Axiomatizing Conditional Independence and Inclusion Dependencies

Transcription

1 Axiomatizing Conditional Independence and Inclusion Dependencies Miika Hannula University of Helsinki Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

2 Introduction We provide an axiomatization for the so-called conditional independence atoms (which correspond to EMVDs + FDs in database theory) and inclusion atoms taken together (a joint work with Kontinen). This work arises from the dependence logic/lax team semantics framework. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

3 Axiomatizating Dependence Logic Dependence Logic (D) cannot be axiomatized: validity problem of D-formulas is of the same complexity as that of full second-order logic formulas. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

4 Axiomatizating Dependence Logic Dependence Logic (D) cannot be axiomatized: validity problem of D-formulas is of the same complexity as that of full second-order logic formulas. We can axiomatize fragments of D (or variants) for which the logical consequence is (at least) semidecidable: for instance, first-order consequences of D-sentences (Kontinen and Väänänen, 2012) and of FO( c )-sentences (H. 2013). Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

5 Axiomatizating Dependence Logic Dependence Logic (D) cannot be axiomatized: validity problem of D-formulas is of the same complexity as that of full second-order logic formulas. We can axiomatize fragments of D (or variants) for which the logical consequence is (at least) semidecidable: for instance, first-order consequences of D-sentences (Kontinen and Väänänen, 2012) and of FO( c )-sentences (H. 2013). Axiomatizations of dependence atoms (+variants): Dependence atoms (Armstrong axioms for FDs, 1974), Pure independence atoms (Geiger, Paz and Pearl for probabilistic independence, 1991), Inclusion atoms (Casanova, Fagin and Papadimitrou for INDs, 1982). Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

6 Conditional Independence Atom A conditional independence atom y x z is defined for sequences of variables x, y, and z (which are not necessarily disjoint or of the same length). Its team semantics is defined as follows: TS-ind: X = y x z if and only if for any two s, s X which assign the same value to x there exists a s X which agrees with s with respect to x and y and with s with respect to z. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

7 Conditional Independence Atom cont. Let X be a team and s, s X such that s(x) = s (x ), and assume that M = X y x z. x y z v s s Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

8 Conditional Independence Atom cont. Let X be a team and s, s X such that s(x) = s (x ), and assume that M = X y x z. Then we find s X as in the picture: x y z v s s s Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

9 Embedded Multivalued Dependency Conditional independence atom generalizes embedded multivalued dependency (EMVD) of database theory. For sets of attributes X, Y, Z, X Y Z is an EMVD, given the following semantic rule r = X Y Z if and only if for every t 0, t 1 r with t 0 [X ] = t 1 [X ] there is t 2 r such that t 2 [X ] = t 0 [X ], t 2 [Y ] = t 0 [Y ] and t 2 [Z \ XY ] = t 1 [Z \ XY ]. Hence X Y Z corresponds to the independence atom Y X Z \ XY. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

10 Functional Dependency A functional dependency (FD) X Y which corresponds to the dependence atom =(X, Y ) can also be expressed as the conditional independence atom Y X Y. X Z Y V s s s s Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

11 Functional Dependency A functional dependency (FD) X Y which corresponds to the dependence atom =(X, Y ) can also be expressed as the conditional independence atom Y X Y. X Z Y V s s s s Hence, conditional independence atoms = EMVDs + FDs Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

12 Inclusion Atom An inclusion atom x y is defined for sequences of variables x and y of the same length (both possibly with repetitions). Its semantic rule is given as follows: TS-inc: X = x y if and only if for all s X there is s X such that s( x) = s ( y). Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

13 Axiomatizing { c, }-atoms Given a set Σ {φ} of { c, }-atoms, the implication problem Σ = φ is to decide whether X = φ for all teams X such that X = Σ. We will present a complete axiomatization of the implication problem for { c, }-atoms. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

14 Axiomatizing { c, }-atoms Given a set Σ {φ} of { c, }-atoms, the implication problem Σ = φ is to decide whether X = φ for all teams X such that X = Σ. We will present a complete axiomatization of the implication problem for { c, }-atoms. Since the implication problem for EMVDs is undecidable (Herrmann, 1995), the same is true for { c, }-atoms taken together (i.e. EMVDs+FDs+INDs in database theory). How can we find complete axioms then? Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

15 Axiomatizing { c, }-atoms Given a set Σ {φ} of { c, }-atoms, the implication problem Σ = φ is to decide whether X = φ for all teams X such that X = Σ. We will present a complete axiomatization of the implication problem for { c, }-atoms. Since the implication problem for EMVDs is undecidable (Herrmann, 1995), the same is true for { c, }-atoms taken together (i.e. EMVDs+FDs+INDs in database theory). How can we find complete axioms then? One could axiomatize some larger class of dependencies where these additional dependencies would be needed at intermediate steps of derivations. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

16 Axiomatizing { c, }-atoms Given a set Σ {φ} of { c, }-atoms, the implication problem Σ = φ is to decide whether X = φ for all teams X such that X = Σ. We will present a complete axiomatization of the implication problem for { c, }-atoms. Since the implication problem for EMVDs is undecidable (Herrmann, 1995), the same is true for { c, }-atoms taken together (i.e. EMVDs+FDs+INDs in database theory). How can we find complete axioms then? One could axiomatize some larger class of dependencies where these additional dependencies would be needed at intermediate steps of derivations. One could (implicitly) use existential/universal quantification. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

