FOL and Prolog

First Order Logic

Chapter 8

Outline

Why FOL?

Syntax and semantics of FOL

Using FOL

Wumpus world in FOL

Knowledge engineering in FOL

Pros and cons of propositional logic

J Propositional logic is declarative

J Propositional logic allows partial/disjunctive/negated information

(unlike most data structures and databases)

J Propositional logic is compositional:

meaning of B_1,1 Ù P_1,2 is derived from meaning of B_1,1 and of P_1,2

J Meaning in propositional logic is context-independent

(unlike natural language, where meaning depends on context)

L Propositional logic has very limited expressive power

(unlike natural language)

E.g., cannot say "pits cause breezes in adjacent squares“

except by writing one sentence for each square

First-order logic

Whereas propositional logic assumes the world contains facts,

first-order logic (like natural language) assumes the world contains

Objects: people, houses, numbers, colors, baseball games, wars, …

Relations: red, round, prime, brother of, bigger than, part of, comes between, …

Functions: father of, best friend, one more than, plus, …

Syntax of FOL: Basic elements

Constants KingJohn, 2, NUS,...

Predicates Brother, >,...

Functions Sqrt, LeftLegOf,...

Variables x, y, a, b,...

Connectives Ø, Þ, Ù, Ú, Û

Equality =

Quantifiers ", $

Atomic sentences

Atomic sentence = predicate (term₁,...,term_n) or term₁ = term₂

Term = function (term₁,...,term_n) or constant or variable

E.g.,

Brother(KingJohn,RichardTheLionheart)

Length(LeftLegOf(Richard)) = Length(LeftLegOf(KingJohn))

Complex sentences

Complex sentences are made from atomic sentences using connectives

ØS, S₁Ù S₂, S₁Ú S₂, S₁Þ S₂, S₁ÛS₂,

E.g. Sibling(KingJohn,Richard) Þ Sibling(Richard,KingJohn)

>(1,2) Ú ≤ (1,2)

>(1,2) Ù Ø >(1,2)

Truth in first-order logic

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements) and relations among them

Interpretation specifies referents for

constant symbols → objects

predicate symbols → relations

function symbols → functional relations

An atomic sentence predicate(term₁,...,term_n) is true

iff the objects referred to by term₁,...,term_n

are in the relation referred to by predicate

Models for FOL: Example

Universal quantification

"<variables> <sentence>

Everyone at NUS is smart:

"x At(x,NUS) Þ Smart(x)

"x P is true in a model m iff P is true with x being each possible object in the model

Roughly speaking, equivalent to the conjunction of instantiations of P

At(KingJohn,NUS) Þ Smart(KingJohn)

Ù At(Richard,NUS) Þ Smart(Richard)

Ù At(NUS,NUS) Þ Smart(NUS)

Ù ...

A common mistake to avoid

Typically, Þ is the main connective with "

Common mistake: using Ù as the main connective with ":

"x At(x,NUS) Ù Smart(x)

means “Everyone is at NUS and everyone is smart”

Existential quantification

$<variables> <sentence>

Someone at NUS is smart:

$x At(x,NUS) Ù Smart(x)

$x P is true in a model m iff P is true with x being some possible object in the model

Roughly speaking, equivalent to the disjunction of instantiations of P

At(KingJohn,NUS) Ù Smart(KingJohn)

Ú At(Richard,NUS) Ù Smart(Richard)

Ú At(NUS,NUS) Ù Smart(NUS)

Ú ...

A common mistake to avoid (2)

Typically, Ù is the main connective with $

Common mistake: using Þ as the main connective with $:

$x At(x,NUS) Þ Smart(x)

is true if there is anyone who is not at NUS!

