Do you know what is a preservation theorem? Stage de M2, MPRI - PowerPoint PPT Presentation

0 Do you know what is a preservation theorem?

Stage de M2, MPRI Aliaume Lopez 11 Juin 2019 Sylvain Schmitz Jean Goubault-Larrecq 1 Preservation Theorems

Motivations

Equivalence : Database Finite Model Evaluation of a query on an incomplete database corresponds to evaluation on a family of structures. 1. Existence of a universal model to answer certain answers is equivalent to a preservation theorem (Used in the Chase algorithm (Deutsch et al., 2008)). 2. Naïve evaluation of a query Q yields certain answers if and only if Q is monotone (Gheerbrant et al., 2014). 2 Database theory 101 Preservation theorem A monotone formula φ ∈ FO [ σ ] is equivalent to a simple formula ψ .

1. Existence of a universal model to answer certain answers is evaluation on a family of structures. equivalent to a preservation theorem (Used in the Chase algorithm (Deutsch et al., 2008)). 2. Naïve evaluation of a query Q yields certain answers if and only if Q is monotone (Gheerbrant et al., 2014). 2 Database theory 101 Preservation theorem A monotone formula φ ∈ FO [ σ ] is equivalent to a simple formula ψ . Equivalence : Database ↔ Finite Model Evaluation of a query on an incomplete database corresponds to

Finite models and logics Motivations

3 1 y y 2 3 A good example is far better than a good precept. Finite structures over σ ≜ {• , , } D ≜ { 1 , 2 , 3 } � • � ≜ { 2 } � ≜ { ( 1 , 2 ) , ( 3 , 3 ) } � � ≜ { ( 1 , 3 ) , ( 3 , 1 ) } � Logical formulas FO [ • , ] , φ := ∃ x .φ | φ ∧ φ | ¬ φ ∃ x . ∀ y . ¬ (( • y ) ∧ ¬ ( x y )) | • x | x | x

4 x 5 Figure 1 – Locality of FO x 9 x 8 x 7 x 6 x 4 x d x 3 x 2 x 1 x 0 The Bible tells us to love our neighbors, and also to love our enemies; probably because generally they are the same people. ∃ x . ∀ y . ( x y ) = ⇒ ( • y ) B ( x 0 , 1 )

4 x 5 Figure 1 – Locality of FO x 9 x 8 x 7 x 6 x 4 x d x 3 x 2 x 1 x 0 The Bible tells us to love our neighbors, and also to love our enemies; probably because generally they are the same people. ∃ x . ∀ y . ( x y ) = ⇒ ( • y ) B ( x 0 , 1 )

Induced substructure Strong Injective Homomorphism Substructure Injective homomorphism Homomorphism Homomorphism 5 Chaos is merely order waiting to be deciphered. Preorders over finite structures ⊆ i ⊆ →

Figure 2 – An investment in knowledge pays the best interest. 6 Orders on finite structures ̸⊆ i , ⊆ , → ̸⊆ i , ̸⊆ , → ⊆ i , ⊆ , →

7 Stone Duality …Lift, Lift, Lift! φ ≜ ∃ x . deg( x ) ≥ 3 Upwards closed ⊆ i ⊆ i ⊆ i Figure 3 – Finite graphs encoded using Σ ≜ { E }

7 Stone Duality …Lift, Lift, Lift! φ ≜ ∃ x . deg( x ) ≥ 3 Upwards closed ⊆ i ⊆ i ⊆ i Figure 3 – Finite graphs encoded using Σ ≜ { E }

7 Stone Duality …Lift, Lift, Lift! φ ≜ ∃ x . deg( x ) ≥ 3 Upwards closed ⊆ i ⊆ i ⊆ i Figure 3 – Finite graphs encoded using Σ ≜ { E }

7 Stone Duality …Lift, Lift, Lift! φ ≜ ∃ x . deg( x ) ≥ 3 Upwards closed ⊆ i ⊆ i ⊆ i Figure 3 – Finite graphs encoded using Σ ≜ { E }

