DatalogDatalogDatalogDatalog

S. CostantiniS. Costantini

S. Costantini / Logical Query Languages

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Logic As a Query Language

• If-then logical rules have been used in many systems.

• Recursive rules extend relational algebra --- have been used to add recursion to SQL-99.

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Datalog• A language for Deductive Databases

– Extensional part: tables– Intensional part: rules

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Logic: Intuition

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Datalog Terminology: Atoms

• An atom is a predicate, or relation name with variables or constants as arguments. E.g.,

in(alan,r123)part_of(r123,cs_building)p(X,a)

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Datalog Terminology: Atoms

• Conventions: – Databases: Predicates begin with a

capital, variables begin with lower-case. Example: P(x,’a’) where ‘a’ is a constant, i.e., a data item.

– Logic Programming: Variables begin with a capital, predicates and constants begin with lower-case. Example: p(X,a).

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Datalog Terminology: Rules

• The head of a rule is an atom; the body of a rule is the AND of one or more atoms. E.g., in(X,Y):- part_of(Z,Y), in(X,Z).where “:-” stands for “”

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Datalog Terminology: Facts

• Rules without body called “facts”: E.g., in(alan,r123).

• Facts constitute the extensional part of a deductive database, i.e., define tables.

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Alternative notation• Explicitly list attributes• E.g., facts:

in(Object:alan,Place:r123) • E.g., rules:

in(Object:X,Place:Y):- part_of(Place: Z, Place:Y),in(Object:X, Place:Z).

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Datalog semantics

• Least Herbrand Model– The head of a rule is in the Least

Herbrand Model only if the body is in the Least Herbrand Model.

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Least Herbrand Model: How to find it

p g,h.g r,s.m p,q.r f.s.f.h.Step 1: facts M0 = {s,f,h}Step 2: M1 = M0 plus what I can derive from facts M1 = {s,f,h,r}Step 3: M2 = M1 plus what I can derive from M1

M2 = {s,f,h,r,g}Step 4: M3 = M2 plus what I can derive from M2

M3 = {s,f,h,r,g,p}If you try to go on, no more added conclusions: fixpoint

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Least Herbrand Model: Intuition

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Least Herbrand Model: Intuition

In the example: Least Herbrand Model

M0 = {in(alan,r123), part_of(r123,cs_building)}

M1= {in(alan,r123), part_of(r123,cs_building), in(alan,cs_building)}

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Datalog and Transitivity• Rule

in(X,Y):- part_of(X,Z),in(Z,Y)defines in as the transitive closure of

part_of

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Arithmetic Subgoals• In addition to relations as predicates,

a predicate of the body can be an arithmetic comparison.– We write such atoms in the usual way,

e.g.: x < y.

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Example: Arithmetic• A beer is “cheap” if there are at least

two bars that sell it for under $2.

Cheap(beer) <- Sells(bar1,beer,p1) ANDSells(bar2,beer,p2) AND p1 < 2.00AND p2 < 2.00 AND bar1 <> bar2

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Negated Subgoals

• We may put NOT in front of an atom, to negate its meaning.

• Example:at_home(alan):- not at_work(alan)

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Negated Subgoals: Intuition

• Given atom A, not A holds if A cannot be proved

• Negation as finite failure

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Safe Rules• A rule is safe if:

1. Each variable in the head2. Each variable in an arithmetic subgoal,3. Each variable in a negated subgoal,

also appears in a nonnegated,subgoal in the body.

• We allow only safe rules.

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Example: Unsafe Rules

• Each of the following is unsafe and not allowed:

1. S(x) <- R(y)2. S(x) <- R(y) AND NOT R(x)3. S(x) <- R(y) AND x < y

• In each case, an infinity of x ’s can satisfy the rule, even if R is a finite relation.

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Datalog Programs• A Datalog program is a collection of

rules.• In a program, predicates can be either

1. EDB = Extensional Database = stored table.

2. IDB = Intensional Database = relation defined by rules.

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Expressive Power of Datalog

• Without recursion, Datalog can express all and only the queries of core relational algebra.

• But with recursion, Datalog can express more than these languages.

• Yet still not Turing-complete.

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Datalog semantics

• With negation: several proposals, in deductive databases answer set semantics.

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Minimal Models

• When there is no negation, a Datalog program has a unique minimal model (one that does not contain any other model).

• But with negation, there can be several minimal models.

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Non-Monotonicità• Aggiungere nuove conoscenze può

portare a ritrarre precedenti conclusioni.

• Vi possono essere conclusioni alternative tramite cicli sulla negazione.

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Non-Monotonicità = Problema?

• No, se gestita con opportuni strumenti semantici

• Non-Monotonic Reasoning (NMR) = campo di ricerca in– Intelligenza Artificiale – Database deduttivi

• si propone di sfruttare la non-monotonicità

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DATALOG senza Negazione

• Semantica del Least Herbrand Model• Paradigma procedurale della ricerca

per tentativi ripetuti • Linguaggio Prolog (numerosi

interpreti fra i quali SICSTUS-Prolog, SWI-Prolog)

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DATALOG + Negazione• Semantica degli Answer Sets• Paradigma dell’Answer Set

Programming (ASP)• Negazione, insiemi-risposta

alternativi• Numerosi solver

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Esempio NMRpersona(anna).persona(carlo).persona(giorgio).malato(giorgio).

a_casa(X):- persona(X), not in_ufficio(X).in_ufficio(X):- persona(X), not a_casa(X).

:- in_ufficio(X),malato(X). % constraint

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Risultato atteso• Anna e Carlo sono o a casa

oppure in ufficio.• Giorgio non può essere in ufficio

perchè è malato e quindi sarà a casa.

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NMR in pratica• Vi sono vari linguaggi logici che

consentono non-monotonicità.• Sono dotati di motori inferenziali

(inference engines, o solvers) in grado di gestirla.

• Possibili soluzioni inesistenti o altrernative.•Uno di essi: smodels

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Risultato di smodels (inference engine per NMR)

• Answer: 1malato(giorgio) a_casa(giorgio) a_casa(carlo)

in_ufficio(anna)

• Answer: 2malato(giorgio) a_casa(giorgio)

in_ufficio(carlo) in_ufficio(anna)

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Risultato di smodels (inference engine per NMR)

• Answer: 3malato(giorgio) a_casa(giorgio)

in_ufficio(carlo) a_casa(anna) • Answer: 4malato(giorgio) a_casa(giorgio) a_casa(carlo)

a_casa(anna)