Logic in computer science ES c233CS F214IS F214 |
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1 | Logic in computer science ES c233/CS | 20 | rules 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 | ||
2 | Motivation. 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 | 21 | Propositional 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 | ||
3 | Motivation. 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? | 22 | Propositional 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 | ||
4 | Motivation. 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 | 23 | Propositional 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 | ||
5 | Motivation. 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 | 24 | Propositional 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 | 25 | Natural Deduction. Construct a | |
proof becomes a much more viable task. | language of reasoning about propositions | ||
6 | Motivation. 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 | ||
7 | Objective of the course. To prepare | a set of premises Be careful though!! | |
the student for using logic as a formal | 26 | Natural Deduction. Constructing a | |
tool in computer science. | proof is much like a programming! It is | ||
8 | Introduction 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 | 27 | Rules 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 | ||
9 | History of Logic. Symbolic Logic (500 | rule) p q q r . . . p r. | |
BC – 19th century) Algebraic Logic (Mid to | 28 | Rules 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 | ||
10 | Fundamental 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 | 29 | Rules 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: | ||
11 | Fundamental 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 | 30 | Rules 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) ^ | ||
12 | Fundamental 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. | ||
13 | Logic in CS. Logic underlies the | 31 | Rules 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 | 32 | Rules 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 | ||
14 | Propositional Logic: Basics. | & FR#4. Get comfortable with FRs! | |
Declarative sentences Non-declarative | 33 | Rules 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 | ||
15 | Propositional 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 | 34 | Rules 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 | ||
16 | Propositional 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 | 35 | Rules 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 | ||
17 | Propositional Logic: Basics. Every | face. | |
logic comprises a (formal) language for | 36 | Rules 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. | ||
18 | Propositional Logic: Basics. Why | 37 | Rules 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 | 38 | Rules 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 | ||
19 | Propositional Logic: Basics. | Fallacy: affirming the consequent! | |
Assertions and Proofs A precise language | 39 | Rules 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 | ||
20 | Propositional 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 | 40 | Rules of Natural Deduction. Sequent | |
which facts follow from the axioms and the | Premises Conclusion. | ||
Logic in computer science ES c233CS F214IS F214.ppt |
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