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

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 results in classical model theory. owenheim in his paper ¨ The first version of DLS is by L¨ Uber M¨ oglichkeiten 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¨ owenheim. His proof used Konig’s lemma (1927) without proving it. 1920s: First fully self-contained proof of L¨ owenheim’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¨ om 1969). A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 2/34

Outline of the talk A. Notions: The Downward L¨ owenheim-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

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 . U A denotes universe of A . | A | denotes size of U A . A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 4/34

A. Notions Formal Methods Update Meet, IIT Mandi, July 18, 2017

The Downward L¨ owenheim-Skolem Property Formal Methods Update Meet, IIT Mandi, July 18, 2017

FO-similarity of structures Q - 3 -1 - 1 0 1 5 2 -2 1 3 2 2 3 R √ √ - 1 1 5 -3 - 3 π -2 - 3 -1 - 3 0 1 2 2 e 3 3 3 2 2 2 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

The Downward L¨ owenheim-Skolem Property Definition We say DLSP holds if A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

The Downward L¨ owenheim-Skolem Property Definition We say DLSP holds if A ∀ A A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

The Downward L¨ owenheim-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

The Downward L¨ owenheim-Skolem Property Definition We say DLSP holds if A ∀ A ∃ B ⊆ A B (i) the size of B is ≤ ω A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

The Downward L¨ owenheim-Skolem Property Definition We say DLSP holds if A ∀ A ≡ ∃ B ⊆ A B (i) the size of B is ≤ ω (ii) B is FO-similar to A ≡ A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

The Downward L¨ owenheim-Skolem Property Definition We say DLSP holds if A ∀ A ≡ ∃ B ⊆ A B (i) the size of B is ≤ ω (ii) B is FO-similar to A ≡ “ A has an FO-similar substructure of size ≤ ω ” A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 6/34

The Downward L¨ owenheim-Skolem theorem Theorem (L¨ owenheim 1915, Skolem 1920s) DLSP is true over all structures. A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 7/34

Downward L¨ owenheim-Skolem theorem in the finite Does not make sense when taken as is. No recursive version of L¨ owenheim’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¨ owenheim-Skolem theorem to finite structures” . A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 8/34

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 over 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

A new logic based combinatorial property of finite structures Formal Methods Update Meet, IIT Mandi, July 18, 2017

FO over finite structures 1 Any finite structure can be captured upto isomorphism by FO. ϕ := ∃ x ∃ y ∃ z ∀ w a � ( x � = y ∧ y � = z ∧ z � = x ) � ( w = x ∨ w = y ∨ w = z ) � c b ( 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 )) A 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

m -similarity of structures ≡ 1 a ≡ 1 A and B are 1-similar, but not 2-similar. c b B A 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

The Equivalent Bounded Substructure Property Definition We say EBSP holds if A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

The Equivalent Bounded Substructure Property Definition We say EBSP holds if A ∀ A ∀ m ∈ N A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

The Equivalent Bounded Substructure Property Definition We say EBSP holds if A ∀ A ∀ m ∈ N ∃ B ⊆ A B A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

The Equivalent Bounded Substructure Property Definition We say EBSP holds if A ∀ A ∀ m ∈ N ∃ B ⊆ A B (i) | B | is bounded in m A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

The Equivalent Bounded Substructure Property Definition We say EBSP holds if A ∀ A ∀ m ∈ N ≡ m ∃ B ⊆ A B (i) | B | is bounded in m (ii) B is m -similar to A ≡ m A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

The Equivalent Bounded Substructure Property Definition We say EBSP holds if A ∀ A ∀ m ∈ N ≡ m ∃ B ⊆ A B (i) | B | is bounded in m (ii) B is m -similar to A ≡ m “ A has a small m -similar substructure” A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

The Equivalent Bounded Substructure Property Definition We say EBSP holds if there exists a witness function θ : N → N such that A ∀ A ∀ m ∈ N ≡ m ∃ B ⊆ A B (i) | B | ≤ θ ( m ) (ii) B is m -similar to A ≡ m “ A has a small m -similar substructure” A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

The Equivalent Bounded Substructure Property Definition We say EBSP holds if there exists a witness function θ : N → N such that A ∀ A ∀ m ∈ N ≡ m ∃ B ⊆ A B (i) | B | ≤ θ ( m ) (ii) B is m -similar to A ≡ m “ A has a small m -similar substructure” A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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 ∀ A ∈ S ∀ m ∈ N ≡ m ∃ B ⊆ A ∃ B ⊆ A , B ∈ S B (i) | B | ≤ θ ( m ) (ii) B is m -similar to A ≡ m A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

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 ∀ A ∈ S ∀ m ∈ N ≡ m ∃ B ⊆ A , B ∈ S ∃ B ⊆ A B (i) | B | ≤ θ ( m ) (ii) B is m -similar to A ≡ m “ A has a small m -similar substructure”– over S A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 12/34

EBSP ( S ) as a finitary analogue of DLSP EBSP ( S ) for a fixed m DLSP A A ≡ m ≡ B B ≡ ≡ m Let p = θ ( m ) ∀ A ∀ A ∈ S ∃ B ⊆ A ∃ B ⊆ A , B ∈ S (i) | B | ≤ ω (i) | B | ≤ p (ii) B is FO-similar to A (ii) B is m -similar to A A. Sankaran Formal Methods Update Meet, IIT Mandi, July 18, 2017 13/34

B. Results Formal Methods Update Meet, IIT Mandi, July 18, 2017

Classes that satisfy EBSP Formal Methods Update Meet, IIT Mandi, July 18, 2017

Posets satisfying EBSP Formal Methods Update Meet, IIT Mandi, July 18, 2017