Bakh Khoussainov Computer Science Department, The ...bjoern/pdf/RandomStructures.pdfBakh Khoussainov Algorithmically random inﬁnite structures. Topology on K! Deﬁnition (Topology)

Algorithmically random infinite structures

Bakh Khoussainov

Computer Science Department, The University of Auckland,New Zealand

February 13, 2015

Partly based on and motivated by the author’s LICS 2014 paper

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Bakh Khoussainov Algorithmically random infinite structures

Plan

MotivationBranching classesTopology, metric, and measure.Martin-Löf randomness.Computable structures and ML-randomness.

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A bit of motivation

The modern history is fascinating; starts with the works ofKolmogorov, Martin-Löf, Chaitin, Schnorr and Levin.The last two decades have witnessed to significantadvances in the area of algorithmic randomness on strings.Monographs by Downey and Hirschfeldt, and Nies.Many notions of randomness, various techniques, andideas have been studied.

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Algorithmic randomness of infinite strings

Consider the Cantor space {0,1}ω and let α ∈ {0,1}ω. Thereare several ways to define algorithmic randomness for α. Themost traditional ones are based on the following ideas:

1 Incompressibilty.2 Effective null cover.3 Computable martingales.

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Semi-informal explanation

1 The string α ∈ 2ω is algorithmically random if all sufficientlylarge segments of α are incompressible (up to a constant).

2 The string α is algorithmically random if no effectivemeasure 0 set contains α.

A set V ⊆ 2ω has effective measure 0 if V is the limit ofembedded sets M0 ⊃ M1 ⊃ M2 ⊃ . . . such that

Each Mi is an open set,Given i we can compute the base open sets that form Mi ,The measure of Mi is bounded by 1/2i .

Effective measure 0 sets are called Martin Löf tests (ML-tests).

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From strings to algebraic structures

We want to investigate the following question:

What is an algorithmically random infinite tree, graph, monoid,or generally, a universal algebra?

In the case of strings, the measure on the Cantor space isfundamental in introducing algorithmic randomness.

Thus, the task is to invent a meaningful measure in the classesof structures.

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What do we expect from an infinitealgorithmically random structure?

Absoluteness: Algorithmic randomness should be anisomorphism invariant property.

Continuum: Random structures should be in abundance,the continuum. This is a property of a collective, the ideathat goes back to Von Mises.

Selection: There should be no effective way to describethe isomorphism type of an infinite part of the structure.

Axiomatization: No set of simple (e.g. universal) axiomsdefine the structure.

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Questions:

Converting into strings: Why don’t we code structures asstrings and transform algorithmic randomness for stringsinto structures?Computability: Can a computable structure beML-random?Immunity: ML-random strings possess immunity property:No ML-random string has a computable subsequence.Does ML-random structure have immunity property?Finite presentability: Can a finitely presented structure,e.g. group, be ML-random?

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Converting into strings

Let G = (ω; E) be a graph. Form the following string αG:

αG(0)αG(1)αG(2) . . . ∈ 2ω,

where αG(i) = 1 iff the i-th pair is an edge in G.

DefinitionThe graph G is string-random if the string αG is ML-random.

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Implications of string-randomness

TheoremIf G is a string-random, then G is Rado graph. Hence,

Any two string-random graphs are isomorphic.The first order theory of the graph is decidable.The string-random graph is axiomatised by extensionaxioms.Any countable infinite graph can be embedded into G.

All of the above defy our intuition that we postulated foralgorithmically random infinite structures.

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Direct limits

DefinitionAn embedded system of structures is a sequence

(A0, f0), (A1, f1), . . . , (Ai , fi), . . .

such that (1) each Ai is a finite structure, and (2) each fi is aproper into embedding from Ai into Ai+1.

The sequence A0,A1, . . . is the base of the system.

Each embedded system determines the limit structure.

DefinitionAn embedded system {(Ai , fi)}i∈ω is strict if its direct limit isisomorphic to the direct limit of any embedded system with thesame base.

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Classes with height function

Let h : K → ω be a computable height function such that:

1 We can compute the cardinality of h−1(i) for every i .

2 For every A ∈ K of height i there is a substructure A[i − 1]of height i − 1 such that all substructures of A of height≤ i − 1 are contained in A[i − 1].

