Formal Methods in Computer Science CS1502 Atomic Sentences

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Formal Methods in Computer ScienceCS1502Atomic
Sentences

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• What logic is about.
• To learn about basic notions in logic
• Models, worlds, FOL, atomic sentences,
  predicates, names
• To gain skills in creating logic models.

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What logic is about

• Logic is the study of claims and reasoning about
  claims.
• Logic is the study of systematic and strategic
  reasoning to determine if a claim is true or
  false, given certain assumptions.

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Example

• Suppose the following.
• Socrates is a man.
• All men are mortal.
• No mortal lives forever.
• Everyone who will eventually die sometimes
  worries about it.
• Can we conclude this?
• Socrates sometimes worries about dying.

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Too easy / Too hard
Hard to comprehend
Our ability with the help of formal logic
Our everyday ability to reason about the world
Easy to comprehend
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What are the basic pieces we need?

• A way to represent English sentences
• Create a mathematical model that can be used to
  represent English sentences
• Logic sentence, Predicates, Names, Terms
• Functions, boolean connectives, quantifiers, etc
• Formalize a procedure for reasoning
• Truth tables
• Inference rules
• Proofs, etc.

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Sentences

• What kind of sentences are we interested in in
  logic?
• Paris is the capital of France
• 22 4
• Toronto is in Asia
• Is Philosophy 101 harder than Gen Chem 101?
• Add sugar, then add salt.
• Declarative Sentences

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Sentences

• Break each sentence into two parts
• Objects that the sentence refer to.
• Names
• Properties/relationships that the sentence refer
  to.
• Predicates
• Form a sentence by combining names and a
  predicate.
• AtSchool(abby)
• AtLibrary(john)

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Worlds and models

• Do our predicate and names apply only to the real
  world?
• AtSchool(abby)
• AtLibrary(john)
• Taller(john,jack)
• It also works on a similar hypothetical world.
• A set of similar worlds is called a model.

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World and Model

• A model describes what worlds are possible within
  the model.
• Thus, a model is a set of worlds.
• A world consists of a number of objects with
  various properties and relationships among them.
• FOL is a family of languages having a similar
  grammar and sharing similar components.
• Such a language used to describe, and make claims
  about worlds in a model.
• Different models have different languages.
• We use Taskis worlds model to study logic.

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A Taskis World
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Example Taskis Worlds Model

• A world in the Taskis worlds model is an
  arrangement of objects in a 8x8 chess board.
• 3 kinds of Objects Cube, Tetrahedron,
  Dodecahedron
• 3 sizes
• Etc.
• Other restrictions
• An object must locate in a square of the board
• No 2 objects can be in the same position.
• An object can have only 1 shape.
• An object can have only 1 size.
• Etc.

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Taskis Model

• We are not interested in
• The precise position of the object.
• The color of the objects.
• (Cubes are always green, )
• The density of the objects.

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Example of Claims

• English
• Object b is a cube.
• Object a is small.
• Object c is to the left of object f.
• Object d and object e have the same shape.(false
  claim)

• FOL
• Cube(b)
• Small(b)
• LeftOf(c,f)
• SameShape(d,e)(false claim)

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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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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
• Sam is taller than Jim
• Sam is taller than Jim and John

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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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Exercise Creating Models

• Divide into small groups
• Create your own model
• Give a description of your model.
• Create your own language
• Predicates
• Names
• Give an example world of your model
• Give a few sentences.
• Indicate if the sentences are true in your world.
• Words home, friend, program, fast, before,
  stone, letters, students, snow, basketball, team,
  logistic, heavy, book.

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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
• starship(son(dr_crusher)) Dr_Crushers sons
  starship

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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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Predicates vs Functions

• Syntax is similar
• Predicate(arg1, arg2,, argn)
• Function(arg1, arg2,, argn)
• Difference
• After applying a predicate to terms, we get a
  sentence
• Cannot be nested
• After applying a function to terms, we get
  another term.
• Can be nested

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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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Functions in Taskis model

• lm(x) Leftmost
• Refer to the leftmost object in the same row as x
  
• It could be x itself
• rm(x) Rightmost
• fm(x) Frontmost
• bm(x) Backmost

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Exercise Creating Models with Functions

• Divide into small groups
• Add some functions to your model created earlier.
• Give an example world of your model
• Give a few sentences with functions.
• Indicate if the sentences are true in your world.
• Words home, friend, program, fast, before,
  stone, letters, students, snow, basketball, team,
  logistic, heavy, book.

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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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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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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
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Goals Have you archived the followings?

• What logic is about.
• To learn about basic notions in logic
• Models, worlds, FOL, atomic sentences,
  predicates, names
• To gain skills in creating logic models.