Notes 8:Predicate logic and inference

ICS 270a Spring 2003

Outline

New ontology objects,relations,properties,functions.

New Syntax Constants, predicates,properties,functions

New semantics meaning of new syntax

Inference rules for Predicate Logic (FOL) Resolution Forward-chaining, Backword-chaining unification

Readings: Nillson’s Chapters 15-16, Russel and Norvig Chapter 8, chapter 9

Propositional logic is not expressive Needs to refer to objects in the world, Needs to express general rules

On(x,y) ~ clear(y) All man are mortal Everyone who passed age 21 can drink One student in this class got perfect score Etc….

First order logic, also called Predicate calculus allows more expressiveness

Syntax

Components Infinite set of object constants, alphanumerc strings

Bb, 12345, Jerusalem, 270a, Irvine Function constants of all aritys (convention: start with lower

case) motherOf, times, greaterThan, color

Relation constants, Predicates of all aritys (start with capital) B21, Parent, multistoried, XYZ, prime

Terms An object constant is a term A function constant of arity n followed by n terms in

parenthesis is a term (called functional expression) motherOf(Silvia), Sam, leftlegof(John), add1(5), times(5,plus(3,6))

Wffs sentences

An atomic sentence is formed from a predicate (relation) symbol followed by a parenthesized list of symbols

Atoms: a relation constant (predicate) followed by n terms in parenthesis seperated by commas.

GreaterThan(6,3), Q, Q(A,B,C,D), Brother(Richard,John) Married(FatherOf(Richard),MotherOf(John))

Complex propositional wffs:

Use logic connectives: Brother(Richard,John) /\ Brother(John,Richard) Older(John,30) V Younger(John,30) Older(John) --> ~Younger(John,30) V P

,,,

Semantics: Worlds

The world consists of objects that has properties. There are relations and functions between these objects Objects in the world, individuals: people, houses,

numbers, colors, baseball games, wars, centuries Clock A, John, 7, the-house in the corner, Tel-Aviv

Functions on individuals: father-of, best friend, third inning of, one more than

Relations: brother-of, bigger than, inside, part-of, has color, occurred after

Properties (a relation of arity 1): red, round, bogus, prime, multistoried, beautiful

Semantics: Interpretation

An interpretation of a wff is an assignment that maps object constants to objects in the worlds, 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 atom has the value “true” in case it denotes a relation that holds for those individuals denoted in the terms. Otherwise it has the value “false”

Example: A,B,C,floor, On, Clear

World: On(A,B) is false, Clear(B) is true, On(C,F1) is true…

Semantics: Models

An interpretation satisfies a wff (sentence) if the wff has the value “true” under the interpretation.

An interpretation that satisfies a wff is a model of that wff

Any wff that has the value “true” under all interpretations is valid

Any wff that does not have a model is inconsistent or unsatisfiable

If a wff w has a value true under all the models of a set of sentences delta then delta logically entails w

Example of models

The formulas: On(A,F1) Clear(B) Clear(B) and Clear(C) On(A,F1) Clear(B) or Clear(A) Clear(B) Clear(C)Possible interpretations which are models:

On = {<B,A>,<A,floor>,<C,Floor>}Clear = {<C>,<B>}

Quantification

Universal and existential quantifiers allow expressing general rules with variables

Universal quantification All cats are mammals

It is equivalent to the conjunction of all the sentences obtained by substitution the name of an object for the variable x.

Syntax: if w is a wff then (forall x) w is a wff.

)( )( xMammalxCatx

,,,,

)()(

)()(

)()(

FelixMammalFelixCat

RebbekaMammalRebbekaCat

SpotMammalSpotCat

Quantification: Existential

Existential quantification : an existentially quantified sentence is true in case one of the disjunct is true

Equivalent to disjunction:

We can mix existential and universal quantification.

