Provenance Semirings. Todd Green Grigoris Karvounarakis Val Tannen. presented by Clemens Ley

Size: px
Start display at page:

Download "Provenance Semirings. Todd Green Grigoris Karvounarakis Val Tannen. presented by Clemens Ley"

Transcription

1 Provenance Semirings Todd Green Grigoris Karvounarakis Val Tannen presented by Clemens Ley

2 place of origin Provenance Semirings Todd Green Grigoris Karvounarakis Val Tannen presented by Clemens Ley

3 place of origin algebraic structure, e.g (N,*,+,0,1) Provenance Semirings Todd Green Grigoris Karvounarakis Val Tannen presented by Clemens Ley

4 Outline Data provenance by example Relational algebra for data provenance Datalog for data provenance

5 Data Provenance

6 Data Provenance Data provenance aims to explain how a particular query result was obtained.

7 Data Provenance R: A B C a b c p S: D B E d b e q join on B Data provenance aims to explain how a particular query result was obtained. R S: A B C D E a b c d e p * q

8 Data Provenance R: A B C a b c p S: D B E d b e q join on B Data provenance aims to explain how a particular query result was obtained. R S: A B C D E a b c d e p * q

9 Data Provenance R: A B C a b c p S: D B E d b e q join on B Data provenance aims to explain how a particular query result was obtained. R S: A B C D E a b c d e p * q means: was obtained from both p and q

10 Data Provenance (2) R: A B C a b c p S: A B C union R S: A B C a b c p + q a b c q

11 Data Provenance (2) R: A B C a b c p means: was obtained from either p or q S: A B C union R S: A B C a b c p + q a b c q

12 Data Provenance (3) R: A B C a b c p a b c q a b c r projection π AB (R): A B a b p + q + r means: was obtained from either p, q, or r

13 Data Provenance (4) Q = σ C=e π AC ( π AC R π BC R π AB R π BC R ) R: A B C a b c p d b e r f g e s Q: A C a c (p 2 + p 2 ) * 0 a e (pr) * 1 d c (rp) * 0 d e (r 2 + rs + r 2 ) * 1 f e (s 2 + rs + s 2 ) * 1

14 Data Provenance (4) Q = σ C=e π AC ( π AC R π BC R π AB R π BC R ) R: A B C a b c p d b e r f g e s Q: A C a c (p 2 + p 2 ) * 0 a e (pr) * 1 d c (rp) * 0 d e (r 2 + rs + r 2 ) * 1 f e (s 2 + rs + s 2 ) * 1 for selection, multiply by 1 or 0.

15 Why would this be useful? Q = σ C=e π AC ( π AC R π BC R π AB R π BC R ) R: A B C a b c p d b e r f g e s Q: A C a c (p 2 + p 2 ) * 0 a e (pr) * 1 d c (rp) * 0 d e (r 2 + rs + r 2 ) * 1 f e (s 2 + rs + s 2 ) * 1 for selection, multiply by 1 or 0.

16 Why would this be useful? Q = σ C=e π AC ( π AC R π BC R π AB R π BC R ) R: A B C a b c p := 2 d b e r := 5 f g e s := 1 Q: A C a c (p 2 + p 2 ) * 0 = 0 a e (pr) * 1 = 10 d c (rp) * 0 = 0 d e (r 2 + rs + r 2 ) * 1 = 55 f e (s 2 + rs + s 2 ) * 1 = 7 for bag semantics, consider annotations as multiplicities

17 Why would this be useful? Q: A C a c (p 2 + p 2 ) * 0 = ((b 1 b 1 ) (b 1 b 1 )) false a e (pr) * 1 = (b 1 b 2 ) true d c (rp) * 0 = (b 2 b 1 ) false d e (r 2 + rs + r 2 ) * 1 = (b 2 (b 2 b 3 ) b 2 ) true R: A B C f e (s 2 + rs + s 2 ) * 1 = (b 3 (b 2 b 3 ) b 3 ) true a b c p := b 1 d b e r := b 2 f g e s := b 3 for incomplete databases, consider annotations as boolean values, * as, + as, 1 as true, and 0 as false

18 Data Structure Relations are mappings from tuples to annotations in K; we require that R(t) 0 for only finitely many tuples t. intuitively, + means alternative use corresponds to union * means joint use and corresponds to join 0 and 1 are special annotations But what is a query languages for such relations?

19 Data Structure Relations are mappings from tuples to annotations in K; we require that R(t) 0 for only finitely many tuples t. intuitively, + means alternative use corresponds to union * means joint use and corresponds to join 0 and 1 are special annotations But what is (K,+,*,0,1) and how are annotations computed?

20 Positive Algebra

21 Positive Algebra Definition 3.2. Let (K, +,, 0, 1) be an algebraic structure with two binary operations and two distinguished elements. The operations of the positive algebra are defined as follows: empty relation For any set of attributes U, there is ; : U-Tup! K such that ;(t) =0. union If R 1,R 2 : U-Tup! K then R 1 [ R 2 : U-Tup! K is defined by (R 1 [ R 2 )(t) def = R 1 (t)+r 2 (t) projection If R : U-Tup! K and V U then V R : V -Tup! K is defined by X R(t 0 ) ( V R)(t) def = t=t 0 on V and R(t 0 )6=0 (here t = t 0 on V means t 0 is a U-tuple whose restriction to V is the same as the V -tuple t; note also that the sum is finite since R has finite support)!

22 X Positive Algebra (2) selection If R : U-Tup! K and the selection predicate P maps each U-tuple to either 0 or 1 then PR : U-Tup! K is defined by ( P R)(t) def = R(t) P(t) Which {0, 1}-valued functions are used as selection predicates is left unspecified, except that we assume that false the constantly 0 predicate, and true the constantly 1 predicate, are always available. natural join If R i : U i -Tup! K i =1, 2 then R 1 1 R 2 is the K-relation over U 1 [ U 2 defined by (R 1 1 R 2 )(t) def = R 1 (t 1 ) R 2 (t 2 ) where t 1 = t on U 1 and t 2 = t on U 2 (recall that t is a U 1 [ U 2 -tuple). renaming If R : U-Tup! K and : U! U 0 is a bijection then R is a K-relation over U 0 defined by ( R)(t) def = R(t )

23 What is K? Proposition 3.4. The following RA identities: union is associative, commutative and has identity ;; join is associative, commutative and distributive over union; projections and selections commute with each other as well as with unions and joins (when applicable); false(r) =; and true(r) =R. hold for the positive algebra on K-relations if and only if (K, +,, 0, 1) is a commutative semiring.

