ai propsitional logic

8/8/2019 AI Propsitional Logic

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Propositional Logic-

Introduction


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Outline

Symbolic AI

Some Applications of Propositional Logic

Propositional Logic

Syntax

Semantic

Truth Table

Inference rules Limitations of Propositional Logic


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Symbolic AI

AI relies on the Physical Symbol System Hypothesis:

Intelligent activity is achieved through the use of

symbol patterns to represent the problem operations on those patterns to generate potential solutions

search to select a solution among the possibilities

An AI representation language must

handle qualitative knowledge

allow new knowledge to be inferred from facts & rules

allow representation of general principles

capture complex semantic meaning

allow for meta-level reasoning

e.g., Predicate Calculus (also, the basis of Prolog)


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Logic

A tool for reasoning

Provides basic concept used in many

computer science fields (Artificial Intelligence,

Information Retrieval, DataBases etc. )


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A logic language consists of semantics and syntax

Semantics:What the sentences mean.

Syntax: How sentences can be assembled.


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Formal Logic

Step 1: Use certain symbols to express the abstract

form of certain statements

Step 2: Use a certain procedure based on these

abstract symbolizations to figure out certain logicalproperties of the original statements.


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Truth Tables

Slow

Systematic

Reveals consequence as well as non-

consequence

Only works for truth-functional logic


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Formal Proofs

Pretty fast (with practice!)

Not systematic

Can only reveal consequence

Can be made into systematic method (that can then

also check for non-consequence) but becomes

inefficient

Can be used for predicate logic


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Truth Trees

Fast

Systematic

Can reveal consequence as well as non-consequence

Can be used for truth-functional as well as

predicate logic


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Propositional Satisfiability

Find the truth assignment that satisfies logical sentence

Propositional SatisfiabilityTesting:

1990: 100 variables/200 clauses (constraints)

1998: 10,000-100,000 variables/10^6 clauses

Some Applications:

Diagnosis, workflow analysis, planning, software/circuit

testing, machine learning, bioinformatics


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Electronic Circuit

&

&

1

1

P

Q

R


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Formal Languages and CommitmentsLanguage Ontological

CommitmentEpistemologicalCommitment

Propositional Logic facts true, false, unknown

First-order Logic facts, objects,

relations

true, false, unknown

Temporal Logic facts, objects,

relations, times

true, false, unknown

Probability Theory facts degree of belief [0,1]

Fuzzy Logic facts with degree of

truth [0,1]

known interval

value


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MotivationMotivation formal methods to perform reasoning are required when dealing

with knowledge

propositional logic is a simple mechanism for basic reasoning tasks

it allows the description of the world via sentences

simple sentences can be combined into more complex ones

new sentences can be generated by inference rules applied to existingsentences

predicate logic is more powerful, but also considerably morecomplex

it is very general, and can be used to model or emulate many othermethods

although of high computational complexity, there is a subclass that can betreated by computers reasonably well


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ObjectivesObjectives know the important aspects of propositional and predicate logic

syntax, semantics, models, inference rules, complexity

understand the limitations of propositional and predicate logic

apply simple reasoning techniques to specific tasks

learn about the basic principles of predicate logic

apply predicate logic to the specification of knowledge-basedsystems

use inference rules to deduce new knowledge from existingknowledge bases


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Evaluation Criteria

check sentences for syntactical correctness

check if a sentence is true or false

formulate simple sentences for simple

problems


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Concerns What does it mean to say a statement is true?

What are sound rules for reasoning

What can we represent in propositional logic?

What is the efficiency?

Can we guarantee to infer all true statements?


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Propositional Logic a relatively simple framework for reasoning

can be extended for more expressiveness at the cost ofcomputational overhead

important aspects

syntax semantics

validity and inference

models inference rules

complexity


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Semantics Model = possible world

x+y = 4 is true in the world x=3, y=1.

x+y = 4 is false in the world x=3, y = 1.

Entailment S1,S2,..Sn |= S means in every world whereS1Sn are true, S is true.

Some cognitive scientists argue that this is the way people

reason.


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Reasoning or Inference Systems

Proof is a syntactic property.

