Logic: Datalog Syntax and Semantics

8/14/2019 Logic: Datalog Syntax and Semantics

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Recap Datalog Syntax Datalog Semantics

Logic: Datalog Syntax and Semantics

CPSC 322 Lecture 22

March 9, 2007Textbook 5

Logic: Datalog Syntax and Semantics CPSC 322 Lecture 22, Slide 1
http://find/http://goback/


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Lecture Overview

1 Recap

2 Datalog Syntax

3 Datalog Semantics



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Top-down definite clause interpreter

To solve the query ?q1 . . . qk:

ac := yes q1 . . . qkrepeat

select atom ai from the body of ac;

choose clause C from KB with ai as head;replace ai in the body of ac by the body of Cuntil ac is an answer.

Dont-care nondeterminism If one selection doesnt lead to asolution, there is no point trying other alternatives. select

Dont-know nondeterminism If one choice doesnt lead to asolution, other choices may. choose



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Representational Assumptions of Datalog

An agents knowledge can be usefully described in terms ofindividuals and relations among individuals.

An agents knowledge base consists ofdefinite and positivestatements.

The environment is static.

There are only a finite number of individuals of interest in thedomain. Each individual can be given a unique name.

= Datalog



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Example Domain for an RRS

in(alan,cs_building)

in(alan,r123).

part_of(r123,cs_building).

in(X,Y)

part_of(Z,Y)

in(X,Z).

alan

r123

r023

cs_building

in( , )part_of( , )

person( )


R D l S D l S i


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Lecture Overview

1 Recap

2 Datalog Syntax

3 Datalog Semantics


R D t l S t D t l S ti


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Syntax of Datalog

Definition (variable)

A variable starts with upper-case letter.

Definition (constant)

A constant starts with lower-case letter or is a sequence of digits.

Definition (term)

A term is either a variable or a constant.

Definition (predicate symbol)

A predicate symbol starts with lower-case letter.


Recap Datalog S ntax Datalog Semantics


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Syntax of Datalog (cont)

Definition (atom)

An atomic symbol (atom) is of the form p or p(t1, . . . , tn) where pis a predicate symbol and ti are terms.

Definition (definite clause)

A definite clause is either an atomic symbol (a fact) or of the form:

a

b1 bm

head body

where a and bi are atomic symbols.

Definition (knowledge base)

A knowledge base is a set of definite clauses.




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Example Knowledge Base

in(alan, R)

teaches(alan, cs322)

in(cs322, R).

grandfather(william, X)

father(william, Y)

parent(Y, X).

slithy(toves) mimsy borogroves

outgrabe(mome, Raths).




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Lecture Overview

1 Recap

2 Datalog Syntax

3 Datalog Semantics




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Semantics: General Idea

Recall: a semantics specifies the meaning of sentences in thelanguage.

ultimately, we want to be able to talk about which sentencesare true and which are false

In propositional logic, all we needed to do in order to come upwith an interpretation was to assign truth values to atoms

For Datalog, an interpretation specifies:

what objects (individuals) are in the worldthe correspondence between symbols in the computer andobjects & relations in world

which constants denote which individualswhich predicate symbols denote which relations (and thus,along with the above, which sentences will be true and whichwill be false)




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Formal Semantics

Definition (interpretation)

An interpretation is a triple I = D, , , where

D, the domain, is a nonempty set. Elements of D areindividuals.

is a mapping that assigns to each constant an element ofD. Constant c denotes individual (c).

is a mapping that assigns to each n-ary predicate symbol a

relation: a function from Dn into {TRUE, FALSE}.




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p g y g

Example Interpretation

Constants: phone, pencil, telephone.Predicate Symbol: noisy (unary), left of (binary).

D = {,,}.

These are actually objects in the world, not symbols

(phone) =, (pencil) =, (telephone) =.

(noisy): FALSE TRUE FALSE(left of):

, FALSE , TRUE , TRUE

, FALSE , FALSE , TRUE, FALSE , FALSE , FALSE




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Important points to note

The domain D can contain real objects. (e.g., a person, aroom, a course). D cant necessarily be stored in a computer.

The constants do not have to match up one-to-one withmembers of the domain. Multiple constants can refer to the

same object, and some objects can have no constants thatrefer to them.

(p) specifies whether the relation denoted by the n-arypredicate symbol p is true or false for each n-tuple ofindividuals.

If predicate symbol p has no arguments, then (p) is eitherTRUE or FALSE.

this was the situation in propositional logic




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Truth in an interpretation

Definition (truth in an interpretation)A constant c denotes in I the individual (c).

Ground (variable-free) atom p(t1, . . . , tn) is

true in interpretation I if (p)(t1

, . . . , tn

) = TRUE, where tidenotes t

iin interpretation I and

false in interpretation I if (p)(t1, . . . , tn) = FALSE.

Ground clause h b1 . . . bm is

false in interpretation I if h is false in I and each bi is true inI, and item true in interpretation I otherwise.A knowledge base, KB, is true in interpretation I if and only

if every clause in KB is true in I.

Notice that truth values are only associated with predicates(atomic symbols; clauses), not variables and constants!




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Example Truths

In the interpretation given before:

noisy(phone)




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Example Truths


noisy(phone) truenoisy(telephone)




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Example Truths


noisy(phone) truenoisy(telephone) true

noisy(pencil)




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Example Truths



noisy(pencil) falseleft of(phone, pencil)




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Example Truths



noisy(pencil) falseleft of(phone, pencil) trueleft of(phone, telephone)




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Example Truths



noisy(pencil) falseleft of(phone, pencil) trueleft of(phone, telephone) falsenoisy(pencil) left of(phone, telephone)




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Example Truths



noisy(pencil) falseleft of(phone, pencil) trueleft of(phone, telephone) falsenoisy(pencil) left of(phone, telephone) truenoisy(pencil) left of(phone, pencil)




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Example Truths



noisy(pencil) falseleft of(phone, pencil) trueleft of(phone, telephone) falsenoisy(pencil) left of(phone, telephone) truenoisy(pencil) left of(phone, pencil) false

noisy(phone) noisy(telephone) noisy(pencil)




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Example Truths



noisy(pencil) falseleft of(phone, pencil) trueleft of(phone, telephone) falsenoisy(pencil) left of(phone, telephone) truenoisy(pencil) left of(phone, pencil) false

noisy(phone) noisy(telephone) noisy(pencil) true




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Variables

How do we determine the truth value of a clause that includesvariables?

Definition (variable assignment)

A variable assignment is a function from variables into the domain.

Given an interpretation and a variable assignment, each termdenotes an individual and each clause is either true or false.

A clause containing variables is true in an interpretation if it is

true for all variable assignments.Variables are universally quantified in the scope of a clause.




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Models and logical consequences

Definition (model)

A model of a set of clauses is an interpretation in which all theclauses are true.

Definition (logical consequence)If KB is a set of clauses and g is a conjunction of atoms, g is alogical consequence of KB, written KB |= g, if g is true in everymodel of KB.

That is, KB |= g if there is no interpretation in which KB istrue and g is false.


Logic: Datalog Syntax and Semantics

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