2012wq171 13 FOL Semantics

8/2/2019 2012wq171 13 FOL Semantics

1/19

Fi rs t -Ord er Log icSemant i cs

Reading: Chapter 8, 9.1-9.2, 9.5.1-9.5.5

FOL Syntax and Semantics read: 8.1-8.2

FOL Knowledge Engineering read: 8.3-8.5

FOL Inference read: Chapter 9.1-9.2, 9.5.1-9.5.5

(Please read lecture topic material before and after eachlecture on that topic)


2/19

Out l i ne

Propositional Logic is Usefu l --- but has L im i ted Exp r ess ive Pow er

First Order Predicate Calculus (FOPC), or First Order Logic (FOL).

FOPC has greatly expanded expressive power, though still limited.

New Ontology

The world consists of OBJECTS (for propositional logic, the world was facts).

OBJECTS have PROPERTIES and engage in RELATIONS and FUNCTIONS.

New Syntax

Constants, Predicates, Functions, Properties, Quantifiers.

New Semantics

Meaning of new syntax.

Knowledge engineering in FOL

Inference in FOL


3/19

You w i l l be expec ted t o kn ow

FOPC syntax and semantics

Syntax: Sentences, predicate symbols, function symbols, constantsymbols, variables, quantifiers

Semantics: Models, interpretations

De Morgans rules for quantifiers

connections between and

Nested quantifiers Difference between x y P(x, y) and x y P(x, y)

x y Likes(x, y)

x y Likes(x, y)

Translate simple English sentences to FOPC and back

x y Likes(x, y) Everyone has someone that they like.

x y Likes(x, y) There is someone who likes every person.


4/19

Out l i ne

Review: KB |= S is equivalent to |= (KB S)

So what does {} |= S mean?

Review: Follows, Entails, Derives

Follows: Is it the case?

Entails: Is it true?

Derives: Is it provable?

Semantics of FOL (FOPC)

Model, Interpretation


5/19

FOL (or FOPC) Ontology:What kind of things exist in the world?What do we need to describe and reason about?Objects --- with their relations, functions, predicates, properties, and general rules.

Reasoning

Representation-------------------A FormalSymbol System

Inference---------------------Formal PatternMatching

Syntax---------What issaid

Semantics-------------What itmeans

Schema-------------Rules ofInference

Execution-------------SearchStrategy


6/19

Rev iew : KB | = S m ea ns | = ( KB S)

KB |= S is read KB entails S.

Means S is true in every world (model) in which KB is true. Means In the world, S follows from KB.

KB |= S is equivalent to |= (KB S)

Means (KB S) is true in every world (i.e., is valid).

And so: {} |= S is equivalent to |= ({} S)

So what does ({} S) mean?

Means True implies S.

Means S is valid. In Horn form, means S is a fact. p. 256 (3rd ed.; p. 281, 2nd ed.)

Why does {} mean True here, but False in resolution proofs?


7/19

Rev iew : ( Tru e S) m eans S i s a fac t .

By convention,

The null conjunct is syntactic sugar for True. The null disjunct is syntactic sugar for False.

Each is assigned the truth value of its identity element.

For conjuncts, True is the identity: (A True) A

For disjuncts, False is the identity: (A False) A

A KB is the conjunction of all of its sentences.

So in the expression: {} |= S

We see that {} is the null conjunct and means True.

The expression means S is true in every world where True is true.

I.e., S is valid.

Better way to think of it: {} does not exclude any worlds (models).

In Conjunctive Normal Form each clause is a disjunct.

So in, say, KB = { (P Q) (Q R) () (X Y Z) }

We see that () is the null disjunct and means False.


8/19

Side Tr ip : Fun c t ion s AND, OR, and n u l l va lu es( No te : These a re syn t act i c suga r i n l og i c.)

f unc t i on AND(arglist) r e t u r n s a truth-valuer e t u r n ANDOR(arglist, True)

f unc t i on OR(arglist) r e t u r n s a truth-value

r e t u r n ANDOR(arglist, False)

f unc t i on ANDOR(arglist, nullvalue) r e t u r n s a truth-value

/* nullvalueis the identity element for the caller. */

i f (arglist= {})t h e n r e t u r n nullvalue

i f ( FIRST(arglist) = NOT(nullvalue) )

t h e n r e t u r n NOT(nullvalue)

r e t u r n ANDOR( REST(arglist) )


9/19

Side Tr ip : W e on ly need one log ica l connect i ve .( No te : AND, OR, NOT a re syn t act i c sugar i n l og ic.)

Both NAND and NOR are log ica l l y com ple te .