Rules for deriving new sentences from old ones.

Sound: any derived sentence is true.

Complete: any true sentence is derivable.

NOTE: Logical Inference is monotonic.


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Forms of Reasoning:

Deduction,Abduction,

InductionTheorem Proving,

Sherlock Holmes,

and All Swans are White...


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Basic Types of Inferences: Induction

Induction: Derive a general rule (axiom) from backgroundknowledge and observations.

Example:

Socrates is a human (background knowledge)

Socrates is mortal (observation/ example)

Therefore, I hypothesize that all humans are mortal (generalization)

Remarks:

Induction means to infer generalized knowledge from exampleobservations: Induction is the inference mechanism for (machine)learning.


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Basic Types of Inferences: Abduction Abduction: From a known axiom (theory) and some

observation, derive a premise.

Example:

All humans are mortal (theory)Socrates is mortal (observation)

Therefore, Socrates must have been a human (diagnosis)

Remarks:

Abduction is typical for diagnostic and expert systems. If one has the flue, one has moderate fewer. Patient X has moderate fewer. Therefore, he has the flue.

Strong relation to causation


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Deduction

Deductive inferences are also called theorem

proving or logical inference.Deduction is truth preserving: If the premises (axioms

and facts) are true, then the conclusion (theorem) is

true.

To perform deductive inferences on a machine, acalculus is needed:

A calculus is a set of syntactical rewriting rules definedfor some (formal) language. These rules must be soundand should be complete.


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Proposition A proposition is a symbolic variables whose value must

be either True or False, and which stands for a natural

language statement which could be either true or false

Examples:

A = Smith has chest pain

B = Smith is depressed

C = it is raining


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Propositional Logic

Representing simple facts

It is raining

RAINING

It is sunny

SUNNY

It is windy

WINDY

If it is raining, then it is not sunny

RAINING SUNNY


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Propositional Logic

VocabularyA set ofpropositionalsymbols

P, Q, R, .

A set of logical connectives

, , ,,

(and) (or) (not) (implication) (equivalence)

Parenthesis (for grouping)( )

Logical constantsTrue,False


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Propositional Logic

Each symbol P, Q, R etc is a (atomic) sentence

Both True and False are (atomic) sentences

A sentence enclosed in parentheses is a sentence

If and are sentences, then so are

conjunction disjunction negation implication

equivalence

The above are complex sentences

Precedence is , , , ,


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Truth Tables for Connectives

PTrue

TrueFalseFalse

P QFalseFalseFalseTrue

P QFalse

TrueTrueTrue

P Q

TrueTrueFalseTrue

P QTrue

FalseFalseTrue

QFalse

TrueFalseTrue

PFalse

FalseTrueTrue


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Sample Sentences

P (P Q) RTrue (P Q) (Q P)(P Q) (P R )

What do the sentences mean?

The meaning depends on user defined semantics. If P is defined asit is hot and Q is defined as it is raining, then

P means it is hotP Q means either is hot or it is raining (or both)Q means that it is not raining


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Well- Formed Formulas Formula

A term (string) in propositional logic.

Well formed formula (WFF)

A term that is constructed correctly according topropositional logic syntax rules.


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WFF Constants: False, True

Variables: P,Q,R

If a and b are WFF, a b are WFF

If a and b are WFF, a b are WFF

If a and b are WFF, ab are WFF

If a and b are WFF, ab are WFF

Any formula that cannot be constructed using these rulesare not WFF.


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Semantics interpretation of the propositional symbols and constants

symbols can stand for any arbitrary fact

sentences consisting of only a propositional symbols are satisfiable, but notvalid

the value of the symbol can be True or False

must be explicitly stated in the model

the constants True and False have a fixed interpretation

True indicates that the world is as stated

False indicates that the world is not as stated

specification of the logical connectives

frequently explicitly via truth tables


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Semantics Interpretation

Evaluation function of a formula Properties of wffs

Valid / tautology

Satisfiable

Contradiction

Equivalent formulas


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Semantics A formula F is a logical consequence of a formula P

A formula F is a logical consequence of a set of formulas P1,Pn

Notation of logical consequence P1,PnF.