24 What is K? Proposition 3.4. The following RA identities: union is associative, commutative and has identity ;; join is associative, commutative and distributive over union; projections and selections commute with each other as well as with unions and joins (when applicable); false(r) =; and true(r) =R. hold for the positive algebra on K-relations if and only if (K, +,, 0, 1) is a commutative semiring. Note that the list does not contain idempotence of union and self join, as these fail for set semantics

25 What is K? Proposition 3.4. The following RA identities: union is associative, commutative and has identity ;; join is associative, commutative and distributive over union; projections and selections commute with each other as well as with unions and joins (when applicable); false(r) =; and true(r) =R. hold for the positive algebra on K-relations if and only if (K, +,, 0, 1) is a commutative semiring. Def. A commutative semiring is a structure (K,+,*,0,1) where + is commutative, associative, with identity 0 * is associative with identity 1 * distributes over + a * 0 = 0 * a = 0 Note that the list does not contain idempotence of union and self join, as these fail for set semantics

26 What is K? Proposition 3.4. The following RA identities: union is associative, commutative and has identity ;; join is associative, commutative and distributive over union; projections and selections commute with each other as well as with unions and joins (when applicable); false(r) =; and true(r) =R. hold for the positive algebra on K-relations if and only if (K, +,, 0, 1) is a commutative semiring. Def. A commutative semiring is a structure (K,+,*,0,1) where + is commutative, associative, with identity 0 * is associative with identity 1 * distributes over + a * 0 = 0 * a = 0

27 well as with unions and joins (when applicable); false(r) =; and true(r) =R. What is K? hold for the positive algebra on K-relations if and only if (K, +,, 0, 1) is a commutative semiring. Def. A commutative semiring is a structure (K,+,*,0,1) where + is commutative, associative, with identity 0 * is associative with identity 1 * distributes over + a * 0 = 0 * a = 0 Examples: the natural numbers: (N, +, *, 0, 1) the booleans: (B,,, true, false) subsets of a set: (P(Ω),,, Ω) the naturals with infinity: (N, +, *, 0, 1) polynomials in X: (N[X], +, *, 0, 1)

28 The fundamental property of RA For!every!query!!!q!!and!every!homomorphism!of! commutamve!semirings!! h : K 1! K 2 the!following! commutes : K 1 Bdata! h! K 2 Bdata! q! q! K 1 Edata! h! K 2 Bdata! Recall, semiring homomorphism is mapping h:k 1 K 2 such that h(1 K1 ) = 1 K2 h(0 K1 ) = 0 K2 h(a + K1 b) = h(a) + K2 h(b) h(a * K1 b) = h(a) * K2 h(b)

29 The fundamental property of RA For!every!query!!!q!!and!every!homomorphism!of! commutamve!semirings!! h : K 1! K 2 the!following! commutes : K 1 Bdata! h! K 2 Bdata! q! q! K 1 Edata! h! K 2 Bdata! Works only if q in RA +. Does not generalize e.g. to negation. Recall, semiring homomorphism is mapping h:k 1 K 2 such that h(1 K1 ) = 1 K2 h(0 K1 ) = 0 K2 h(a + K1 b) = h(a) + K2 h(b) h(a * K1 b) = h(a) * K2 h(b)

30 Which semiring do we choose? Definition 4.1. Let X be the set of tuple ids of a (usual) database instance I. The positive algebra provenance semiring for I is the semiring of polynomials with variables (a.k.a. indeterminates) from X and coe cients from N, with the operations defined as usual 4 : (N[X], +,, 0, 1).

31 Which semiring do we choose? Definition 4.1. Let X be the set of tuple ids of a (usual) database instance I. The positive algebra provenance semiring for I is the semiring of polynomials with variables (a.k.a. indeterminates) from X and coe cients from N, with the operations defined as usual 4 : (N[X], +,, 0, 1). But why?

32 A nice property of N[X] If!!!K!!is!a!commutaMve!semiring,!then!any!funcMon! on!tokens,!!!f!:!x!! K extends!uniquely!to!a!! homomorphism!!! h!: N[X]!!! K. Example: K= (N, +, *, 0, 1), X={p,q}, f:{p,q} K be f(p)=3 and f(q)=5. Then the extension of f is h: N[X] K and evaluates the polynomial (e.g. h(p+2q)=13)

33 Nice + Fundamental R: A B C a b c p a d e q homomorphism h R: A B C a b c 3 a d e 5 π A (R): query π A A a p + q homomorphism h Example: K= (N, +, *, 0, 1), X={p,q}, f:{p,q} K be f(p)=3 and f(q)=5. Then the extension of f is h: N[X] K and evaluates the π A (R): polynomial (e.g. h(p+2q)=13) query π A A a 8

34 Nice + Fundamental R: A B C a b c p homomorphism h R: A B C a b c 3 a d e q a d e 5 π A (R): query π A A Example: K= (N, +, *, 0, 1), X={p,q}, f:{p,q} K be f(p)=3 and f(q)=5. Then the extension of f is h: N[X] K and evaluates the polynomial (e.g. h(p+2q)=13) π A (R): query π A A a p + q homomorphism h a 8

35 Free the semiring! Nice implies: For every commutative semiring K, and every K-relation R, there is abstractly tagged N[X]-relation R and a homomorphism Eval v from R to R.

36 Free the semiring! Nice implies: For every commutative semiring K, and every K-relation R, there is abstractly tagged N[X]-relation R and a homomorphism Eval v from R to R. R: A B a b 3 a d 5 query q q(r): A B a b 15

37 Free the semiring! Nice implies: For every commutative semiring K, and every K-relation R, there is abstractly tagged N[X]-relation R and a homomorphism Eval v from R to R. R: A B R: A B a b p a d q Eval v a b 3 a d 5 query q q(r): A B a b 15

38 Free the semiring! Nice implies: For every commutative semiring K, and every K-relation R, there is abstractly tagged N[X]-relation R and a homomorphism Eval v from R to R. R: A B R: A B a b p a d q Eval v a b 3 a d 5 query q query q q(r): A B q(r): A B a b pq Eval v a b 15

39 Free the semiring! Nice implies: For every commutative semiring K, and every K-relation R, there is abstractly tagged N[X]-relation R and a homomorphism Eval v from R to R. R: A B R: A B a b p a d q Eval v a b 3 a d 5 query q Theorem 4.3. For any RA + query q we have q(r) = Eval v q( R) query q q(r): A B q(r): A B a b pq Eval v a b 15

40 Instantiation of Positive Algebra (B,,, true, false) Set semantics (N, +, *, 0, 1) Bag semantics (P(Ω),,, Ω) Probabilistic events (BoolExp(P),,, true, false) (A, min, max, 0, P) where A = P < C < S < T < 0 Conditional tables Access control levels

41 More nice Example:!2x 2 y!+!xy!+!5y 2!+!z! drop!coefficients! x 2 y!+!xy!+!y 2!+!z! B[X]! N[X]! Trio(X)! drop!exponents! 3xy!+!5y!+!z! drop!both!exp.!and!coeff.!!!!!!!!!!xy!+!y!+!z! collapse!terms! xyz! Lin(X)! Why(X)! PosBool(X)! apply!absorpmon! (ab!+!b!=!b)! y!+!z! A!path!downward!from!K 1!to!K 2!indicates!that!there!exists!an! onto!(surjec1ve))semiring)homomorphism)))!h!:!k 1!!K 2!!!!!!!

42 More nice drop!coefficients! x 2 y!+!xy!+!y 2!+!z! B[X]! drop!both!exp.!and!coeff.!!!!!!!!!!xy!+!y!+!z! collapse!terms! xyz! Example:!2x 2 y!+!xy!+!5y 2!+!z! Lin(X)! N[X]! Why(X)! Trio(X)! PosBool(X)! drop!exponents! 3xy!+!5y!+!z! apply!absorpmon! (ab!+!b!=!b)! y!+!z! most!informamve! least!informamve! A!path!downward!from!K 1!to!K 2!indicates!that!there!exists!an! onto!(surjec1ve))semiring)homomorphism)))!h!:!k 1!!K 2!!!!!!!