7 Stone Duality …Lift, Lift, Lift! φ ≜ ∃ x . deg( x ) ≥ 3 Upwards closed ⊆ i ⊆ i ⊆ i Figure 3 – Finite graphs encoded using Σ ≜ { E }

Preservation theorems Motivations

8 EFO EPFO Ordre Fragment Link between syntax and semantics Known results Str( σ ) FinStr( σ ) Łós-Tarski ✓ ⊆ i ✗ Tait (1959) ⊆ Tarski-Lyndon ✓ EPFO ̸ = ✗ Ajtai and Gurevich (1994) → H.P.T. ✓ ✓ Rossman (2008)

8 EFO EPFO Ordre Fragment Link between syntax and semantics Known results Str( σ ) FinStr( σ ) Łós-Tarski ✓ ⊆ i ✗ Tait (1959) ⊆ Tarski-Lyndon ✓ EPFO ̸ = ✗ Ajtai and Gurevich (1994) → H.P.T. ✓ ✓ Rossman (2008)

8 EFO EPFO Ordre Fragment Link between syntax and semantics Known results Str( σ ) FinStr( σ ) Łós-Tarski ✓ ⊆ i ✗ Tait (1959) ⊆ Tarski-Lyndon ✓ EPFO ̸ = ✗ Ajtai and Gurevich (1994) → H.P.T. ✓ ✓ Rossman (2008)

Preservation theorems do not relativise to subclasses. 9 Relativisation Lemma

Classical results Motivations

Proof Compactness M . Par l’absurde, cette théo- traire une théorie finie incohérente. Or, T est incohérente. Donc en utilisant le théo- semble fini incohérent. Comme T est cohérente, on a donc une 10 , ce qui est absurde. Ainsi, M possède un modèle N , par construction M i . N , N donc M T , et Par la suite, rème de compacité, on déduit que celle-ci possède un sous en- formule dans T qui est équivalente à . donc M est incohérente. . Ainsi, . Considérons T et universelle . Par construction, T . Par construction, cela veut dire que Soit M un modèle de T , montrons que M est un modèle de Pour cela, considérons rie est incohérente, le théorème de compacité permet d’en ex- M est stable par conjonction finie et est cohérente. Ainsi, il existe une formule M telle que Proof of a classical result Łós-Tarski's theorem Let φ be a closed formula, preserved under induced substructure. There exists a closed existential formula ψ such that φ ⇐ ⇒ ψ .

10 rie est incohérente, le théorème de compacité permet d’en ex- rème de compacité, on déduit que celle-ci possède un sous en- conjonction finie et est cohérente. Ainsi, il existe une formule Proof of a classical result Łós-Tarski's theorem Let φ be a closed formula, preserved under induced substructure. There exists a closed existential formula ψ such that φ ⇐ ⇒ ψ . Proof Compactness Considérons T ∀ ≜ { θ | φ ⊢ θ et θ universelle } . Par construction, φ ⊢ T ∀ . Soit M un modèle de T ∀ , montrons que M est un modèle de φ . Pour cela, considérons { φ } ∪ Diag( M ) . Par l’absurde, cette théo- traire une théorie finie incohérente. Or, Diag( M ) est stable par θ ∈ Diag( M ) telle que { θ, φ } est incohérente. Par construction, cela veut dire que φ ⊢ ¬ θ . Ainsi, ¬ θ ∈ T ∀ , et donc M | = ¬ θ , ce qui est absurde. Ainsi, { φ }∪ Diag( M ) possède un modèle N , par construction M ⊆ i N , N | = φ donc M | = φ . Par la suite, {¬ φ }∪ T ∀ est incohérente. Donc en utilisant le théo- semble fini incohérent. Comme T ∀ est cohérente, on a donc une formule dans T ∀ qui est équivalente à φ .

