Homomorphism Preservation Theorem. Albert Atserias Universitat Politècnica de Catalunya Barcelona, Spain

Transcription

1 Homomorphism Preservation Theorem Albert Atserias Universitat Politècnica de Catalunya Barcelona, Spain

2 Structure of the talk 1. Classical preservation theorems 2. Preservation theorems in finite model theory and CSPs 3. Rossman s Theorem 4. More on preservation theorems in finite model theory

3 Classical Preservation Theorems

4 Existential Formulas (1) The smallest class of formulas that contains all atomic formulas Ê Ü ½ Ü Ö µ and Ü ½ Ü ¾, contains all negated atomic formulas Ê Ü ½ Ü Ö µ and Ü ½ Ü ¾, is closed under conjunction ³ ½ ³ ¾, is closed under disjunction ³ ½ ³ ¾, is closed under existential quantification ܵ ³ ܵµ, is closed under universal quantification ܵ ³ ܵµ.

5 Existential Formulas (2) In the vocabulary of graphs : Example 1: There is an induced pentagon. Ü ½ µ Ü µ Ü ½ Ü ¾ µ Ü ¾ Ü µ Ü Ü ¾ µ Ü Ü µµ Example 2: Either there is an induced pentagon or an induced co-pentagon.

6 Existential Formulas (3) Easy observation: if ³ and is an induced substructure of, then ³. Proof by induction on the structure of ³. Question: Are existential formulas the only first-order formulas that are preserved under extensions?

7 Existential Formulas (4) Łoś-Tarski Theorem Let ³ be a first-order formula. The following are equivalent: ³ is preserved under extensions, ³ is equivalent to an existential formula. Proof uses the Compactness Theorem for first-order logic.

8 Existential Positive Formulas (1) The smallest class of formulas that contains all atomic formulas Ê Ü ½ Ü Ö µ and Ü ½ Ü ¾, contains all negated atomic formulas Ê Ü ½ Ü Ö µ and Ü ½ Ü ¾, is closed under conjunction ³ ½ ³ ¾, is closed under disjunction ³ ½ ³ ¾, is closed under existential quantification ܵ ³ ܵµ, is closed under universal quantification ܵ ³ ܵµ.

9 Existential Positive Formulas (2) In the vocabulary of graphs : Example 1: There is a homomorphic copy of the pentagon. Ü ½ µ Ü µ Ü ½ Ü ¾ µ Ü ¾ Ü µ Ü Ü ½ µµ Example 2: Either there is a homomorphic copy of a triangle or a homomorphic copy of a pentagon.

10 Existential Positive Formulas (3) Easy observation: if ³ and there is a homomorphism, then ³. Proof by induction on the structure of ³. Question: Are existential positive formulas the only first-order formulas that are preserved under homomorphisms?

11 Existential Positive Formulas (4) Lyndon-Łoś-Tarski Theorem Let ³ be a first-order formula. The following are equivalent: ³ is preserved under homomorphisms, ³ is equivalent to an existential positive formula. Proof uses the Compactness Theorem for first-order logic. (Rossman gave a new proof! Coming soon)

12 Primitive Positive Formulas (1) The smallest class of formulas that contains all atomic formulas Ê Ü ½ Ü Ö µ and Ü ½ Ü ¾, contains all negated atomic formulas Ê Ü ½ Ü Ö µ and Ü ½ Ü ¾, is closed under conjunction ³ ½ ³ ¾, is closed under disjunction ³ ½ ³ ¾, is closed under existential quantification ܵ ³ ܵµ, is closed under universal quantification ܵ ³ ܵµ.

13 Primitive Positive Formulas (2) Easy observation: if ³ and there is a homomorphism, then ³, if ³ and ³, then ³. Proof by induction on the structure of ³. Note: Primitive positive formulas (in prenex normal form) are precisely the conjunctive queries.

14 Primitive Positive Formulas (3) Corollary to Lyndon-Łoś-Tarski Let ³ be a first-order formula. The following are equivalent: ³ is preserved under products and homomorphisms, ³ is equivalent to a primitive positive formula. Proof : As ³ is preserved under homomorphisms, we have Then Ñ ½ are models of ³, so Ñ ½ ³. ½ Ñ for some, and for every, so Hence, ³ ½ Ñ Hence ³. ¾ ³ µ µ ³

15 Primitive Positive Formulas (3) Corollary to Lyndon-Łoś-Tarski Let ³ be a first-order formula. The following are equivalent: ³ is preserved under products and homomorphisms, ³ is equivalent to a primitive positive formula. Proof : As ³ is preserved under homomorphisms, we have Then Ñ ½ are models of ³, so Ñ ½ ³. ½ Ñ for some, and for every, so Hence, ³ ½ Ñ Hence ³. ¾ ³ µ µ ³

16 Other Classical Preservation Theorems Many other similar results are known: Preserved under surjective hom. : positive formulas (Lyndon). Preserved under unions of chains : ¾ formulas. Preserved under reduced products : Horn formulas....