43 Datalog

44 Syntax and Semantics edb R: aa 2 Q: ab 3 bb 4 (b) (a) aa 2 2=4 idb ab = 18 bb 4 4 = 16 (c) Figure 6: Datalog with bag semantics

45 Syntax and Semantics edb R: aa 2 Q: ab 3 bb 4 (b) (a) aa 2 2=4 idb ab = 18 bb 4 4 = 16 (c) Figure Q(a,b) 6: Datalog with bag semantics R(a,a) R(a,b)

46 Syntax and Semantics edb R: aa 2 Q: ab 3 bb 4 (b) (a) aa 2 2=4 idb ab = 18 bb 4 4 = 16 (c) Figure Q(a,b) 6: Datalog with bag semantics Q(a,b) R(a,a) R(a,b) R(a,b) R(b,b)

47 Syntax and Semantics edb R: aa 2 Q: ab 3 bb 4 (b) (a) aa 2 2=4 idb ab = 18 bb 4 4 = 16 (c) Figure Q(a,b) 6: Datalog with bag semantics Q(a,b) R(a,a) R(a,b) R(a,b) R(b,b) Q(a,a) Q(b,b) R(a,a) R(a,a) R(b,b) R(b,b)

48 Datalog with Bag Semantics R: aa 2 ab 3 bb 4 (b) Q: (a) aa 2 2=4 ab = 18 bb 4 4 = 16 (c) Figure Q(a,b) 6: Datalog with bag semantics Q(a,b) R(a,a) R(a,b) R(a,b) R(b,b) Q(a,a) Q(b,b) R(a,a) R(a,a) R(b,b) R(b,b)

49 Datalog with Bag Semantics Q(a,b) R(a,a) R(a,b) R: aa 2 ab 3 bb 4 (b) Q: (a) aa 2 2=4 ab = 18 bb 4 4 = 16 (c) Figure 6: Datalog with bag semantics Q(a,b) R(a,b) R(b,b) Q(a,a) Q(b,b) R(a,a) R(a,a) R(b,b) R(b,b)

50 Datalog with Bag Semantics Q(a,b) R(a,a) R(a,b) R: aa 2 ab 3 bb 4 (b) Q: (a) aa 2 2=4 ab = 18 bb 4 4 = 16 (c) Figure 6: Datalog with bag semantics Q(a,b) R(a,b) R(b,b) Q(a,a) Q(b,b) R(a,a) R(a,a) R(b,b) R(b,b)

51 What annotations do we need? Q(a,b) R(a,a) R(a,b) Q(a,a) R: aa 2 Q: ab 3 bb 4 (b) (a) aa 2 2=4 ab = 18 bb 4 4 = 16 (c) Figure 6: Datalog with bag semantics Q(a,b) R(a,b) R(b,b) Q(b,b) R(a,a) R(a,a) R(b,b) R(b,b)

52 What annotations do we need? Q(a,b) R(a,a) R(a,b) Q(a,a) R: aa 2 Q: ab 3 bb 4 (b) (a) aa 2 2=4 ab = 18 bb 4 4 = 16 (c) Figure 6: Datalog with bag semantics How about: the tag of an answer tuple is the sum over all derivation trees and the product of the tags of each leaf. q(r)(t) = X yields t Y t 0 2leaves( ) R(t 0 ) Q(a,b) R(a,b) R(b,b) Q(b,b) R(a,a) R(a,a) R(b,b) R(b,b)

53 What annotations do we need? Q(a,b) R(a,a) R(a,b) Q(a,a) R: aa 2 Q: ab 3 bb 4 (b) (a) aa 2 2=4 ab = 18 bb 4 4 = 16 (c) Figure 6: Datalog with bag semantics How about: the tag of an answer tuple is the sum over all derivation trees and the product of the tags of each leaf. q(r)(t) = X yields t Y t 0 2leaves( ) R(t 0 ) Q(a,b) R(a,b) R(b,b) Q(b,b) R(a,a) R(a,a) Problem: A tuple may have infinitely many derivation trees. Hence we need to work in semirings in which infinite sums are defined. R(b,b) R(b,b)

54 ω-continuos semirings Def. (Natural preorder) x y iff there is z such that x+z=y. Def. (Naturally ordered semiring) if the natural pre-order is an order. Def. (ω-complete) when x 1 x 2 x 2 have suprema. In naturally ordered semirings, we can make sense of infinite sums: X n2n a n 8 def = sup ( m2n mx a i ) Def. (ω-continous) when * and + preserve suprema. ( e.g. sup(a i + b i ) = sup(a i ) + b i ). Lemma. Over ω-continuos semirings, functions defined by polynomials have least fixed points. i=0

55 ω-continuos semirings Def. (Natural preorder) x y iff there is z such that x+z=y. Def. (Naturally ordered semiring) if the natural pre-order is an order. Def. (ω-complete) when x 1 x 2 x 2 have suprema. In naturally ordered semirings, we can make sense of infinite sums: X n2n a n 8 def = sup ( m2n mx a i ) Def. (ω-continous) when * and + preserve suprema. ( e.g. sup(a i + b i ) = sup(a i ) + b i ). Lemma. Over ω-continuos semirings, functions defined by polynomials have least fixed points. i=0 Preorder: reflexive and transitive. Not necessarily anti-symmetric (x y and y x implies x=y)

56 ω-continuos semirings Def. (Natural preorder) x y iff there is z such that x+z=y. Def. (Naturally ordered semiring) if the natural pre-order is an order. E.g. Z is not naturally ordered because -5 5, 5-5, but -5 5 Def. (ω-complete) when x 1 x 2 x 2 have suprema. In naturally ordered semirings, we can make sense of infinite sums: X n2n a n 8 def = sup ( m2n mx a i ) Def. (ω-continous) when * and + preserve suprema. ( e.g. sup(a i + b i ) = sup(a i ) + b i ). Lemma. Over ω-continuos semirings, functions defined by polynomials have least fixed points. i=0

57 ω-continuos semirings Def. (Natural preorder) x y iff there is z such that x+z=y. Def. (Naturally ordered semiring) if the natural pre-order is an order. Def. (ω-complete) when x 1 x 2 x 2 have suprema. In naturally ordered semirings, we can make sense of infinite sums: X n2n a n 8 def = sup ( m2n mx a i ) Def. (ω-continous) when * and + preserve suprema. ( e.g. sup(a i + b i ) = sup(a i ) + b i ). Lemma. Over ω-continuos semirings, functions defined by polynomials have least fixed points. i=0

58 Semantics of annotated Datalog Definition 5.1. Let (K, +,, 0, 1) be a commutative!-continuous semiring. To keep notation simple let q be a datalog query with one argument (it is easy to generalize to multiple arguments). For any K-relation R define q(r)(t) = X Y R(t 0 ) yields t t 0 2leaves( ) where ranges over all q-derivation trees for t and t 0 ranges over all the leaves of.

