtz randomness. Theoretical Computer

Science , 321(2-3):249–270.

Droste, M. and Kuske, D. (2003). On random relational structures. Journal of Combinatorial

Theory Series A, 102:241–254. ACM ID: 859229.

Fouche, W. L. (1994). Identifying randomness given by high descriptive complexity. Acta Appli-candae Mathematicae , 34(3):313–328.

Fouche, W. L. and Potgieter, P. H. (1998). Kolmogorov complexity and symmetric relationalstructures. The Journal of Symbolic Logic , 63(3):1083–1094.

Komjath, P. and Pach, J. (1991). Universal elements and the complexity of certain classes of infinite graphs. Discrete Mathematics , 95(1-3):255–270.

Kurtz, S. (1981). Randomness and Genericity in the Degrees of Unsolvability . PhD thesis,University of Illinois.

Pach, J. (1981). A problem of Ulam on planar graphs. European J of Combinatorics , 2:357–361.

Potgieter, P. H. (1996). Representations of binary sequences of high Kolmogorov complexity .PhD thesis, University of Pretoria.

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