INDEPENDENCE LOGIC AND ABSTRACT INDEPENDENCE RELATIONS 5 1 0 GIANLUCAPAOLINI 2 y a M Abstract. Wecontinuetheworkontherelationsbetweenindependencelogic and the model-theoretic analysis of independence, generalizing the results of 1 [16]totheframeworkofabstractindependencerelationsforanarbitraryAEC. 2 Wegiveamodel-theoreticinterpretationoftheindependenceatomandcharac- terizeunderwhichconditionswecanproveacompletenessresultwithrespect ] tothedeductivesystemthataxiomatizesindependenceinteamsemanticsand O statistics. L . h Contents t a 1. Introduction 1 m 2. Abstract Independence Relations 3 [ 2.1. Abstract Independence Relations in First-Order Theories 3 5 2.2. Abstract Independence Relations in Abstract Elementary Classes 4 v 3. Federation 6 7 3.1. Federated Pregeometries 7 0 3.2. Federated Sequences in Stable Theories 9 9 4. Independence Logic 10 6 . 4.1. Atomic Independence Logic 11 1 4.2. Abstract Independence Relation Atomic Independence Logic 13 0 5. Conclusion 20 4 1 References 20 : v i X r 1. Introduction a Inmathematicsandmodeltheorytheconceptsofdependenceandindependence areofcrucialimportance,itisinfactalwaysinfunctionofanindependencecalculus that a classificationtheory for a class of classes ofstructures is developed. For this reason,the notionsofdependence andindependence areobjectsofintense studyin the model-theoretic community. Three main frameworks in which (in)dependence has been studied are: pregeometries, first-order theories and abstract elementary classes (AECs). Table 1 lists the most important cases of (in)dependence studied in these contexts. The research of the author was supported by the Finnish Academy of Science and Letters (Vilho,YrjöandKalleVäisäläfoundation)andgrantTM-13-8847ofCIMO.Theauthorwouldlike tothankaboveallTapaniHyttinen,butalsoJohnBaldwin,˚AsaHirvonen,andJoukoVäänänen forusefulconversationsrelatedtothispaper. Theauthorwouldalsoliketothanktherefereefor hiscarefulreadingofthepaper,hiscorrectionsandhissuggestions. 1 2 GIANLUCAPAOLINI Recently, Vaä¨nänen [20] developed a logical approach to the notions of dependence and independence, establishing a generaltheory of(in)dependence that goes under the name of dependence logic. Dependence logic provides an abstract char- acterization of (in)dependence, which accounts for the way dependence and independence behave in several disciplinary fields, e.g. database theory and statistics. Pregeometries Forking Indep. Indep. in AECs Vector ω-stable ℵ -stable 0 spaces theories homogeneous Alg. closed Stable Excellent fields theories classes Graphs Simple Finitary theories AECs Table 1. Independence in model theory In[16]thecasesof(in)dependenceoccurringinpregeometriesandω-stabletheo- rieswerealsoshownto be instances ofthis theory. We now generalizethese results to the other cases of independence listed in Table 1. We work in the framework of abstract independence relations for an arbitrary abstract elementary class, which subsumes most of the cases of independence of interest in model theory. Thekeyfeatureofthefamilyoflogicsstudiedindependencelogicisthepresence of logical atoms different from the equational one. Each kind of atom corresponds toadifferentnotionof(in)dependence,andeachlogicinthefamilyischaracterized bythelogicalatomspresentinthesyntax. Thismakesthestudyoftheatomiclevel of the (in)dependence logics of great relevance, as indeed this is the added layer of expressivity that these systems have at disposal. This study often results in the analysisofthe implicationproblemforasetofatomsofaparticularform. Thatis, the searchfor a complete deductive system for these atoms. Emblematic examples are the axiomatizations of functional dependence and stochastic independence due to Armstrong [2] and Geiger, Paz and Pearl[6], respectively. Our specific aim in this paper is the solution of the implication