59 Semantics of annotated Datalog Definition 5.1. Let (K, +,, 0, 1) be a commutative!-continuous semiring. To keep notation simple let q be a datalog query with one argument (it is easy to generalize to multiple arguments). For any K-relation R define q(r)(t) = X Y R(t 0 ) yields t t 0 2leaves( ) where ranges over all q-derivation trees for t and t 0 ranges over all the leaves of. For!every!query!!!q!!and!every!homomorphism!of! commutamve!semirings!! h : K 1! K 2 the!following! commutes : K 1 Bdata! h! K 2 Bdata! q! q! K 1 Edata! h! K 2 Bdata!

60 Semantics of annotated Datalog Definition 5.1. Let (K, +,, 0, 1) be a commutative!-continuous semiring. To keep notation simple let q be a datalog query with one argument (it is easy to generalize to multiple arguments). For any K-relation R define q(r)(t) = X Y R(t 0 ) yields t t 0 2leaves( ) where ranges over all q-derivation trees for t and t 0 ranges over all the leaves of. Datalog ω-continuos For!every!query!!!q!!and!every!homomorphism!of! commutamve!semirings!! h : K 1! K 2 the!following! commutes : K 1 Bdata! h! K 2 Bdata! q! q! K 1 Edata! h! K 2 Bdata!

61 The Datalog provenance Semiring Problem: there can be infinitely many derivation trees for one tuple infinite sums in annotations In particular two kinds of infinite summations infinitely many copies of the same monomial coefficients in N = N { } infinitely many copies of different monomials formal power series K[[X]]

62 The Datalog provenance Semiring Problem: there can be infinitely many derivation trees for one tuple infinite sums in annotations In particular two kinds of infinite summations infinitely many copies of the same monomial coefficients in N = N { } infinitely many copies of different monomials formal power series K[[X]] Formal power series: basically polynomials with infinite summation

63 The Datalog provenance Semiring Problem: there can be infinitely many derivation trees for one tuple infinite sums in annotations In particular two kinds of infinite summations X infinitely many copies of the same monomial coefficients in N = N { } infinitely many copies of different monomials formal power series K[[X]] Definition 6.1. Let X be the set of tuple ids of a database instance I. The datalog provenance semiring for I is the commutative!-continuous semiring of formal power series N 1 [[X]].

64 Fixed Point Semantics Q(x, y) :-R(x, y) Q(x, y) :-Q(x, z), Q(z, y) R: ab ac cb bd dd ab 2 ac 3 cb 2 bd 1 dd 1 ab 8 ac 3 cb 2 bd 1 dd 1 ad 1 (a) (b) (c) m n p r s R: Q(R): ab ac cb bd dd ad x y z u v w Q(R): x = m + yz y = n z = p u = r + uv v = s + v 2 w = xu + wv (d) (e) (f) Transform immediate consequence operator of Q into a union of conjunctive queries; here R (Q 2=1 Q) Apply this RA query to R and Q. Equate! This leads to system of equations of polynomials in N 1 [[m, n, p, r, s]][x, y, z, u, v, w] As N [[m,n,p,r,s]] is omega continuos, these equations { have least fixed points that can be computed.

65 Decidability A tuple can have an annotations in any of the classes below. It is decidable in which class the annotation of a tuple is. N [[X]] N [X] N[[X]] N[X]

66 Decidability: Case N[X] Claim: Let Q be a Datalog program, D a database, and R an relation in the intensional schema of P. R(t) N[X] iff t has a derivation tree T w.r.t. Q and D of height less than (# of atoms +2) that has a path with two occurrences of the same atom a. N [[X]] N [X] N[[X]] N[X]

67 Decidability: Case N[X] Claim: Let Q be a Datalog program, D a database, and R an relation in the intensional schema of P. R(t) N[X] iff t has a derivation tree T w.r.t. Q and D of height less than (# of atoms +2) that has a path with two occurrences of the same atom a. Proof: Assume such a tree exists: a a

68 Decidability: Case N[X] Claim: Let Q be a Datalog program, D a database, and R an relation in the intensional schema of P. R(t) N[X] iff t has a derivation tree T w.r.t. Q and D of height less than (# of atoms +2) that has a path with two occurrences of the same atom a. Proof: Assume such a tree exists: a a a a a Then this is also a derivation tree: a

69 Decidability: Case N[X] Claim: Let Q be a Datalog program, D a database, and R an relation in the intensional schema of P. R(t) N[X] iff t has a derivation tree T w.r.t. Q and D of height less than (# of atoms +2) that has a path with two occurrences of the same atom a. Proof: In particular, trees of height (# of atoms +1) have no path with two repeated atoms.

70 Decidability: Case N[X] Claim: Let Q be a Datalog program, D a database, and R an relation in the intensional schema of P. R(t) N[X] iff t has a derivation tree T w.r.t. Q and D of height less than (# of atoms +2) that has a path with two occurrences of the same atom a. Proof: In particular, trees of height (# of atoms +1) have no path with two repeated atoms. Hence, by the pigeon hole principle, there are no derivation trees of height equal to (# of atoms +1).

71 Decidability: Case N[X] Claim: Let Q be a Datalog program, D a database, and R an relation in the intensional schema of P. R(t) N[X] iff t has a derivation tree T w.r.t. Q and D of height less than (# of atoms +2) that has a path with two occurrences of the same atom a. Proof: In particular, trees of height (# of atoms +1) have no path with two repeated atoms. Hence, by the pigeon hole principle, there are no derivation trees of height equal to (# of atoms +1).

72 Decidability: Case N[X] Claim: Let Q be a Datalog program, D a database, and R an relation in the intensional schema of P. R(t) N[X] iff t has a derivation tree T w.r.t. Q and D of height less than (# of atoms +2) that has a path with two occurrences of the same atom a. Proof: In particular, trees of height (# of atoms +1) have no path with two repeated atoms. Hence, by the pigeon hole principle, there are no derivation trees of height equal to (# of atoms +1). greater than or equal

73 Decidability: Case N[X] Claim: Let Q be a Datalog program, D a database, and R an relation in the intensional schema of P. R(t) N[X] iff t has a derivation tree T w.r.t. Q and D of height less than (# of atoms +2) that has a path with two occurrences of the same atom a. Proof: In particular, trees of height (# of atoms +1) have no path with two repeated atoms. Hence, by the pigeon hole principle, there are no derivation trees of height equal to (# of atoms +1). greater than or equal Thus there are only finitely many derivation trees.

74 Also decidable: given t q(i), and a monomial μ, the coefficient of μ in the power series that is the provenance of t is computable (including when it is ). testing whether all coefficients are. Not decidable: testing whether all coefficients are 1.

75 Conclusion A versatile framework for provenance computation. Specializes to many known systems for provenance. In a sense most general within frameworks that use Semirings. Provides semantics for positive datalog under rich semantics (e.g. bag semantics).