Properties of quantifiers

"x "y is the same as "y "x

$x $y is the same as $y $x

$x "y is not the same as "y $x

$x "y Loves(x,y)

“There is a person who loves everyone in the world”

"y $x Loves(x,y)

“Everyone in the world is loved by at least one person”

Quantifier duality: each can be expressed using the other

"x Likes(x,IceCream) Ø$x ØLikes(x,IceCream)

$x Likes(x,Broccoli) Ø"x ØLikes(x,Broccoli)

Equality

term₁ = term₂ is true under a given interpretation if and only if term₁ and term₂ refer to the same object

E.g., definition of Sibling in terms of Parent:

"x,y Sibling(x,y) Û [Ø(x = y) Ù $m,f Ø (m = f) Ù Parent(m,x) Ù Parent(f,x) Ù Parent(m,y) Ù Parent(f,y)]

Using FOL

The kinship domain:

Brothers are siblings

"x,y Brother(x,y) Þ Sibling(x,y)

One's mother is one's female parent

"m,c Mother(c) = m Û (Female(m) Ù Parent(m,c))

“Sibling” is symmetric

"x,y Sibling(x,y) Û Sibling(y,x)

Using FOL

The set domain:

"s Set(s) Û (s = {} ) Ú ($x,s₂ Set(s₂) Ù s = {x|s₂})

Ø$x,s {x|s} = {}

"x,s x Î s Û s = {x|s}

"x,s x Î s Û [ $y,s₂} (s = {y|s₂} Ù (x = y Ú x Î s₂))]

"s₁,s₂ s₁ Í s₂ Û ("x x Î s₁ Þ x Î s₂)

"s₁,s₂ (s₁ = s₂) Û (s₁ Í s₂ Ù s₂ Í s₁)

"x,s₁,s₂ x Î (s₁ Ç s₂) Û (x Î s₁ Ù x Î s₂)

"x,s₁,s₂ x Î (s₁ È s₂) Û (x Î s₁ Ú x Î s₂)

Interacting with FOL KBs

Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5:

Tell(KB,Percept([Smell,Breeze,None],5))

Ask(KB,$a BestAction(a,5))

I.e., does the KB entail some best action at t=5?

Answer: Yes, {a/Shoot} ← substitution (binding list)

Given a sentence S and a substitution σ,

Sσ denotes the result of plugging σ into S; e.g.,

S = Smarter(x,y)

σ = {x/Hillary,y/Bill}

Sσ = Smarter(Hillary,Bill)

Ask(KB,S) returns some/all σ such that KB╞ σ

KB for the wumpus world

Perception

"t,s,b Percept([s,b,Glitter],t) Þ Glitter(t)

Reflex

"t Glitter(t) Þ BestAction(Grab,t)

Deducing hidden properties

"x,y,a,b Adjacent([x,y],[a,b]) Û

[a,b] Î {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}

Properties of squares:

"s,t At(Agent,s,t) Ù Breeze(t) Þ Breezy(s)

Squares are breezy near a pit:

Diagnostic rule - infer cause from effect

"s Breezy(s) Þ $r Adjacent(r,s) Ù Pit(r)

Causal rule - infer effect from cause

"r Pit(r) Þ ["s Adjacent(r,s) Þ Breezy(s) ]

Knowledge engineering in FOL

Identify the task

Assemble the relevant knowledge

Decide on a vocabulary of predicates, functions, and constants

Encode general knowledge about the domain

Encode a description of the specific problem instance

Pose queries to the inference procedure and get answers

Debug the knowledge base

The electronic circuits domain

One-bit full adder

The electronic circuits domain

Identify the task

Does the circuit actually add properly? (circuit verification)

Assemble the relevant knowledge

Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)

Irrelevant: size, shape, color, cost of gates

Decide on a vocabulary

Alternatives:

Type(X₁) = XOR

Type(X₁, XOR)

XOR(X₁)

The electronic circuits domain

Encode general knowledge of the domain

"t₁,t₂ Connected(t₁, t₂) Þ Signal(t₁) = Signal(t₂)

"t Signal(t) = 1 Ú Signal(t) = 0

1 ≠ 0

"t₁,t₂ Connected(t₁, t₂) Þ Connected(t₂, t₁)

"g Type(g) = OR Þ Signal(Out(1,g)) = 1 Û $n Signal(In(n,g)) = 1

"g Type(g) = AND Þ Signal(Out(1,g)) = 0 Û $n Signal(In(n,g)) = 0

"g Type(g) = XOR Þ Signal(Out(1,g)) = 1 Û Signal(In(1,g)) ≠ Signal(In(2,g))

"g Type(g) = NOT Þ Signal(Out(1,g)) ≠ Signal(In(1,g))

The electronic circuits domain

Encode the specific problem instance

Type(X₁) = XOR Type(X₂) = XOR

Type(A₁) = AND Type(A₂) = AND

Type(O₁) = OR

Connected(Out(1,X₁),In(1,X₂)) Connected(In(1,C₁),In(1,X₁))

