CIS341 Artificial Intelligence

1
CIS341 Artificial Intelligence

• Weeks 8 11
• Predicate calculus and tableau proofs
• Natural language syntax and parsing
• Search and problem solving
• Parsing as proof, proof as a search problem
• Coursework 2 proposed dates
• Given out Friday March 13, due in Friday April
  3rd

2
Week 8 logic and inference revisited

• Revision of propositional logic
• Brief overview of proof techniques
• Introduction to predicate calculus
• Demonstration of tableau proofs on whiteboard

3
Automated reasoning and formal logic

• Logic is concerned with validity of inference
• A central concern of 20th-century mathematical
  logic was to develop systematic, mechanical
  techniques for proving logical consequence
• The AI subfields of logic programming and
  automated reasoning have built on these efforts
  to develop systems to perform logical inferences.
• Considerations of speed and efficiency put
  limitations on the kinds of inference that can be
  implemented.

4
Overview of classical logic

• Just as many people understand the word music to
  refer to European classical music composed
  between the 17th and 20th centuries, so the word
  logic is often shorthand for a variant called
  classical logic.
• The remainder of these slides provide an overview
  of classical logic, comprising
• Boolean or propositional logic
• First-order predicate calculus (the words
  first-order are often omitted)
• There are many non-classical logics but they do
  not come into this course.

5
Some important terminology

• A proposition is something that can be true or
  false
• A proposition is consistent if it is possible for
  it to be true
• A proposition is inconsistent or contradictory if
  it is not possible for it to be true
• A proposition is a necessary truth or a tautology
  if it is not possible for it to be false.
• Class exercise 1 think up examples in each of
  these categories.

6
Boolean or propositional logic

• Boolean logic is so-called because logical
  formulas are interpreted in terms of Boolean
  truth values
• True or False
• T or F
• 1 or 0
• The semantics of Boolean logic tells us how to
  calculate the truth values of complex formulas as
  a function of the truth values of their
  constituent literals using truth tables
• Simple propositions generally written P, Q etc.

7
Syntax of propositional logic

• Formulas of Boolean logic are made up of
  propositional letters P, Q, R etc and the symbols
• or ? conjunction, read as and
• ? inclusive disjunction, read as or
• ? or negation, read as not
• ? or ? implication, read as implies or if
  then
• ? or ? biconditional, read as if and only if
  or iff

8
Truth tables
9
Truth table for
10
Truth table for implication
11
Inference involving implication

• If the solution is acid, the paper will turn
  red.
• True ? True the solution is acid, the paper
  turns red. The implication is verified.
• True ? False the solution is acid, the paper
  does not turn red. The implication is falsified.
• False ? True the solution is not acid, the paper
  still turns red.
• False ? False the solution is not acid, the
  paper does not turn red.
• Is the implication verified or falsified in cases
  3 and 4?

12
Class/self-study exercise 2

• Construct truth tables for
• ?P ? Q
• ?(P Q)
• ?(P ? Q)
• ?(P ? Q)

13
Validity of inference

• An inference is valid if it is not possible for
  the premises to be true and the conclusion false.
• Valid
• P, P ?Q / Q
• ?(P ? Q) / ?P
• Invalid
• ?(P Q) / ?P
• Q, P ?Q / P
• Validity in propositional logic can be checked
  using truth tables.

14
Some proof techniques

• Modus ponens
• P ? Q P / Q
• Modus tollens
• P ? Q ?Q / ?P
• Provide a short cut for calculating inferences
  without constructing truth tables.
• A proof theory consists of a set of such rules
• There are proof theories for propositional logic
  which allow all valid inferences to be proved.

15
Resolution rule

• Equivalent of modus ponens
• ?P ? Q P / Q
• Basis of logic programming
• Requires disjunctions with at most one positive
  literal, for example
• (P Q) ? R
• ? (P Q) ? R
• ?P ? ? Q ? R

16
Tableau proof

• A graphic proof technique based on tree
  diagrams called semantic tableaux
• Will be covered in later slides
• Good tutorial in Wilfred Hodges, Logic
• Useful online tutorial at http//logic.philosophy.
  ox.ac.uk/main.htm designed for Oxford philosophy
  students
• Key point there are many proof techniques which
  are all equally sound and complete, i.e. they
  will prove all and only valid arguments.