76

77 Thank You!

Provenance Semirings

Provenance Semirings Provenance Semirings Todd J. Green tjgreen@cis.upenn.edu Grigoris Karvounarakis gkarvoun@cis.upenn.edu Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104,

More information

PUG: A Framework and Practical Implementation for Why & Why-Not Provenance (extended version)

PUG: A Framework and Practical Implementation for Why & Why-Not Provenance (extended version) arxiv:808.05752v [cs.db] 6 Aug 208 PUG: A Framework and Practical Implementation for Why & Why-Not Provenance (extended version) Seokki Lee, Bertram Ludäscher, Boris Glavic IIT DB Group Technical Report

More information

Provenance for Aggregate Queries

Provenance for Aggregate Queries Provenance for Aggregate Queries Yael Amsterdamer Tel Aviv University and University of Pennsylvania yaelamst@post.tau.ac.il Daniel Deutch Ben Gurion University and University of Pennsylvania deutchd@cs.bgu.ac.il

More information

7 RC Simulates RA. Lemma: For every RA expression E(A 1... A k ) there exists a DRC formula F with F V (F ) = {A 1,..., A k } and

7 RC Simulates RA. Lemma: For every RA expression E(A 1... A k ) there exists a DRC formula F with F V (F ) = {A 1,..., A k } and 7 RC Simulates RA. We now show that DRC (and hence TRC) is at least as expressive as RA. That is, given an RA expression E that mentions at most C, there is an equivalent DRC expression E that mentions

More information

Database Design and Implementation

Database Design and Implementation Database Design and Implementation CS 645 Data provenance Provenance provenance, n. The fact of coming from some particular source or quarter; origin, derivation [Oxford English Dictionary] Data provenance

More information

TROPICAL SCHEME THEORY

TROPICAL SCHEME THEORY TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),

More information

Axioms of Kleene Algebra

Axioms of Kleene Algebra Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.

More information

Version January Please send comments and corrections to

Version January Please send comments and corrections to Mathematical Logic for Computer Science Second revised edition, Springer-Verlag London, 2001 Answers to Exercises Mordechai Ben-Ari Department of Science Teaching Weizmann Institute of Science Rehovot

More information

Provenance-Based Analysis of Data-Centric Processes

Provenance-Based Analysis of Data-Centric Processes Noname manuscript No. will be inserted by the editor) Provenance-Based Analysis of Data-Centric Processes Daniel Deutch Yuval Moskovitch Val Tannen the date of receipt and acceptance should be inserted

More information

Additional Practice Lessons 2.02 and 2.03

Additional Practice Lessons 2.02 and 2.03 Additional Practice Lessons 2.02 and 2.03 1. There are two numbers n that satisfy the following equations. Find both numbers. a. n(n 1) 306 b. n(n 1) 462 c. (n 1)(n) 182 2. The following function is defined

More information

An Overview of Residuated Kleene Algebras and Lattices Peter Jipsen Chapman University, California. 2. Background: Semirings and Kleene algebras

An Overview of Residuated Kleene Algebras and Lattices Peter Jipsen Chapman University, California. 2. Background: Semirings and Kleene algebras An Overview of Residuated Kleene Algebras and Lattices Peter Jipsen Chapman University, California 1. Residuated Lattices with iteration 2. Background: Semirings and Kleene algebras 3. A Gentzen system

More information

CS632 Notes on Relational Query Languages I

CS632 Notes on Relational Query Languages I CS632 Notes on Relational Query Languages I A. Demers 6 Feb 2003 1 Introduction Here we define relations, and introduce our notational conventions, which are taken almost directly from [AD93]. We begin

More information

With Question/Answer Animations. Chapter 2

With Question/Answer Animations. Chapter 2 With Question/Answer Animations Chapter 2 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Sequences and Summations Types of

More information

Databases 2011 The Relational Algebra

Databases 2011 The Relational Algebra Databases 2011 Christian S. Jensen Computer Science, Aarhus University What is an Algebra? An algebra consists of values operators rules Closure: operations yield values Examples integers with +,, sets

More information

1 Predicates and Quantifiers

1 Predicates and Quantifiers 1 Predicates and Quantifiers We have seen how to represent properties of objects. For example, B(x) may represent that x is a student at Bryn Mawr College. Here B stands for is a student at Bryn Mawr College

More information

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

More information

Parallel-Correctness and Transferability for Conjunctive Queries

Parallel-Correctness and Transferability for Conjunctive Queries Parallel-Correctness and Transferability for Conjunctive Queries Tom J. Ameloot1 Gaetano Geck2 Bas Ketsman1 Frank Neven1 Thomas Schwentick2 1 Hasselt University 2 Dortmund University Big Data Too large

More information

A Dichotomy. in in Probabilistic Databases. Joint work with Robert Fink. for Non-Repeating Queries with Negation Queries with Negation

A Dichotomy. in in Probabilistic Databases. Joint work with Robert Fink. for Non-Repeating Queries with Negation Queries with Negation Dichotomy for Non-Repeating Queries with Negation Queries with Negation in in Probabilistic Databases Robert Dan Olteanu Fink and Dan Olteanu Joint work with Robert Fink Uncertainty in Computation Simons

More information

Fuzzy Sets. Mirko Navara navara/fl/fset printe.pdf February 28, 2019

Fuzzy Sets. Mirko Navara   navara/fl/fset printe.pdf February 28, 2019 The notion of fuzzy set. Minimum about classical) sets Fuzzy ets Mirko Navara http://cmp.felk.cvut.cz/ navara/fl/fset printe.pdf February 8, 09 To aviod problems of the set theory, we restrict ourselves

More information

CS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:

CS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic: x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which

More information

Algebraic Expressions

Algebraic Expressions Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly

More information

Informal Statement Calculus

Informal Statement Calculus FOUNDATIONS OF MATHEMATICS Branches of Logic 1. Theory of Computations (i.e. Recursion Theory). 2. Proof Theory. 3. Model Theory. 4. Set Theory. Informal Statement Calculus STATEMENTS AND CONNECTIVES Example

More information

Lecture 2: Syntax. January 24, 2018

Lecture 2: Syntax. January 24, 2018 Lecture 2: Syntax January 24, 2018 We now review the basic definitions of first-order logic in more detail. Recall that a language consists of a collection of symbols {P i }, each of which has some specified

More information

Propositional Logic: Models and Proofs

Propositional Logic: Models and Proofs Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505

More information

Annotation algebras for RDFS

Annotation algebras for RDFS University of Edinburgh Semantic Web in Provenance Management, 2010 RDF definition U a set of RDF URI references (s, p, o) U U U an RDF triple s subject p predicate o object A finite set of triples an

More information

FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY REVIEW for MIDTERM 1 THURSDAY Feb 6 Midterm 1 will cover everything we have seen so far The PROBLEMS will be from Sipser, Chapters 1, 2, 3 It will be

More information

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that:

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: Equivalence MA Relations 274 and Partitions Scott Taylor 1. EQUIVALENCE RELATIONS Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: (1) is reflexive. That is, (2) is

More information

Sets and Motivation for Boolean algebra

Sets and Motivation for Boolean algebra SET THEORY Basic concepts Notations Subset Algebra of sets The power set Ordered pairs and Cartesian product Relations on sets Types of relations and their properties Relational matrix and the graph of

More information

Approximate Lineage for Probabilistic Databases. University of Washington Technical Report UW TR: #TR

Approximate Lineage for Probabilistic Databases. University of Washington Technical Report UW TR: #TR Approximate Lineage for Probabilistic Databases University of Washington Technical Report UW TR: #TR2008-03-02 Christopher Ré and Dan Suciu {chrisre,suciu}@cs.washington.edu Abstract In probabilistic databases,

More information

3 Propositional Logic

3 Propositional Logic 3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists

More information

A Logic Primer. Stavros Tripakis University of California, Berkeley. Stavros Tripakis (UC Berkeley) EE 144/244, Fall 2014 A Logic Primer 1 / 35

