Harvard-MIT Division of Health Sciences and Technology
HST.952: Computing for Biomedical Scientists
Data and Knowledge
Representation
Lecture 2
Last Time We Talked About
z Why
is knowledge/data representation
important
z Propositional Logic
z Operators
z WFF
z Truth
table
Today We Will Talk About
z Boolean
Algebra
z Predicate Logic (First order logic)
Exercise
z
Generate a truth table for
( a ∧ b ) ∨ ( a ∧ ¬b )
Boolean Algebra
z Named
in honor of George Boole
z http://www.digitalcentury.com/encyclo/updat
e/boole.html
z Another
way of reasoning with proposition
logic
z Is a form of equational reasoning
Laws of Boolean Algebra
Operations with Constants
z a Λ False = False
za
V True = True
z a Λ True = a
z a V False = a
Laws of Boolean Algebra
Basic properties of Λ and V
z a→aVb
z aΛb→a
z aΛa=a
z
z
z
z
z
aVa=a
aΛb=bΛa
aVb=bVa
(a Λ b) Λ c = a Λ (b Λ c)
(a V b) V c = a V (b V c)
Note the difference
between “→” and “=“
a→aVb
a
b
aVb
a→aVb
0
0
0
1
0
1
1
1
1
0
1
1
1
1
1
1
aΛb→a
a
b
aΛb
aΛb→a
0
0
0
1
0
1
0
1
1
0
0
1
1
1
1
1
Laws of Boolean Algebra
Distributive and DeMorgan’s Laws
z a Λ (b V c) = (a Λ b) V (a Λ c)
z a V (b Λ c) = (a V b) Λ (a V c)
z ¬ (a Λ b) = ¬ a V ¬b
z ¬ (a V b) = ¬a Λ ¬b
Laws of Boolean Algebra
Laws on Negation
z ¬True = False
z ¬False = True
z a Λ ¬a = False
z a V ¬a = True
z ¬(¬a) = a
Laws of Boolean Algebra
Laws on Implication
z (a Λ (a → b)) → b
z ((a → b) Λ ¬b) → ¬a
z ((a V b) Λ ¬a) → b
z ((a → b) Λ (b → c)) → (a → c)
z ((a → b) Λ (c → d)) → ((a Λ c) → (b Λ d))
z ((a → b) → c) → (a → (b → c))
z ((a Λ b) → c) = (a → (b → c))
z a → b = ¬a V b
z a → b = ¬b → ¬a
z (a → b) Λ (a → ¬ b) = ¬a
a→b = ¬aVb
a
b
a→b
¬a
¬aVb
a→b =
¬aVb
0
0
1
1
1
1
0
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
1
1
(a → b) Λ (a → ¬ b) = ¬a
( a → b ) ∧ ( a → ¬b )
= ( ¬ a ∨ b ) ∧ ( ¬a ∨ ¬b )
= ¬a ∨ (b ∧ ¬b)
= ¬a ∨ False
= ¬a
Laws of Boolean Algebra
Equivalence
a ↔ b = (a → b) Λ (b → a)
Examples
z Please
write a propositional logic WFF for
the following eligibility criteria:
“a women over 50-year old and not on
hormonal therapy”
w Λ old Λ ¬ (hrt)
Need to define w, old, and hrt
Examples
z Please
write a propositional logic WFF for
the following guideline:
“All women over 30 years old should have
pap smear every year”
w Λ Over30 →
pap_yearly
How to convey “all”?
Predicate Logic
z Extension
to propositional logic
z Augmented with variables, predicates and
quantifiers
Predicate
zA
predicate is a statement that an object x
has certain property
z E.g.
F (x): F is the predicate and x is the
variable F applies to
z Any
term in the form F(x), where F is a
predicate name and x is a variable name,
is a well-formed formula. Similarly, F(X1,
X2………Xk) is a well-formed formula; this
is a predicate containing k variables
Examples
z Women
over 70 years old
Women (x) Ξ x is female
senior (x) Ξ x is over 70 years old
Quantifiers
z
There are two quantifiers in predicate logic:
z
The universal quantification:
z
The existential quantification:
∀
∃
Universal Quantification
z If
F(x) is a well-formed formula containing
the variable x, then ∀ x . F ( x ) is a wellformed called a universal quantification.