3 For all A ∈ K of height i and C ⊆ A \ A[i − 1], the height ofthe substructure C ∪ A[i − 1], where C 6= ∅, is i in case thesubstructure belongs to K.

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Properties of classes with height function

LemmaFor all A,B ∈ K , the structures A and B are isomorphic iffh(A) = h(B) and A[j] = B[j] for all j ≤ h(A).

Lemma

Every embedded system of structures from the class K isstrict.

DefinitionThe class K is a branching class, or B-class for short, if for allA ∈ K of height i there exist distinct structures B, C ∈ K suchthat h(B) = h(C) > h(A) and B[i] = C[i] = A.

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Examples of B-classes

Example 1. Pointed connected graphs (G,p) of boundeddegree. The height function is the max distance from p tovertices of G.

Example 2. Trees of bounded degree. The height function isthe height of the tree.

Example 3. Relational structures whose Gaifman graph is aconnected graph of a bounded degree.

Example 4. Partially ordered sets (P;≤,C,p), where p is theleast element, C(x , y) is the cover relation, and each x in P hasat most d covers.

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Examples of B-classes

Example 5. Two generated, by generators a and b, partialmonoids. The height function is the max of the shortest wordsrepresenting monoid elements.

Example 6. The class of binary rooted ordered trees.

Example 7. The class of n-generated universal partialalgebras. The hight function is the max among the heights ofthe shortest terms representing the elements of the algebras.

Example 8. The class of (a,b)-sparse graphs. A connectedpointed graph is (a,b)-sparce if every subgraph of G with mvertices has at most am + b edges.

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Definition of tree T (K )

Let K be a B-class. Define T (K ) as follows:

1 The root is ∅. This is level −1.

2 The nodes of at level n ≥ 0 are structures of height n.

3 Let B be a structure of height n. Its successor is anystructure C of height n + 1 such that B = C[n].

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Properties of the tree T (K ):

Let K be a B-class. Set

Kω = {A | A is the direct limit of structures from K}.

1 Given any node x of the three T (K ), we can effectivelycompute the structure Bx associated with the node x .

2 Each x in T (K ) has an immediate successor. We cancompute the number of immediate successors of x .

3 Each path η = B0,B1, . . . determines the limit structureBη = ∪iBi ∈ Kω.

4 The mapping η → Bη is a bijection from [T (K )] to Kω.

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Topology on Kω

Definition (Topology)

Let B be a structure of height n. The cone of B is:

Cone(B) = {A | A ∈ Kω, and A[n] = B for all n}.

Declare the cones Cone(B) to be the base open sets of thetopology on Kω. We refer to B as the base of the cone.

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Measure

Definition (Measure)The measure of the cone based at the root is 1.Assume that the measure µ(Cone(Bx )) has been defined.Let ex be the number of immediate successors of x . Thenfor any immediate successor y of x the measure ofCone(By ) is

µ(Cone(By )) =µ(Cone(Bx ))

ex.

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Metric

Definition (Metric)For A,B ∈ Kω, let n be the maximal level at which A[n] = B[n].The distance d(A,B) is then: d(A,B) = µm(Cone(A[n])).

Lemma

The function d is a metric in the space Kω .

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Basic properties of the space Kω

Fact1 Kω is compact.2 The set K is countable and dense in Kω.3 Finite unions of cones form clo-open sets in the topology.4 The set of all µ-measurable sets is a σ-algebra.

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ML-random structures

Definition

A structure A ∈ Kω is ML-random if it passes every ML-test.

Corollary

Let K be a B-class. The number of ML-random structures in theclass Kω is continuum. In particular, for all the examples ofB-classes K we considered, each of the classes Kω containscontinuum ML-random structures.

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Computable structures

DefinitionAn infinite structure A is computable if it is isomorphic to astructure with domain ω such that all atomic operations andrelations of the structure are computable.

DefinitionA computable structure A from Kω is strictly computable if thesize of the substructure A[i] can be computed for all i ∈ ω.

