# Logic databases 1. 2 Overview motivation and informal introduction of deductive databases components of deductive databases domains and axioms queries.

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Logic databases 1

2 Overview motivation and informal introduction of deductive databases components of deductive databases domains and axioms queries updates integrity constraints further issues

Logic databases 3 Relational databases Data select / filter relations specified extensionally by evaluating a (relational algebra) formula Required result

Logic databases 4 Example - database (extension) PersonParent

Logic databases 5 Example - query #1 Find who is Dean Kents mother SELECT Person.Name FROM Person, Parent WHEREPerson.Sex = F AND Person.Name = Parent.Name AND Parent.Child = Dean Kent ;

Logic databases 6 Example - query #2 Find who is Dean Kents grandmother, on his mothers side SELECT Person.Name FROMPerson, Parent WHEREPerson.Sex = F AND Person.Name = Parent.Name AND Parent.Child IN (SELECT Person.Name FROMPerson, Parent WHEREPerson.Sex = F AND Person.Name = Parent.Name AND Parent.Child = Dean Kent);

Logic databases 7 Example - query #3 Find all mothers (and their addresses) who have at least one daughter SELECT Name, Address FROMPerson WHERESex = F AND EXISTS (SELECT * FROMParent WHEREPerson.Name = Parent.Name AND Parent.Child IN (SELECT * FROM Person WHERESex = F)) ;

Logic databases 8 Example - query #4 Find all grandmothers (and their addresses) who have at least one grand daughter homework

Logic databases 9 Example #1 - conclusion the answer to grandmother queries would have been easier if a Grandparent relation had existed improper solution if the relation would have been defined extensionally - redundancies (exemplified in the following slides) therefore intensional definitions are required (exemplified on the following slides) could the query language be simpler? yes; Datalog (Prolog like), for instance

Logic databases 10 Grandparent relation - extensionally ParentGrandparent

Logic databases 11 not really a good solution

Logic databases 12 Grandparent relation - intensionally Parent A is Grandparent of B IFF /* there is a C such that */ A is the Parent of C AND C is the Parent of B

Logic databases 13 Activity Person Reconsider the issues discussed so far on the Person relation below, which substitutes the previous (page 4) Person and Parent relations.

Logic databases 14 Example - database (extension) #2 PersonParent

Logic databases 15 Example - query #5 Find who Bill Dyers ancestors are. recursive query not possible to answer in the relational model another formalism is necessary - which allows recursive definitions

Logic databases 16 Recursive definition A is ancestor of P IFF A is parent of P A is ancestor of P IFF B (B is ancestor of P AND A is parent of B)

Logic databases 17 Intermediate conclusion a mechanism is required to allow relations to be intensionally defined allow new facts to be deduced from the database based on the defined facts and rules by means of some reasoning methods note the motivation for deductive databases is more complex (i.e. not reduced to the above two aspects)

Logic databases 18 Deductive database - informal definition a database that supports the definition of both facts (extensionally defined relations) and rules (intensionally defined relations) and which supplies a reasoning mechanism by means of which existing or new facts can be deduced from the database a deductive database uses logic (first order predicate logic) as its representational formalism

Logic databases 19 Relational vs Deductive databases extensionally defined relations evaluating a formula result ground axioms deductive axioms reasoning mechanism result model theoretic proof theoretic

Logic databases 20 Deductive databases - domains name(Linda Fox). name(John Fox). name(Mary Fox). name(June Dyer). name(Bill Dyer). name(John Hunt).... sex(m). sex(f). sex(other). month(1). month(2).... month(12).

Logic databases 21 Deductive databases - ground axioms person(Linda Fox, dob(10, 12, 1823), f). person(John Fox, dob(2, 11, 1811, m). person(Mary Fox, dob(12, 4, 1850), f). person(John Hunt, dob(30, 1, 1875), m). person(Rose Hunt, dob(25, 3, 1877), f).... parent(Linda Fox, Mary Fox). parent(John Fox, Mary Fox). parent(Mary Fox, John Hunt). parent(Mary Fox, Rose Hunt).... married_to(Linda Fox, John Fox).%having a child together married_to(John Hunt, Christa Wolf).%does not mean married...%powerful representation!

Logic databases 22 Deductive databases - deductive axioms ancestor(X, Y, 1) :- parent(X, Y). ancestor(X, Y, Distance) :- parent(X, Z), ancestor(Z, Y, Init_dist), Distance is Init_dist +1.%built in arithmetics brother_or_sister(X, Y) :- parent(Z, X), parent(Z, Y).%can be refined using sex unfaithful(X, Y) :- married_to(X, Y), parent(X, Z), not parent(Y, Z).%negation as failure

Logic databases 23 Language elements constants, terms (structured objects), variables assertions (unnamed attributes) definitions generalised Horn clauses closed world assumption negation as finite failure built in Arithmetic (scalar functions) how to design the database? depends on what we want to express example: reification -> a more a powerful representation

Logic databases 24 Normal forms - CNF conjunctive normal form (CNF) a conjunction of disjunctions of literals e.g. (A 1 A 2 A 3 ) (B 1 B 2 B 3 ) ( C 1 C 2 ) every formula can be brought to CNF

Logic databases 25 Normal forms - clausal clausal form equivalent to CNF each disjunction (in CNF) is represented separately as an implication; e.g.: (A 1 A 2 A 3 ) (B 1 B 2 B 3 ) ( C 1 C 2 ) becomes A 1, A 2, A 3 (reads: it is true that A 1 A 2 A 3 ) B 1 B 2, B 3 (reads: B 1 if B 2 B 3 ) C 1,C 2 (reads: C 1 C 2 is false) all literals are positive every formula can be brought to the clausal form

Logic databases 26 Normal forms - Horn clauses DEF1: a clause with just one literal (implicitly positive) in the conclusion B 1 B 2, B 3 is a Horn clause B 1, B2 B 3 is not a Horn clause DEF2: a disjunction with one positive literal only B 1 B 2 B 3 is a Horn clause B 1 B 2 B 3 is not a Horn clause

Logic databases 27 Logic database In the rest of this lecture, a logic database is considered to be a set of Horn clauses enhanced with negation as finite failure (generalised Horn clauses)

Logic databases 28 Queries Is Bill Dyer a man? Yes/No queries :- person(Bill Dyer, _, m). Find value(s) queries - use of only ground axioms Who is Linda Fox married to? :- married_to(Linda Fox, X).

