A Finitary Analogue of the Downward L owenheim-Skolem Property - - PowerPoint PPT Presentation

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A Finitary Analogue of the Downward L owenheim-Skolem Property - - PowerPoint PPT Presentation

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A Finitary Analogue of the Downward L owenheim-Skolem Property Abhisekh Sankaran IMSc, Chennai Formal Methods Update Meet IIT Mandi July 18, 2017 Introduction The Downward L owenheim-Skolem theorem (DLS) is amongst the earliest

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SLIDE 1

A Finitary Analogue of the Downward L¨

  • wenheim-Skolem Property

Abhisekh Sankaran IMSc, Chennai Formal Methods Update Meet IIT Mandi July 18, 2017

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SLIDE 2

Introduction

The Downward L¨

  • wenheim-Skolem theorem (DLS) is amongst

the earliest results in classical model theory. The first version of DLS is by L¨

  • wenheim in his paper ¨

Uber M¨

  • glichkeiten im Relativkalk¨

ul (1915) and reads as follows: If a first order sentence over a countable vocabulary has an infinite model, then it has a countable model. Historically,

1915: First version of DLS by L¨

  • wenheim. His proof used

Konig’s lemma (1927) without proving it. 1920s: First fully self-contained proof of L¨

  • wenheim’s

statement and various generalizations of DLS by Skolem 1936: The most general version of DLS by Mal’tsev

DLS + compactness = first order logic (Lindstr¨

  • m 1969).
  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 2/34

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SLIDE 3

Outline of the talk

  • A. Notions:

The Downward L¨

  • wenheim-Skolem Property: DLSP

The Equivalent Bounded Substructure Property: EBSP

  • B. Results:

Classes of finite structures satisfying EBSP Closure properties of EBSP Techniques and f.p.t. algorithms Connection with Nature

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 3/34

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SLIDE 4

Some assumptions and notation for the talk

Assumptions: First order (FO) logic Finite relational vocabularies (i.e. only predicates) Notation: A ⊆ B means A is a substructure of B. UA denotes universe of A. |A| denotes size of UA.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 4/34

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  • A. Notions

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 6

The Downward L¨

  • wenheim-Skolem Property

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 7

FO-similarity of structures

1 2

5 3

1

2

  • 1
  • 2
  • 1

3

  • 3

2

1 2 5 3

1 2

  • 1
  • 2
  • 1

3

  • 3

2 √ 2

  • √

3 3 e

  • 3
  • 3π

2

Q R Q and R are FO-similar

We say structures A and B are FO-similar, denoted A ≡ B, if A and B agree on all properties that can be expressed in FO.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 5/34

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SLIDE 8

The Downward L¨

  • wenheim-Skolem Property

Definition

We say DLSP holds if

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

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SLIDE 9

The Downward L¨

  • wenheim-Skolem Property

Definition

We say DLSP holds if

∀ A A

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

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SLIDE 10

The Downward L¨

  • wenheim-Skolem Property

Definition

We say DLSP holds if

∀ A A ∃ B ⊆ A B

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

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SLIDE 11

The Downward L¨

  • wenheim-Skolem Property

Definition

We say DLSP holds if

∀ A A ∃ B ⊆ A (i) the size of B is ≤ ω B

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

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The Downward L¨

  • wenheim-Skolem Property

Definition

We say DLSP holds if

∀ A A ∃ B ⊆ A (i) the size of B is ≤ ω (ii) B is FO-similar to A B ≡ ≡

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

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SLIDE 13

The Downward L¨

  • wenheim-Skolem Property

Definition

We say DLSP holds if

∀ A A ∃ B ⊆ A (i) the size of B is ≤ ω (ii) B is FO-similar to A B ≡ ≡

“A has an FO-similar substructure of size ≤ ω”

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

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The Downward L¨

  • wenheim-Skolem theorem

Theorem (L¨

  • wenheim 1915, Skolem 1920s)

DLSP is true over all structures.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 7/34

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Downward L¨

  • wenheim-Skolem theorem in the finite

Does not make sense when taken as is. No recursive version of L¨

  • wenheim’s statement – there is no

recursive function bounding the size of a small model of an FO sentence. Grohe showed a stronger negative result: For every recursive function f : N → N, there is an FO sentence ϕ and n ≥ f (|ϕ|), such that ϕ has a model of each size ≥ n but no model of size < n. Quoting Grohe, the above counterexample“refutes almost all possible extensions of the classical L¨

  • wenheim-Skolem

theorem to finite structures” .