17 Preservation Theorems in Finite Model Theory and CSP

18 Compactness Theorem Fails on Finite Models Compactness Theorem Let be a collection of first-order formulas. If every finite subset of has a model, then has a model. Failure on the finite: ³ Ò Ü ½ µ Ü Ò µ Î Ü Ü µ. ³ Ò Ò ½. Every finite subset has a finite model, but all models of are infinite.

19 But do Preservation Theorems Fail? Most fail, indeed: Tait (1959!): Preservation under extensions fails. Ajtai-Gurevich (1987): Preservation under surjective hom. fails.... Preservation under homomorphisms: does it hold? Remained open for many years... solved in 2005 (holds) by Rossman.

20 While Waiting for Ben... Let us restrict attention to classes of the form ËÈ Àµ À Note: ËÈ Àµ is automatically closed under homomorphisms: If À and, then À. Question: Suppose ËÈ Àµ is first-order definable. Is it definable by an existential-positive formula?

21 Preservation Theorem for CSPs Theorem (A. 2005) Yes. Much weaker result than Rossman s. But proof much simpler as well. Proof uses a key result in graph-theoretic approach to CSP. Proof builds on well-known previous work....

22 Overview of Proof Fix À and suppose ³ is a first-order formula that defines ËÈ Àµ. Ingredients: Random preimage lemma. Scattered sets on large-girth structures. Density lemma. Putting it all together.

23 Definitions Definitions: Let be a finite structure: is a minimal model of ³ if ³ and no proper substructure is a model of ³. Gaifman graph µ of : vertices are elements of, edges are pairs of elements that appear together in some tuple of. A -scattered set is a set of elements that are pairwise at distance in the Gaifman graph.

24 Ingredient 1 : Random Preimage Lemma Interesting history: [Erdös 1959] There exist graphs of arbitrary large girth and arbitrary large chromatic number. [Nešetřil-Rödl 1979] Sparse Incomparability Lemma: for every non-bipartite graph À there is an incomparable ¼ with ¼ À and large girth. [Feder-Vardi ] Same proof technique as in Sparse Incomparability Lemma but slightly different statement and for arbitrary structures. We call it the Random Preimage Lemma.

25 Random Preimage Lemma Random Preimage Lemma (Feder-Vardi ) For every and and for every finite structure, there exists a finite structure ¼ such that: (preimage) ¼, (equivalence) ¼ À iff À, whenever À, (large girth) the girth of ¼ as a hypergraph is at least.

26 Why is the Random Preimage Lemma true? 1. Replace every point of by a big independent set (a potato). 2. Throw random edges between independent sets that are edges in. 3. Remove one edge from every short cycle to force large girth

27 Why is the Random Preimage Lemma true? 1. Replace every point of by a big independent set (a potato). 2. Throw random edges between independent sets that are edges in. 3. Remove one edge from every short cycle to force large girth

28 Why is the Random Preimage Lemma true? 1. Replace every point of by a big independent set (a potato). 2. Throw random edges between independent sets that are edges in. 3. Remove one edge from every short cycle to force large girth

29 Pigeonhole Argument Every homomorphism to a small À, À, must be constant on a large portion of every independent set. Hence, ¼ À implies À. À implies ¼ À.

30 Ingredient 2 : Scattered Sets Lemma: Let, Ñ and be integers. For every large finite structure of girth at least, there exists such that Ò contains a -scattered of size Ñ. Case of small degree is easy. Case of large degree:

31 Ingredient 3: Density Lemma Density Lemma (Ajtai-Gurevich 1988) Let ³ be preserved under homomorphisms on finite models. There exist and Ñ such that if is a minimal model of ³, then has no -scattered set of size Ñ. Moreover, neither does Ò for every ¾. Proof depends on Gaifman s Locality Theorem.

32 Ingredient 4 : Putting it all together Suppose ËÈ Àµ definable by ³. We prove ³ has finitely many minimal models ½ Ñ. This is enough because then ³ ½ Ñ We bound the size of minimal models. Suppose is a very large minimal model.

33 Using Ingredient 1 large minimal model, in particular À (Ingredient 1: random preimage) there is a large minimal model ¼¼ ¼ of large girth

34 Using Ingredient 2 ¼¼ has large girth (Ingredient 2: scattered sets) there is an ¾ ¼¼ such that ¼¼ Ò has a large scattered set

35 Using Ingredient 3 there is an ¾ ¼¼ such that ¼¼ Ò has a large scattered set (Ingredient 3: density lemma) ¼¼ is not a minimal model. Contradiction. ¾

36 Rossman s Theorem

37 Finite Homomorphism Preservation Theorem Theorem (Rossman 2005) Let ³ be a first-order formula. The following are equivalent: ³ is preserved under homomorphisms on finite structures, ³ is equivalent to an existential positive formula on finite structures. Proof (cook) plan: Ingredient 1: Existential positive types. Ingredient 2: Existential positive saturation. Ingredient 3: Finitization and back and forth argument.