A Logic Primer. Stavros Tripakis University of California, Berkeley. Stavros Tripakis (UC Berkeley) EE 144/244, Fall 2014 A Logic Primer 1 / 35 EE 144/244: Fundamental Algorithms for System Modeling, Analysis, and Optimization Fall 2014 A Logic Primer Stavros Tripakis University of California, Berkeley Stavros Tripakis (UC Berkeley) EE 144/244,

More information

ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes)

ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes) ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes) Steve Vickers CS Theory Group Birmingham 2. Theories and models Categorical approach to many-sorted

More information

Complete Partial Orders, PCF, and Control

Complete Partial Orders, PCF, and Control Complete Partial Orders, PCF, and Control Andrew R. Plummer TIE Report Draft January 2010 Abstract We develop the theory of directed complete partial orders and complete partial orders. We review the syntax

More information

Database Theory VU , SS Complexity of Query Evaluation. Reinhard Pichler

Database Theory VU , SS Complexity of Query Evaluation. Reinhard Pichler Database Theory Database Theory VU 181.140, SS 2018 5. Complexity of Query Evaluation Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 17 April, 2018 Pichler

More information

On the satisfiability problem for a 4-level quantified syllogistic and some applications to modal logic

On the satisfiability problem for a 4-level quantified syllogistic and some applications to modal logic On the satisfiability problem for a 4-level quantified syllogistic and some applications to modal logic Domenico Cantone and Marianna Nicolosi Asmundo Dipartimento di Matematica e Informatica Università

More information

GAV-sound with conjunctive queries

GAV-sound with conjunctive queries GAV-sound with conjunctive queries Source and global schema as before: source R 1 (A, B),R 2 (B,C) Global schema: T 1 (A, C), T 2 (B,C) GAV mappings become sound: T 1 {x, y, z R 1 (x,y) R 2 (y,z)} T 2

More information

University of Aberdeen, Computing Science CS2013 Predicate Logic 4 Kees van Deemter

University of Aberdeen, Computing Science CS2013 Predicate Logic 4 Kees van Deemter University of Aberdeen, Computing Science CS2013 Predicate Logic 4 Kees van Deemter 01/11/16 Kees van Deemter 1 First-Order Predicate Logic (FOPL) Lecture 4 Making numerical statements: >0, 1,>2,1,2

More information

On Factorisation of Provenance Polynomials

On Factorisation of Provenance Polynomials On Factorisation of Provenance Polynomials Dan Olteanu and Jakub Závodný Oxford University Computing Laboratory Wolfson Building, Parks Road, OX1 3QD, Oxford, UK 1 Introduction Tracking and managing provenance

More information

A Logic Primer. Stavros Tripakis University of California, Berkeley

A Logic Primer. Stavros Tripakis University of California, Berkeley EE 144/244: Fundamental Algorithms for System Modeling, Analysis, and Optimization Fall 2015 A Logic Primer Stavros Tripakis University of California, Berkeley Stavros Tripakis (UC Berkeley) EE 144/244,

More information

LESSON 7.1 FACTORING POLYNOMIALS I

LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of

More information

Lecture Notes in Real Analysis Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay

Lecture Notes in Real Analysis Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay Lecture Notes in Real Analysis 2010 Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay August 6, 2010 Lectures 1-3 (I-week) Lecture 1 Why real numbers? Example 1 Gaps in the

More information

Introduction to Kleene Algebra Lecture 13 CS786 Spring 2004 March 15, 2004

Introduction to Kleene Algebra Lecture 13 CS786 Spring 2004 March 15, 2004 Introduction to Kleene Algebra Lecture 13 CS786 Spring 2004 March 15, 2004 Models of KAT In this lecture we show that the equational theories of KAT, KAT (the star-continuous Kleene algebras with tests),

More information

Packet #1: Logic & Proofs. Applied Discrete Mathematics

Packet #1: Logic & Proofs. Applied Discrete Mathematics Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should

More information

Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)

Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a) Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5

More information

Data Cleaning and Query Answering with Matching Dependencies and Matching Functions

Data Cleaning and Query Answering with Matching Dependencies and Matching Functions Data Cleaning and Query Answering with Matching Dependencies and Matching Functions Leopoldo Bertossi 1, Solmaz Kolahi 2, and Laks V. S. Lakshmanan 2 1 Carleton University, Ottawa, Canada. bertossi@scs.carleton.ca

More information

CHAPTER 4. βs as a semigroup

CHAPTER 4. βs as a semigroup CHAPTER 4 βs as a semigroup In this chapter, we assume that (S, ) is an arbitrary semigroup, equipped with the discrete topology. As explained in Chapter 3, we will consider S as a (dense ) subset of its

More information

Algebras. Larry Moss Indiana University, Bloomington. TACL 13 Summer School, Vanderbilt University

Algebras. Larry Moss Indiana University, Bloomington. TACL 13 Summer School, Vanderbilt University 1/39 Algebras Larry Moss Indiana University, Bloomington TACL 13 Summer School, Vanderbilt University 2/39 Binary trees Let T be the set which starts out as,,,, 2/39 Let T be the set which starts out as,,,,

More information

Datalog and Constraint Satisfaction with Infinite Templates

Datalog and Constraint Satisfaction with Infinite Templates Datalog and Constraint Satisfaction with Infinite Templates Manuel Bodirsky 1 and Víctor Dalmau 2 1 CNRS/LIX, École Polytechnique, bodirsky@lix.polytechnique.fr 2 Universitat Pompeu Fabra, victor.dalmau@upf.edu

More information

The non-logical symbols determine a specific F OL language and consists of the following sets. Σ = {Σ n } n<ω

The non-logical symbols determine a specific F OL language and consists of the following sets. Σ = {Σ n } n<ω 1 Preliminaries In this chapter we first give a summary of the basic notations, terminology and results which will be used in this thesis. The treatment here is reduced to a list of definitions. For the

More information

U + V = (U V ) (V U), UV = U V.

U + V = (U V ) (V U), UV = U V. Solution of Some Homework Problems (3.1) Prove that a commutative ring R has a unique 1. Proof: Let 1 R and 1 R be two multiplicative identities of R. Then since 1 R is an identity, 1 R = 1 R 1 R. Since

More information

Completeness of Star-Continuity

Completeness of Star-Continuity Introduction to Kleene Algebra Lecture 5 CS786 Spring 2004 February 9, 2004 Completeness of Star-Continuity We argued in the previous lecture that the equational theory of each of the following classes

More information

1 First-order logic. 1 Syntax of first-order logic. 2 Semantics of first-order logic. 3 First-order logic queries. 2 First-order query evaluation

1 First-order logic. 1 Syntax of first-order logic. 2 Semantics of first-order logic. 3 First-order logic queries. 2 First-order query evaluation Knowledge Bases and Databases Part 1: First-Order Queries Diego Calvanese Faculty of Computer Science Master of Science in Computer Science A.Y. 2007/2008 Overview of Part 1: First-order queries 1 First-order