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 8/34

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Classical theorems over classes of finite structures

Most theorems from classical model theory fail over all finite structures (DLS, preservation theorems, interpolation theorems, etc.) Active research in last 15 years to“recover”classical theorems

  • ver classes interesting from structural and algorithmic

perspectives. Acyclic, bounded degree, wide, bounded tree-width –

  • Lo´

s-Tarski pres. theorem In addition to the above, quasi-wide classes, classes excluding atleast one minor – homomorphism pres. theorem No such studies in the literature for the DLS theorem.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 9/34

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A new logic based combinatorial property

  • f finite structures

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 18

FO over finite structures

1 Any finite structure can be captured upto isomorphism by FO.

A

a b c

ϕ := ∃x ∃y ∃z ∀w (E(x, x) ∧ E(x, y) ∧ E(x, z) ∧ E(y, z)) ¬(E(y, y) ∨ E(z, z) ∨ E(y, x) ∨ E(z, y) ∨ E(z, x))

  • (x = y ∧ y = z ∧ z = x)

(w = x ∨ w = y ∨ w = z)

2 Then FO-similarity = isomorphism in the finite. 3 Define a weaker version of FO-similarity by considering FO

sentences of a fixed quantifier nesting depth.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 10/34

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m-similarity of structures

A B a b c

A and B are 1-similar, but not 2-similar. ≡1 ≡1

We say structures A and B are m-similar, denoted A ≡m B, if A and B agree on all properties that can be expressed using FO sentences having quantifier nesting depth m.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 11/34

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SLIDE 20

The Equivalent Bounded Substructure Property

Definition

We say EBSP holds if

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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The Equivalent Bounded Substructure Property

Definition

We say EBSP holds if

∀ A ∀m ∈ N A

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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SLIDE 22

The Equivalent Bounded Substructure Property

Definition

We say EBSP holds if

∀ A ∀m ∈ N A ∃ B ⊆ A B

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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SLIDE 23

The Equivalent Bounded Substructure Property

Definition

We say EBSP holds if

∀ A ∀m ∈ N A ∃ B ⊆ A B (i) |B| is bounded in m

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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SLIDE 24

The Equivalent Bounded Substructure Property

Definition

We say EBSP holds if

∀ A ∀m ∈ N A ∃ B ⊆ A (ii) B is m-similar to A B ≡m ≡m (i) |B| is bounded in m

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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SLIDE 25

The Equivalent Bounded Substructure Property

Definition

We say EBSP holds if

∀ A ∀m ∈ N A ∃ B ⊆ A (ii) B is m-similar to A B ≡m ≡m (i) |B| is bounded in m

“A has a small m-similar substructure”

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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SLIDE 26

The Equivalent Bounded Substructure Property

Definition

We say EBSP holds if there exists a witness function θ : N → N such that

∀ A ∀m ∈ N A ∃ B ⊆ A (i) |B| ≤ θ(m) (ii) B is m-similar to A B ≡m ≡m

“A has a small m-similar substructure”

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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SLIDE 27

The Equivalent Bounded Substructure Property

Definition

We say EBSP holds if there exists a witness function θ : N → N such that

∀ A ∀m ∈ N A ∃ B ⊆ A (ii) B is m-similar to A B ≡m ≡m (i) |B| ≤ θ(m)

“A has a small m-similar substructure”

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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The Equivalent Bounded Substructure Property

Definition

Given a class S of finite structures, we say EBSP(S) holds if there is a witness function θ : N → N such that

∃ B ⊆ A (ii) B is m-similar to A B ≡m ≡m (i) |B| ≤ θ(m) ∀ A ∈ S ∀m ∈ N ∃ B ⊆ A, B ∈ S

