First Order Logic

COMP30411: Knowledge Representation

University of Manchester, 2006 1

CS1081

First Order Logic:Syntax and Semantics

COMP30411Sean Bechhofer

[email protected]

2COMP30411: Knowledge Representation

Logic Recap

You should already know the basics of First Order Logic (FOL) Its a prerequisite of this course!

The next two lectures provide a brief refresher But we wont go over everything in detail

Youre strongly advised to do some background reading if this isntfamiliar territory Much of the remainder of the course will rely on an

understanding of this material. E.g. The Language of First Order Logic, Jon Barwise & John

Etchemendy, CSLI Lecture Notes, 1993




What is a Logic?

When people talk about logic they often mean propositional or first-order predicate logic.

However there are many other logics, for example modal andtemporal logics and description logics. Well meet some of these later on.

A logic usually has a well defined syntax, semantics and proof theory. The syntax of a logic defines the syntactically acceptable objects of

the logic, or well-formed formulae. The semantics of a logic associates each formula with a meaning. The proof theory is concerned with manipulating formulae

according to certain rules.


Coming Up

Over the course of the next few lectures, well look in some detail atthe syntax, semantics and proof theory of propositional logic.

Well then move on to predicate logic, seeing how this extendspropositional logicand touching on some of the key characteristics ofsuch languages.




An Example: Cloth Weaves[Maier & Warren, Computing with Logic, 1988]

An example showing how we can represent the qualities andcharacteristics of cloth types using a simple propositional logicknowledge base.


Cloth

Woven fabrics consist of two sets of threads interlaced at rightangles.

The warp threads run the length of the fabric The weft (fill, pick or woof) threads are passed back and forth between

the warp threads. When weaving, the warp threads are raised or lowered in patterns,

leading to different weaves. Factors include:

The pattern in which warps and wefts cross Relative sizes of threads Relative spacing of threads Colours of threads




Plain Weave

Over and under in aregular fashion


Twill Weave

Warp end passes overmore than one weft Known as floats

Successive threadsoffset by 1




Satin Weave

Longer floats Offsets larger than 1


Classifying Cloth

The example provides a number of rules that describe how particularkinds of cloth are described.

alternatingWarp ! plainWeave If a piece of cloth has alternating warp, then its a plain weave.

hasFloats, warpOffsetEq1 ! twillWeave If a piece of cloth has floats and a warp offset of 1, then its a twill

weave. There are many other properties concerning the colour of threads,

spacings etc.




Using the Rules

We could use these rules to build a system that would be able torecognise different kinds of cloth through recognising the individualcharacteristics.

The example given shows that once we have recognised the followingcharacteristics diagonalTexture floatGTSink colouredWarp whiteFill

Then we can determine that this cloth is denim.


Knowledge Representation

Although this is relatively simple (in terms of both the expressivity ofthe language used and the number of facts), this really is an exampleof Knowledge Representation. The rules represent some knowledge about cloth A machine can make use of this knowledge to deduce

consequences.

This particular example makes use of propositional logic.




Propositional Logic

The syntax of propositional logic is constructed from propositions andconnectives.

A proposition is a statement that is either true or false but not both. Examples: the following are propositions

Walter Smith is Scotland Manager 2 + 3 = 5 Steve MacLaren is Scotland Manager 2 + 3 = 6 the reactor is in a stable state

whereas the following are not What's the time? Fantastic Goal! 2+3 antelope


Propositions

In general, we can determine whether any given statement is aproposition by prefixing it with It is true that

and seeing whether the result makes grammatical sense. Propositions are often abbreviated using propositional variables eg p,

q, r. Thus we must associate the propositional variable with its meaning

i.e. Let p be Tony Blair is Prime Minister. Alternatively we might write something like

reactor_is_in_a_stable_state so that the intended meaning of thepropositional variable is clearer. But we need to be very careful here that we dont encode too

much information in the names of the propositions.




Connectives

Connectives allow us to form compound propositions by combiningpropositions.