More information

MATH 145 LECTURE NOTES. Zhongwei Zhao. My Lecture Notes for MATH Fall

MATH 145 LECTURE NOTES. Zhongwei Zhao. My Lecture Notes for MATH Fall MATH 145 LECTURE NOTES Zhongwei Zhao My Lecture Notes for MATH 145 2016 Fall December 2016 Lecture 1, Sept. 9 Course Orientation and Organization About the Professor Stephen New MC 5419 Ext 35554 Email:

More information

Herbrand Theorem, Equality, and Compactness

Herbrand Theorem, Equality, and Compactness CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Herbrand Theorem, Equality, and Compactness The Herbrand Theorem We now consider a complete method for proving the unsatisfiability of sets of first-order

More information

We define the multi-step transition function T : S Σ S as follows. 1. For any s S, T (s,λ) = s. 2. For any s S, x Σ and a Σ,

We define the multi-step transition function T : S Σ S as follows. 1. For any s S, T (s,λ) = s. 2. For any s S, x Σ and a Σ, Distinguishability Recall A deterministic finite automaton is a five-tuple M = (S,Σ,T,s 0,F) where S is a finite set of states, Σ is an alphabet the input alphabet, T : S Σ S is the transition function,

More information

Queries and Materialized Views on Probabilistic Databases

Queries and Materialized Views on Probabilistic Databases Queries and Materialized Views on Probabilistic Databases Nilesh Dalvi Christopher Ré Dan Suciu September 11, 2008 Abstract We review in this paper some recent yet fundamental results on evaluating queries

More information

Foundations of Logic Programming

Foundations of Logic Programming Foundations of Logic Programming Deductive Logic e.g. of use: Gypsy specifications and proofs About deductive logic (Gödel, 1931) Interesting systems (with a finite number of axioms) are necessarily either:

More information

Query answering using views

Query answering using views Query answering using views General setting: database relations R 1,...,R n. Several views V 1,...,V k are defined as results of queries over the R i s. We have a query Q over R 1,...,R n. Question: Can

More information

SCTT The pqr-method august 2016

SCTT The pqr-method august 2016 SCTT The pqr-method august 2016 A. Doledenok, M. Fadin, À. Menshchikov, A. Semchankau Almost all inequalities considered in our project are symmetric. Hence if plugging (a 0, b 0, c 0 ) into our inequality

More information

An Algebraic Approach to Energy Problems I -Continuous Kleene ω-algebras

An Algebraic Approach to Energy Problems I -Continuous Kleene ω-algebras Acta Cybernetica 23 (2017) 203 228. An Algebraic Approach to Energy Problems I -Continuous Kleene ω-algebras Zoltán Ésika, Uli Fahrenberg b, Axel Legay c, and Karin Quaas d Abstract Energy problems are

More information

5 Set Operations, Functions, and Counting

5 Set Operations, Functions, and Counting 5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,

More information

First-Order Theorem Proving and Vampire

First-Order Theorem Proving and Vampire First-Order Theorem Proving and Vampire Laura Kovács 1,2 and Martin Suda 2 1 TU Wien 2 Chalmers Outline Introduction First-Order Logic and TPTP Inference Systems Saturation Algorithms Redundancy Elimination

More information

Topics in Probabilistic and Statistical Databases. Lecture 2: Representation of Probabilistic Databases. Dan Suciu University of Washington

Topics in Probabilistic and Statistical Databases. Lecture 2: Representation of Probabilistic Databases. Dan Suciu University of Washington Topics in Probabilistic and Statistical Databases Lecture 2: Representation of Probabilistic Databases Dan Suciu University of Washington 1 Review: Definition The set of all possible database instances:

More information

Chapter 1. Sets and Numbers

Chapter 1. Sets and Numbers Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write

More information

review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17

review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17 1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

More information

Scalable Uncertainty Management

Scalable Uncertainty Management Scalable Uncertainty Management 05 Query Evaluation in Probabilistic Databases Rainer Gemulla Jun 1, 2012 Overview In this lecture Primer: relational calculus Understand complexity of query evaluation

More information

A Type Theory for Probabilistic and Bayesian Reasoning

A Type Theory for Probabilistic and Bayesian Reasoning A Type Theory for Probabilistic and Bayesian Reasoning Robin Adams 1 and Bart Jacobs 1 1 Institute for Computing and Information Sciences, Radboud University, the Netherlands {r.adams,bart}@cs.ru.nl Abstract

More information

Logic and Databases. Phokion G. Kolaitis. UC Santa Cruz & IBM Research Almaden. Lecture 4 Part 1

Logic and Databases. Phokion G. Kolaitis. UC Santa Cruz & IBM Research Almaden. Lecture 4 Part 1 Logic and Databases Phokion G. Kolaitis UC Santa Cruz & IBM Research Almaden Lecture 4 Part 1 1 Thematic Roadmap Logic and Database Query Languages Relational Algebra and Relational Calculus Conjunctive

More information

The Complexity of Computing the Behaviour of Lattice Automata on Infinite Trees

The Complexity of Computing the Behaviour of Lattice Automata on Infinite Trees The Complexity of Computing the Behaviour of Lattice Automata on Infinite Trees Karsten Lehmann a, Rafael Peñaloza b a Optimisation Research Group, NICTA Artificial Intelligence Group, Australian National

More information

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In

More information

Applied Logic. Lecture 1 - Propositional logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw

Applied Logic. Lecture 1 - Propositional logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw Applied Logic Lecture 1 - Propositional logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018

More information

DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY

DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY HANDOUT ABSTRACT ALGEBRA MUSTHOFA DEPARTMENT OF MATHEMATIC EDUCATION MATHEMATIC AND NATURAL SCIENCE FACULTY 2012 BINARY OPERATION We are all familiar with addition and multiplication of two numbers. Both

More information

2MA105 Algebraic Structures I

2MA105 Algebraic Structures I 2MA105 Algebraic Structures I Per-Anders Svensson http://homepage.lnu.se/staff/psvmsi/2ma105.html Lecture 12 Partially Ordered Sets Lattices Bounded Lattices Distributive Lattices Complemented Lattices

More information

Qualifying Exam Logic August 2005

Qualifying Exam Logic August 2005 Instructions: Qualifying Exam Logic August 2005 If you signed up for Computability Theory, do two E and two C problems. If you signed up for Model Theory, do two E and two M problems. If you signed up

More information

THE EXT ALGEBRA AND A NEW GENERALISATION OF D-KOSZUL ALGEBRAS

THE EXT ALGEBRA AND A NEW GENERALISATION OF D-KOSZUL ALGEBRAS THE EXT ALGEBRA AND A NEW GENERALISATION OF D-KOSZUL ALGEBRAS JOANNE LEADER AND NICOLE SNASHALL Abstract. We generalise Koszul and D-Koszul algebras by introducing a class of graded algebras called (D,

More information

Path Queries under Distortions: Answering and Containment

Path Queries under Distortions: Answering and Containment Path Queries under Distortions: Answering and Containment Gosta Grahne Concordia University Alex Thomo Suffolk University Foundations of Information and Knowledge Systems (FoIKS 04) Postulate 1 The world

More information

Introduction to Kleene Algebras

Introduction to Kleene Algebras Introduction to Kleene Algebras Riccardo Pucella Basic Notions Seminar December 1, 2005 Introduction to Kleene Algebras p.1 Idempotent Semirings An idempotent semiring is a structure S = (S, +,, 1, 0)