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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The Equivalent Bounded Substructure Property

Definition

Given a class S of finite structures, we say EBSP(S) holds if there is a witness function θ : N → N such that

∃ B ⊆ A (ii) B is m-similar to A B ≡m ≡m (i) |B| ≤ θ(m) ∀ A ∈ S ∀m ∈ N ∃ B ⊆ A, B ∈ S

“ A has a small m-similar substructure”– over S

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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EBSP(S) as a finitary analogue of DLSP

A A DLSP EBSP(S) for a fixed m ∀ A (i) |B| ≤ ω (ii) B is FO-similar to A Let p = θ(m) ∃ B ⊆ A B B ∀ A ∈ S (i) |B| ≤ p (ii) B is m-similar to A ∃ B ⊆ A, B ∈ S ≡m ≡m ≡ ≡

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 13/34

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SLIDE 31
  • B. Results

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 32

Classes that satisfy EBSP

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 33

Posets satisfying EBSP

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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Words and trees (unordered, ordered, ranked)

Classically studied structures

a b b a b b a a b b a a b b a b b a Unordered Σ-tree Ordered Σ-tree Ordered Σ-tree ranked by ρ ; ρ = {a → 2 ,b → 1)} b Σ = {a, b} Σ = {a, b} Σ = {a, b} a b a Σ-word Σ = {0, 1}

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 14/34

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Nested words

Introduced by Alur and Madhusudan in 2004 as joint generalization of words and ordered unranked trees.

b b a a a a b a b

1 2 3 4 5 6 7 8 9

a a bb a ab b a

W = (abaababba , ) = {(2, 8), (4, 7)}

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 15/34

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Regular languages of words, trees and nested words

A regular language of words/trees/nested words is a class of words/trees/nested words that can be recognized by a finite word/tree/nested word automaton. Recall: EBSP(S) says for each m, that a large S-structure contains a small m-similar S-substructure. Theorem Let S be a regular language of words, trees (unordered, ordered or ranked) or nested words. Then EBSP(S) holds with a computable witness function (which is non-elementary, in general).

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 16/34

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SLIDE 37

Graphs satisfying EBSP

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 38

m-partite cographs

Hlin˘ en´ y, Ne˘ set˘ ril, et al. introduced in 2012, the class of m-partite cographs. An m-partite cograph G is a graph that has an m-partite cotree representation t:

fx = fz = 0 fy = 1 fv( , ) = 1, else 0 fw( , ) = 1, else 0 c b e f d a

t

d f a c b e x y z v w 2 2 1 2 1 1

G

Label set = {1, 2} 2 2 1 1

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 17/34

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SLIDE 39

Important subclasses of m-partite cographs

Cographs (1-partite cographs): complete graphs, complete k-partite graphs, threshold graphs, Turan graphs, etc. Bounded tree-depth graphs Bounded shrub-depth graphs All of the above classes are of active current interest for their excellent algorithmic and logical properties!

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 18/34

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m-partite cographs and its subclasses satisfy EBSP

Theorem Let S be a hereditary subclass of any of the following graph

  • classes. Then EBSP(S) holds with a computable witness function.

For classes with bounded parameters as below, there exist elementary witness functions.

1 the class of m-partite cographs 2 any graph class of bounded shrub-depth 3 any graph class of bounded tree-depth 4 the class of cographs

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 19/34

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SLIDE 41

EBSP and well-quasi-orders

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 42

A property of binary strings

Consider the binary strings u = 00, v = 010 and w = 111. We say u is a substring of v since u “embeds inside”v. Observe that neither of v or w is a substring of the other. For each n, there exists a set of n strings such that no string in the set is a substring of another.