)(),( xCatspotxxSister

d)...Cat(Richarhard,Spot)Sister(Ric

Cat(Felix)ix,Spot)Sister(Fel

a)Cat(Rebeccecca,Spot)Sister(Reb

Cat(Spot)Spot)tSister(Spo

,

Useful Equivalences

De Morgan laws

Inference rules: Universal instantiation: foall x f(x) |-- f(alpha) Existential generalization: from f(alpha) |-- exist x f(x)

Modeling a domain; Conceptualization The kinship domain:

object are people Properties include gender and they are related by relations

such as parenthood, brotherhood,marriage predicates: Male, Female (unary)

Parent,Sibling,Daughter,Son... Function:Mother Father

Resolution in the predicate calculus Unification

Algorithm unify

Using unification in predicate calculus resolution Completeness and soundness Converting a wff to clause form The mechanics of resolution Answer extraction

Inference Rules in FOL

All the inference rules for propositional logic applies and additional three rules that use substitution; assigning constants to variables

Subst(x/constant) Subst({x/Sam,y/Pam},Likes(x,y)) = Likes(Sam,Pam)

Universal elimination From fromall x Likes(x,IceCream) we can substitute {x/Ben} and

infer Likes(Ben,IceCream) Existential Elimination

For any sentence and for any symbol k that does not appear elsewhere in the knowledge-base

exists x Kill(x,Victim) we can infer Kill(Murderer,victim) As long as Murdered soes not appear anywhere

Existential introduction From Likes(Jerry,IceCream) we can infer exists x Likes(x,IceCream)

Predicate calculus resolution

Suppose the 1 and 2 are two clauses represented as

set of literals. If there is an atom in 1 and a literal ~

in 2 such as and have a common unifier

Then these two clauses have a resolvent row. It is obtained by applying the substitution to 1 2

leaving out complementary literals

Example: xyfsubst

CxRAxPCBQAyfP

)(

)},(),({)},,()),(({

Converting to clause form

),(

)27,(

)28,()27,(

)(),(

),()28,()27,()()( ,

ABS

BI

AIAI

BPAP

yxSyIxIyPxPyx

Example: Answer Extraction

Example: Resoluction Refutation Prove I(A,27)

Rule-Based Systems

Rule-based Knowledge Representation and Inference combines rules

if DOG(x) then TAIL(X) with working memory,

DOG(SPOT) DOG(LUNA)

is not a formal knowledge representation scheme like logic somewhat like a combination of propositional and FOPL uses simple methods for inference often combined with other mechanisms such as inheritance motivated by practical use rather than theoretical properties also has some basis in cognitive models

Forward Chaining Procedure

While (rules can still fire or goal is not reached)for each rule in the rulebase which has not fired

1. try to bind premises by matching to assertions in WM

2. if all premises in a rule are supported then assert the conclusion and add to WM

Repeat 1. and 2. for all matching/ instantiation alternativesendcontinue

Example: Zoo Rulebase

Want to identify animals based on their characteristics could have rules like

if A1 and A2 and A3.....and A27 then animal is a monkey i.e., one very specific rule per animal

Better to use intermediate concepts e.g., the intermediate concept of a mammal if ?x is a mammal and B1...and B3 then ?x is a monkey

Why are intermediate concepts useful? rules are easier to create rulebase is easier to maintain easier to perform inference however: finding good intermediate concepts is non-trivial!

Where do the rules come from ?

Manual Knowledge Acquisition interview experts expert describes the rules process of translating expert knowledge into a formal representation is

also known as knowledge engineering this is notoriously difficult

experts are often very poor at explicitly describing what they do e.g., ask Michael Jordan how he plays basketball or ask Tiger Woods how he

plays golf

Automated Knowledge Acquisition derive rules automatically from formal specifications

e.g., by automatic analysis of solution manuals machine learning

learn rules from data

Inference Network Representation

We can represent forward chaining with rules as a network:has hair

eats meat

tawny color

black stripes

has a tail

is a tiger

is a carnivore

is a mammal

Inference using Backward Chaining

Establish a hypothesis e.g., Tony is a tiger a hypothesis is as assertion whose truth we wish to test

i.e., assert the hypothesis see if that is consistent with other data

Work backwards, i.e., match the hypothesis to the conclusion of the rules look at the premise conditions of the rule

if these are unknown, declare these as intermediate hypotheses backward chain recursively (depth-first manner)

