Logical Query Languages Datalog Example Motivatio n beer Likesdrinker beer price Sellsbar Logical rules extend more naturally to bar Frequentsdrinker recursive queries than do es relational algebra ✦ Used in SQL recursion Happyd Frequentsdbar AND Logical rules form the basis for many Likesdbeer AND informationintegration systems and Sellsbarbeerp applications Ab oveisarule Left side head Right side body AND of subgoals Head and subgoals are atoms ✦ Atom predicate and arguments ✦ Predicate relation name or arithmetic predicate eg ✦ Arguments are variables or constants Subgoals not head may optionally be negated by NOT Meaning of Rules Evaluation of Rules Head is true of its arguments if there exist values Two dual approaches for local variables those in b o dy not in head that Variablebased Consider all p ossible make all of the subgoals true assignments of values to variables If all If no negation or arithmetic comparisons just subgoals are true add the head to the result natural join the subgoals and pro ject onto the relation head variables Tuplebased Consider all assignments of tuples to subgoals that makeeach subgoal Example true If the variables are assigned consistent Ab ove rule equivalent to Happyd values add the head to the result Frequents Likes Sells dr ink er Example VariableBased Assignment Sxy Rxz AND Rzy AND NOT Rxy R B A Only assignments that make rst subgoal true Example TupleBased Assignment Trick start with the p ositive not negated x z relational not arithmetic subgoals only x z Sxy Rxz AND Rzy In case y makes second subgoal true AND NOT Rxy Since is not in R the third subgoal is R also true B A ✦ Thus add x y to relation S In case no value of y makes the second subgoal true Thus S B A Four assignments of tuples to subgoals R x z R z y Only the second gives a consistentvalue to z That assignmentalsomakes NOT Rxy true Thus is the only tuple for the head Safety Datalog Programs A rule can make no sense if variables app ear in A collect i on of rules is a Datalogprogram funnyways Predicatesrelati ons divide into two classes Examples ✦ EDB extensional database relation stored in DB Sx Ry ✦ IDB intensional database relation Sx NOT Rx dened by one or more rules Sx Ry AND x y A predicate must b e IDB or EDB not b oth In each of these cases the result is innite even if ✦ Thus an IDB predicate can app ear in the the relation R is nite b o dy or head of a rule EDB only in the To make sense as a database op eration we body need to require three things of a variable x denition of safety If x app ears in either The head A negated subgoal or An arithmetic comparison then x must also app ear in a nonnegated ordinary relational subgoal of the b o dy We insist that rules b e safe henceforth Example ExpressivePower of Datalog Convert the following SQL Find the Nonrecursive Datalog classical relational manufacturers of the b eers Jo e sells algebra manf Beersname ✦ See discussion in text beer price Sellsbar Datalog simulates SQL selectfromwhere SELECT manf without aggregation and grouping FROM Beers Recursive Datalog expresses queries that WHERE name IN cannot b e expressed in SQL SELECT beer FROM Sells But none of these languages havefull WHERE bar Joes Bar expressivepower Turing completeness to a Datalog program JoeSellsb SellsJoes Bar b p Answerm JoeSellsb AND Beersbm Note Beers Sells EDBJoeSells Answer IDB Recursion Iterative FixedPoint Evaluates Recursive Rules IDB predicate P depends on predicate Q if there is a rule with P in the head and Q in a subgoal Start Draw a graph no des IDB predicates arc IDB P Q means P dep ends on Q Cycles i recursive Recursive Example Apply rules to IDB EDB Sibxy Parxp AND Paryp AND x y Cousinxy Sibxy Cousinxy Parxxp Change AND Paryyp done to IDB yes no AND Cousinxpyp Sib Cousin Example Initial EDB Par Round b c c e add g h j k a d Round b c c e add g h j k b c e Round f g f h add g i h i i k f g h Round k k add i j j k i Note b ecause of symmetry Sib and Cousin facts app ear in pairs so we shall mention only x y when b oth x y and y x are meant Stratied Negation Problem with Recursive Negation Consider Negation wrapp ed inside a recursion makes no sense Px Qx AND NOT Px Even when negation and recursion are Q EDBf g separated there can b e ambiguity ab out what Compute IDB P iterativel y the rules mean and some one meaning must b e selected ✦ Initial ly P Stratiednegation is an additional restrainton ✦ Round P f g recursive rules like safety that solves b oth ✦ Round P etc etc problems It rules out negation wrapp ed in recursion When negation is separate from recursion it yields the intuitivel y correct meaning of rules the stratiedmodel Strata Example Intuitively stratum of an IDB predicate Which target no des cannot b e reached from any maximum numb er of negations you can pass source no de through on the way to an EDB predicate Reachx Sourcex Reachx Reachy AND Arcyx Must not b e in stratied rules NoReachx Targetx Dene stratum graph AND NOT Reachx ✦ No des IDB predicates ✦ Arc P Q if Q app ears in the b o dy of a NoReach rule with head P ✦ Lab el that arc if Q is in a negated Reach subgoal Example Px Qx AND NOT Px P Computing Strata Example Stratum of an IDB predicate A maximum Reachx Sourcex number of arcs on anypathfromA in the Reachx Reachy AND Arcyx stratum graph NoReachx Targetx Examples AND NOT Reachx For rst example stratum of P is EDB For second example stratum of Reach is ✦ Source fg stratum of NoReach is ✦ Arc f g Stratied Negation ✦ Target f g A Datalog program is stratied if every IDB predicate has a nite stratum Stratied Mo del source target target If a Datalog program is stratied we can compute First compute Reach f g stratum the relations for the IDB predicates lowest stratumrst Next compute NoReach fg Is the Stratied Solution Obvious Not really There is another mo del that makes the rules true no matter what values we substitute for the variables ✦ Reach f g ✦ NoReach Rememb er the only way to make a Datalog rule false is to nd values for the variables that make the b o dy true and the head false ✦ For this mo del the heads of the rules for Reach are true for all values and in the rule for NoReach the subgoal NOT Reachx assures that the b o dy cannot b e true .