Connected(Out(1,X₁),In(2,A₂)) Connected(In(1,C₁),In(1,A₁))

Connected(Out(1,A₂),In(1,O₁)) Connected(In(2,C₁),In(2,X₁))

Connected(Out(1,A₁),In(2,O₁)) Connected(In(2,C₁),In(2,A₁))

Connected(Out(1,X₂),Out(1,C₁)) Connected(In(3,C₁),In(2,X₂))

Connected(Out(1,O₁),Out(2,C₁)) Connected(In(3,C₁),In(1,A₂))

The electronic circuits domain

Pose queries to the inference procedure

What are the possible sets of values of all the terminals for the adder circuit?

$i₁,i₂,i₃,o₁,o₂ Signal(In(1,C₁)) = i₁ Ù Signal(In(2,C₁)) = i₂ Ù Signal(In(3,C₁)) = i₃ Ù Signal(Out(1,C₁)) = o₁ Ù Signal(Out(2,C₁)) = o₂

Debug the knowledge base

May have omitted assertions like 1 ≠ 0

Summary

First-order logic:

objects and relations are semantic primitives

syntax: constants, functions, predicates, equality, quantifiers

Increased expressive power: sufficient to define wumpus world

PROgramming in LOGic

A crash course in Prolog

Slides edited from William Clocksin’s versions at Cambridge Univ.

What is Logic Programming?

A type of programming consisting of facts and relationships from which the programming language can draw a conclusion.

In imperative programming languages, we tell the computer what to do by programming the procedure by which program states and variables are modified.

In contrast, in logical programming, we don’t tell the computer exactly what it should do (i.e., how to derive a conclusion). User-provided facts and relationships allow it to derive answers via logical inference.

Prolog is the most widely used logic programming language.

Prolog Features

Prolog uses logical variables. These are not the same as variables in other languages. Programmers can use them as ‘holes’ in data structures that are gradually filled in as computation proceeds.

Unification is a built-in term-manipulation method that passes parameters, returns results, selects and constructs data structures.

Basic control flow model is backtracking.

Program clauses and data have the same form.

A Prolog program can also be seen as a relational database containing rules as well as facts.

Example: Concatenate lists a and b

Outline

General Syntax

Terms

Operators

Rules

Queries

Syntax

.pl files contain lists of clauses

Clauses can be either facts or rules

male(bob).

male(harry).

child(bob,harry).

son(X,Y):-

male(X),child(X,Y).

Complete Syntax of Terms

Compound Terms

Examples of operator properties

Prolog has shortcuts in notation for certain operators (especially arithmetic ones)

Position Operator Syntax Normal Syntax

Prefix: -2 -(2)

Infix: 5+17 +(17,5)

Associativity: left, right, none.

X+Y+Z is parsed as (X+Y)+Z

because addition is left-associative.

Precedence: an integer.

X+Y*Z is parsed as X+(Y*Z)

because multiplication has higher precedence.

Rules

Rules combine facts to increase knowledge of the system

son(X,Y):-

male(X),child(X,Y).

X is a son of Y if X is male and
X is a child of Y

Interpretation of Rules

Rules can be given a declarative reading or a procedural reading.

Queries

Prolog is interactive; you load a KB and then ask queries

Composed at the ?- prompt

Returns values of bound variables and yes or no

?- son(bob, harry).

yes

?- king(bob, france).

no

Another example

Quantifiers

Points to consider

Variables are bound by Prolog, not by the programmer

You can’t assign a value to a variable.

Successive user prompts ; cause the interpreter to return all terms that can be substituted for X.

They are returned in the order found.

Order is important

PROLOG adopts the closed-world assumption:

All knowledge of the world is present in the database.

If a term is not in the database assume is false.

Prolog’s ‘yes’ = I can prove it, ‘no’ = I can’t prove it.

Queries

Can bind answers to questions to variables

Who is bob the son of? (X=harry)

?- son(bob, X).

Who is male? (X=bob, harry)

?- male(X).

Is bob the son of someone? (yes)

?- son(bob, _).

No variables bound in this case!