More information

The Structure of Inverses in Schema Mappings

The Structure of Inverses in Schema Mappings To appear: J. ACM The Structure of Inverses in Schema Mappings Ronald Fagin and Alan Nash IBM Almaden Research Center 650 Harry Road San Jose, CA 95120 Contact email: fagin@almaden.ibm.com Abstract A schema

More information

Closure Properties of Regular Languages. Union, Intersection, Difference, Concatenation, Kleene Closure, Reversal, Homomorphism, Inverse Homomorphism

Closure Properties of Regular Languages. Union, Intersection, Difference, Concatenation, Kleene Closure, Reversal, Homomorphism, Inverse Homomorphism Closure Properties of Regular Languages Union, Intersection, Difference, Concatenation, Kleene Closure, Reversal, Homomorphism, Inverse Homomorphism Closure Properties Recall a closure property is a statement

More information

Data Cleaning and Query Answering with Matching Dependencies and Matching Functions

Data Cleaning and Query Answering with Matching Dependencies and Matching Functions Data Cleaning and Query Answering with Matching Dependencies and Matching Functions Leopoldo Bertossi Carleton University Ottawa, Canada bertossi@scs.carleton.ca Solmaz Kolahi University of British Columbia

More information

Part I: Propositional Calculus

Part I: Propositional Calculus Logic Part I: Propositional Calculus Statements Undefined Terms True, T, #t, 1 False, F, #f, 0 Statement, Proposition Statement/Proposition -- Informal Definition Statement = anything that can meaningfully

More information

INTRODUCTION TO RELATIONAL DATABASE SYSTEMS

INTRODUCTION TO RELATIONAL DATABASE SYSTEMS INTRODUCTION TO RELATIONAL DATABASE SYSTEMS DATENBANKSYSTEME 1 (INF 3131) Torsten Grust Universität Tübingen Winter 2017/18 1 THE RELATIONAL ALGEBRA The Relational Algebra (RA) is a query language for

More information

Propositional Logic: Part II - Syntax & Proofs 0-0

Propositional Logic: Part II - Syntax & Proofs 0-0 Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems

More information

A GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING. David F. Anderson and Elizabeth F. Lewis

A GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING. David F. Anderson and Elizabeth F. Lewis International Electronic Journal of Algebra Volume 20 (2016) 111-135 A GENERAL HEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUAIVE RING David F. Anderson and Elizabeth F. Lewis Received: 28 April 2016 Communicated

More information

CHAPTER 10. Gentzen Style Proof Systems for Classical Logic

CHAPTER 10. Gentzen Style Proof Systems for Classical Logic CHAPTER 10 Gentzen Style Proof Systems for Classical Logic Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning

More information

Database Theory VU , SS Ehrenfeucht-Fraïssé Games. Reinhard Pichler

Database Theory VU , SS Ehrenfeucht-Fraïssé Games. Reinhard Pichler Database Theory Database Theory VU 181.140, SS 2018 7. Ehrenfeucht-Fraïssé Games Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 15 May, 2018 Pichler 15

More information

Algebra I. Book 2. Powered by...

Algebra I. Book 2. Powered by... Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........

More information

Semantic Metatheory of SL: Mathematical Induction

Semantic Metatheory of SL: Mathematical Induction Semantic Metatheory of SL: Mathematical Induction Preliminary matters: why and when do we need Mathematical Induction? We need it when we want prove that a certain claim (n) holds for all n N (0, 1, 2,

More information

Equivalence of SQL Queries In Presence of Embedded Dependencies

Equivalence of SQL Queries In Presence of Embedded Dependencies Equivalence of SQL Queries In Presence of Embedded Dependencies Rada Chirkova Department of Computer Science NC State University, Raleigh, NC 27695, USA chirkova@csc.ncsu.edu Michael R. Genesereth Department

More information

Any Wizard of Oz fans? Discrete Math Basics. Outline. Sets. Set Operations. Sets. Dorothy: How does one get to the Emerald City?

Any Wizard of Oz fans? Discrete Math Basics. Outline. Sets. Set Operations. Sets. Dorothy: How does one get to the Emerald City? Any Wizard of Oz fans? Discrete Math Basics Dorothy: How does one get to the Emerald City? Glynda: It is always best to start at the beginning Outline Sets Relations Proofs Sets A set is a collection of

More information

A Weak Bisimulation for Weighted Automata

A Weak Bisimulation for Weighted Automata Weak Bisimulation for Weighted utomata Peter Kemper College of William and Mary Weighted utomata and Semirings here focus on commutative & idempotent semirings Weak Bisimulation Composition operators Congruence

More information

i=1 β i,i.e. = β 1 x β x β 1 1 xβ d

i=1 β i,i.e. = β 1 x β x β 1 1 xβ d 66 2. Every family of seminorms on a vector space containing a norm induces ahausdorff locally convex topology. 3. Given an open subset Ω of R d with the euclidean topology, the space C(Ω) of real valued

More information

On the Boolean Algebra of Shape Analysis Constraints

On the Boolean Algebra of Shape Analysis Constraints On the Boolean Algebra of Shape Analysis Constraints Viktor Kuncak and Martin Rinard Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA 02139, USA

More information

Predicate Calculus. Formal Methods in Verification of Computer Systems Jeremy Johnson

Predicate Calculus. Formal Methods in Verification of Computer Systems Jeremy Johnson Predicate Calculus Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. Motivation 1. Variables, quantifiers and predicates 2. Syntax 1. Terms and formulas 2. Quantifiers, scope

More information

POSITIVE POLYNOMIALS LECTURE NOTES (09: 10/05/10) Contents

POSITIVE POLYNOMIALS LECTURE NOTES (09: 10/05/10) Contents POSITIVE POLYNOMIALS LECTURE NOTES (09: 10/05/10) SALMA KUHLMANN Contents 1. Proof of Hilbert s Theorem (continued) 1 2. The Motzkin Form 2 3. Robinson Method (1970) 3 3. The Robinson Form 4 1. PROOF OF

More information

Logic Part I: Classical Logic and Its Semantics

Logic Part I: Classical Logic and Its Semantics Logic Part I: Classical Logic and Its Semantics Max Schäfer Formosan Summer School on Logic, Language, and Computation 2007 July 2, 2007 1 / 51 Principles of Classical Logic classical logic seeks to model

More information

CS 154, Lecture 4: Limitations on DFAs (I), Pumping Lemma, Minimizing DFAs

CS 154, Lecture 4: Limitations on DFAs (I), Pumping Lemma, Minimizing DFAs CS 154, Lecture 4: Limitations on FAs (I), Pumping Lemma, Minimizing FAs Regular or Not? Non-Regular Languages = { w w has equal number of occurrences of 01 and 10 } REGULAR! C = { w w has equal number

More information

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter

More information

Subdirectly Irreducible Modes

Subdirectly Irreducible Modes Subdirectly Irreducible Modes Keith A. Kearnes Abstract We prove that subdirectly irreducible modes come in three very different types. From the description of the three types we derive the results that

More information

UNIT-III REGULAR LANGUAGES

UNIT-III REGULAR LANGUAGES Syllabus R9 Regulation REGULAR EXPRESSIONS UNIT-III REGULAR LANGUAGES Regular expressions are useful for representing certain sets of strings in an algebraic fashion. In arithmetic we can use the operations

More information