11 Involved counter-example The sad truth The family S of simple planar graphs using only two labels does not satisfy a preservation theorem for ⊆ i . Adaptation Can be (using some tricks) adapted to ⊆ .

12 Figure 4 – The graph G 5 Hide this family that I shall not see

Two sides of a same coin. Motivations

13 First example : Finite paths Lemma The family P = { P k | k ∈ N ≥ 1 } of finite paths satisfies a preservation theorem for ⊆ i .

14 . . . First example : Finite paths Monotone ⊆ i ✓ ⊆ i ✗ ⊆ i ✗ ⊆ i ✗ ⊆ i ✗ Figure 5 – Evaluation of a monotone formula φ over P

14 . . . First example : Finite paths Monotone ⊆ i ✓ ⊆ i ✗ ⊆ i ✗ ⊆ i ✗ ⊆ i ✗ Figure 5 – Evaluation of a monotone formula φ over P

Notes (i) The order i is total and well founded over (1) (ii) No property of FO were ever used! 15 First example : Finite paths Lemma A formula φ preserved under ⊆ i on P is equivalent to ∃ x 1 , . . . , ∃ x k . x 1 ̸ = x 2 ̸ = · · · ̸ = x k

(1) (ii) No property of FO were ever used! 15 First example : Finite paths Lemma A formula φ preserved under ⊆ i on P is equivalent to ∃ x 1 , . . . , ∃ x k . x 1 ̸ = x 2 ̸ = · · · ̸ = x k Notes (i) The order ⊆ i is total and well founded over P

Application U Figure 6 – Every non empty upwards closed set U has a non empty finite basis of (finite) minimal elements. preservation (2) 16 First example : Generalisation Well Quasi Order / wqo (e.g. Kruskal, 1972) B

Application U Figure 6 – Every non empty upwards closed set U has a non empty finite basis of (finite) minimal elements. preservation (2) 16 First example : Generalisation Well Quasi Order / wqo (e.g. Kruskal, 1972) B

U Figure 6 – Every non empty upwards closed set U has a non empty finite basis of (finite) minimal elements. preservation (2) 16 First example : Generalisation Well Quasi Order / wqo (e.g. Kruskal, 1972) B Application wqo = ⇒

17 Second example : Finite cycles Lemma The family C = { C k | k ∈ N ≥ 3 } of finite cycles satisfies a preservation theorem for ⊆ i .

18 Second example : Finite cycles Locality ̸⊆ i ̸⊆ i ̸⊆ i ̸⊆ i ̸⊆ i · · · i ̸⊇ i ̸⊇ i ̸⊇ i ̸⊇ i ̸⊇ ✗ ✗ ✓ ✓ ✓ ⊆ i Figure 7 – Evaluation of a monotone formula φ over C

18 Second example : Finite cycles Locality ̸⊆ i ̸⊆ i ̸⊆ i ̸⊆ i ̸⊆ i · · · i ̸⊇ i ̸⊇ i ̸⊇ i ̸⊇ i ̸⊇ ✗ ✗ ✓ ✓ ✓ ⊆ i Figure 7 – Evaluation of a monotone formula φ over C

18 Second example : Finite cycles Locality ̸⊆ i ̸⊆ i ̸⊆ i ̸⊆ i ̸⊆ i · · · i ̸⊇ i ̸⊇ i ̸⊇ i ̸⊇ i ̸⊇ ✗ ✗ ✓ ✓ ✓ ⊆ i Figure 7 – Evaluation of a monotone formula φ over C

18 Second example : Finite cycles Locality ̸⊆ i ̸⊆ i ̸⊆ i ̸⊆ i ̸⊆ i · · · i ̸⊇ i ̸⊇ i ̸⊇ i ̸⊇ i ̸⊇ ✗ ✗ ✓ ✓ ✓ ⊆ i Figure 7 – Evaluation of a monotone formula φ over C