Announcements

1
Announcements

• Reading Assignment
• Nilsson chapters 13-14
• Announcements
• LISP and Extra Credit Project Assigned
• Todays Handouts
• Outline for Class 33
• Web Site
• www.mil.ufl.edu/eel5840
• Software and Notes

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Todays Menu

• Informal Introduction to the Predicate Calculus
• The Propositional Calculus
• Expressing Constraints in Feature Values
• The Language
• Rules of Inference
• Proofs
• Semantics
• Interpretations
• Truth Tables
• Satisfiability and Models
• Validity
• Equivalence
• Entailment

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Informal Introduction to PC

• Predicate Calculus (PC)
• Plusses ()
• Concise rich notation. Universally understood
• A lot is known about its limitations
  decidability, admissibility
• We can place bounds on expected performance
• Minuses (-)
• May direct attention away from the problem
  specification phase
• Many problems do not map well to mathematical
  analysis
• Concepts
• Notation e.g., ?x ?y P(x)?Q(y)
• Proofs by Refutation
• Rules of Inference
• modus ponens
• modus tolens
• resolution
• Operators
• Alternatives for Proofs constraint propagation
  and justification

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Informal Introduction to PC

• Predicate Calculus (PC)
• A mathematical language that allows us to specify
  Rules of Inference as ways (algorithms) to
  reason.
• Example 1 It is a bird it it has feathers, or it
  flies and lays eggs
• Rule 1a If animal has feathers then animal is
  bird
• Rule 1b If animal flies and animal lays eggs
  then
• animal is bird
• In Robotics we need to capture the idea that
  something has properties
• Example 2 In the domain of birds these are true
  predicates
• Feathers(Robin)
• Bird(Robin)

5
Informal Introduction to PC

• Predicate Calculus (PC)
• To say that Feathers(Julie) is true means that
  Julie is constrained to what it stands for. Other
  constraints are Flies(Julie)Lays-Eggs(Julie)
• Expressions joined by or by ? are called
  conjunctions
• Expressions joined by or by ? are called
  disjunctions
• Logical Connectives are , (?), (?), and ?
  (? )
• Example 3 Feathers(Sue)
• Sue is an object for which Feathers(Sue) is not
  satisfiable

6
Informal Introduction to PC

• PREDICATE CALCULUS (PC)
• Example 4 Feathers(Squicky) ? Bird(Squicky)
• also F ? G ? F ? G ? ?F ? G
• or better yet Feathers(X) ? Bird(X)
• X is an object in the domain of interest
  constrained to satisfy the following definition
  If Feathers(X) then Bird(X) else true
• Example 4 is true if Feathers(X) is true and if
  Bird(X) is true
• However the definition allows Feathers(X) and
  Bird(X) to be both false
• and the definition allows Feathers(X) is false
  and Bird(X) to be true
• If Feathers(X) is true and Bird(X) is false then
  Example 4 is false
• A K-Map of E1 ? E2 demonstrates that it is
  equivalent to ?E1 ? E2
• 0 1
• 0 1 0 E1 ? E2 ? ? E1 ? E2
• 1 1 1

E1
E2
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Informal Introduction to PC

• Predicate Calculus (PC)
• Precedence Rules Use parentheses ( ) to clarify
• has the highest precedence
• (?), (?) equal precedence and lower than
• ? (? ) lowest precedence
• Logical Connectors are commutative E1?E2?E2?E1
  E1?E2?E2?E1
• Logical Connectors are distributive
• E1?(E2?E3)?(E1?E2)?(E1?E3)
• E1?(E2?E3)?(E1?E2)?(E1?E3)
• Logical Connectors are associative
• E1?(E2?E3)?(E1?E2)?E3
• E1?(E2?E3)?(E1?E2)?E3
• Identity Property E1 E1
• De Morgans Law (E1?E2)? E1? E2
  (E1?E2)? E1 ? E2

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Informal Introduction to PC

• Predicate Calculus (PC)
• Quantifiers (?, ?) are used to determine when
  things are true
• Example 5 (?x)Feathers(x) ? Bird(x)
• Quantifier variable
• For all objects x in the domain of interest
  (x?D) we get a true expression when we substitute
  any object x inside the square brackets. Notice a
  sort of domain independence, i. e., Anything
  that has feathers is a bird (in the domain of
  interest)
• The expression surrounded by square brackets
  associated with a quantifier is said to lie in
  the scope of the quantifier. (In other words
  Feathers(x) ? Bird(x) lies in the scope of
  (?x) )
• ? is called the universal quantifier, it is true
  for all objects
• ? is called the existential quantifier, it is
  true for some objects (at least 1 object).
  It is pronounced there exists.

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Informal Introduction to PC

• Predicate Calculus (PC)
• (?x)Bird(x) ? there exists an x (at least one)
  for which Bird(x) is true
• Vocabulary of the PC
• Terms Domain Objects (A,B, Robin, etc.)
• Variables (x, y, z, w, ...) they range over
  domain objects
• Functions of objects that return objects, e.
  g., f(x)
• Arguments to functions, e.g., (x), (x,y),
  (x,y,z), etc.
• Arguments to predicates, e.g., (x), (h(x)),
  (x,f(z)), etc.
• Atomic Formulas (Atf) the atoms T or F
• Predicates with Arguments, e.g., P(x), Q(w,y)
• lttermgtlttermgt or lttermgt?lttermgt
• Literals Atf or Atf
• Well-Formed-Formula (wff) is any literal or
• ltwffgt (?), (?), ? (? ) ltwffgt
• (?vi)ltwffgt (?vi)ltwffgt
• ltwffgt