$iffif and only if,

!impliesif then)

~negationnot

|disjunctionor

&conjunctionand


And (Conjunction)

The conjunction p and q, written p q of two propositions is truewhen both p and q are true, false otherwise.

Its Monday and its rainingp q

Its rainingq

Its Mondayp




Truth Tables

Truth tables provide a summary of a connective. The truth table shows all possible combinations of the inputs to the

connective, along with the outcome. Rows in the table give all possible settings of the propositions to

true (T) or false (F). For a connective with n inputs, we need 2n rows in our truth table. A truth table for is:

FFF

FTF

FFT

TTT

p qqp


Or (Disjunction)

The disjunction p or q, written p q, of two propositions is truewhen p or q (or both) are true, false otherwise.

This is sometimes called inclusive or, as we dont require that only oneof the propositions is true.

FFF

TTF

TFT

TTT

p qqp

Its Monday or its rainingp q

Its rainingq

Its Mondayp




Not (Negation)

The negation not p of a proposition p is true when p is false and isfalse otherwise. Can also be read as it is false that p

TF

FT

pp

Its false that logic is easyIts not the case that logic is easyLogic isnt easy

p

Logic is easyp


If then (Implication)

The implication p implies q, written p q, of two propositions istrue when either p is false or q is true, and false otherwise.

TFF

TTF

FFT

TTT

p qqp

If I study hard then, I get richWhen I study hard, I get rich

p q

I get richq

I study hardp




If then (Implication)

We need to be careful with as it may not quite capture ourintuitions about implication.

In particular (taking the previous example), p q is true in thefollowing situations: I study hard and I get rich; or I don't study hard and I get rich; or I don't study hard and I don't get rich.

Note the last two situations, where the implication is true regardlessof the truth of p.

The one thing we can say is that if I've studied hard but failed tobecome rich then the proposition is clearly false.


Iff (Bi-implication)

The bi-implication p iff q (p if and only if q), written as p q, of twopropositions is true when both p and q are true or when both p andq are false, and false otherwise.

TFF

FTF

FFT

TTT

p qqp

Sean is happy iff its the weekendp q

Its the weekendq

Sean is happyp




Propositional Logic: Moreformally

The language of propositional logic consists of the following symbols. A set PROP, of proposition symbols, p, q, r etc A set of propositional connectives:-

nullary connectives true and false; unary connective binary connectives , , ,

The symbols ( and ) these are used to avoid ambiguity.


Well Formed Formulae

The set of Well-Formed Formulae (WFF) is defined as: Any propositional symbols is in WFF The nullary connectives true and false are in WFF If A and B are in WFF then so are

A

(A) A B A B A B

A B




Examples

p (p) (()) true p true p q

p q (p q) true false p q (p q) r p q r


Truth Tables

We can extend the basic truth tables to show interpretations ofcompound propositions.

To draw up a truth table, construct a column for each propositioninvolved.

We need 2n rows for n propositions in order to record all possibleways of setting the propositions to T and F.

If we have 3 propositions, , i.e. we need 23 = 8 rows. Construct a column for each connective, the most deeply nested first. Evaluate each column using values for propositions or previous

columns.




Example

TTTTFFTFTFFFTFFFFFFF

FTFTr

TTTTp q

FFTTFTFTTTTT(p q) rqp

(p q) r

I eat toast for breakfastr

I eat cereal for breakfastq

I eat eggs for breakfastp

Compound proposition is true if I eat eggs, cereal and toast; or I eat eggs and toast; or I eat cereal and toast


Interpretations

Given a particular formula, can you tell if it is true or not? The truth of a compound formula depends on the truth values of the

component propositions. A truth valuation is a function:

Iv: PROP {T,F}

which assigns a truth value to each atomic proposition Given a truth valuation, we can define a interpretation I that gives a

truth value for a (well-formed) compound formula using the truthvaluation for the atomic propositions and the truth tables for theconnectives involved.