Theorem. Formula F is a logical consequence of a set of formulas P1,Pn if the formula P1,Pn F isvalid.

Theorem. Formula F is a logical consequence of a setof formulas P1,Pn if the formula P1 Pn ~F is acontradiction.


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Some terms: summary

The meaning or semantics of a sentence determinesits interpretation.

Given the truth values of all symbols in a sentence, itcan be evaluated to determine its truth value

(True or False).

A model for a KB is a possible world (assignmentof truth values to propositional symbols) in whicheach sentence in the KB is True.


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More terms A valid sentence or tautology is a sentence that is True

under all interpretations, no matter what the world is actuallylike or how the semantics are defined. Example: Its raining or

its not raining.

An inconsistent sentence or contradiction is a sentencethat is False under all interpretations. The world is never like

what it describes, as in Its raining and its not raining.

P entails Q, written P |= Q, means that whenever P is True,

so is Q. In other words, all models of P are also models of Q.

M d l


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Models

if there is an interpretation for a sentence such that

the sentence is true in a particular world, that world is

called a model

refers to specific interpretations

models can also be thought of as mathematical objects these mathematical models can be viewed as equivalence

classes for worlds that have the truth values indicated by

the mapping under that interpretation

a model then is a mapping from proposition symbols to

True or False


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Propositional Logic: Semantics A modelMM of a propositional language consists of

a collection of atoms, say B = { bi : 0 < i }, where _|_ is

excluded from B, and

a partial mapping M from

A = { ai : 0 < i } to B = { bi : 0 < i }.

IfM(ai) B, we say that ai is true inM.M.

We write ai is true inMM asMM |= ai. (ReadMM satisfies ai).

|= is referred to as the satisfaction relation.


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Propositional Logic: Example B = { a1, a3} where M given as

M(a1) = a1 and M(a2) = a2 has the following properties.

MM |= a1

MM |= a1 a3MM |= a2MM |= a2 a4

MMdoes not satisfy the following propositions.

MM |= a4

MM |= a1 a4


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Models and Entailment a sentence is entailed by a knowledge base KB if the models of

the knowledge base KB are also models of the sentence

KB |= inference rules allow the construction of new sentences from

existing sentences

an inference procedure generates new sentences on the basis ofinference rules

if all the new sentences are entailed, the inference procedure iscalled sound or truth-preserving


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Tautology and Contradiction

Tautology: proposition that is always trueMale V ~Male

Contradiction: proposition that is always false.

Healthy ~Healthy


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Suppose we want to prove (P Q) (P Q)

An inference procedure for proposition logic

Construct a truth table, if(P Q) (P Q) is true for allvalues of P and Q, then we have proved it.

For any sentence, no matter how complex, we can always

prove or disprove it this way. In other words, truth tableconstruction is complete.


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Equivalence rules

P)(QQ)(PQPnimplicatiodoubleEliminate

QP~QPnImplicatioEliminate

Q~P~Q)(P~Q~P~Q)(P~MorganDe

R)(PQ)(PR)(QPR)(PQ)(PR)(QPveDistributi

PQQPPQQPPQQPComutative

R)(QPRQ)(PR)(QPRQ)(PAsociative

PPPPPPIdempotent


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Inference RulesSome patterns of reasoning are so common that instead of creating a truth tableeach time we see them, we can just establish their truth once, then reuse the

pattern in any situation.

IrishHot, Irish |- HotIf we know Irish implies hot is true and know


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If we know Irish implies hot is true, and knowIrish is true, we can infer Hot is true. Read as

Irish implies Hot, Irish, Therefore Hot

Irish Blue |- BlueIf we know Irish and Blue is true, we can inferthat Blue is true. Read as

Irish and Blue Therefore Blue

Irish, Red |- Irish RedIf we know Irish is true, and we know Red is true,

we can infer that Irish and Red is true. Read asIrish , Red Therefore Irish and Red


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Irish |- Irish GreenIf we know Irish is true, then we know that Irishor Green is true. Read as

Irish, Therefore Irish or Green

Irish |- IrishIf we know not not Irish is true, we can infer thatIrish is true. Read as

Not not Irish, Therefore Irish

Irish Red, Red Fast |- Irish FastIf we know Irish or Red is true, and we know not Red or Fast is true, wecan infer that Irish or Fast must be true. Read as

Irish or Red, not Red or Fast, Therefore Irish or Fast

Suppose the knowledge base consists of the facts


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Suppose the knowledge base consists of the facts

S T (P R)STR

And we need to prove P is entailed. We can use the rules ofinference to do this..