NAND is a l so ca l led t he She f fe r s t rok e

NOR is a lso ca l led Pie rce s a r row

(NOT A) = (NAND A TRUE) = (NOR A FALSE)

(AND A B) = (NAND TRUE (NAND A B))

= (NOR (NOR A FALSE) (NOR B FALSE))

(OR A B) = (NAND (NAND A TRUE) (NAND B TRUE))=(NOR FALSE (NOR A B))


10/19

Rev iew : Schem at i c fo r Fo l l ow s , En t a i l s, and Der i v es

If KB is true in the real world,

then any sentence entailedby KB

and any sentence derivedfrom KB

by a sound inference procedureis also true in the real world.

Sentences SentenceDerivesInference


11/19

Schem at ic Ex am ple : Fo l low s, Ent a i l s , and Der iv es

Inference

Mary is Sues sister and

Amy is Sues daughter. Mary is

Amys aunt.Representation

Derives

Entails

FollowsWorld

Mary Sue

Amy

Mary is Sues sister and

Amy is Sues daughter.

An aunt is a sister

of a parent.

An aunt is a sisterof a parent.

Sister

Daughter

Mary

Amy

Aunt

Mary is

Amys aunt.

Is it provable?

Is it true?

Is it the case?


12/19

Rev iew : Model s ( and i n FOL, I n t e rp re t a t i ons )

Models are formal worlds in which truth can be evaluated

We say mis a model ofa sentence if is true in m

M() is the set of all models of

Then KB iffM(KB) M() E.g. KB, = Mary is Sues sister

and Amy is Sues daughter. = Mary is Amys aunt.

Think of KB and as constraints,and of models m as possible states.

M(KB) are the solutions to KBand M() the solutions to .

Then, KB , i.e., (KB a) ,when all solutions to KB are also solutions to .


13/19

Sem ant i cs : W or ld s

Th e world cons is ts o f objects t h a t h a v e properties.

There a r e relations an d functions b e t w e en t h e se o b j e ct s

Ob j ects i n t h e w o r ld , i nd i v idua ls : people, houses, numbers,colors, baseball games, wars, centuries

Clock A, John, 7, the-house in the corner, Tel-Aviv, Ball43

Func t ions on individuals: father-of, best friend, third inning of, one more than

Rela t ions:

brother-of, bigger than, inside, part-of, has color, occurredafter

Prope r t i es ( a re la t i on o f a r i t y 1 ) :

red, round, bogus, prime, multistoried, beautiful


14/19

Se m a n t i cs: I n t e r p r e t a t i o n

An i n t e r p r e t a t i o n of a sentence (wff) is an assignment thatmaps

Object constant symbols to objects in the world,

n-ary function symbols to n-ary functions in the world,

n-ary relation symbols to n-ary relations in the world

Given an interpretation, an atomic sentence has the valuetrue if it denotes a relation that holds for those individualsdenoted in the terms. Otherwise it has the value false.

Example: Kinship world:

Symbols = Ann, Bill, Sue, Married, Parent, Child, Sibling,

World consists of individuals in relations:

Married(Ann,Bill) is false, Parent(Bill,Sue) is true,


15/19

Tru th i n f i r st - o rde r l og i c

Sentences are true with respect to a model and an interpretation

Model contains objects (domainelements) and relations among them

Interpretation specifies referents for

constantsymbols objects

predicatesymbols relations

functionsymbols functional relations

An atomic sentence predicate(term1, . . . , termn) is trueiff the objects referred to by term1, . . . , termnare in the relation referred to by predicate


16/19

Sem ant ics: Mode ls

A n i n t e r p r e t a t i on satisfies a w f f ( se n t e n ce ) i f t h e w f f h a st h e v a lu e t r u e u n d e r t h e i n t e r p r et a t i on .

Model : A dom a in and an i n te rp re ta t i on t ha t sa t i s f ies aw f f i s a model o f t h at w f f

Va li d it y : A n y w f f t h a t h a s t h e v a lu e t r u e u n d e r a l l

i n t e rp re ta t i ons i s valid Any w f f t h a t does no t have a m ode l is i nconsi sten t o r

unsa t i s f iab le

I f a w f f w h a s a v al u e t r u e u n d e r a l l t h e m o d e ls o f a se to f sen t ences KB then KB log ica l l y en t a i l s w


17/19

Models fo r FOL: Ex am ple


18/19

FOL (or FOPC) Ontology:What kind of things exist in the world?What do we need to describe and reason about?Objects --- with their relations, functions, predicates, properties, and general rules.

Reasoning

Representation-------------------

A FormalSymbol System

Inference---------------------

Formal PatternMatching

Syntax---------What issaid

Semantics-------------What itmeans

Schema-------------Rules ofInference

Execution-------------SearchStrategy


19/19

S u m m a r y

First-order logic:

Much more expressive than propositional logic Allows objects and relations as semantic primitives

Universal and existential quantifiers

Syntax: constants, functions, predicates, equality, quantifiers

Nested quantifiers

Translate simple English sentences to FOPC and back

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2012wq171 13 FOL Semantics