17 Axioms for FDs+INDs The latter line is used in Mitchell s axiomatization (1983) for FDs and inclusion dependencies (INDs) taken together. The corresponding implication problem is known to be undecidable but the following finite set of rules is still sound and complete. Complete axioms for FDs alone (3). Complete axioms for INDs alone (3). (Identity) From AB CC and σ derive τ, where τ is obtained from σ by substituting A for one or more occurrences of B. (Pullback) From UV XY and X Y derive U V, where X = U. (Collection) From UV XY, UW XZ and X Y derive UVW XYZ, where X = U. (Attribute Introduction) From U V and V B derive UA VB. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

18 Axioms for FDs+INDs cont. Note that Attribute Introduction from U V and V B derive UA VB, is not sound in the usual sense. For instance the following relation U V A B t t satisfies U V and V B, but not UA VB. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

19 Axioms for FDs+INDs cont. Note that Attribute Introduction from U V and V B derive UA VB, is not sound in the usual sense. For instance the following relation U V A B t t satisfies U V and V B, but not UA VB. Soundness is obtained if we assume that the attribute A does not appear earlier in the proof, and no new variables of this kind are allowed to appear at the final step of a derivation. Hence, A should be thought of as implicitly existentially quantified. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

20 Axioms for { c, }-atoms I In this style we can axiomatize { c, }-atoms also: 1 Reflexivity: x x. 2 Projection and Permutation: if x 1... x n y 1... y n, then x i1... x ik y i1... y ik, for each sequence i 1,..., i k of integers from {1,..., n}. 3 Transitivity: if x y y z, then x y. 4 Identity Rule: if ab cc φ, then φ, where φ is obtained from φ by replacing any number of occurrences of a by b. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

21 Axioms for { c, }-atoms II 5 Inclusion Introduction: if a b, then ax bc, where x is a new variable. 6 Start Axiom: a c a x b a x a x a c where x is a sequence of pairwise distinct new variables. 7 Chase Rule: if y x z a b x y a c x z, then a b c x y z. 8 Final Rule: if a c a x b a x a b x a b c, then b a c. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

22 Axioms for { c, }-atoms III A deduction from Σ is a sequence of formulas (φ 1,..., φ n ) such that: 1 Each φ i is either an element of Σ, an instance of Reflexivity or Start Axiom, or follows from one or more formulas of Σ {φ 1,..., φ i 1 } by one of the rules. 2 If φ i is an instance of Start Axiom (or follows from Σ {φ 1,..., φ i 1 } by Inclusion Introduction), then the new variables of x (or the new variable x) must not appear in Σ {φ 1,..., φ i 1 }. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

23 Axioms for { c, }-atoms IV We say that φ is provable from Σ, written Σ φ, if there is a deduction (φ 1,..., φ n ) from Σ with φ = φ n and such that no variables in φ are new in φ 1,..., φ n. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

24 On Soundness: Inclusion Introduction Definition If a b, then ax bc, where x is a new variable. This rule is sound if we interpret x as existentially quantified. For instance X = {s 0, s 1, s 2, s 3 } satisfies a b, and it can be extended to satisfy ax bc. a b c s s s s Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

25 On Soundness: Inclusion Introduction Definition If a b, then ax bc, where x is a new variable. This rule is sound if we interpret x as existentially quantified. For instance X = {s 0, s 1, s 2, s 3 } satisfies a b, and it can be extended to satisfy ax bc. a b c x s s s s Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

26 On Soundness: Start Axiom Definition a c a x b a x a x a c where x is a sequence of pairwise distinct new variables. This rule is sound if we interpret x as existentially quantified in the (lax) team semantics sense: Definition (Ex. Quant. in TS) M = X vψ if and only if there is a function F : X P(M)\{ } such that M = X [F /v] ψ, where X [F /v] = {s[m/v] : s X, m F (s)}. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

27 On Soundness: Start Axiom cont. Definition a c a x b a x a x a c where x is a sequence of pairwise distinct new variables. Let X be a team. If we extend each assignment s X with a set of values for x as below, then X satisfies ac ax b a x ax ac. a b c... s Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

28 On Soundness: Start Axiom cont. Definition a c a x b a x a x a c where x is a sequence of pairwise distinct new variables. Let X be a team. If we extend each assignment s X with a set of values for x as below, then X satisfies ac ax b a x ax ac. a b c x.... s {s (c) : s X, s (a) = 1}.... Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

29 Soundness: Conclusion Theorem Let Σ {φ} be a finite set of conditional independence and inclusion atoms. Then Σ = φ if Σ φ. Proof. Idea. Assume that X = Σ; we have to show that X = φ. By the assumption there is a deduction (φ 1,..., φ n ) from Σ with φ = φ n and such that no variables in φ are new in φ 1,..., φ n. By the previous considerations we find a team X = φ which extends X with values for the new variables. Since no new variables appear in φ, we obtain that X = φ. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20

30 On completeness Theorem Let Σ {φ} be a finite set of conditional independence and inclusion atoms. Then Σ φ if Σ = φ. Proof. Idea. A chase characterization of the implication problem is presented: for each Σ {φ}, we inductively construct an infinite graph G Σ,φ, consisting of nodes and labeled edges, and such that Σ = φ exactly when we find a node v G Σ,φ which is connected to the initial stage in a certain way. The construction of G Σ,φ can be simulated in our axiom system from which the result will follow. Miika Hannula (University of Helsinki) Axiomatizing Conditional Independence and Inclusion Dependencies / 20