112 lines
4.9 KiB
Plaintext
112 lines
4.9 KiB
Plaintext
- 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,
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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),
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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
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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
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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)
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evaluate to true.
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- Support
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- Support: The set of values of Y for which R(Y)
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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
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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)
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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
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support: Because Z is safe in S, it does not need to be
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safe in R.
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- In general, a variable is safe in a conjunction of terms
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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
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(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]
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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
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Z < 10, Multiply multiplicity by X }
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- Sequence of transformations, each modifying the
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output of the last
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- Pipelining technique sometimes referred to as a
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“Monad"
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- ^— all actually used in practice. DBToaster’s AGCA and
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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,
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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
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Product)
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