17
Shortcomings of propositional logic

• A simple proof
• Fred is ill.
• If Fred is ill, he should not drive.
• So, Fred should not drive.
• In symbolic form
• P Fred is ill''
• Q Fred should not drive
• Modus ponens
• P ? Q, P / Q

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Shortcomings of propositional logic (cont.)

• So far, so good. But what if Charlie is ill?
• Do we construct a separate rule for every
  individual?
• P2 Charlie is ill''
• Q2 Charlie should not drive
• P2 ? Q2

19
From propositional logic to predicate calculus

• Solution is to decompose statements into
  predicates and arguments
• P(x) x is ill''
• Q(x) x should not drive
• P(x) ? Q(x)
• Proof will look more like this
• P(x) ? Q(x)
• P(fred)
• P(fred) ? Q(fred)
• ? Q(fred)

20
From propositional logic to predicate calculus
(cont.)

• Boolean logic is the logic of propositions.
• Predicate calculus introduces predicates and
  arguments.
• Arguments can be constant terms or variables.
• Predicates have to be constant terms in the
  first-order predicate calculus.
• There are higher-order logics with predicate
  variables but we will not be concerned with them.

21
Quantifiers in predicate calculus I

• Implicational rules are generally written like
  this
• ?x(P(x) ? Q(x))
• The upsidedown A is called the universal
  quantifier this means that the implication is
  true for all values of x (i.e. for any
  individual).
• Read as for all x, P(x) implies Q(x)
• Can also be read as all Ps are Q or all Ps Q
• ?x(fish(x) ? swim(x))
• ?x(cow(x) ? mammal(x))

22
Quantifiers in predicate calculus II

• The existential quantifier ? means that statement
  that follows it is true of some individual,
  though we may not know which.
• ?x(Px Qx)
• Read as
• there exists an x such that P(x) and Q(x)
• some P is a Q
• some Ps Q
• ?x(mammal(x) lays-eggs(x))

23
Quantifiers in predicate calculus III

• You may have noticed that ? generally goes with
  the conjunction and ? goes with the
  implication ?
• This is very important. What would the following
  mean? Are they true?
• ?x(fish(x) ? sings(x))
• ?x(cow(x) mammal(x))
• This is probably what beginning logic students
  most often get wrong.

24
Quantifiers in predicate calculus IV

• Multiple quantifiers can be combined in the same
  formula
• ?x(student(x) ? ?y (tutor(y,x)))
• Every student has a tutor
• ?x(professor(x) ??y(course(y) teach(x,y)))
• There is a professor who does not teach any
  courses

25
Duality of first-order quantifiers

• The quantifiers ? and ? can be defined in terms
  of each other, using Boolean equivalences
• ?x(P(x) ? Q(x))
• ??x?(P(x) ? Q(x))
• ??x(P(x) ?Q(x))
• ?x(P(x) Q(x))
• ??x?(P(x) Q(x))
• ??x(P(x) ? ?Q(x))
• Class/self-study exercise 3
• Convince yourself that the above equivalences are
  correct.

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Class/self-study exercise 3

• Convert the following to predicate calculus
• All mammals are vertebrates.
• Not all vertebrates are mammals.
• Only fish swim.
• Some primates have tails.
• No primates have feathers.
• Rhinos eat either leaves or grass.

27
Inference in predicate calculus

• Techniques such as modus ponens and tableau proof
  are applicable to predicate calculus, with extra
  complications caused by the use of variables.
• Example given the premises
• ?x(P(x) ? Q(x))
• ?y(Q(y) ? R(y))
• P(fred)
• we can infer via modus ponens or using tableaux
  (several steps omitted)
• Q(fred) R(fred)

28
Beginning tableau proofs

• The Tableau method uses a standard technique of
  proof by contradiction
• To show whether an argument is valid, attempt to
  show that the premises are inconsistent with the
  negation of the conclusion
• To show whether a complex proposition is a
  tautology, attempt to show that its negation is
  inconsistent
• (see whiteboard and handouts)

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Useful further reading

• Books
• A very short introduction to logic, Graham
  Priest, OUP. A clear and stimulating guide to
  basic concepts of logic.
• Logic, Wilfred Hodges, Penguin. Goes into a lot
  more detail than Priest, useful for
  reference/revision.
• Website
• Introduction to logic, http//logic.philosophy.ox.
  ac.uk/main.htm
• Tutorial on propositional and predicate logic
  aimed at Oxford philosophy students. May not
  display correctly in all browsers. Seems to work
  best in Internet Explorer.