of Review Logic as a Query Language
Datalog programs are collections of rules, which are
Horn clauses or if-then expressions.
Example
The following rules express what is needed to
make" a le. It assumes these relations or EDB
extensional database predicates are available:
1. sourceF : F is a source le, i.e., stored in
the le system.
2. includesF; G: le F includes le G.
3. createF; P; G: we create le F by applying
process P to le G.
reqF,F
reqF,G
reqF,G
reqF,G
Rules
::::-
sourceF
includesF,G
createF,P,G
reqF,H & reqH,G
Head :- Body
is read if"
Atom = predicate applied to arguments.
Head is atom.
Body is logical AND of zero or more atoms.
Atoms of body are called subgoals.
Head predicate is IDB intensional database =
predicate de ned by rules. Body subgoals may
have IDB or EDB predicates.
Datalog program = collection of rules. One
IDB predicate is distinguished and represents
result of program.
:-
Meaning of Rules
The head is true for its arguments whenever there
exist values for any local variables those that
appear in the body, but not the head that make
all the subgoals true.
Extensions
1. Negated subgoals. Example:
cycleF :- reqF,F & NOT sourceF
2. Constants as arguments. Example:
1
reqF,"stdio.h" :- typeF,"cCode"
3. Arithmetic subgoals. Example:
compositeA :- dividesB,A &
B 1&B
A
3
Opposite of an arithmetic atom is a
relational atom.
Applying Rules Naive Evaluation"
Given an EDB:
1. Start with all IDB relations empty.
2. Instantiate with constants variables of all
rules in all possible ways. If all subgoals
become true, then infer that the head is true.
3. Repeat 2 in rounds," as long as new IDB
facts can be inferred.
2 makes sense and is nite, as long as
rules are safe = each variable that appears
anywhere in the rule appears in some
nonnegated, nonarithmetic subgoal of the
body.
Limit of 1 3 = Least xed point of the
rules and EDB.
Seminaive Evaluation
More e cient approach to evaluating rules.
Based on principle that if at round i a fact is
inferred for the rst time, then we must have
used a rule in which one or more subgoals
were instantiated to facts that were inferred
on round i , 1.
Thus, for each IDB predicate p, keep both
relation P and relation P ; the latter represents
the new facts for p inferred on the most recent
round.
Outline of SNE Algorithm
1. Initialize IDB relations by using only those
rules without IDB subgoals.
2. Initialize the -IDB relations to be equal to
the corresponding IDB relations.
3. In one round, for each IDB predicate p:
2
a Compute new P by applying each rule
for p, but with one subgoal treated as a
-IDB relation and the others treated as
the correct IDB or EDB relation. Do for
all possible choices of the -subgoal.
b Remove from new P all facts that are
already in P .
c P := P P .
4. Repeat 3 until no changes to any IDB
relation.
Example
1
2
3
4
reqF,F
reqF,G
reqF,G
reqF,G
::::-
sourceF
includesF,G
createF,P,G
reqF,H & reqH,G
Assume EDB relations S , I , C and IDB
relation R, with obvious correspondence to
predicates.
Initialize: R := R := 1=2 S S
I 1 3C .
Iterate until R = ;:
1. R := 1 3R . R R . R
2. R := R , R
3. R := R R
;
;
Models
Model of rules + EDB facts = set of ground
atoms selected to be true such that
1. An EDB fact is selected true i it is in the
given EDB relation.
2. All rules become true under any instantiation
of the variables.
3 Facts not stated true in the model are
assumed false.
3 Only way to falsify a rule is to make each
subgoal true and head the false.
Minimal model = model + no proper subset is
a model.
3
For a Datalog program with only nonnegated,
relational atoms in the bodies, the unique
minimal model is what naive or seminaive
evaluation produces, i.e., the IDB facts we are
forced to deduce.
Moreover, this LFP is reached after a nite
number of rounds, if the EDB is nite.
Function Symbols
Terms built from
1. Constants.
2. Variables.
3. Function symbols applied to terms as
arguments.
3 Example:
,
addr streetmaple; number101
Example
Binary trees de ned by
isTreenull
isTreenodeL,T1,T2 :labelL &
isTreeT1 &
isTreeT2
If labela and labelb are true, infers facts like
,
isTree
nodea; null; null
,
isTree node b; null; nodea; null; null
Application of rules as for Datalog: make all
possible instantiations of variables and infer
head if all subgoals are true.
LFP is still unique minimal model, as long as
subgoals are relational, nonnegated.
But LFP may be reached only after an in nite
number of rounds.
Problems for Datalog With Negation
Recall extra safety condition: variables in
a negated subgoal must appear also in a
nonnegated subgoal.
4
Apply rule as without negation: search for
substitutions that make all subgoals true.
Resulting head is true.
3 But | a subgoal NOT S is true i S is
false.
Example
Failed attempt to express X is a bachelor i there
does not exists a person Y such that X is married
to Y ":
Neither safe, nor correct.
bachelorX :- personX &
NOT marriedX,Y
Suppose fa; b; cg are persons, and marrieda; b.
Substitution X ! a, Y ! c makes both subgoals
true and lets us infer bachelora incorrectly."
The following is a safe" version of the
incorrect program, which makes it clearer why
the above interpretation is right:
bachelorX :- personX &
personY &
NOT marriedX,Y
Correct version:
spouseX :- marriedX,Y
bachelorX :- personX &
NOT spouseX
Multiple Minimal Models
EDB = redX; Y , greenX; Y .
IDB = greenPathX; Y , monopolyX; Y .
1 greenPathX,Y :- greenX,Y
2 greenPathX,Y :greenPathX,Z &
greenPathZ,Y
3 monopolyX,Y :- redX,Y &
NOT greenPathX,Y
EDB data: red1; 2, red2; 3, green1; 2.
green
1
red
2
5
red
3
Model1: greenPath1; 2 + monopoly2; 3
+ EDB.
Model2: greenPath1; 2 + greenPath2; 3
+ greenPath1; 3 + EDB.
Both are minimal.
Dependency Graph
Nodes = predicates.
Arc p ! q if there is a rule with predicate p in
the head and predicate q in some subgoal.
Arc p ! q with label |" if there is a rule
with p in the head and a negated subgoal with
q.
Example
Arcs for monopoly" program:
monopoly
greenPath
green
red
Strati ed Logic Models
Stratum of a predicate p = largest number
of | arcs on a path in dependency graph
originating at p.
Thus, If p depends negatively on q, then
stratump stratumq.
If there are no cycles involving negation i.e.,
no recursive negation, then all strata are
nite.
If a logic program has no recursive negation, it
is strati ed.
Example: The Win" Program
This Datalog program represents winning positions
in a board game, e.g., Nim, where you win by
6
giving your opponent a position with no legal
move.
winX :- moveX,Y & NOT winY
Win" is not strati ed.
win
move
Strati ed Models
For strati ed programs, the strati ed model is
computed bottom-up."
3 Work from lowest strata to highest.
3 Compute the LFP for a stratum assuming
subgoal NOT pX1 ; : : :; X is true i
pX1 ; : : :; X is false in the LFP for the
stratum of p.
For monopoly," Model 1 is the strati ed
model.
Win" has no strati ed model.
n
n
7