First Order Logic


Logic is a mathematical attempt to formalize the
way we think.
First-order predicate calculus was created in an
attempt to mechanize reason.
First Order Logical Representations

All zombies eat brains.

This assumption can be represented in
propositional logic by the propositional variable p.

In first order logic, the expression can be broken
down




X (zombie(X) eatsbrains(X))
X (zombie(X) eats(X, brains))
X Y (zombie(X) eats(X, Y) ^ brains(Y))
Some people with no heartbeats are zombies.

X (person(X) ^ zombie(X) ^ ¬ heartbeat(X))
Natural Deductive Rules


Natural deductive rules are logical rules
accepted as being intuitively true.
Universal Instantiation: If a variable is assigned
a value, it will always have that value.


If X = joe, X (zombie(X) eatsbrains(X)) is the
same as (zombie(joe) eatsbrains(joe))
Modus Ponens: For some pair of expressions p
and q, p ^ (p q)  q.

If zombie(X), (zombie(X) eatsbrains(X)) becomes
(eatsbrains(X))
Truth Tables
Truth tables can be constructed from first order
logic by transforming expressions into atomic
formulas (something with value true or false).

Atomic formulas can be constructed from first
order logic by either quantifying a formula or
mapping it to a value (analogous to valuation).

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Interpretations are analogous to truth tables.
Truth Tables
zombie(joe)
zombie(X)
eatsbrains(joe)
eatsbrains(X) ...
0
?
0
?
0
?
1
By mapping (valuation) of
X to fred
?
zombie(joe)
0
zombie(fred) eatsbrains(joe)
eatsbrains(fred)
0
0
0
0
0
1
1
Brief Intro: Herbrand Base

The Herbrand base is the set of ground atomic
formulas.


In the previous example, this is the set
{zombie(joe), zombie(fred), eatsbrains(joe),
eatsbrains(fred)}.
A model is satisfiable if and only if it has a
Herbrand Interpretation (a valid truth
assignment given its Herbrand base).
Function Symbols


Function symbols map an individual in the
domain to another in the domain.
Everyone loves their mother.


In the expression X loves(X, mother(X)), mother(X)
maps X to the element in the domain.
Peano's Axioms for Arithmetic

In the expression
X (number(X) number(S(X)),
the occurance of number in number(X) is a function
symbol. In the second occurrence, number is a
predicate and S(X) is a function symbol.
Relational Databases
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Assume there is a relational database with a
patient table, visit table, and a prescription
table.
The patient table has fields ID, Date of Birth,
and Gender.
The visit table has fields ID, Visit Date, and
Diagnosis Code.
The prescription table has fields ID, Date, and
Drug.
Relational Databases
Patient
ID
1
2
Date of Birth
01/01/01
02/02/02
Gender
male
female
Visit
ID
1
2
Visit Date
11/03/09
03/05/06
Diagnosis Code
111
123
Prescription
ID
1
2
Date
11/03/09
03/05/06
Drug
A
B
Relational Databases

In a relational database, the table can be
represented by an atomic formula of a predicate
objects composed of constants and/or function
symbols. One such formula exists for each
patient.

Patient table: patient(id, date of birth, gender)

Visit table: visit(id, date, diagnosis code)

Prescription table: prescription(id, date, drug)
Relational Databases

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Any query can be represented as a formula.
To find all male patients with diagnosis code
111, use the query formula
P (interest(P)patient(P, B, male) ^ visit(P, D, 111)

The query result can define a new table
containing the desired data.

Extensional: Write out entries in a table.

Intensional: Write out tables via formula
First Order Logical Inference-Datalog


Datalog is a syntactic variation on relational
database queries. It forms an intensional
representation to the query equivalent to those
formed by relational algebra.
All quanitifiers are assumed universal and out
front of the formula. All formulas are atomic or
implications. No function symbols are used.
(interest(P)patient(P, B, male), visit(P, D, 111)
Queries in Datalog

Datalog queries can be called with a goal.
male-flu(P, D)(patient(P, B, male), visit(P, D, 111)

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Goals can be satisfied by finding all bindings on
the goal variables and using pattern matching.
The process of matching all atomic formulas by
binding the variables to force matching is known
as Unification.
Queries in Datalog


Unification is most frequently executed by
running a backtracking search for proofs
satisfying all subgoals.
Running unification on the male-flu example, if
there exists patient(1, 1-1-01, male) and visit(1,
3-3-03, 111), there will be a new row binding in
the resulting male-flu table with P=1, D=3-3-03.
Prolog

All formulas have the format:
atomic formula  atomic formula [, atomic formula(s)]
written in Prolog as:
atomic formula :- atomic formula [, atomic formula(s)].

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Prolog uses only universally quantified
variables.
Some arithmetic functions are allowed, such as
W is X*Y

Specification of correct output is allowed.
Prolog Lists

cons(H, T) constructs the list with H as the head
and T as the tail. This is equivalent to [H|T].
[1,2,3]=[1|[2, 3]]=[1|2|3|nil]]]=cons(1, cons(2, cons(3, nil)))
where nil is the empty list.

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All functions are evaluated based on unification.
Conventionally, arguments to a Prolog
expression are given with inputs first, then
outputs.
Prolog is Turing-complete so long as function
symbols are allowed.
Defining Prolog Functions
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Member function:
member(H, [H|T]).
member(H, [S|T])  member(H,T).
Insertion Sort:
sort([ ], [ ]).
sort([H|T], L)  sort(T, T2), insert(H, T2, L).
insert(H, [A|T], [H,A|T] H < A.
insert(H, [A|T], [A|L])  H >= A, insert(H, T, L).
Running Prolog
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On CSL machines, Prolog is run via the yap
interpreter.
To load a file into the interpreter:
?- consult('filename').
Once a file has been loaded, definitions
contained in that file may be executed. The code
will then be run top down until a solution is found
and then bound to the output.
?- sort([5, 1, 10, 3, 9], X). %Bind sorted list to X