Logic databases 29 Queries Find value(s) queries - use of deductive axioms Who are the brothers and sisters of John Hunt? :- brother_or_sister(John Hunt, X). %one sol at a time :- forall(brother_or_sister(John Hunt, X)). %all sols Find value(s) queries - use of recursive deductive axioms Who are the ancestors of John Hunt? :- ancestor(John Hunt, X). %one sol. at a time :- forall(ancestor(John Hunt, X)). %all sols.

Logic databases 30 Reasoning mechanism resolution A 1, A 2,..., A n, B 1, B 2,..., B m are all positive or negative literals (X A 1 A 2... A n ) ( X B 1 B 2... B m ) (A 1 A 2... A n B 1 B 2... B m )

Logic databases 31 Resolution for Horn clauses X A 1 A 2... A k... A n A k B 1 B 2... B m X A 1 A 2... B 1... B m... A n

Logic databases 32 Basic manipulation primitives assert(person(Brooklin Peckham, dob(1, 3, 1899), m)). assert(parent(Brooklin Peckham, David Peckham)). assert(parent(Brooklin Peckham, Gingy Spicy)). assert(married_to(David Peckham, Gingy Spicy)).... retract(person(Linda Fox, _, _, _)). retract(married_to(Linda Fox, John Fox))....

Logic databases 33 Updating the databse via assert and retract equivalent to delete / insert note the same formalism for data definition and data manipulation

Logic databases 34 Integrity constraints integrity constraints and rules in deductive databases are syntactically indistinguishable rules - are sentences which define the data and are part of the database (Kowalski) integrity constraints - are sentences which constrain the data and are not part of the database (Kowalski) the same sentence can be regarded, in different contexts, as rule or integrity constraint

Logic databases 35 Sentence as rule person(Linda Fox, dob(10, 12, 1823), f). person(John Fox, dob(2, 11, 1811, m).... parent(Linda Fox, Mary Fox). parent(John Fox, Mary Fox).... %two different persons are married if they have a child together % parent(X, Z) parent(Y, Z) different(X,Y) married_to(X, Y) %definition - part of the database (definition) married_to(X, Y) :- parent(X, Z), parent(Y, Z), not X = Y.

Logic databases 36 Sentence as a constraint person(Linda Fox, dob(10, 12, 1823), f).... parent(Linda Fox, Mary Fox). parent(John Fox, Mary Fox).... married_to(Linda Fox, John Fox).... %if two different persons have a child together then they must be %married %constraint - not part of the database (definition) %parent(X, Z) parent(Y, Z) different(X,Y) married_to(X, Y) :- parent(X, Z), parent(Y, Z), not X = Y, not married_to(X, Y).

Logic databases 37 Constraints enforcement #1 %a simple checking mechanism at the time of update make_parent(X, Y) :- parent(X, Y). make_parent(X, Y) :- not parent(X, Y), %X is bound from the head; it is an update not (parent(Z, Y), not married_to(X, Z)), assert(parent(X, Y)).

Logic databases 38 Constraints enforcement #2 %automatic consistency maintenance is_parent_of(X, Y) :- parent(X, Y). is_parent_of(X, Y) :- not parent(X, Y), %X is bound from the head; it is an update parent(Z, Y), not married_to(X, Z), assert(married_to(X, Z)), % automatic consistency maintenance assert(parent(X, Y)). is_parent_of(X, Y) :- not parent(X, Y), %X is bound from the head; it is an update not (parent(Z, Y), not married_to(X, Z)), assert(parent(X, Y)).

Logic databases 39 Integrity constraints what about retract? homework! there are other mechanisms (even other logic models) of integrity maintenance note the syntactic equivalence (same formalism) between data definition and integrity constraints definition (and data manipulation)

Logic databases 40 Constraints enforcement #3 %constraints declaration (part of the db); checked after every update % operation constraint(:- parent(X, Z), parent(Y, Z), not X = Y, not married_to(X, Y).). constraint(…). %constraints declaration (part of the db); constraints are linked to % update operations - their checking is triggered by corresponding updates trigger(make_parent(X, Y), :- parent(X, Z), parent(Y, Z), not X = Y, not married_to(X, Y).). %etc. %preconditions...

Logic databases 41 Datalog and Prolog we have used a combination of Datalog and Prolog Datalog incorporates the forall predicate - for set processing Datalog is insensitive to the order of the clauses special control predicates (e.g. cut) do not exist in Datalog Datalog does not contain function symbols however, extensions to pure Datalog exist that include: built in predicates (e.g. arithmetics), negation, functions (for complex objects).

Logic databases 42 Approaches to deductive databases evolutionary approaches add a Prolog like system to a relational database revolutionary approaches propose new architectures; e.g. Datalog

Logic databases 43 Further issues persistency transaction support security / recovery / concurrency...

Logic databases 44 Advantages of deductive databases representational uniformity operational uniformity semantic modelling extended application

Logic databases 45 Conclusions deductive databases uniform representation formalism powerful still problems when dealing with large applications terminology knowledge base management systems - preferred

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