Artificial Intelligence

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Artificial Intelligence
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• Ahmed M. Zeki
• Sem 2, Nov 2004

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Logic

• Knowledge bases consist of sentences.
• These sentences are expressed according to the
  syntax of the representation language, which
  specifies all the sentences that are well formed.
• The notion of syntax is clear enough in ordinary
  arithmetic x y 4 is a well-formed
  sentence, whereas x2y is not.

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Logic

• The logic must also define the semantics of the
  language. Semantics has to do with the meaning
  of sentences.
• The semantics of the language defines the truth
  of each sentence with respect to each possible
  world (or model).
• For example, the usual semantics adopted for
  arithmetic specifies that the sentence x y
  4 is true in a world where x is 2 and y is 2,
  but false in a world where x is 1 and y is 1.
• In standard logics, every sentence must be either
  true or false in each possible world, there is no
  in-between.

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Logic

• The relation of logical entailment between
  sentences the idea that a sentence follows
  logically from another sentence.
• a ? b means that sentence a entails the sentence
  b.
• The formal definition of entailment is
• a ? b if and only if, in every model in which a
  is true, b is also true.
• a ? b if a is true, then b must also be true.
• a ? b the truth of b is contained in the truth of
  a.

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Propositional Logic

• The syntax of propositional logic defines the
  allowable sentences.
• The atomic sentences -the indivisible syntactic
  elements- consists of a single proposition
  symbol.
• Each such symbol stands for a proposition that
  can be true or false.
• Uppercase names will be used for symbols (e.g. P,
  Q and R).

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Propositional Logic

• There are two proposition symbols with fixed
  meanings
• True is the always-true proposition.
• False is the always-false proposition.
• Complex sentences are constructed from simpler
  sentences using logical connectives.

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The five connectives

• not
• The sentence such as W is called the negation of
  W.
• ? and
• A sentence whose main connective is ?, such as W
  ? P, is called conjunction its parts are the
  conjuncts.
• V or
• A sentence using V, such as W V P is a
  disjunction of the disuncts W and P.
• ? implies
• A sentence such as (W ? P) ? W is called an
  implication (or conditional). Its premise or
  antecedent is (W ? P) and its conclusion or
  consequent is W.
• ? if and only if
• The sentence W ?? P is a biconditional

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Propositional Logic

• The grammar is very strict about parentheses
  every sentence constructed with binary
  connectives must be enclosed in parentheses.
• The order of precedence in propositional logic is
  from higher to lowest , ?, V, ? and ? .
• e.g The sentence
• P V Q ? R ? S
• is equivalent to the sentence
• ((P) V (Q ? R)) ? S

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Propositional Logic

• Precedence does not resolve ambiguity in
  sentences such as A ? B ? C which could be read
  as ((A ? B) ? C) or as (A ? (B ? C)).
• Sentences such as A ? B ? C are not allowed
  because the two readings have different meanings,
  hence we insist on parentheses in this case.

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Propositional Logic

• The semantics defines the rules for determining
  the truth of a sentence with respect to a
  particular model.
• In propositional logic, a model simply fixes the
  truth values true or false- for every
  proposition symbol.
• For example, if the sentences in the knowledge
  base make use of the proposition symbols P, Q and
  S, then one possible model is
• m1 P false, Q false, S true)
• With three proposition symbols, there are 23 8
  possible models.

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Propositional Logic

• The semantics for propositional logic must
  specify how to compute the truth value of any
  sentence, given a model.
• All sentences are constructed from atomic
  sentences and the five connectives.
• We need to specify
• how to compute the truth of atomic sentences
• How to compute the truth of sentences formed with
  each of the five connectives.

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Propositional Logic

• Atomic sentences are easy
• True is true in every model and false is false in
  every model.
• The truth value of every other proposition symbol
  must be specified directly in the model. For
  example in the model m1, P is false.
• m1 P false, Q false, S true)
• For complex sentences, we have rules such as
• For any sentence s and any model m, the sentence
  s is true in m if and only if s is false in m.

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Truth Tables

• Such rules reduce the truth of a complex sentence
  to the truth of simpler sentences.
• The rules for each connective can be summarized
  in a truth table that specifies the truth value
  of a complex sentence for each possible
  assignment of truth values to its components.
• Using the truth tables in the next slide, the
  truth value of any sentence s can be computed
  with respect to any model m by simple process of
  recursive evaluation.