Can use search algorithms to control how backward chaining proceeds

Inference Network Representation of Backward Chaining

• We can represent backward chaining as a network:

has hair

eats meat

tawny color

black stripes

is a tiger

is a carnivore

is a mammal

forward pointing eyeshas clawshas teeth

Backward Chaining ProcedureWhile (unsupported hypotheses exist or goal is not reached)

for each hypothesis Hfor each rule R whose conclusion matches with H

try to support each rule’s premises by1. matching assertions in WM2. recursively applying backward chaining

if rule’s premises are supported conclude that H is true

endend

continue

Forward Chaining or Backward Chaining:which is better? Which direction is better depends on the circumstances

4 main factors of relevance the number of possible start states and number of possible

goal states smaller is better

the branching factor in each direction smaller is better

what type of explanation the user wants more natural explanations may be important

the typical action which triggers a rule i.e., data-driven or hypothesis-driven?

What is a rule-based expert system (ES)?

ES performs a task in limited domains that require human expertise

medical diagnosis; fault diagnosis; status monitoring; data interpretation; mineral exploration; credit checking; tutoring; computer configuration

ES solves problems by application of domain specific knowledge rather than weak methods.

Domain knowledge is acquired by interviewing human experts

ES can explain a problem solution in human-understandable terms

ES cannot operate in situations that call for common sense

Examples of Expert Systems1965 DENDRAL Stanford analyze mass spectrometry data1965 MACSYMA MIT symbolic mathematics problems1972 MYCIN Stanford diagnosis of blood diseases1972 Prospector SRI Mineral Exploration1975 Cadeceus U. of PITT Internal Medicine1978 DIGITALIS MIT digitalis therapy advise1979 PUFF Stanford obstructive airway diseases1980 R1 CMU computer configuration1982 XCON DEC computer configuration1983 KNOBS MITRE mission planning1983 ACE AT&T diagnosis faults in telephone cables1984 FAITH JPL spacecraft diagnosis1986 ACES Aerospace satellite anomaly diagnosis1986 Cambpell diagnose cooker malfunctions1986 DELTA/CATS GE diagnosis of diesel locomotives1987 AMEX credit authorization1992 MAX NYNEX telephone network troubleshooting1995 Caltech PacBell network management1997 UCI planning drug treatment for HIV

Mycin: Diagnosis and treatment of blood diseasesRULESPREMISE ($AND (SAME CNTXT GRAM GRAMNEG) (SAME CNTXT MORH ROD) (SAME CNTXT AIR AEROBIC))ACTION: (CONCLUDE CNTXT CLASS ENTEROBACTERIACEAE .8)

If the stain of the organism is gramneg, and the morphology of the organism is rod, and

the aerobicity of the organism is aerobicTHEN there is strongly suggestive evidence (.8) that

the class of organism is enterocabateriaceae

DATAORGANISM-1:

GRAM = (GRAMNEG 1.0)MORP = (ROD .8) (COCCUS .2)AIR = (AEROBIC .6)(FACUL .4)

The XCON Application

A rule-based expert system expert in the sense that the rules capture expert design knowledge addresses a design/configuration problem

Digital Equipment Corporation (or Compaq by now!) XCON = rule-based expert system for computer configuration decides how peripherals are configured on new orders has 10,000 rules developed in early 1980’s based on “reactive rule-based systems”

if antecedents then action forward chaining in style

estimated to have saved DEC several hundred million $’s

Limitations of Rule-Based Representations

Can be difficult to create the “knowledge engineering” problem

Can be difficult to maintain in large rule-bases, adding a rule can cause many unforeseen

interactions and effects => difficult to debug

Many types of knowledge are not easily represented by rules uncertain knowledge: “if it is cold it will probably rain” information which changes over time procedural information (e.g. a sequence of tests to diagnose a

disease)

Summary

Knowledge Representation methods formal logic rule-based systems (less formal)

Rule-based representations are composed of Working memory: Rules

Inference occurs by forward or backward chaining

Rule-based expert systems useful for certain classes of problems which do not have direct algorithmic solutions

have their limitations