For all x in the universe, the predicate F(x)
holds. In other words: every x has the
property F.
z ∀x. F ( x) = F ( x ) ∧ F ( x ) ∧ F ( x ) K ∧ F ( x )
1
2
2
n
Existential Quantification
z If
F(x) is a well-formed formula containing
the variable x, then ∃x. F ( x) is a wellformed called a existential quantification.
For one or more x in the universe, the
predicate F(x) holds. In other words:
some x has the property F.
∃x. F ( x) = F ( x1 ) ∨ F ( x2 ) ∨ F ( x2 ) K ∨ F ( xn )
Universe of Discourse
z Also
called universe or U
z A set of possible values that the variables
can have
z Let U be all female, C (x, “xx”) mean x has
XX chromosomes, then ∀x C ( x, " xx" ) is true
z Let U be all patients, C (x, “xx”) mean x
has XX chromosomes, then ∀x C ( x, " xx" ) is
false
Scope of Variables Bindings
z Quantifiers
bind variables by assigning
them values from a universe
z This formula ∀x. F ( x) ∨ E ( x)
Can be interpreted as (∀x. F ( x)) ∨ E ( x)
Or ∀x. ( F ( x) ∨ E ( x))
z Use
parentheses if not clear
Translation
A ≡ a patient is female
B ≡ a patient is pregnant
C ≡ a pregnancy test is positive
What does B → A mean?
What does A Λ C → B mean?
But is C a test the patient in A took?
Translation
A( x) ≡ x is female
B( x) ≡
x is pregnant
C ( x, y ) ≡ x is a pregnancy test of y
D( x) ≡ x is positive
What does ∀x. A( x) → B( x) mean?
What does ∀x.∃y.A(x) ∧ C( y, x) ∧ D( y) → B(x)
mean?
Can we say some pregnant female does not have
positive pregnancy test results?
Algebraic Laws of Predicate Logic
∀x. f ( x) → f (c)
f (c) → ∃x. f ( x)
• x is any element in the universe.
• The element c is any fixed element
of the universe.
Algebraic Laws of Predicate Logic
∀ x . ¬ f ( x ) = ¬∃ x . f ( x )
∃ x . ¬ f ( x ) = ¬∀ x . f ( x )
∀ x . f ( x ) ∧ q = ∀ x .( f ( x ) ∧ q )
∀ x . f ( x ) ∨ q = ∀ x .( f ( x ) ∨ q )
∃ x . f ( x ) ∧ q = ∃ x .( f ( x ) ∧ q )
∃ x . f ( x ) ∨ q = ∃ x .( f ( x ) ∨ q )
• q is a proposition that does not contain x
Algebraic Laws of Predicate Logic
∀x. f ( x) ∧ ∀x.g ( x) = ∀x.( f ( x) ∧ g ( x))
∀x. f ( x) ∨ ∀x.g ( x) → ∀x.( f ( x) ∨ g ( x))
∃x.( f ( x) ∨ g ( x)) → ∃x. f ( x) ∨ ∃g ( x)
∃x. f ( x) ∨ ∃x.g ( x) = ∃x.( f ( x) ∨ g ( x))
• Please note the difference between “→” and “=“
Equational Reasoning
z Showing
two values are the same by
building up chains of equalities
z Substitute equals for equals
Example
( False ∧ P) ∨ Q
= ( P ∧ False) ∨ Q
= False ∨ Q
= Q ∨ False
=Q
Example
Prove ¬∃x.( f ( x) → g ( x)) ∧ ∀x.¬f ( x) = False
¬∃x.( f ( x) → g ( x)) ∧ ∀x.¬f ( x)
= ∀x.¬( f ( x) → g ( x)) ∧ ∀x.¬f ( x)
= ∀x.(¬( f ( x) → g ( x)) ∧ ¬f ( x))
= ∀x.(¬(¬f ( x) ∨ g ( x)) ∧ ¬f ( x))
= ∀x.(¬(¬f ( x)) ∧ ¬g ( x) ∧ ¬f ( x))
= ∀x.( f ( x) ∧ ¬g ( x) ∧ ¬f ( x))
= ∀x.((¬g ( x) ∧ f ( x)) ∧ ¬f ( x))
= ∀x.(¬g ( x) ∧ ( f ( x) ∧ ¬f ( x)))
= ∀x.(¬g ( x) ∧ False)
= False
Exercise
z
Prove Pˬ(QVP) = False
P ∧ ¬(Q ∨ P)
= P ∧ ( ¬ Q ∧ ¬P )
= P ∧ ( ¬ P ∧ ¬Q )
