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Logic in computer science ES c233CS F214IS F214

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1Logic in computer science ES c233/CS 20rules of inference. When using logic as a
F214/IS F214. Prof. Navneet Goyal, CSIS proof system, one is not concerned with
Department, BITS-Pilani. the meaning of the statements that are
2Motivation. Logic became popular in manipulated, but with the arrangement of
the early 20th century among philosophers these statements, and specifically,
and mathematicians What constitutes a whether proofs or refutations can be
correct proof in Mathematics? Some constructed.
‘correct’ proofs were later disproved by 21Propositional Logic. Declarative
other mathematicians Concept of logic sentences in English ? string of symbols
helps us to figure out what constitutes a Compressed but complete encoding of
correct argument and what constitutes a declarative statements Allows us to
wrong argument Euclid’s parallel postulate concentrate on the mere mechanics of our
& Fermat’s Last theorem are classic argumentation Specifications of systems or
examples! software are sequence of such declarative
3Motivation. Faults (bugs) have been statements Automatic manipulation of such
detected in proofs (programs) Bugs are statements, something that machines love
hard to detect! Notion of correct argument to do.
Formal Logic as foundation to Mathematics? 22Propositional Logic. Atomic or
Mathematics does rest on one strong indecomposable sentences The number 5 is
foundation – Set Theory! Set theory is even Composition of atomic sentences p: I
based on First-order Logic! won the lottery last week q: I purchased a
4Motivation. Questions related to lottery ticket r: I won the last week’s
automation or mechnizability of proofs sweepstakes (horse race) ¬ p: I did not
needs to be answered These questions are win the lottery last week p v r: atleast
relevant & important for present day one of them is true. Disjunction. I won
computer science! They form the basis for the lottery last week or I won the last
automatic theorem proving David Hilbert week’s sweepstake (not to be confused with
asked the important question, as to English OR).
whether all mathematics, if reduced to 23Propositional Logic. p ^ r:
statements of symbolic logic, can be conjunction. Last week I won the lottery
derived by a machine. and the sweepstakes p q: implication. If I
5Motivation. Can the act of won the lottery last week, then I
constructing a proof be reduced to the purchased a lottery ticket. p is called
manipulation of statements in symbolic the assumption and q is called conclusion.
logic? Logic enabled mathematicians to 24Propositional Logic. p q: implication:
point out why an alleged proof is wrong, Some interpretations p implies q If p then
or where in the proof, the reasoning has q p only if q p is a sufficient condition
been faulty. By symbolising arguments for q q is a necessary condition for p q
rather than writing them out in some if p q follows from p q provided p q is a
natural language (which is fraught with consequence of p q whenever p.
ambiguity), checking the correctness of a 25Natural Deduction. Construct a
proof becomes a much more viable task. language of reasoning about propositions
6Motivation. Since the latter half of Set of rules which allow us to draw a
the twentieth century logic has been used conclusion given a set of premises PROOF
in computer science for various purposes RULES Allow us to infer formulas from
ranging from program specification and other formulas Applying these rules is
verification to theorem-proving. succession, we may infer a conclusion from
7Objective of the course. To prepare a set of premises Be careful though!!
the student for using logic as a formal 26Natural Deduction. Constructing a
tool in computer science. proof is much like a programming! It is
8Introduction to Logic. Logic is called not obvious which rules to apply and in
the CALCULUS of Computer Science! LOGIC: what order to obtain the desired
CS CALCULUS: Physical sciences & conclusion Careful choice of proof rules!
Engineering Disciplines CS areas where we 27Rules of Natural Deduction. Rules of
use LOGIC Architecture (logic gates) inference/natural deduction specify which
Software Engineering (Specification & conclusions may be inferred legitimately
Verification) Programming Languages ( from assertions known, assumed, or
Semantics & Logic Programming) AI previously established Fundamental rule 1
(automatic theorem proving) Algorithms (modus ponens or rule of detachment) p p q
(complexity) Databases (SQL). . . . q Fundamental rule 2 (transitive
9History of Logic. Symbolic Logic (500 rule) p q q r . . . p r.
BC – 19th century) Algebraic Logic (Mid to 28Rules of Natural Deduction.
late 19th century) Mathematical Logic Fundamental rule 1 (modus ponens or rule
(19th century to 20th century) Logic in of detachment) p p q . . . q The rule is a
Computer Science. valid inference because [p ^ (p q)] q is a
10Fundamental of Logic. Two famous laws tautology! (use truth tables or
of classical logic Law of the excluded abbreviated truth tables to show that a
middle Law of contradiction Declarative proposition is a tautology).
statements Truth values – T or F 29Rules of Natural Deduction. Example:
Propositions For every proposition p, if it is 11:00 o’ clock in Tallahassee if
either p is T or p is F For every it is 11:00 o’ clock in Tallahassee, then
proposition p, it is not the case that p it is 11:00 o’ clock in New Orleans then
is both T and F. by rule of detachment, we must conclude:
11Fundamental of Logic. We are it is 11:00 o’ clock in New Orleans
interested in precise declarative Difference between implication and
statements about computer systems and inference! the truth of an implication p q
programs We not only want to specify such does not guarantee the truth of either p
statements, but also want to check whether or q. But the truth of both p and p q does
a given program or system fulfils guarantee the truth of q.
specifications at hand Need to develop a 30Rules of Natural Deduction.
calculus of reasoning which allows us to Fundamental rule 2 (transitive rule) p q q
draw conclusions Derive new facts from r . . . p r This is a valid rule of
given facts. inference because the implication (p q) ^
12Fundamental of Logic. 5 basic (q r) (p r) is a tautology! Generalization
connectives And Or If…then If and only if is possible De Morgan’s law: FR #3 Law of
Not. contraposition: FR #4.