Example

Proposition:(p q) r

Truth valuation:Iv(p) = F, Iv(q) = T, Iv(r) = F

Applying the truth valuation gives us:I((p q) r)

= (Iv(p) Iv(q)) Iv(r))= (F T) F= T F

= F

FFF

TTF

TFT

TTT

p qqp

TFF

TTF

FFT

TTT

p qqp


Semantics of PropositionalLogic

Given an truth valuation Iv:PROP ! {T,F}, we can extend this toprovide us with a interpretation I that assigns either T or F to anystatement of the language.

The symbol is used to represent the relationship betweeninterpretations and formulae:

I A if and only if I Bif, and only ifI A , BIf I A then I Bif, and only ifI A ) BI A or I Bif, and only ifI A BI A and I Bif, and only ifI A BI 2 Aif, and only ifI A

if, and only if Iv(p) = TI pI 2 falseI true




Tautologies and Satisfiability

A formula is valid or is a tautology iff it is true under everyinterpretation. If A is a tautology, this is written

A A formula is said to be satisfiable (or consistent) iff it is true under at

least one interpretation. A formula is said to be unsatisfiable (or inconsistent or contradictory) iff

there is no interpretation under which it is true If a formula A is a tautology then A is unsatisfiable.


Truth Tables

Each line of a truth table corresponds to a interpretation. So we can use truth tables to determine whether or not

formulae are valid. Example: (p ) q) (q ) p)

TTTF

p ) q

TFTT

q ) p

TFF TTF TTT TFT

(p ) q) (q ) p)qp

Tautology




Truth Tables

Example: (p ) q) p

TTTF

p ) q

FFF FTF TTT FFT

(p ) q) pqp

Not a tautology, but satisfiable


Translations

An alternative to providing a direct semantics (e.g. Interpretations asweve seen here) is to provide a translation of the connectives

For example, we can define p ! q

as p q

If we look at the truth tables for each of these formulae, then we cansee that for any valuations of p and q, the result is the same.




Defining New connectives

Using this approach of translation rather than a direct semantics, wecan define new connectives.

Recall the truth table for or p q is true when either p or q is true.

Its also true when both p and q are true For some situations, we might want a new connective that tells us

when only one of p or q is true: also known as exclusive or andsometimes written p q

FFF

TTF

TFT

TTT

p qqp


Defining New connectives

We could define p q using a truth table:

Alternatively, we can give a translation to an equivalent expressionusing the other connectives:

p q (p q) (p q)

FFF

TTF

TFT

FTT

p qqp




Functional Completeness

The above example illustrates that we dont actually need the exclusiveor connective It can be defined in terms of , and

So how many connectives do we really need? We say a set of connectives is functionally complete if there is no

truth table that can not be expressed as a formula involving onlythese connectives.

For example, the set {, , } is functionally complete In fact {,} is functionally complete

De Morgans Laws


Logical Consequences

We have provided definitions of the syntax and semantics ofpropositional logic.

Given a set of formulae, we want to know which formulae are logicalconsequences of that set

B is a consequence of A if its the case that whenever B is true, thenA must be true also. We write:

A B For example, if we know that both p and p ) q hold, then we might

want to be able to conclude that q also holds.





We can do this by constructing truth tables. If we want to know whether B follows from A1, A2, . An, then we

could construct a truth table forA1 A2 An ) B

and check to see is this is a tautology. E.g. is it the case that q follows from (p ) q) and p?

Construct a truth table for ((p ) q) p) ) q

TTFT(p ) q)

FFFT(p ) q) p

TFFTTFTFTTTT((p ) q) p) ) qqp


Knowledge Bases

We can now use thise notion of logical consequence. A Knowledge Base K = {p1, , pn} is a collection of propositions that

describe facts about a situation of affairs that we are trying to model We can then derive consequences from this knowledge base where q

is a consequence ifp1,,pn q

For example, the rules encapsulating the different characteristics andqualities of different cloth weaves.





As we saw before, we could test for logical consequence byconstructing truth tables.

If we want to know whether B follows from A1, A2, . An, then wecould construct a truth table for

A1 A2 An ) B

and check to see is this is a tautology. E.g. is it the case that q follows from (p ) q) and p?