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Informal Introduction to PC

• Predicate Calculus (PC)
• Sentences are closed wffs ? a wff that has no
  free variables
• Example 6 6a (?x)Feathers(x) ? Bird(x)
• 6b Feathers(Oriole) ? Bird(Oriole)
• x in 6a is said to be bound, that is, it appears
  within the scope of its corresponding quantifier.
  Further, variables that are not bound are free.
  Both 6a and 6b are sentences.
• Example 7 (?x)Feathers(x) ? Feathers(y)
• This example is not a sentence. Can you explain
  why?
• If variables are allowed to represent only
  objects the logic is 1st order. It is called the
  First Order Predicate Calculus.
• Second Order Predicate Calculus variables
  allowed to represent predicates and object
  functions.
• ZOPC/Boolean Algebra/Propositional Calculus - No
  variables

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Informal Introduction to PC

• Predicate Calculus (PC)
• Clause A wff consisting of a disjunction of
  literals, e.g., (?x)F(x) ? B(x) (?P)P(x) ?
  Q(x) (?f)P(x) ? H(x, f(x))
• Interpretations Tie logic symbols to words. The
  goal is to say something about the world, e.g.,
  ON(A,B). We do this by associating functions,
  predicates and objects with tangible things.
  Thus, objects in some domain D correspond to
  object symbols in logic.
• Example 9
• Logic Side Real World
• Symbolic Object A Block A
• Symbolic Object B Block B
• _
• ON(B,A) implies the relation that block B is
  on block A
• P(x,y) is read as x P y

P(A,B)
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Informal Introduction to PC
B

• Predicate Calculus (PC)
• Functions Bwhats_on(A)
• Bis_on(A) Is_whats_on(B,A)
• Def Interpretation A triple written as ID, Iv,
  Ic which is a full accounting of the
  correspondence between the logic world and the
  real world. D is the domain of interest, Iv the
  assignments to variables and Ic is the assignment
  to constants.
• Real World Logic
• objects ? object symbols
• relations ? predicates
• object functions? symbolic object functions
• BIG CONCEPT Proofs tie axioms to consequences
• Suppose we are given a set of wffs which are
  assumed to be true, this set is called axioms (or
  given/assumed) set.

A
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Informal Introduction to PC

• Predicate Calculus (PC)
• Example 9 9a Feathers(Julie)
• 9b (?x)Feathers(x) ? Bird(x)
• When we say these are axioms we are restricting
  the interpretations to the objects, symbols,
  predicates and functions for which the implied
  imaginable world holds. These interpretations for
  which the axiom set holds are said to models of
  the wffs.
• Suppose we are asked to show that all
  interpretations that make the axioms true make
  some other given wff true also. If we succeed, we
  say we have proven that this extra wff is a
  theorem with respect to the axioms.
• We prove that a wff is a theorem when we show
  that the theorem MUST be true give that the
  axioms are true.
• We prove that a wff is a theorem when every model
  for the axioms is also a model for the wff. If
  so, we say the wff logically follows from the
  axioms or alternatively, the axioms logically
  imply the wff.

?
14
Informal Introduction to PC

• Predicate Calculus (PC)
• Given the set of axioms in Ex. 9 can we prove
  Bird (Julie) is a theorem w.r.t. the axioms? Yes!
• HOW?
• We need a procedure or algorithm to do it. A
  proof is a procedure consisting of manipulations
  based on equivalences which are called Sound
  Rules of Inference which produce a new wff from
  old ones guaranteeing that models which make the
  old wffs true also make the new (derived) wffs
  true.
• A straightforward proof procedure applies sound
  rules of inference to the axioms recursively
  until the desired wff is produced.
• Do you like this? Why or why not?

15
Informal Introduction to PC

• Predicate Calculus (PC)
• Proving a theorem is not the same as showing that
  a wff is valid. Why? A valid wff is true
  independent of interpretation
• Proving a theorem is not the same as showing that
  a wff is satisfiable. Why? A satisfiable wff is
  true for some interpretation
• Modus Ponens the most straightforward sound
  rule of inference when applied recursively or
  successively.
• Def Given the set ? E1 ? E2 E1 then E2
  is a theorem w.r.t. ? thus, we can add E2
  to the axioms if it was obtained by mp
• Using mp prove Bird(Julie)
• Example 10 10a Feathers(Julie)
• 10b (?x)Feathers(x) ? Bird(x)
• Since ? is true for all x?D we can let xJulie
  and obtain 10b Feathers(Julie) ? Bird(Julie) and
  therefore Bird(Julie) m.p.

?
16
Informal Introduction to PC

• Predicate Calculus (PC)
• We observe that this proof was merely a syntactic
  (mechanical) procedure. We obtained Bird(Julie)
  because it follows from the axioms logically. The
  possibility exists that the given axioms or
  derived theorem may clash with our sense of the
  world.
• Modus Tolens ? A sound rule of inference when
  applied recursively or successively produces a
  derived wff as follows
• Def Given the set ? E1 ? E2 E2 then E1
  is a theorem w.r.t. ? ? we can add E1 to
  the axioms if it was obtained by mt
• Example 11 11a Bird(Larry)
• 11b (?x)Feathers(x) ? Bird(x)
• Since ? is true for all x?D we can let xLarry
  and obtain 11b Feathers(Larry) ? Bird(Larry) and
  therefore Feathers(Larry) m.t.

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• The End!