13Logic in CS. Logic underlies the 31Rules of Natural Deduction. De
reasoning in mathematical statements Morgan’s law: FR #3 ~(p v q) = (~p) ^ (~q)
Objective is to develop languages to model ~(p ^ q) = (~p) v (~q) Law of
the situations that we encounter in CS contrapositive: FR #4 p q = (~q ~p) Double
Reasoning about situations formally Negation ~(~p) =p Implication p q = (~p) v
Constructing arguments about them q.
Arguments should be valid and can be 32Rules of Natural Deduction. Most
defended rigorously Can be executed on a arguments in Mathematics are based on FR#1
machine. & FR#2, with occasional use of FR#3
14Propositional Logic: Basics. & FR#4. Get comfortable with FRs!
Declarative sentences Non-declarative 33Rules of Natural Deduction. Fallacies:
sentences Go and attend classes Don’t take 3 forms of faulty inferences! Fallacy of
make-ups Examples of declarative affirming the consequent Fallacy of
statements Goldbach’s conjecture All denying the antecedent Non-sequitur
BITSIANs are intelligent A is older than B fallacy Fallacy of affirming the
There is ice in the glass. consequent p q q . . . p If the prices of
15Propositional Logic: Basics. It’s a gold are rising, then inflation is surely
language! Propositional logic is based on coming. Inflation is surely coming.
propositions or declarative statements Therefore, the price of gold is rising.
Propositions or declarative statements can Check: [(p q) ^ q] p for tautology!
be mapped onto Boolean values T or F 34Rules of Natural Deduction. Fallacy of
Statements about computer systems or denying the antecedent (affirming the
programs We also want to check whether a opp.) p q ~p . . . ~q Since the opp. of p
computer program or a system satisfies the q is ~p ~q, this fallacy is the same as
specifications. affirming the opp. Non-sequitur (it does
16Propositional Logic: Basics. not follow) p If Socrates is a man, then
Propositional logic describes ways to Socrates is mortal . . . q Socrates is a
combine true statements by means of man Therefore, Socrates is mortal.
connectives to produce other true Socrates is a man Therefore, Socrates is
statements. If it is asserted that `Jack mortal.
is taller than Jill' and `Jill can run 35Rules of Natural Deduction. Examples
faster than Jack' are T `Jack is taller of Arguments If a baby is hungry, then the
than Jill and Jill can run faster than baby cries. If the baby is not mad, then
Jack'. However, if Jill is actually taller he does not cry. If a baby is mad, then he
than Jack, then the 1st statement is F and has a red face. Therefore, if a baby is
the combined statement is false as well. hungry, then he has a red face. Model this
Propositional logic allows us to formalize problem!! h: a baby is hungry c: a baby
such statements In concise form: A ^ B. cries m: a baby is mad r: a baby has a red
17Propositional Logic: Basics. Every face.
logic comprises a (formal) language for 36Rules of Natural Deduction. Examples
making statements about objects and of Arguments If a baby is hungry, then the
reasoning about properties of these baby cries. If the baby is not mad, then
objects. We will restrict our attention to he does not cry. If a baby is mad, then he
mathematical objects, programs, and data has a red face. Therefore, if a baby is
structures in particular Statements in a hungry, then he has a red face. Model this
logical language are constructed according problem!! h: a baby is hungry c: a baby
to a predefined set of formation rules cries m: a baby is mad r: a baby has a red
called ‘syntax rules’. face.
18Propositional Logic: Basics. Why 37Rules of Natural Deduction. Examples
English or any other natural language of Arguments If Nixon is not reelected,
can’t be used? English is a rich language then Tulsa will lose its air base Nixon
which cant be formally described Meaning will be reelected iff Tulsa votes for him
of an English sentence can be ambiguous, If Tulsa keeps its air base, Nixon will be
subject to different interpretations reelected Therefore, Nixon will be
depending on the context and implicit relelected Model this problem!! R: Nixon
assumptions Another important factor is will be reelected T: Tulsa votes for Nixon
conciseness. Natural languages tend to be A: Tulsa keeps its air base.
verbose, and even fairly simple 38Rules of Natural Deduction. Examples
mathematical statements become exceedingly of Arguments If angles A & B are rt
long (and unclear) when expressed in them. angles, then they are equal The angles A
The logical languages that we shall define & B are equal Hence, the angles A
contain special symbols used for & B must be rt angles R: A & B are
abbreviating syntactical constructs. at rt angles E: A & B are equal
19Propositional Logic: Basics. Fallacy: affirming the consequent!
Assertions and Proofs A precise language 39Rules of Natural Deduction. What
is required whose syntax can be completely remains when arguments are symbolized is
described in a few simple rules semantics the bare logical skeleton, the mere form
can be defined unambiguously A logical of argument which many arguments may have
language can be used in different ways in common regardless of the context of the
Deduction system or proof system. sentences It is this form that enables us
20Propositional Logic: Basics. This use to analyze the inference, for deduction
of a logical language is called proof has more to do with forms of the
theory. A set of facts ‘called’ axioms and propositions in an argument than with
a set of deduction rules (inference rules) their meanings.
are given, and the object is to determine 40Rules of Natural Deduction. Sequent
which facts follow from the axioms and the Premises Conclusion.
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