Formal Aspects of Computer Science - Week 5Logic and Reasoning

Lee McCluskey, room 2/07

Email lee@hud.ac.uk

Logic and Reasoning in AI

Recap

• Fundamental to logic languages is the idea of INTERPRETATIONS - mapping predicates and constants to some conceptualization of the world.

• A well formed sentence in Logic is called a Wff.

Wff2 LOGICALLY FOLLOWS from Wff1 if and only if every interpretation that makes Wff1 true also makes Wff2 true.

Wff2 is LOGICALLY EQUIVALENT to Wff1 if and only if every interpretation that makes Wff1 true also makes Wff2 true AND vice-versa.

Logic and Reasoning in AI

Meaning of QuantifiersConsider a Universe with individuals a,b,c,…

Ax P(x) = P(a) & P(b) & P(c) & ….

Ex P(x) = P(a) V P(b) V P(c) V ….

Ax Ay R(x,y) = R(a,a) & R(a,b) & R(a,c) &…

& R(b,a) & R(b,b) & R(b,c) & …

Ax Ey R(y,x) = Ey R(y,a) & Ey R(y,b) & Ey R(y,c) &…

= (R(a,a) V R(b,a) V R(c,a) V …) &

(R(a,b) V R(b,b) V R(c,b) V …) &

(R(a,c) V R(b,c) V R(c,c) V …) & ….

Logic and Reasoning in AI

Meaning of Connectives The Connectives

&, V, ~, ->, <-> , <- (NB alternative syntax =>, , <= ETC)

Get their meaning via propositional truth tables –

P Q P V Q ETC

T T T

T F T

F F F

F T T

Logic and Reasoning in AI

“Laws”These are some well known equivalent FORMS in FOL

called laws ( De Morgans laws etc)

¬ ( P & Q ) = ¬P V ¬Q

¬ ( P V Q ) = ¬P & ¬Q

P=>Q = ¬P V Q

¬ ¬ P = P

P Q = (P=>Q)&(Q=>P)

etc

Logic and Reasoning in AI

Quantifiers + Negation LAWS

1. ¬ Ax P(x) = Ex ¬ P(x)

2. ¬ Ex P(x) = Ax ¬ P(x)

Similary (and abstractly)

¬ A E = E A ¬

Logic and Reasoning in AI

Interpretations revisitedAx Ey R(y,x)

Mother_of

persons

“Given any person there isSomeone who is their mother”

Greater_than

numbers

“Given any number thereIs some number greater than it”

NB Ax Ey … =/= Ey Ax

These 2 Interpretations SATISFY this WFF

WFF = WFF =

Logic and Reasoning in AI

Example

“Every student is an academic.

Everybody who teaches an academic is an academic.

Jeff teaches Fred who is a student.”

What can we say about the statement “Jeff is an academic”

Translate to FOL:

S = student, D = academic, T = teaches

Ax S(x)=>D(x) Ax (Ey T(x,y) & D(y)) => D(x)

S(Fred) T(Jeff,Fred)

Goal: D(Jeff)

How can we get agents to automatically deduce such facts??

Logic and Reasoning in AI

Another Example

Imagine Deep Space 1 travels to Mars and observes many things about the Martians, including the fact that some seem very hostile towards

humans. Concrete observations are as follows:

(a) All green Martians have antennae.

(b) A Martian is friendly to humans if all of its children have antennae.

(c) A Martian is green if at least one of its parents is green.

On its way back from Mars the robot is hotly pursued

by a spacecraft containing green Martians only. Should the robot

suspect it is being attacked? Or can the robot reason with its observations to answer the question: `Are all green Martians friendly?''

and hence avert an inter-planetary conflict.

Logic and Reasoning in AI

Deduction

We deduce using sound inference rules

A Rule (Law) of Inference is a method for producing a new wff from

parents is SOUND if it only ever produces wffs that also

logically follow from the parents.

Logic and Reasoning in AI

Natural Deduction

The most famous Laws of Inference is known by its Latin name “Modus Ponens”

From wffs OF THE FORM…

P(a)

Ax P(x)=>Q(x)

We can deduce the following Wff

Q(a)

Example: Socrates is a Man, All Men are Mortal

Deduce: Socrates is Mortal

Logic and Reasoning in AI

Natural Deduction

Another is called “Modus Tollens”

From wffs OF THE FORM…

¬Q(a)

Ax P(x)=>Q(x)

We can deduce the following Wff

¬P(a)

Example: If a thing is smoking then it is on fire. I am not on fire.

Deduce: I am not smoking

Logic and Reasoning in AI

Unsound DeductionExample: If a person is the murderer then that

person must have bloody hands.

The Butler has bloody hands.

Deduce: The Butler is the murderer

This is UNSOUND!!!

BH(butler)

Ax M(x)=>BH(x)

We can’t deduce anything from this!!

Logic and Reasoning in AI

One “Inference Rule” to Rule them all...A COMPLETE proof procedure is one that,

given a wff w does follow from wff W, it will always generate a proof.

There is a single inference rule which can be used to create a complete proof procedure. “You will need no other..!

It is called ....

“The Law of Resolution”(wow)

Logic and Reasoning in AI

Automated DeductionAutomating deduction using RESOLUTION

requires Wffs to be translated to a “clausal form”. Prolog statements are in a type of clausal form.

The most common clausal form is when a Wff is expressed as a set of clauses, where each clause in the set is a disjunction of literals, and where any variables are universally quantified. EG

{ BH(butler),

~M(x) V BH(x) }

Logic and Reasoning in AI

SummaryFOL is equipped with a form of reasoning

called deduction that can be automated

Next lecture I will cover resolution refutation, a very efficient way to automate deduction.