problem for the independence atom under a model-theoretic interpretation. This analysis was initiated in [16], where several (in)dependence atoms were shown to have natural model-theoretic counterparts. In the present study we deal exclusively with the independenceatom~x⊥~y and,onlymarginally,withitsconditionalversion~x⊥ ~y. ~z In Section 2 we set the stage, defining what is an abstract elementary class and what is the axiomatization of independence to which we refer. We also give the principalexamplesofindependence, amongwhichforking independenceinasimple theory,andpregeometric independence inanAECwithauniformpregeometricop- erator. InSection3weintroduceaparticularclassofindependencerelations,which wecallfederated. We showthatthesearea generalizationofthe wayindependence behave in vector spaces, algebraically closed fields, and abelian groups. We then focus on its pregeometric version, and show that any ω-homogenous non-trivial pregeometryis federated (modulo a finite localization). Thus, we use this result to deduce that in any first-order stable theory that admits non-trivial regular types forking independence is federated (over some set of parameters). In Section 4 we usethetheorydevelopedinSection3tocharacterizeunderwhichconditionswecan INDEPENDENCE LOGIC AND ABSTRACT INDEPENDENCE RELATIONS 3 prove a completeness result with respect to the deductive system that axiomatizes independence in team semantics and statistics, giving a complete answer to the motivating question of the paper. 2. Abstract Independence Relations To makeclearthe levelsofgeneralizationatwhichwe work,wefirstdefine what is anabstractindependence relationinthe contextoffirst-ordertheories,andthen generalize this definition to the context of abstract elementary classes. 2.1. Abstract Independence Relations in First-Order Theories Werefertotheframeworkof[3]and[1]. Wefixsomenotation. AB isshorthand forA∪B. Fora complete first-ordertheoryT,wedenote by Mits monstermodel. Definition 2.1. Let T be a complete theory and | a ternary relation between ⌣ (bounded) subsets of the monster model M. We say that | is a pre-independence ⌣ relation if it satisfies the following axioms. (a.) (Invariance) If A | B and f ∈Aut(M), then f(A) | f(B). ⌣C ⌣f(C) (b.) (Existence) A | B, for any A,B ⊆M. ⌣A (c.) (Monotonicity) If A | B and D ⊆A, then D | B. ⌣C ⌣C (d.) (Base Monotonicity) Let D ⊆C ⊆B. If A | B, then A | B. ⌣D ⌣C (e.) (Symmetry) If A | B, then B | A. ⌣C ⌣C (f.) (Transitivity) Let D ⊆C ⊆B. If B | A and C | A, then B | A. ⌣C ⌣D ⌣D (g.) (Normality) If A | B, then AC | B. ⌣C ⌣C (h.) (Finite Character) If A | B for all finite A ⊆A, then A | B. 0 ⌣C 0 ⌣C (i.) (Anti-Reflexivity) If A | A, then A | C for any C ⊆M. ⌣B ⌣B If in addition | satisfies the following two axioms, then we say that | is an ⌣ ⌣ independence relation. (j.) (Extension) If A | B and B ⊆ D, then there is f ∈ Aut(M) fixing BC ⌣C pointwise such that f(A) | D. ⌣C (k.) (Local Character) There is a cardinal κ(T) such that for every~a ∈ M<ω and B ⊆M there is C ⊆B with |C|<κ(T) and~a | B. ⌣C In this context we do not distinguish between finite sets and finite sequences. Thus, if A = {a ,...,a }, B = {b ,...,b } and C = {c ,...,c }, we may 0 n−1 0 m−1 0 k−1 write a ···a | b ···b instead of A | B. By Transitivity we will 0 n−1 ⌣c0···ck−1 0 m−1 ⌣C refer to the following (a-priori) stronger property. Proposition 2.2 (Transitivity). A | B andA | D ifandonlyifA | BD. ⌣C ⌣CB ⌣C Proof. For the direction (⇒), suppose that A | BD. We have that A | B by ⌣C ⌣C Monotonicity. Furthermore,bySymmetryandNormalitywehavethatA | BCD ⌣C and so, by Base Monotonicity, A | BCD. Thus, by Monotonicity, A | D. ⌣CB ⌣CB For the direction (⇐), suppose that A | B and A | D. By Symmetry, ⌣C ⌣CB B | A and D | A, so, by Normality, CB | A and DCB | A. Thus, ⌣C ⌣CB ⌣C ⌣CB by Transitivity (the