Lists

The first element of a list can be separated from the tail

using operator |

Example:

Match the list [tom,dick,harry,fred] to

[X|Y] then X = tom and Y = [dick,harry,fred]

[X,Y|Z] then X = tom, Y = dick, and Z = [harry,fred]

[V,W,X,Y,Z|U] will not match

[tom,X|[harry,fred]] gives X = dick

Example: List Membership

We want to write a function member that works as follows:

?- member(a,[a,b,c,d,e])

yes

?- member(a,[1,2,3,4])

no

?- member(X,[a,b,c])

X = a

;

X = b

;

X = c

;

no

Function Membership Solution

Define two predicates:

member(X,[X|T]).

member(X,[Y|T]) :- member(X,T).

A more elegant definition uses anonymous variables:

member(X,[X,_]).

member(X,[_|T]) :- member(X,T).

Again, the symbol _ indicates that the contents of that variable is unimportant.

Notes on running Prolog

You will often want to load a KB on invocation of Prolog

Use “consult(‘mykb.pl’).” at the “?-” prompt.

Or add it on the command line as a standard input “pl < mykb.pl”

If you want to modify facts once Prolog is invoked:

Use “assert(p).”

Or “retract(p).” to remove a fact

Prolog Summary

A Prolog program is a set of specifications in FOL. The specification is known as the database of the system.

Prolog is an interactive language (the user enters queries in response to a prompt).

PROLOG adopts the closed-world assumption

How does Prolog find the answer(s)? We return to this next week in Inference in FOL


	Why FOL?
	Syntax and semantics of FOL
	Using FOL
	Wumpus world in FOL
	Knowledge engineering in FOL


J Propositional logic is declarative
J Propositional logic allows partial/disjunctive/negated information
	(unlike most data structures and databases)
J Propositional logic is compositional:
	meaning of B_1,1 Ù P_1,2 is derived from meaning of B_1,1 and of P_1,2
J Meaning in propositional logic is context-independent
	(unlike natural language, where meaning depends on context)
L Propositional logic has very limited expressive power
	(unlike natural language)
	E.g., cannot say "pits cause breezes in adjacent squares“
		except by writing one sentence for each square


	Whereas propositional logic assumes the world contains facts,
	first-order logic (like natural language) assumes the world contains
		Objects: people, houses, numbers, colors, baseball games, wars, …
		Relations: red, round, prime, brother of, bigger than, part of, comes between, …
		Functions: father of, best friend, one more than, plus, …


	Constants KingJohn, 2, NUS,...
	Predicates Brother, >,...
	Functions Sqrt, LeftLegOf,...
	Variables x, y, a, b,...
	Connectives Ø, Þ, Ù, Ú, Û
	Equality =
	Quantifiers ", $


	Atomic sentence = predicate (term₁,...,term_n) or term₁ = term₂

	Term = function (term₁,...,term_n) or constant or variable

	E.g.,
	Brother(KingJohn,RichardTheLionheart)
	Length(LeftLegOf(Richard)) = Length(LeftLegOf(KingJohn))


	Complex sentences are made from atomic sentences using connectives
	ØS, S₁Ù S₂, S₁Ú S₂, S₁Þ S₂, S₁ÛS₂,

	E.g. Sibling(KingJohn,Richard) Þ Sibling(Richard,KingJohn)
	>(1,2) Ú ≤ (1,2)
	>(1,2) Ù Ø >(1,2)


	Sentences are true with respect to a model and an interpretation

	Model contains objects (domain elements) and relations among them

	Interpretation specifies referents for
		constant symbols → objects
		predicate symbols → relations
		function symbols → functional relations

	An atomic sentence predicate(term₁,...,term_n) is true
	iff the objects referred to by term₁,...,term_n
	are in the relation referred to by predicate


	"<variables> <sentence>

	Everyone at NUS is smart:
	"x At(x,NUS) Þ Smart(x)

	"x P is true in a model m iff P is true with x being each possible object in the model

	Roughly speaking, equivalent to the conjunction of instantiations of P
			At(KingJohn,NUS) Þ Smart(KingJohn)
			Ù At(Richard,NUS) Þ Smart(Richard)
			Ù At(NUS,NUS) Þ Smart(NUS)
			Ù ...


