Harvard-MIT Division of Health Sciences and Technology HST.952: Computing for Biomedical Scientists. Data and Knowledge Representation Lecture 2

Transcription

1 Harvard-MIT Division of Health Sciences and Technology HST.952: Computing for Biomedical Scientists Data and Knowledge Representation Lecture 2

2 Last Time We Talked About Why is knowledge/data representation important Propositional Logic Operators WFF Truth table

3 Today We Will Talk About Boolean Algebra Predicate Logic First order logic

4 Eercise Generate a truth table for a b a b

5 Boolean Algebra Named in honor of George Boole digitalcentury.com/.com/encyclo/updat e/boole boole.html Another way of reasoning with proposition logic Is a form of equational reasoning

6 Laws of Boolean Algebra Operations with Constants a Λ False = False a V True = True a Λ True = a a V False = a

7 Laws of Boolean Algebra Basic properties of Λ and V a a V b a Λ b a a Λ a = a a V a = a a Λ b = b Λ a a V b = b V a 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 =

8 a a V b a b a V b a a V b

9 a Λ b a a b a Λ b a Λ b a

10 Laws of Boolean Algebra Distributive and DeMorgan s Laws a Λ b V c = a Λ b V a Λ c a V b Λ c = a V b Λ a V c a Λ b = a V b a V b = a Λ b

11 Laws of Boolean Algebra Laws on Negation True = False False = True a Λ a a = False a V a a = True a = a

12 Laws of Boolean Algebra Laws on Implication a Λ a b b a b Λ b a a V b Λ a b a b Λ b c a c a b Λ c d a Λ c b Λ d a b c a b c a Λ b c = a b c a b = a a V b a b = b a a b Λ a b = a

13 a b b = avb a 0 b 0 a b a avb a b b = avb

14 a b Λ a b = a a b a b = a b a b = a b b = a = a False

15 Laws of Boolean Algebra Equivalence a b = a b Λ b a

16 Eamples 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 hrt Need to define w, old, and hrt

17 Eamples 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?

18 Predicate Logic Etension to propositional logic Augmented with variables, predicates and quantifiers

19 Predicate A predicate is a statement that an object has certain property E.g. F : F is the predicate and is the variable F applies to Any term in the form F, where F is a predicate name and is a variable name, is a well-formed formula. Similarly, FX, X 2 X k is a well-formed formula; this is a predicate containing k variables

20 Eamples Women over 70 years old Women Ξ is female senior Ξ is over 70 years old

21 Quantifiers There are two quantifiers in predicate logic: The universal quantification: The eistential quantification:

22 Universal Quantification If F is a well-formed formula containing the variable, then. F is a well - formed called a universal quantification. For all in the universe, the predicate F holds. In other words: every has the property F.. F = F F 2 F 2 K F n

23 Eistential Quantification If F is a well-formed formula containing the variable, then. F is a well - formed called a eistential quantification. For one or more in the universe, the predicate F holds. In other words: some has the property F.. F = F F 2 F 2 K F n

24 Universe of Discourse Also called universe or U A set of possible values that the variables can have Let U be all female, C, mean has XX chromosomes, then C, " " is true Let U be all patients, C, mean has XX chromosomes, then C, " " is false

25 Scope of Variables Bindings Quantifiers bind variables by assigning them values from a universe This formula. F E Can be interpreted as Or. F E. F E Use parentheses if not clear

26 Translation A B C a patient is female a patient is pregnant 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?

27 Translation A is female B is pregnant C, y is a pregnancy test of y D is positive What does. A B mean? What does. y. A C y, D y B mean? Can we say some pregnant female does not have positive pregnancy test results?

28 Algebraic Laws of Predicate Logic. f f c. f c f is any element in the universe. The element c is any fied element of the universe.

29 Algebraic Laws of Predicate Logic Algebraic Laws of Predicate Logic q f q f q f q f q f q f q f q f f f f f = = = = = = q is a proposition that does not contain

30 Algebraic Laws of Predicate Logic Algebraic Laws of Predicate Logic g f g f g f g f g f g f g f g f = = Please note the difference between and =

31 Equational Reasoning Showing two values are the same by building up chains of equalities Substitute equals for equals

32 Eample False P Q = P False Q = False Q = Q False = Q

33 Eample Prove. f g. f = False. f g. f =. f g. f =. f g f =. f g f =. f g f =. f g f =. g f f =. g f f =. g False = False

34 Eercise Prove PΛ QVP P = False P Q = = = = P P False Q False P Q P P Q = P P Q

35 Eercise Prove. f g =. f. g. f g =. f. g =. f.g

36 Eercise Translate the following into Predicate Logic: Some patients only speak Spanish. All doctors speak English, while some can also speak Spanish. Spanish speaking doctors should be assigned to patients who only speak Spanish

37 Alternatives in Modeling X has Diabetes: Diabetes Has_Diagnosis, Diabetes Has, Diagnosis, Diabetes Trade off between efficiency and epressiveness Has, y, Diabetes

38 Relationship to OO model Representing patient X has Diabetes in OO model: Diabetes Object has an attribute called Diabetes Has_Diagnosis, Diabetes Object has an attribute called Has_Diagnosis which can have value Diabetes Has, Diagnosis, Diabetes Object has an attribute called Has which can have value observation, which is an object with attributes observation type Diagnosis and observation value Diabetes

39 Relationship to DB Representing patient X has Diabetes in a table: Diabetes A table called Diabetes with column s identifying patient and a column of the value of Diabetes Has_Diagnosis, Diabetes A table called Diagnosis with column s identifying patient, and diagnosis y and a column of the value of Has_Diagnosis, y Has, Diagnosis, Diabetes A table called observation with column s identifying patient, observation type y and observation value z and a column of the value of Has, y, z

40 Different Representation of First- Order Logic Conceptual Graph CG Knowledge Interchange Format KIF Conceptual Graph Interchange Format CDIF..

41 Eamples of Medical Knowledge 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.

42 Limitation Real world sometimes can not be represented using logic Induction and deduction model Uncertainty and probability Contet and eception

43 Alternatives Case-based reasoning Analogy Fuzzy logic Nonmonotonic logic

44 Etra Reading Aho s book chapter 4 Sowa s book p