S T (P R) , S , T And-Introduction

S T (P R) , S T Double Negation EliminationS T (P R) , (S T) Modus ponens(P R) And-EliminationP Double Negation Elimination

P

So the rules of inference allow us to (sometimes) bypass having to

build truth tables.


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Logical Inference also referred to as deduction

implements the entailment relation for sentences

validity a sentence is valid if it is true under all possible interpretations in all possible

world states

independent of its intended or assigned meaning

independent of the state of affairs in the world under consideration valid sentences are also called tautologies

satisfiability

a sentence is satisfiable if there is some interpretation in some world state (a

model) such that the sentence is true a sentence is satisfiable iff its negation is not valid

a sentence is valid iff its negation is not satisfiable


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Computational Inference computers cannot reason informally (common sense)

they dont know the interpretation of the sentences they usually dont have access to the state of the real world to

check the correspondence between sentences and facts

computers can be used to check the validity of sentences if the sentences in a knowledge base are true, then the

sentence under consideration must be true, regardless of itspossible interpretations

can be applied to rather complex sentences

Computational Approaches


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Computational Approaches

to Inference model checking based on truth tables

generate all possible models and check them for validity or satisfiability

exponential complexity, NP-complete

all combinations of truth values need to be considered

search

use inference rules as successor functions for a search algorithm

also exponential, but only worst-case

in practice, many problems have shorter proofs

only relevant propositions need to be considered


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Validity and Inference truth tables can be used to test sentences for

validityone row for each possible combination of truth

values for the symbols in the sentence

the final value must be True for every sentencea variation of the model checking approach

not very practical for large sentences

sometimes used with customized improvements inspecific domains, such as VLSI design


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Validity and Computers the computer has no access to the real world, and cant

check the truth value of individual sentences (facts)

humans often can do that, which greatly decreases thecomplexity of reasoning

humans also have experience in considering only important

aspects, neglecting others if a conclusion can be drawn from premises, independent of

their truth values, then the sentence is valid

usually too tedious for humans

may exclude potentially interesting sentences

some, but not all interpretations are true


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Propositional Logic: Proofs

What formulas hold in all models ?

i.e. can we check if a given proposition is true

in all models without going through all

possible models?

Need proofs to answer this question.

We use Natural Deduction proofs.


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Propositional Proofs: Examples Prove: ( A B ) (A B)

Notice:The outermost connective is. Hence, the last step ofthe proof must be an implication introduction.

That means, we must assume ( A B ) and prove(A B), and then discharge the assumption by using introduction rule.

In order to prove (A B) from ( A B ), we must useintroduction, and hence must prove either A or Bfrom ( A B ).This plan forms a skeleton of a proof.

P P f E l C i d


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Prop. Proof: Example Continued

Prove: ( A B ) (A B)(A ^ B)

A ^ elimination

A v B v introduction

( A ^ B ) => (A v B) => introduction

Proofs are analyzed backwards, i.e. start unraveling the

logical structure of the conclusion and work backwards tothe assumptions.


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Derived Rules These are rules derived from other rules.

Example:A ^ B

B ^ A

Here is the derivation:A ^ B B ^ A

B A^

elimination

B ^ A ^ introduction

Inference rules


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Inference rules

Logical inference is used to create new sentences that

logically follow from a given set of predicate calculus

sentences (KB).

An inference rule is sound if every sentence X produced

by an inference rule operating on a KB logically follows

from the KB. (That is, the inference rule does not create

any contradictions)

An inference rule is complete if it is able to produce

every expression that logically follows from (is entailed by)

the KB. (Note the analogy to complete search algorithms.)