axiom), DCB | A and so, by Monotonicity and Symmetry, ⌣C A | BD. ⌣C The following principle will be of crucial importance in Section 4. Corollary 2.3 (Exchange). If A | B and AB | C, then A | BC. ⌣D ⌣D ⌣D 4 GIANLUCAPAOLINI Proof. A | B and AB | C ⌣D ⌣D ⇓ A | B and C | AB ⌣D ⌣D ⇓ A | B and C | A [by Transitivity] ⌣D ⌣DB ⇓ A | B and A | C [by Symmetry] ⌣D ⌣DB ⇓ A | BC [by Transitivity]. ⌣D In the following two orthogonal examples (not generalizing each others), that cover a broad class of first-order theories. The second one is by far the most important example of independence that has ever been formulated, the original definition is due to Shelah [18]. Example 2.4 (Independence in o-minimal theories [17]). Let T be an o-minimal theory. For A,B,C ⊆ M, define A |acl C if for every ~a ∈ A we have that ⌣B dim (~a/B∪C)=dim (~a/B). Then |acl is a pre-independence relation. acl acl ⌣ Example 2.5 (Forking in simple theories [14]). Let T be a simple theory. For A,B,C ⊆ M, define A |f C if for every ~a ∈ A we have that tp(~a/B ∪C) is a ⌣B non-forking extension of tp(~a/B). Then |f is an independence relation. ⌣ 2.2. Abstract Independence Relations in Abstract Elementary Classes The axiomatization of independence that we gave in the previous section does not refer to any intrinsically first-order property, it thus makes sense to generalize it to the context of abstract elementary classes [19]. First of all we define what an abstract elementary class is and what are the analog of the first-order notions of amalgamationand joint embedding. Definition 2.6 (Abstract Elementary Class [19]). Let K be a class of structures in the signature L. We say that (K,4) is an abstract elementary class (AEC) if the following conditions are satisfied. (1) K and 4 are closed under isomorphism. (2) If A4B, then A⊆B. (3) The relation 4 is a partial order on K. (4) If (A ) is an increasing 4-chain, then: i i<δ (4.1) A ∈K; i<δ i S (4.2) for each j <δ, A 4 A ; j i<δ i S (4.3) if each A 4M, then A 4M. j i<δ i S (5) If A,B,C ∈K, A4C, B 4C and A⊆B, then A4B. (6) There is a Lo¨wenheim-Skolem number LS(K) such that if A∈K and B ⊆A, then there is A′ ∈K such that B ⊆A′, A′ 4A and |A′|=|B|+|L|+LS(K). Definition 2.7. If M,N ∈ K and f : M → N is an embedding such that f(M)4N, then we say that f is a 4-embedding. Let κ be a cardinal, we let K ={A∈K||A|=κ}. κ Definition 2.8. Let (K,4) be an AEC. INDEPENDENCE LOGIC AND ABSTRACT INDEPENDENCE RELATIONS 5 (i) Wesaythat(K,4)hastheamalgamationproperty(AP)ifforanyA,B ,B ∈ 0 1 K with A4B for i<2, there are C ∈K and 4-embeddings f :B →C for i i i i<2, such that f ↾A=f ↾A. 0 1 (ii) Wesaythat(K,4)hasthejointembeddingproperty(JEP)ifforanyB ,B ∈ 0 1 K there are C ∈K and 4-embeddings f :B →C for i<2. i i (iii) Wesaythat(K,4)hasarbitrarilylargemodels(ALM)ifforeveryκ>LS(K), K 6=∅. κ If (K,4) has AP, JEP and ALM, then, using the same technique as in the elementary case, we can build a monster model for (K,4). Consistent with the notation used for the elementary case, we denote this model by M. We are now in the position to generalize Definition 2.6 to the context of AECs. Also in this case we distinguish between pre-independence and independence relations. In our study we will work only at the level of pre-independence, but we consider worth mentioning what are (some of) the further axioms that are required in order to develop a classification theory for the AEC under examination. Definition 2.9. Let(K,4)beanAECwithAP,JEPandALM,and | aternary ⌣ relation between (bounded) subsets of the monster model M. We say that | is a ⌣ pre-independence relation if it satisfies the following axioms. (a.) (Invariance) If A | B and f ∈Aut(M), then f(A) | f(B). ⌣C ⌣f(C) (b.) (Existence) A | B, for any A,B ⊆M. ⌣A (c.) (Monotonicity) If A | B and D ⊆A, then D | B. ⌣C ⌣C (d.) (Base Monotonicity) Let D ⊆C ⊆B. If A | B, then A | B. ⌣D ⌣C (e.) (Symmetry) If A | B, then B | A. ⌣C ⌣C (f.) (Transitivity) Let D ⊆C ⊆B. If B | A and C | A, then B | A. ⌣C ⌣D ⌣D (g.) (Normality) If A | B, then AC | B. ⌣C ⌣C (h.) (Finite Character) If A | B for all finite A ⊆A, then A | B. 0 ⌣C 0 ⌣C (i.) (Anti-Reflexivity) If A | A, then A | C for any C ⊆M. ⌣B ⌣B If in addition | satisfies the following two axioms, then we say that | is an ⌣ ⌣ independence relation. (j.) (Extension) If A | B and B ⊆ D, then there is f ∈ Aut(M) fixing BC ⌣C pointwise such that f(A) | D. ⌣C (k.) (Local Character) There is a cardinal κ(K) such that for every~a∈M<ω and B ⊆M there is C ⊆B with |C|<κ(K) and~a | B. ⌣C Asintheprevioussection,byTransitivitywewillrefertothefollowing(a-priori) stronger property. Proposition2.10(Transitivity). A | BandA | DifandonlyifA | BD. ⌣C ⌣CB ⌣C Proof. As in Proposition 2.2. Corollary 2.11 (Exchange). If A | B and AB | C, then A | BC. ⌣D ⌣D ⌣D Proof. As in Corollary 2.3. IfT isacompletefirst-ordertheoryandwedenoteby4therelationofelementary substructure,thenthepair(Mod(T),4)isanAECwithAP,JEPandALM. Thus allthe cases ofindependence examinedin the previous sectionareinstances ofthis 6 GIANLUCAPAOLINI more general definition. Furthermore, the generality at which we work allow us to subsume also the non-elementary cases of independence. Example 2.12 (Independence in Pregeometries [8]). Let (K,4) be an AEC with AP, JEP and ALM, and cl : M → M a pregeometric operator. For A,B,C ⊆ M, define A |cl C if forevery~a∈A we havedim (~a/B∪C)=dim (~a/B). Then |cl ⌣B cl cl ⌣ is a pre-independence relation. Example2.13(HilbertSpaces). LetKbetheclassofHilbertSpacesoverR(resp. C)and4theclosedlinearsubspacerelation,then(K,4)isanAECwithAP,JEP and ALM. Given a closed linear subspace C ⊆M and a∈M, we denote by P (a) C the orthogonal projection of a onto C. For D,B ⊆M, we then say that D |ort B ⌣A if for every a ∈ D and b ∈ B we have PA⊥(a) ⊥ PA⊥(b). Then ⌣|ort is orthogonal over A for any A⊆M. Example 2.14 (Independence in Finitary AECs). See [12]. 3. Federation We introduce two fundamental notions: independent sequences and algebraic tuples. Independent sequences play a fundamental role in classification theory, where they often occur in the form of sequences of indiscernibles. Definition 3.1 (Independent Sequence). Let | be a pre-independence relation ⌣ and (I,<) a linear order. Let A ⊆ M and (a | i ∈ I) ∈ MI injective. We say i that (a | i ∈ I) is an | -independent sequence over A if for all j ∈ I, we have i ⌣ (a | i < j) | a . We say that (a | i ∈ I) is an | -independent sequence if it is i ⌣A j i ⌣ an | -independent sequence over ∅. ⌣ Definition3.2(AlgebraicTuple). Let | apre-independencerelation. Wesaythat ⌣ ~e∈M<ω is | -algebraic over A if~e | ~e. We say that~e∈M<ω is | -algebraic if ⌣ ⌣A ⌣ it is | -algebraic over ∅. ⌣ When it is clear to which pre-independence relation | we refer, we just talk of ⌣ independent sequences and algebraic tuples. Lemma 3.3. Let | be a pre-independence relation and (a | i ∈ I) ∈ MI an ⌣ i independent sequence over A. Then for all~a,~b ⊆ (a | i ∈ I) ∈ MI with~a∩~b = ∅ i we have~a | ~b. ⌣A Proof. It suffices to show that for~a = (a ,...,a ) and~b = (a ,...,a ) with k0 kn−1 j0 jm−1 k < ··· < k , j < ··· < j and~a∩~b = ∅, we have that~a | ~b. We prove 0 n−1 0 m−1 ⌣A this by induction on max(k ,j )=t. n−1 m−1 t=0. If this is the case,then either a=∅ or b=∅ because~a∩~b=∅. Suppose the first,theothercaseissymmetrical. ByExistenceA | b,andso,byMonotonicity, ⌣A ∅ | b. ⌣A t>0. Suppose that t=j , the other case is symmetrical. By the independence m−1 of the sequence and Monotonicity, it follows that a ···a b ···b | b . k0 kn−1 j0 jm−2 ⌣ jm−1 A Noticenowthatmax(k ,j )

Independence Logic and Abstract Independence Relations

Similar Independence Logic and Abstract Independence Relations