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Datalog
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Propositional Calculus (0th order logic)
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Facts (P), Basic operations: (Not, And, Or), Implication
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Example Facts: AliceWentToTheStore, BobWentToTheStore, AliceWentToHome
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Implication:
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(if P and Q then R)
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=== R \or \not P \or \not Q
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=== “Horn Clause”
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1st order logic
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Goal: Quantification
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Challenge, need way to enumerate classes/sets of facts
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Groups of facts: WentTo(Store, Alice), WentTo(Store, Bob), WentTo(Home, Alice)
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For all facts in a group (\forall) a property holds
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There exists a fact in a group (\exists) such that a property holds
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New way to discuss implication:
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Given one or more facts P(X,Y), Q(X), …., “infer” a new fact R(Y)
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If WentTo(X, Y) and ShoppingAt(X) then ShoppingDone(Y)
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\forall Y, \exists X: If P(X,Y) and Q(X) then R(Y)
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\forall Y, \exists X: \not P(X,Y) \or \not Q(X) \or R(Y)
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Which elements of R must be true?
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SELECT Y FROM P NATURAL JOIN Q INTO R
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Datalog
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R(Y) -= P(X,Y), Q(X)
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Head, Body
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Find values of Y for which R is true?
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Find a value of X for which P(X,Y) and Q(X) are true
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What about R(Y, Z) -= P(X,Y), Q(X)
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Alternative View:
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R(Y) is a function
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Dom => Bool
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Given Y, find a value of X for which P(X,Y) and Q(X) evaluate to true.
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Support
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Support: The set of values of Y for which R(Y) evaluates to true
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Finite Support: The support set has a fixed size
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If P(X,Y), Q(X) have finite support, so does R(Y)
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actually, we can do a bit better… to be discussed shortly
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Natural consequence:
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R(Y,Z) is true for any value of Z as long as R(Y) would be true.
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R(Y,Z) has an infinite support!
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Z is “unsafe” or “unbound"
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Y is “bound” or “safe”
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Safety and Support
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Assume we have a S(Y,Z) with finite support.
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R(Y,Z) dies not have finite support
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What about ( S(Y,Z) and R(Y,Z) )
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Interestingly enough, this actually does have finite support: Because Z is safe in S, it does not need to be safe in R.
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In general, a variable is safe in a conjunction of terms IFF it is safe in at least one of the terms.
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What else can you do with this idea?
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Functions F(X) -> ???
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What about other functions?
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F(X,Y) -> true if (X < Y)
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How about to natural numbers?
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Simple way to express Bags!
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R(X) -> N = number of instances of X in R (multiplicity)
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Leads to some interesting math:
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R(X) U S(X) === R(X) + S(X)
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R(X) |><| S(X) === R(X) * S(X)
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Q(X) -= R(X,Y) * S(X, Y) === Aggregation: Sum[Y] R(X,Y) * S(X,Y)
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SELECT COUNT(*) FROM R NATURAL JOIN S
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How about real numbers?
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R(X) -> Multiplicity
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F(X) -> {X}
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Q() -= SUM[X] ( R(X) * {X} )
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SELECT SUM(X) FROM R
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Q() -= SUM[X] ( R(X,Y) * S(Y,Z) * {Z < 10} * {X} )
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SELECT SUM(R.X) FROM R NATURAL JOIN S WHERE S.Z < 10
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OR: Aggregate { Start with R, Join with S, Filter on Z < 10, Multiply multiplicity by X }
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Sequence of transformations, each modifying the output of the last
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Pipelining technique sometimes referred to as a “Monad"
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^— all actually used in practice. DBToaster’s AGCA and LogicBlox LogiQL
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Connections to Rings:
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Semiring:
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Set/Type, +: SxS->S, 0, *: SxS->, 1
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+/* associative, commutative
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+ distributive over *
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0+x = x
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0*x = 0
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1*x = x
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Ring = Semiring + Invertor:
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-x + x = 0
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U = + // |><| = *
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Monad Algebra
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Problem: Nested data! We want to transform it in bulk
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Same idea of pipelining operations:
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Applied to
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Overview
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Types: Dom, Tuple(X,Y), List(X)
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Basic Ops: Const, Emptyset, Singleton, Map, Union, Tuple, DeTuple, f(DOM)
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Key Features: Fold
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If/Then/Else
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Key Features: Tensor Strength — PairWith (equiv: Cross Product)
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