Construct a truth table for ((p ) q) p) ) q)

TTFT(p ) q)

FFFT(p ) q) p

TFFTTFTFTTTT((p ) q) p) ) q)qp

Problem: This is potentiallyexpensive: we need 2n rowsfor n propositions, so ourtruth tables may becomelarge.


Proof Rules

An alternative is to use some proof rules or proof theory. Proof rules provide us with a mechanism for determining logical

consequences through some syntactic manipulation of the formulae. A rule stating that B follows (or is provable from) A1,,An is written

as:

A well known proof rule is modus ponens:

where A and B are any WFF.

B

A1 An

B

A ) B A




More Proof Rules

A common rule, known as -elimination is:

or

We can read the first as If A and B hold, then A must also hold.

The -introduction rule tells us:

or

Again, we can read the first as: If A holds, then A B must also hold.

A

A B

B

A B

A B

B

A B

A


Proof

By combining proof rules together, we can deduce consequencesfrom premises.

Ex: from p q and q ) r, can we prove r?

1. p q [given]2. q ) r [given]3. q [1 -elimination]

4. r [2,3 modus ponens]




Proof Theory

This reasoning about statements of the logic is being done withoutconsidering the interpretations Were not drawing up truth tables to work out if the

consequences hold. Proof rules show us, given true statements, how to generate further

true statements. Axioms describe universal truths of the logic

For example, p p is an axiom of propositional logic


Proofs

If A1,An,B are WFF, then there is a proof of B from A1,,An iffthere exists some sequence of formulae C1,,Cn such that Cn = Band each formula Ck for 1 k < n is either an axiom, one of theformulae A1,An or is the conclusion of a rule whose premisesappeared earlier in the sequence.

We write A1,,An ` B to show that B is provable from A1,,An(given some set of proof rules and axioms).

Ex: from p ) q, ( r q) ) (s p), q can we prove s q?1. p ) q [given]2. ( r q) ) (s p) [given]3. q [given]4. s q [3, -introduction]

Alternative would be truth table for (p ) q ( r q) ) (s p) q ) ) (s q)




Relating Proofs and Semantics

So far weve considered notions of validity and provability Validity is related to semantics. A propositional formula is valid if it is

satisfied under all possible interpretations We draw up a truth table and check that every line evaluates to

true. Provability is related to the proof theory. A propositional formula is

provable if it is an axiom or it is provable from other provableformulae We construct a proof applying proof rules at each step.

Whats the connections between the two? Soundness and Completeness


Soundness

We write A1,,An ` B to denote that B is provable from A1,,Anusing a set of proof rules

A Soundness theorem states that:If A1,,An ` Bthen A1 An B (i.e. A1 An ) B is valid)

Informally, soundness gives us an assurance that our proof theory isproducing correct answers




Completeness

A Completeness theorem states that:If A1 An B (i.e. A1 An ) B is valid)then A1,,An ` B

Informally, completeness gives us an assurance that if a formula isvalid, we can construct a proof of it using our proof theory.

Note that completeness doesnt say anything at all about how wemight go about constructing such a proof just that the proofexists.

Note that both soundness andcompleteness are with respectto a particular set of axiomsand proof rules. `

Completeness

Soundness


Soundness and Completeness

An example of an unsound rule is:

Using this rule, given any p and q, we can deduce an arbitrary r.However p q ) r is not a valid formula. Thus including this rule inour proof rules would not give us soundness.

If we only have rules for modus ponens and -elimination, we do nothave completeness. With only those rules, we cant prove

p ` p q

even thoughp ) (p q)

is valid

C

A B




Summary

Weve seen a (brief) overview of propositional logic, its syntax,semantics and proof theory. We havent seen all the details of the proof rules for

propositional logic though. These can be found in anyintroductory logic text.

Weve seen the notions of soundness and completeness, relating theinterpretation semantics and the proof theory.

Next, well look at predicate calculus, which gives us a much moreexpressive language than propositional logic.

Documents

First Order Logic