  • Eg. n = 4 : 0000, 0101, 1010, 1111

However this cannot be done if n becomes infinite! Theorem (Higman 1952) For every infinite set {w1, w2, . . .} of binary strings, there exist i, j such that wi is a substring of wj . In other words, binary strings are well-quasi-ordered (w.q.o.) under the substring relation.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 20/34

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Well-quasi-ordering and EBSP

Definition A class S of structures is said to be w.q.o. under embedding if for every infinite set {A1, A2, . . .} of structures of S, there exist i, j such that Ai is embeddable in Aj . Theorem Let S be w.q.o. under embedding. Then EBSP(S) is true. Applications: The following classes satisfy EBSP(S): k-letter graphs for each k (e.g. threshold graphs, unbounded interval graphs) k-uniform graphs for each k

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 21/34

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Constructing new classes satisfying EBSP

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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Unary operations on structures

a b c d e1 e2 e3 e4 e1 e2 e3 e4 line a b c d a b c d complement a b c d a b c d transpose

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 22/34

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SLIDE 46

Binary operations on structures

a b c d e a b c d e G1 G2 G1 ⊔ G2 disjoint union a b c d e a b c d e G1 G2 G1 ⊲ ⊳ G2 join

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 23/34

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SLIDE 47

Binary operations on structures

parallel connect series connect a b c d f e a b c e a b c f e G1 G2 G1 G2 G1 + G2

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 23/34

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SLIDE 48

Binary operations on structures

(a, 1) (a, 2) (a, 3) (b, 2) (b, 3) (a, 4) (b, 4) (b, 1) (a, 1) (b, 3) (b, 4) (a, 2) (a, 3) (b, 2) (b, 1) (a, 4) G1 G2 G1 ⊗ G2 cartesian product G1 × G2 tensor product 1 2 3 4 a b

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 23/34

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Generating graphs using trees of operations

× K2

  • ⊗

⊗ K2

  • line

K2 ⊲ ⊳ ⊔ K1 K1 K2

  • K2

K2 line ⊲ ⊳ K1 ⊔ K2 K1

K1 = single vertex; K2 = single edge

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 24/34

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Closure of EBSP under unary operations

Theorem Given a class S, let Z be any one of the following classes.

1 Complement(S) 2 Transpose(S) 3 Line(S)

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 25/34

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SLIDE 51

Closure of EBSP under unary operations

Theorem Given a class S, let Z be any one of the following classes.

1 Complement(S) 2 Transpose(S) 3 Line(S)

Then the following are true: EBSP(S) → EBSP(Z)

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 25/34

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SLIDE 52

Closure of EBSP under unary operations

Theorem Given a class S, let Z be any one of the following classes.

1 Complement(S) 2 Transpose(S) 3 Line(S)

Then the following are true: EBSP(S) → EBSP(Z) If EBSP(S) holds with a computable/elementary witness function, then so does EBSP(Z).

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 25/34

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Closure of EBSP under binary operations

Theorem Given classes S1 and S2, let Z be any one of the following classes.

  • 1. Disjoint-union(S1, S2)
  • 3. Series-connect(S1, S2)
  • 5. Cartesian-product(S1, S2)
  • 2. Join(S1, S2)
  • 4. Parallel-connect(S1, S2)
  • 6. Tensor-product(S1, S2)

Then the following are true:

  • EBSP(S1) ∧ EBSP(S2)
  • → EBSP(Z)

If the conjuncts in the antecedent hold with computable/ elementary witness functions, then so does the consequent.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 26/34

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SLIDE 54

An overview of classes satisfying EBSP

EBSP

Posets

  • 1. Words
  • 2. Trees
  • a. Unordered
  • b. Ordered
  • c. Ranked
  • 3. Nested words

Graphs

  • 1. Cographs
  • 2. Bounded tree-depth graphs
  • 3. Bounded shrub-depth graphs
  • 4. m-partite cographs
  • 5. Graphs w.q.o. under embedding

Classes generated using

  • 1. Unary operations
  • a. complement
  • b. transpose
  • c. line
  • 2. Binary operations
  • a. disjoint union
  • b. join
  • c. series-connect
  • d. parallel-connect
  • e. cartesian product
  • f. tensor product
  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 27/34

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SLIDE 55

Techniques used to prove EBSP for a class

Key technical features we make use of:

Most structures A seen so far have tree representations tA. The representations are“good”- the operations used in tA satisfy a Feferman-Vaught kind composition property. The index of the“m-similarity”equivalence relation is finite.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 28/34

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SLIDE 56

Techniques used to prove EBSP for a class

Key technical features we make use of:

Most structures A seen so far have tree representations tA. The representations are“good”- the operations used in tA satisfy a Feferman-Vaught kind composition property. The index of the“m-similarity”equivalence relation is finite.