	Typically, Þ is the main connective with "
	Common mistake: using Ù as the main connective with ":
		"x At(x,NUS) Ù Smart(x)
		means “Everyone is at NUS and everyone is smart”


	$<variables> <sentence>

	Someone at NUS is smart:
	$x At(x,NUS) Ù Smart(x)

	$x P is true in a model m iff P is true with x being some possible object in the model

	Roughly speaking, equivalent to the disjunction of instantiations of P
		At(KingJohn,NUS) Ù Smart(KingJohn)
		Ú At(Richard,NUS) Ù Smart(Richard)
		Ú At(NUS,NUS) Ù Smart(NUS)
		Ú ...


	Typically, Ù is the main connective with $

	Common mistake: using Þ as the main connective with $:
	$x At(x,NUS) Þ Smart(x)
	is true if there is anyone who is not at NUS!


	"x "y is the same as "y "x
	$x $y is the same as $y $x

	$x "y is not the same as "y $x
	$x "y Loves(x,y)
		“There is a person who loves everyone in the world”
	"y $x Loves(x,y)
		“Everyone in the world is loved by at least one person”

	Quantifier duality: each can be expressed using the other
	"x Likes(x,IceCream) Ø$x ØLikes(x,IceCream)
	$x Likes(x,Broccoli) Ø"x ØLikes(x,Broccoli)


	term₁ = term₂ is true under a given interpretation if and only if term₁ and term₂ refer to the same object

	E.g., definition of Sibling in terms of Parent:
		"x,y Sibling(x,y) Û [Ø(x = y) Ù $m,f Ø (m = f) Ù Parent(m,x) Ù Parent(f,x) Ù Parent(m,y) Ù Parent(f,y)]


	The kinship domain:
	Brothers are siblings
		"x,y Brother(x,y) Þ Sibling(x,y)
	One's mother is one's female parent
		"m,c Mother(c) = m Û (Female(m) Ù Parent(m,c))
	“Sibling” is symmetric
		"x,y Sibling(x,y) Û Sibling(y,x)


	The set domain:
	"s Set(s) Û (s = {} ) Ú ($x,s₂ Set(s₂) Ù s = {x\|s₂})
	Ø$x,s {x\|s} = {}
	"x,s x Î s Û s = {x\|s}
	"x,s x Î s Û [ $y,s₂} (s = {y\|s₂} Ù (x = y Ú x Î s₂))]
	"s₁,s₂ s₁ Í s₂ Û ("x x Î s₁ Þ x Î s₂)
	"s₁,s₂ (s₁ = s₂) Û (s₁ Í s₂ Ù s₂ Í s₁)
	"x,s₁,s₂ x Î (s₁ Ç s₂) Û (x Î s₁ Ù x Î s₂)
	"x,s₁,s₂ x Î (s₁ È s₂) Û (x Î s₁ Ú x Î s₂)


	Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5:

		Tell(KB,Percept([Smell,Breeze,None],5))
		Ask(KB,$a BestAction(a,5))

	I.e., does the KB entail some best action at t=5?

	Answer: Yes, {a/Shoot} ← substitution (binding list)

	Given a sentence S and a substitution σ,
	Sσ denotes the result of plugging σ into S; e.g.,
		S = Smarter(x,y)
		σ = {x/Hillary,y/Bill}
		Sσ = Smarter(Hillary,Bill)

	Ask(KB,S) returns some/all σ such that KB╞ σ


	Perception
		"t,s,b Percept([s,b,Glitter],t) Þ Glitter(t)

	Reflex
		"t Glitter(t) Þ BestAction(Grab,t)


"x,y,a,b Adjacent([x,y],[a,b]) Û
[a,b] Î {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}

Properties of squares:
"s,t At(Agent,s,t) Ù Breeze(t) Þ Breezy(s)

Squares are breezy near a pit:
	Diagnostic rule - infer cause from effect
		"s Breezy(s) Þ $r Adjacent(r,s) Ù Pit(r)
	Causal rule - infer effect from cause
		"r Pit(r) Þ ["s Adjacent(r,s) Þ Breezy(s) ]


	Identify the task
	Assemble the relevant knowledge
	Decide on a vocabulary of predicates, functions, and constants
	Encode general knowledge about the domain
	Encode a description of the specific problem instance
	Pose queries to the inference procedure and get answers
	Debug the knowledge base


