Upload
krutarth-patel
View
220
Download
0
Embed Size (px)
Citation preview
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)
8/2/2019 2012wq171 13 FOL Semantics
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
8/2/2019 2012wq171 13 FOL Semantics
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.
8/2/2019 2012wq171 13 FOL Semantics
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
8/2/2019 2012wq171 13 FOL Semantics
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
8/2/2019 2012wq171 13 FOL Semantics
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?
8/2/2019 2012wq171 13 FOL Semantics
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.
8/2/2019 2012wq171 13 FOL Semantics
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) )
8/2/2019 2012wq171 13 FOL Semantics
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))
8/2/2019 2012wq171 13 FOL Semantics
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
8/2/2019 2012wq171 13 FOL Semantics
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?
8/2/2019 2012wq171 13 FOL Semantics
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 .
8/2/2019 2012wq171 13 FOL Semantics
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
8/2/2019 2012wq171 13 FOL Semantics
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,
8/2/2019 2012wq171 13 FOL Semantics
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
8/2/2019 2012wq171 13 FOL Semantics
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
8/2/2019 2012wq171 13 FOL Semantics
17/19
Models fo r FOL: Ex am ple
8/2/2019 2012wq171 13 FOL Semantics
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
8/2/2019 2012wq171 13 FOL Semantics
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