First Order Logic

1
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.

2
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))?

3
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))?

4
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).
• Interpretations are analogous to truth tables.

5
Truth Tables
zombie(joe) zombie(X) eatsbrains(joe) eatsbrai
ns(X) ...
0 ? 0 ? 1 ? 0 ?
By mapping (valuation) of X to fred
zombie(joe) zombie(fred) eatsbrains(joe)
eatsbrains(fred)?
0 0 0 0 1 0 0 1
6
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).

7
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.

8
Relational Databases

• 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.

9
Relational Databases
Patient
Visit
Prescription
10
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)?

11
Relational Databases

• Any query can be represented as a formula.
• To find all male patients with diagnosis code
  111, use the query formula ?P (interest(P)?patien
  t(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

12
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)

13
Queries in Datalog

• Datalog queries can be called with a
  goal. male-flu(P, D)?(patient(P, B, male),
  visit(P, D, 111)?
• 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.

14
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 P1,
  D3-3-03.

15
Prolog

• All formulas have the format atomic formula ?
  atomic formula , atomic formula(s)written in
  Prolog as atomic formula - atomic formula ,
  atomic formula(s).
• Prolog uses only universally quantified
  variables.
• Some arithmetic functions are allowed, such as
  W is XY
• Specification of correct output is allowed.

16
Prolog Lists

• cons(H, T) constructs the list with H as the head
  and T as the tail. This is equivalent to
  HT.1,2,312, 3123nilcons(1,
  cons(2, cons(3, nil))) where nil is the empty
  list.
• 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.

17
Defining Prolog Functions

• Member function member(H, HT).member(H,
  ST) ? member(H,T).
• Insertion Sortsort( , ).sort(HT, L) ?
  sort(T, T2), insert(H, T2, L).insert(H, AT,
  H,AT ? H lt A.insert(H, AT, AL) ? H gt
  A, insert(H, T, L).

18
Running Prolog

• 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