Perform appropriate prunings and graftings iteratively in tA to create a subtree that is“rainbow”in the following sense:

Height: No two nodes in any path from root to leaf represent structures of the same m-similarity class. Degree: For a node a with children bi where 1 ≤ i ≤ n, let ti be the subtree rooted at a, from which the subtrees rooted at bi+1, . . . , bn, are deleted. Then no two ti’s represent structures of the same m-similarity class.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 28/34

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SLIDE 57

Techniques used to prove EBSP for a class

Key technical features we make use of:

Most structures A seen so far have tree representations tA. The representations are“good”- the operations used in tA satisfy a Feferman-Vaught kind composition property. The index of the“m-similarity”equivalence relation is finite.

Perform appropriate prunings and graftings iteratively in tA to create a subtree that is“rainbow”in the following sense:

Height: No two nodes in any path from root to leaf represent structures of the same m-similarity class. Degree: For a node a with children bi where 1 ≤ i ≤ n, let ti be the subtree rooted at a, from which the subtrees rooted at bi+1, . . . , bn, are deleted. Then no two ti’s represent structures of the same m-similarity class.

The rainbow subtree represents a small m-similar substructure.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 28/34

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SLIDE 58

Algorithmic meta-theorems for EBSP classes

The aforementioned prunings and graftings can be implemented in time linear in the size of the tree. The rainbow subtree thus obtained in linear time, represents a small uniform kernel for all FO [m] properties of the original structure. FO properties of the kernel can be checked in constant time. Theorem Let S be a class of structures admitting good tree representations. Then there exists a linear time f.p.t. algorithm for FO model checking over S, provided input structures are given in the form of their tree representations.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 29/34

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Connection with Nature

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 60

Fractals

Mathematical objects that exhibit self-similarity at all scales. Appear widely in Nature. Fern leaf

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 30/34

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SLIDE 61

Fractals

Mathematical objects that exhibit self-similarity at all scales. Appear widely in Nature. Conch shell

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 30/34

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SLIDE 62

Fractals

Mathematical objects that exhibit self-similarity at all scales. Appear widely in Nature. Romanesco cauliflower

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 30/34

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SLIDE 63

A strengthening of EBSP

Definition

Given a class S of finite structures, we say EBSP#(S) holds if

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 31/34

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SLIDE 64

A strengthening of EBSP

Definition

Given a class S of finite structures, we say EBSP#(S) holds if

1 2 3 4

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 31/34

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SLIDE 65

A strengthening of EBSP

Definition

Given a class S of finite structures, we say EBSP#(S) holds if

∀ A ∈ S ∀m ∈ N A

1 2 3 4

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 31/34

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SLIDE 66

A strengthening of EBSP

Definition

Given a class S of finite structures, we say EBSP#(S) holds if

∀ A ∈ S ∀m ∈ N A If |A| ∈ i, then ∀j < i

1 2 3 4

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 31/34

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SLIDE 67

A strengthening of EBSP

Definition

Given a class S of finite structures, we say EBSP#(S) holds if

∀ A ∈ S ∀m ∈ N A ∃ Bj ⊆ A, Bj ∈ S B3 If |A| ∈ i, then B2 B1 ∀j < i

1 2 3 4

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 31/34

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SLIDE 68

A strengthening of EBSP

Definition

Given a class S of finite structures, we say EBSP#(S) holds if

∀ A ∈ S ∀m ∈ N A ∃ Bj ⊆ A, Bj ∈ S (i) |Bj| ∈ j B3 If |A| ∈ i, then B2 B1 ∀j < i

1 2 3 4

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 31/34

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SLIDE 69

A strengthening of EBSP

Definition

Given a class S of finite structures, we say EBSP#(S) holds if

∀ A ∈ S ∀m ∈ N A ∃ Bj ⊆ A, Bj ∈ S (i) |Bj| ∈ j (ii) Bj is m-similar to A B3 ≡m ≡m If |A| ∈ i, then B2 B1