Sound rules of inference


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Sound rules of inference

Here are some examples of sound rules of inference

A rule is sound if its conclusion is true whenever the premise is true

Each can be shown to be sound using a truth table

RULE PREMISE CONCLUSION

Modus Ponens A, A B B

And Introduction A, B A B

And Elimination A B A

Double Negation A AUnit Resolution A B, B A

Resolution A B, B C A C

Soundness of modus ponens


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Soundness of modus ponens

A B A

B OK?

True True True True False False

False True True False False True

Soundness of the


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resolution inference rule


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Proving things A proofis a sequence of sentences, where each sentence iseither a premise or a sentence derived from earlier sentencesin the proof by one of the rules of inference.

The last sentence is the theorem (also called goal or query)that we want to prove.

Example for the weather problem given above.

1 Humid Premise It is humid

2 HumidHot Premise If it is humid, it is hot

3 Hot Modus Ponens(1,2) It is hot

4 (HotHumid)Rain Premise If its hot & humid, its raining

5 HotHumid And Introduction(1,2) It is hot and humid

6 Rain Modus Ponens(4,5) It is raining


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Horn sentences A Horn sentence or Horn clause has the form:

P1 P2 P3 ... Pn Qor alternatively

P1 P2 P3 ... Pn Q

where Ps and Q are non-negated atoms To get a proof for Horn sentences, apply Modus

Ponens repeatedly until nothing can be done

(PQ) = (PQ)


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Entailment and derivation Entailment: KB |= Q

Q is entailed by KB (a set of premises or assumptions)if and only if there is no logically possible world inwhich Q is false while all the premises in KB are true.

Or, stated positively, Q is entailed by KB if and only if

the conclusion is true in every logically possible worldin which all the premises in KB are true.

Derivation: KB |- Q

We can derive Q from KB if there is a proofconsisting of a sequence of valid inference stepsstarting from the premises in KB and resulting in Q

Two important properties for inference


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Two important properties for inference

Soundness: If KB |- Q then KB |= Q

If Q is derived from a set of sentences KB using a

given set of rules of inference, then Q is entailed byKB.

Hence, inference produces only real entailments, orany sentence that follows deductively from the

premises is valid.Completeness: If KB |= Q then KB |- Q

If Q is entailed by a set of sentences KB, then Q canbe derived from KB using the rules of inference.

Hence, inference produces all entailments, or all validsentences can be proved from the premises.

Soundness and Completeness


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Soundness and Completeness

A rule A1 An is said to be sound if for every B

model in which all ofA1 An are true, then so is B. i.e. if

MM |=A1 , , MM |=An, then MM |=B. A collection of rules are sound if all rules in the collection is

sound.

A collection of rules is complete ifMM |=A for all models M,M,then A is provable. I.e. there is a proof of A using the given set ofrules. (Denoted |R-- A ) where Ris the set of rules.


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Complexity the truth-table method to inference is complete

enumerate the 2n rows of a table involving n symbols

computation time is exponential

satisfiability for a set of sentences is NP-complete

so most likely there is no polynomial-time algorithm

in many practical cases, proofs can be found with moderateeffort

there is a class of sentences with polynomial inferenceprocedures (Horn sentences or Horn clauses)

P1 P2 ... Pn Q

We can represent facts in Propositional Logic, and we have a


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sound and complete procedure for inference.But Propositional Logic has some weaknesses...

The size of the truth tables grows exponentially. So we may

run out of space and time before we can answer a question.

Suppose we want to encode the fact that all men are mammals,we have no choice but to list that fact for each individual man in

the knowledge baseP means Paul is a mammalQ means Quentin is a mammal

R means Robert is a mammal

S means Steve is a mammaletc etc

What we really need is a compact way represent these kinds offacts.


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Validity != Provability Goldbachs conjecture: Every even number (>2) is

the sum of 2 primes.

This is either valid or not.

It may not be provable.

Godel: No axiomization of arithmetic will becomplete, i.e. always valid statements that are not

provable.