Identify the task
	Does the circuit actually add properly? (circuit verification)

Assemble the relevant knowledge
	Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)
	Irrelevant: size, shape, color, cost of gates

Decide on a vocabulary
	Alternatives:
		Type(X₁) = XOR
		Type(X₁, XOR)
		XOR(X₁)


	Encode general knowledge of the domain
		"t₁,t₂ Connected(t₁, t₂) Þ Signal(t₁) = Signal(t₂)
		"t Signal(t) = 1 Ú Signal(t) = 0
		1 ≠ 0
		"t₁,t₂ Connected(t₁, t₂) Þ Connected(t₂, t₁)
		"g Type(g) = OR Þ Signal(Out(1,g)) = 1 Û $n Signal(In(n,g)) = 1
		"g Type(g) = AND Þ Signal(Out(1,g)) = 0 Û $n Signal(In(n,g)) = 0
		"g Type(g) = XOR Þ Signal(Out(1,g)) = 1 Û Signal(In(1,g)) ≠ Signal(In(2,g))
		"g Type(g) = NOT Þ Signal(Out(1,g)) ≠ Signal(In(1,g))


	Encode the specific problem instance
		Type(X₁) = XOR Type(X₂) = XOR
		Type(A₁) = AND Type(A₂) = AND
		Type(O₁) = OR

		Connected(Out(1,X₁),In(1,X₂)) Connected(In(1,C₁),In(1,X₁))
		Connected(Out(1,X₁),In(2,A₂)) Connected(In(1,C₁),In(1,A₁))
		Connected(Out(1,A₂),In(1,O₁)) Connected(In(2,C₁),In(2,X₁))
		Connected(Out(1,A₁),In(2,O₁)) Connected(In(2,C₁),In(2,A₁))
		Connected(Out(1,X₂),Out(1,C₁)) Connected(In(3,C₁),In(2,X₂))
		Connected(Out(1,O₁),Out(2,C₁)) Connected(In(3,C₁),In(1,A₂))


	Pose queries to the inference procedure
		What are the possible sets of values of all the terminals for the adder circuit?
	$i₁,i₂,i₃,o₁,o₂ Signal(In(1,C₁)) = i₁ Ù Signal(In(2,C₁)) = i₂ Ù Signal(In(3,C₁)) = i₃ Ù Signal(Out(1,C₁)) = o₁ Ù Signal(Out(2,C₁)) = o₂

	Debug the knowledge base
		May have omitted assertions like 1 ≠ 0


	First-order logic:
		objects and relations are semantic primitives
		syntax: constants, functions, predicates, equality, quantifiers

	Increased expressive power: sufficient to define wumpus world


	A crash course in Prolog

	Slides edited from William Clocksin’s versions at Cambridge Univ.


	A type of programming consisting of facts and relationships from which the programming language can draw a conclusion.
		In imperative programming languages, we tell the computer what to do by programming the procedure by which program states and variables are modified.
		In contrast, in logical programming, we don’t tell the computer exactly what it should do (i.e., how to derive a conclusion). User-provided facts and relationships allow it to derive answers via logical inference.

	Prolog is the most widely used logic programming language.


	Prolog uses logical variables. These are not the same as variables in other languages. Programmers can use them as ‘holes’ in data structures that are gradually filled in as computation proceeds.
	Unification is a built-in term-manipulation method that passes parameters, returns results, selects and constructs data structures.
	Basic control flow model is backtracking.
	Program clauses and data have the same form.
		A Prolog program can also be seen as a relational database containing rules as well as facts.


	.pl files contain lists of clauses
	Clauses can be either facts or rules


	male(bob).
	male(harry).
	child(bob,harry).
	son(X,Y):-
	male(X),child(X,Y).


	Prolog has shortcuts in notation for certain operators (especially arithmetic ones)

	Position Operator Syntax Normal Syntax
	Prefix: -2 -(2)
	Infix: 5+17 +(17,5)

	Associativity: left, right, none.
	X+Y+Z is parsed as (X+Y)+Z
	because addition is left-associative.

	Precedence: an integer.
	X+YZ is parsed as X+(YZ)
	because multiplication has higher precedence.


	Rules combine facts to increase knowledge of the system

	son(X,Y):-
	male(X),child(X,Y).

	X is a son of Y if X is male and X is a child of Y