≡m ≡m

∀j < i

1 2 3 4

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 31/34

slide-70
SLIDE 70

EBSP# – a fractal-like property

∀ A ∈ S ∀m ∈ N A ∃ Bj ⊆ A, Bj ∈ S (i) |Bj| ∈ j (ii) Bj is m-similar to A B3 ≡m ≡m If |A| ∈ i, then B2 B1

≡m ≡m

∀j < i

1 2 3 4

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 32/34

slide-71
SLIDE 71

EBSP# – a fractal-like property

∀ A ∈ S ∀m ∈ N A ∃ Bj ⊆ A, Bj ∈ S (i) |Bj| ∈ j (ii) Bj is m-similar to A B3 ≡m ≡m If |A| ∈ i, then B2 B1

≡m ≡m

∀j < i

1 2 3 4

EBSP# indeed asserts logical self-similarity at all scales.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 32/34

slide-72
SLIDE 72

EBSP# – a fractal-like property

∀ A ∈ S ∀m ∈ N A ∃ Bj ⊆ A, Bj ∈ S (i) |Bj| ∈ j (ii) Bj is m-similar to A B3 ≡m ≡m If |A| ∈ i, then B2 B1

≡m ≡m

∀j < i

1 2 3 4

EBSP# indeed asserts logical self-similarity at all scales. All the classes seen so far can be shown to satisfy EBSP#.

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 32/34

slide-73
SLIDE 73

EBSP# – a fractal-like property

∀ A ∈ S ∀m ∈ N A ∃ Bj ⊆ A, Bj ∈ S (i) |Bj| ∈ j (ii) Bj is m-similar to A B3 ≡m ≡m If |A| ∈ i, then B2 B1

≡m ≡m

∀j < i

1 2 3 4

EBSP# indeed asserts logical self-similarity at all scales. All the classes seen so far can be shown to satisfy EBSP#. Whereby all these classes can be regarded as

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 32/34

slide-74
SLIDE 74

EBSP# – a fractal-like property

∀ A ∈ S ∀m ∈ N A ∃ Bj ⊆ A, Bj ∈ S (i) |Bj| ∈ j (ii) Bj is m-similar to A B3 ≡m ≡m If |A| ∈ i, then B2 B1

≡m ≡m

∀j < i

1 2 3 4

EBSP# indeed asserts logical self-similarity at all scales. All the classes seen so far can be shown to satisfy EBSP#. Whereby all these classes can be regarded as logical fractals!

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 32/34

slide-75
SLIDE 75

Conclusion

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 76

Summary of the talk

  • A. Notions:

The Downward L¨

  • wenheim-Skolem Property: DLSP

The Equivalent Bounded Substructure Property: EBSP

  • B. Results:

Classes of finite structures satisfying EBSP Closure properties of EBSP Techniques and f.p.t. algorithms Connection with Nature

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 33/34

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SLIDE 77

A challenging open problem

Is there a structural characterization of EBSP/logical fractals?

  • A. Sankaran

Formal Methods Update Meet, IIT Mandi, July 18, 2017 34/34

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SLIDE 78

Dhanyav¯ ad!

Formal Methods Update Meet, IIT Mandi, July 18, 2017

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SLIDE 79

References

  • A. Sankaran. A finitary analogue of the downward

L¨

  • wenheim-Skolem property. Proceedings of CSL 2017,

Stockholm, Sweden, August 20-24, 2017, to appear.

  • A. Sankaran, B. Adsul, and S. Chakraborty. A generalization of

the Lo´ s-Tarski preservation theorem. Annals of Pure and Applied Logic, 167(3):189 - 210, 2016.

  • A. Sankaran, B. Adsul, and S. Chakraborty. A generalization of

the Lo´ s-Tarski preservation theorem over classes of finite

  • structures. In Proceedings of MFCS 2014, Budapest, Hungary,

August 25-29, 2014, Part I, pages 474 - 485, 2014.

Formal Methods Update Meet, IIT Mandi, July 18, 2017