CS1502 Formal Methods in Computer Science

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CS1502 Formal Methods in Computer Science

• Lecture Notes 1
• Course Information
• Introduction to Logic
• Part 1

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Content

• 70 logic and proofs
• Pervades computer science, e.g., hardware circuit
  design, artificial intelligence, knowledge
  representation, database systems, programming
  languages, software engineering (design,
  verification, specification),
• Will help you understand proofs in math, science,
  theoretical CS

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Content

• 30 Abstract models of computation
• Needed for hardware design, compilers,
  computational complexity analysis,
• Interesting proofs you will use what you learn
  in the first part of the course to understand
  them

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Logistics

• Prerequisites
• CS441 Discrete Structures for Computer Science
• CS445 Data Structures
• Materials on course website, reachable from
  www.cs.pitt.edu/wiebe
• Schedule of readings, homeworks, exams
• Lecture notes, available before class
• Support for lectures filled in in class

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Logistics

• The TA will go over exercises and answer
  questions in recitation.
• He will also cover logistics of submitting
  homeworks, and will post solutions to the
  homework.

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Logistics

• Workload is steady throughout the course
• 3 exams
• Weekly homeworks (no projects)
• Many are graded electronically
• You can submit solutions as many times as you
  like
• Get feedback without having to wait for
  instructor grading
• Purpose is to help you master the material for
  the exams
• Exams 90 Homework 10

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Logistics

• Exams
• Challenging, but not devious
• All of the questions will be related to an
  example worked out in the text, a homework
  exercise, or a problem we did in class.

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Logistics

• Work is steady throughout the course
• If you keep up, you will likely do OK
• The work gets harder!
• Please ask questions (lectures, recitations,
  office hours)
• My office hours are T,TH 130-230 and by
  appointment just send mail to set up a time

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Logistics

• Do readings before lecture
• Do the you try it exercises as you read the
  logic text

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Software with LPL TextFitch

• Named for Frederic Fitch
• Construct formal proofs
• Prove an argument is valid

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Software with LPL TextBoole

• Named for George Boole
• Construct truth tables
• Verify a sentence is a tautology
• Verify two sentences are tautologically
  equivalent
• Prove an argument is valid

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Software with the TextTarskis World

• Named for Alfred Tarski
• Construct sentences in first-order logic
• Determine if a sentence is true wrt a world
• Create a world that shows that an argument is
  invalid

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Software with TextSubmit

• Verify your solutions are correct (without
  instructor seeing)
• Submit homework for final grading to TA (by due
  date)

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More Notes on Homework

• Many exercises will be submitted to the grade
  grinder
• Please request that your grade assessments be
  sent to the TA (not to me!)
• You may submit your solutions as many times as
  you like, but please send your TA only a single
  report per assignment
• Now, lets look at the webpage and syllabus

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Introduction to Logic
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Examples
circle(a) small(a)
All x All y ((cube(x) tet(y)) ? (leftof(x,y)
frontof(x,a)))
feed(max,scruffy)
All x All y ((pet(x,y) hungry(y)) ? feed(x,y))
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Logic

• A simple grammar. Each sentence has a single
  interpretation (unlike English!)
• Used to describe a world, which we define.
• Once we define the world, we can say what things
  names refer to, and whether a logical sentence is
  true or false.

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Names

• Constants are used to name existing objects
• a, b, c, d, e, f
• max, claire, carl
• No constant can name more than one object
• An object can have more than one name or no name
  at all

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Predicates

• A property possessed by an object
• Shape (e.g., Tet, Cube)
• Size (e.g., Small, Large)
• A relationship among objects
• Shape relationship (e.g., SameShape)
• Size relationship (e.g., Smaller)
• Positional relationship (e.g., Between, LeftOf)
• Equality

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Quick Example

• Ackermanns sentences and world in Tarski
• Properties
• Cube, Tet (4 faces), Dodec (12 faces), Medium
• Relations
• Backof, Leftof
• (Click verify, and you see that one of the
  sentences is false about the world)

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Predicates

• Each predicate has a fixed number of arguments or
  arity. This is the number of constants the
  predicate needs to form a sentence.
• In English, OK, but not in logic
• Susanna is taller than Dimitri
• Susanna is taller than Dimitri and Jerome

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Predicates

• Predicates must be determinate
• Suppose p is an n-ary predicate.
• For every n-tuple lto1,o2,,oNgt of objects,
  p(o1,o2,,oN) is true or false (not kind of
  true).
• Whats an n-tuple?
• An n-tuple is a collection of n objects where
  order matters. Duplicates are allowed. In
  contrast, sets may not have duplicates, and the
  members of sets are not ordered.

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Atomic Sentences (so far)

• A sentence formed by a single predicate followed
  by one or more names
• Max is tall Tall(max)
• e is larger than b Larger(e,b)
• e is identical to a e a
• A sentence expresses a claim that is either true
  or false

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Atomic Sentences (so far)

• Predicate(arg1, arg2,, argn)
• Predicates have names beginning with an uppercase
  letter or are represented by an operator symbol
• The number of arguments is called the predicates
  arity
• The order of the arguments is importantLarger(e,c
  ) e is larger than cLarger(c,e) c is larger
  than eBetween(a,b,e) a is between b and
  eBetween(b,a,e) b is between a and e
• (a,b)
• a and b are identical
• Usually, written in infix form a b

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Function Symbols

• A function is used to express complex names (a
  reference to an individual without using a name)
• father(b) bs father
• Used in a sentence Tall(father(b))
• password(c) cs password
• Used in a sentence Long(password(c))
• A function may be nested
• father(father(max))
• Used in a sentence Short(father(father(max)))
• Cant nest predicates
• Tall(Tall(max)) not a legal sentence

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Functional Expressions

• Function(arg1, arg2,, argn)
• Function names begin with a lowercase letter or
  are expressed with a symbol
• father(max) Maxs father
• father(mother(max)) Maxs mothers father
• youngestChild(max,ann) Max and Anns youngest
  child
• (5,(2,4)) 30

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Atomic Sentences (so far)

• A sentence formed by a single predicate followed
  by one or more terms
• A term is either a constant or a functional
  expression

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Example Atomic Sentences (which are functions
and which are predicates?)

• Predicate(term1,term2,,term-n)
• Happy(bossof(sally))
• Father(bill)
• Tall(fatherOf(motherOf(sally)))
• Happier(motherOf(bill),bossof(fatherOf(max)))
• Fake(santa(rossParkMall)))
• Real(santa(robinsonTownCenter)))

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ConnectivesApply to sentences tocreate more
complex sentences.

• Not ?
• And, Or ?, ?
• Material Conditional ?
• Biconditional ?

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Examples

• ?Larger(e,c)
• Cube(b) ? Large(b)
• SameRow(e,c) ? BackOf(e,b)

e is not larger than c
b is a cube or b is large
e and c are in the same row and e is in back of b
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First Order Logic

• Names
• Predicates
• Functions
• Connectives

Are there more?
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Example FOL
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Translation

• Brando is Nancys favorite actor.

• brando favoriteActor(nancy)

• BetterActor(favoriteActor(nancy),
  favoriteActor(max))

• Nancys favorite actor is better than Maxs
  favorite actor.

• sean favoriteActor(sean)

• Sean is his own favorite actor.

• Brando is someones favorite actor.

• ?x(brando favoriteActor(x))

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Quantifiers and Variables

• For every x ?x
• ?x (man(x) ? mortal(x))
• There exists y ?y
• ?x(brando favoriteActor(x))

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First Order Logic

• Names
• Predicates
• Functions
• Connectives
• Quantifiers and variables

Revised List