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Strict computability

The following are true:1 Every computable finitely generated algebra is strictly

computable.2 A computable pointed graph G of bounded degree is

strictly computable iff there is an algorithm that given avertex v of G computes degree(v).

3 A computable rooted tree T of bounded degree is strictlycomputable iff there is an algorithm that given a nodev ∈ T computes the number of immediate successors of v .

4 A computable d-bounded partial order with the leastelement is strictly computable iff there is an algorithm thatfor every v of the partial order computes all covers of v .

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Strict computability and randomness

TheoremIf A is strictly computable then A is not ML-random.

CorollaryLet A be either an infinite pointed graph or tree or partial orderof bounded degree. If A is computable and its ∃-diagram, thatis the set

{φ(a) | a ∈ A and A |= φ(a) and φ(x) is an existential first-orderformula},

is decidable then A is not ML-random.

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Computability in the Halting set

TheoremFor every B-class K there exist an H-computable ML-randomstructure in class Kω.

Thus, we have the following corollary:

Corollary

For all the examples of B-classes K that we have consideredeach of the classes Kω contains ML-random H-computablestructures.

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The mapping from [T (K )] to Kω

Construction of Aη from η is computable in η. Hence, if η iscomputable then so is Aη.

How about the opposite:

How complex is that to compute η from Aη?

Answer:

To compute η, we need to compute Aη[i] for each i . ComputingAη[i] requires the jump of the open diagram of Aη.

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ML-randomness and computability

TheoremThere exists a B-class K such that Kω contains an ML-randomyet a computable structure.

Proof (idea). A binary ordered tree B belongs to S if:1 All leaves of B are of the same height,2 If v in B has the right child then all nodes left of v on the

v ’s level-order including v have both children,3 At each level i there is at most one node such that it is the

left child of its parent that does not have a right child.

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The structure of T (S)

Lemma

If B belongs to S and has height n then there are exactly twonon-isomorphic extensions of B of height n + 1 both in S.Hence, the tree T (S) is isomorphic to the infinite binary tree.

LemmaFor every n ≥ 0, the set of all trees in S of height n form a chainof embedded structures.

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Embedding Lemma

So, we naturally identify the infinite binary tree {0,1}? withT (S) through the lemmas above.

Lemma

Let x � y, where � is the lexicographical order on binarystrings. Then:

1 If |x | ≤ |y | then Ax is embedded into Ay .2 If |x | > |y | then Ax is embedded into Ayz for all z such that|x | ≤ |yz|

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Computable structure lemma

LemmaThe B-class S possesses a computable ML-random structure.

Proof (idea). Consider the tree T (S). Let η be the ML-randompath obtained as the leftmost path that avoids the universalML-test. Because of the lemmas above, the structure Aη canbe constructed computably.

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Nontrivial varieties

DefinitionA class of universal algebras is a variety if its closed undersub-algebras, homomorphisms, and ultra-products.

A class of algebras is variety if and only if is axiomatised by aset E of universally quantified equitations.

An equation is p(x) = q(x) where p and q are terms.The equation p(x) = q(x) non-trivial if at least one of the termscontains a variable and p 6= q syntactically.

If E contains at least one non-trivial equation then we call thevariety of algebras satisfying E a non-trivial variety.

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The measure of nontrivial varieties

TheoremThe class of all infinite n-generated algebras that belong to anon-trivial variety has an effective measure zero. Hence, nofinitely presented algebra of a non-trivial variety is ML-random.

CorollaryNo finitely generated ML-random algebra exists that satisfies anontrivial set of equations. Hence, no ML-random group,monoid, or lattice exist.

CorollaryA finitely axiomatised variety V has either an effective measure0 or its measure is a rational number > 0. The latter caseoccurs iff the variety is axiomatised by a trivial set of equations.Moreover, µ(V ) > 0 iff V is a finite union of cones.

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Open questions

1 Are there ML-random c.e. or co-c.e. universal algebras?

2 Build ML-random finitely generated groups and rings.

3 Is ML-randomness quasi-isometry invariant in the class ofgraphs?

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Bakh Khoussainov Computer Science Department, The ...bjoern/pdf/RandomStructures.pdfBakh Khoussainov Algorithmically random inﬁnite structures. Topology on K! Deﬁnition (Topology)