= ( P ∧ ¬ P ) ∧ ¬Q
= False ∧ ¬Q
= False
Exercise
z
Prove
∀x.( f ( x) ∧ ¬g ( x)) = ∀x. f ( x) ∧ ¬∃x.g ( x)
∀x.( f ( x) ∧¬g( x))
= ∀x. f (x) ∧ (∀x.¬g(x))
= ∀x. f (x) ∧¬∃x.g(x)
Exercise
z Translate
Logic:
z Some
the following into Predicate
patients only speak Spanish.
z All doctors speak English, while some can also
speak Spanish.
z Spanish speaking doctors should be assigned
to patients who only speak Spanish
Alternatives in Modeling
zX
has Diabetes:
z Diabetes
(x)
z Has_Diagnosis (x, “Diabetes”)
z Has (x, “Diagnosis”, “Diabetes”)
z Trade
off between efficiency and
expressiveness
z Has
(x, y, “Diabetes”)
Relationship to OO model
z
Representing patient X has Diabetes in OO
model:
z
Diabetes (x)
z
z
Has_Diagnosis (x, “Diabetes”)
z
z
Object x has an attribute called Diabetes
Object x has an attribute called Has_Diagnosis which can
have value “Diabetes”
Has (x, “Diagnosis”, “Diabetes”)
z
Object x has an attribute called Has which can have value
observation, which is an object with attributes observation
type “Diagnosis” and observation value “Diabetes”
Relationship to DB
z
Representing patient X has Diabetes in a table:
z
Diabetes (x)
z
z
Has_Diagnosis (x, “Diabetes”)
z
z
A table called Diabetes with column (s) identifying patient x
and a column of the value of Diabetes (x)
A table called Diagnosis with column (s) identifying patient x,
and diagnosis y and a column of the value of Has_Diagnosis
(x, y)
Has (x, “Diagnosis”, “Diabetes”)
z
A table called observation with column (s) identifying patient
x, observation type y and observation value z and a column
of the value of Has (x, y, z)
Different Representation of FirstOrder Logic
z Conceptual
Graph (CG)
z Knowledge Interchange Format (KIF)
z Conceptual Graph Interchange Format
(CDIF)
z ………..
Examples of Medical Knowledge
z
z
z
z
z
Nitrates are a safe and effective treatment that can be used in
patients with angina and left ventricular systolic dysfunction.
On the basis of currently published evidence, amlodipine is the
calcium channel antagonist that it is safest to use in patients with
heart failure and left ventricular systolic dysfunction.
Coronary artery bypass grafting may be indicated, in some, for
relief of angina
All patients with heart failure and angina should be referred for
specialist assessment.
Patients with angina and mild to moderately symptomatically
severe heart failure that is well controlled, and who have no other
contraindications to major surgery, should be considered for
coronary artery bypass grafting on prognostic (as well as
symptomatic) grounds.
Limitation
z Real
world sometimes can not be
represented using logic
z Induction
and deduction model
z Uncertainty and probability
z Context and exception
Alternatives
z Case-based
z Fuzzy
reasoning (Analogy)
logic
z Nonmonotonic logic
Extra Reading
z Aho’s
book chapter 14
z Sowa’s book p467-488