Limitation of Propositional logic


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Limitation of Propositional logic

Theorem proving is decidable

The propositional calculus has its limitations cannot deal properly with general statements of the

form:-

All men are mortal. You cannot derive from the conjunction of this and

Devindra is a man that..

Devindra is mortal. Cannot represent objects and quantification


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Limitations If all men are mortal =P

Devindra is a man = Q

Devindra is mortal =R

Then (P & Q) R is not valid

To do this, you need to analyze propositions intopredicates and arguments and delay explicitly with

quantification.

Limitations of Propositional Logic


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Limitations of Propositional Logic

number of propositions

since everything has to be spelled out explicitly, the number of rules is

immense dealing with change (monotonicity)

even in very simple worlds, there is change

the agents position changes

time-dependent propositions and rules can be used

even more propositions and rules

propositional logic has only one representational device, theproposition

difficult to represent objects and relations, properties, functions,

variables, ...


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Implies => If 2+2 = 5 then monkeys are cows. TRUE

If 2+2 = 5 then cows are animals. TRUE

Indicates a difference with natural reasoning. Singleincorrect or false belief will destroy reasoning. No

weight of evidence.


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Inference Does s1,..sk entail s?

Say variables (symbols) v1vn. Check all 2^n possible worlds.

In each world, check if s1..sk is true, that s is true.

Approximately O(2^n).

Complete: possible worlds finite for propositionallogic, unlike for arithmetic.


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What cant we say? Quantification: every student has a father.

Relations: If X is married to Y, then Y is married toX.

Probability:There is an 80% chance of rain.

Combine Evidence: This car is better than thatone because

Uncertainty: Maybe John is playing golf.

Propositional logic is a weak language


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p g g g

Hard to identify individuals (e.g., Mary, 3)

Cant directly talk about properties of individuals or

relations between individuals (e.g., Bill is tall) Generalizations, patterns, regularities cant easily be

represented (e.g., all triangles have 3 sides)

First-Order Logic (abbreviated FOL or FOPC) isexpressive enough to concisely represent this kind of

information

FOL adds relations, variables, and quantifiers, e.g.,Every elephant is gray: x (elephant(x) gray(x))

There is a white alligator: x (alligator(X) ^ white(X))

Example


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p

Consider the problem of representing the

following information:Every person is mortal.

Confucius is a person.

Confucius is mortal.

How can these sentences be represented so

that we can infer the third sentence from the

first two?


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Example II In PL we have to create propositional symbols to stand for all or part

of each sentence. For example, we might have:

P = person; Q = mortal; R = Confucius

so the above 3 sentences are represented as:

P Q; R P; R Q

Although the third sentence is entailed by the first two, we needed anexplicit symbol, R, to represent an individual, Confucius, who is amember of the classes person and mortal

To represent other individuals we must introduce separate symbolsfor each one, with some way to represent the fact that all individualswho are people are also mortal


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Some Applications ofPropositional Logic

SUDOKU


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1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9

Mathematical models


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Dutch Soccer League


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if Eindhoven and Amsterdam play on the same day the TV income is

x

If Eindhoven and Amsterdam play on two different days, the incomeis 2x

if a team plays on Wednesday champions league it doesnt play on

Friday

there are at most 3 plays on Friday

.. in sum several thousand constraints over LP and Boolean

variables

League is modelled by the Barcelogic tool

Transition Systems


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Example an Editor that can

hi hli h h i


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highlight the semantic error

Summary


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The process of deriving new sentences from old one is called inference.

Sound inference processes derives true conclusions given true premises

Complete inference processes derive all true conclusions from a set ofpremises

A valid sentence is true in all worlds under all interpretations

If an implication sentence can be shown to be valid, thengiven itspremiseits consequent can be derived

Different logics make different commitments about what the world is

made of and what kind of beliefs we can have regarding the facts Logics are useful for the commitments they do not make because lack of

commitment gives the knowledge base engineer more freedom

Propositional logic commits only to the existence of facts that may ormay not be the case in the world being represented

It has a simple syntax and simple semantics. It suffices to illustrate the processof inference

Propositional logic quickly becomes impractical, even for very small worlds

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