38 we write ÔÔ Ò Ingredient 1 : Existential positive types Definitions: Let be a structure, let be a -tuple, let Ò be an integer. the quantifier rank of a formula is the nesting-depth of quantifers. Ó Ò : collection of all first-order formulas of quantifier rank Ò. ÔÔ Ò : collection of all primitive positive formulas of quantifier rank Ò. Ó Ò µ: collection of all ³ ¾ Ó Ò such that ³ µ. ÔÔ Ò µ: collection of all ³ ¾ ÔÔ Ò such that ³ µ. we write Ó Ò if Ó Ò µ Ó Ò µ. if ÔÔ Ò µ ÔÔ Ò µ.

39 Main Property of Types Fact: Let be a class of structures. The following are equivalent: is preserved under ÔÔ Ò, is definable by a finite disjunction of formulas in ÔÔ Ò. Fact: Let be a class of structures. The following are equivalent: is preserved under Ó Ò, is definable by a finite disjunction of formulas in Ó Ò. Proofs: Follows easily from the number of formulas of quantifier rank Ò, up to logical equivalence, is finite.

40 ÔÔ Ò ¼ Ó Ò Goal ³ Ó Given Ò preserved under homomorphisms, we Ò want to ¾ find so ¼ that for every and with, there exist and so that ¼ ÔÔ Ò

41 ÔÔ Ò Ingredient 2 : Existential Positive Saturation Definition: Let be a structure and let Ò and be integers. We say that is existential-positively Ò-saturated if for every Ò we have ¾ ½ µ ¾ µ Ì ÔÔ Ò µ Ì ÔÔ Ò µµ ¾ µ ÔÔ Ò µ Ì µ Main Property: If and are existential-positively Ò-saturated, then µ Ó Ò Proof : By induction on Ò prove, for every ¾ and ¾ : ÔÔ Ò µ ÔÔ Ò µ µ Ó Ò µ Ó Ò µ

42 But can we saturate structures anyway? Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction:

43 But can we saturate structures anyway? Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction:

44 But can we saturate structures anyway? Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction:

45 But can we saturate structures anyway? Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction:

46 But can we saturate structures anyway? Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction:

47 But can we saturate structures anyway? Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction:

48 But can we saturate structures anyway? Theorem: Yes. But oops, they become infinite! Proof by iterated ear-construction:

49 ÔÔ Ò ¼ Ó Ò Ingredient 3 : Finitization and back and forth Theorem: For every Ò, Ò there exists such that if and are finite ¼ Ò structures with, then there exist finite and such that ¼ ÔÔ Ò Note: and are sufficiently saturated.

50 More preservation theorems in finite model theory

51 Restricted Classes of Finite Structures Let Ì be the class of finite oriented forests. Suppose ³ is a first-order formula that is preserved under homomorphisms on Ì. Is ³ equivalent, on Ì, to an existential-positive formula? Note: Doesn t seem to follow from Rossman s result. But the answer is yes. Why?: every large forest has a large scattered set after removing (at most) one point, so it cannot be a minimal model of ³ by the Density Lemma. ¾

52 Restricted Classes of Finite Structures Let Ì be the class of finite structures whose Gaifman graphs have treewidth at most. Theorem (A., Dawar and Kolaitis 2004) The homomorphism preservation property holds on Ì. More generally. Theorem (A., Dawar and Kolaitis 2004) The homomorphism preservation property holds on any class of finite structures whose Gaifman graphs exclude some minor and is closed under substructures and disjoint unions.

53 Preservation under Extensions? Theorem (A., Dawar and Grohe 2005) The extension preservation property holds on the following classes: graphs of bounded degree, acyclic structures, Ì for every ½, Remarkably, it fails for the class of planar graphs.

54 Counterexample on Planar Graphs there are at least two different white points such that either some point is not connected to both, or every black point has exactly two black neighbors.

55 Conclusions

56 Conclusions Homomorphisms are flexible enough to carry over model-theoretic constructions, and in the context of CSPs, the finite model theory is even easier. Open Question 1: Can we finitize Rossman s saturation using randomness? Open Question 2: Does Ä È ÀÇÅ Ø ÐÓ? Does Ä È co- ËÈ Ø ÐÓ? It is known that: Ø ÐÓ µ ÀÇÅ Ø ÐÓ (Feder-Vardi 2003)