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The Truth Tables for the five logical connectives
XOR.
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Example

• m1 P false, Q false, S true)
• The sentence P ? (Q V S) evaluated in m1, gives
• T ? (F V T) T ? T T
• It can also be solved using truth tables.

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Example

• Apply the given connectives on the following
  sentences
• P 5 is odd
• Q Kuala Lumpur is the Capital of Malaysia

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Examples

• Construct a truth table for the proposition
• ( Q ? P ) ? P
• Construct a truth table for the proposition
• ( P ? Q) V R ? P

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Tautology and Contradiction

• A tautology is a propositional form whose truth
  values is true for all possible values of its
  propositional variables, e.g., P V P. It is also
  known as valid sentence, i.e a sentence is valid
  if it is true in all models.
• A contradiction is a propositional form which is
  always false, such as P ? P.
• A propositional form which is neither a tautology
  nor a contradiction is called a contingency.

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Propositional Logic

• Previously we said that a knowledge base consists
  of a set of sentences.
• We can now see that a logical knowledge base is a
  conjunction of those sentences.
• That is, if we start with an empty KB and do
• TELL(KB S1)
• TELL(KB Sn)
• then KB S1 ? ? Sn.
• This means that we can treat knowledge bases and
  sentences interchangeably.

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Equivalence

• Two sentences A and B are logically equivalent if
  they are true in the same set of models. A ? B
• Example Prove that A ? B and B ? A are logically
  equivalent.

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Logical Equivalences

• (A ? B) ? (B ? A) Commutativity of ?
• (A V B) ? (B V A) Commutativity of V
• (( A ? B) ? C) ? ( A ? (B ? C)) Associativity of
  ?
• (( A V B) V C) ? ( A V (B V C)) Associativity of
  V
• (P) ? P Double-Negation Elimination

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Logical Equivalences

• (A ? B) ? ( B ? A) Contraposition
• (A ? B) ? (A V B) Implication elimination
• (A ? B) ? (( A ? B) ? (B ? A)) biconditional
  elimination
• ( A ? B ) ? (A V B) de Morgan
• ( A V B ) ? (A ? B) de Morgan
• (A ? ( B V C)) ? ((A ? B ) V (A ? C ))
  Distributivity of ? over V
• (A V ( B ? C)) ? ((A V B ) ? (A V C ))
  Distributivity of V over ?

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Example

• Simplify the following propositional form
• (A ? B) V (A ? D) ? (BVD)
• Answer
• (A V B) V (A V D) ? (B V D)
• A V (B V D) ? (B V D)
• A V (B V D) V (B V D)
• A ? (B V D) V (B V D)
• (A V B V D) ? (B V D) V (B V D)
• (A V B V D) ? 1
• A V B V D

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Reasoning Patterns in Propositional Logic

• The standard patterns of inference that can be
  applied to derive chains of conclusions that lead
  to the desired goal are called inference rules.
• The best-known rule is called Modus Ponens and is
  written as follows
• A ? B, A then B
• The notation means that, if (A ? B) and A are
  true then B should be true.
• Another useful inference rule is And-Elimination,
  which says that, from a conjunction, any of the
  conjuncts can be inferred
• If (A ? B) is true then A is true
• All of the logical equivalences can be used as
  inference rules.

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Reasoning Patterns in Propositional Logic

• If A ? B then (A ? B) ? (B ? A)
• If (A ? B) ? (B ? A) then A ? B

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Summary

• Intelligent agents need knowledge about the world
  in order to reach good decisions.
• Knowledge is contained in agents in the form of
  sentences in a knowledge representation language
  that are stored in a knowledge base.
• A knowledge-based agent is composed of a
  knowledge base and an inference mechanism. It
  operates by storing sentences about the world in
  its knowledge base, using the inference mechanism
  to infer new sentences, and using these sentences
  to decide what action to take.

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Summary

• A representation language is defined by its
  syntax, which specifies the structure of
  sentences, and its semantics, which defines the
  truth of each sentence in each possible world or
  model.
• The relationship of entailment between sentences
  is crucial to our understanding of reasoning. A
  sentence A is entails another sentence B if B is
  true in all worlds where A is true. Equivalent
  definitions include the validity of the sentence
  A ? B and the unsatisfiability of the sentence A
  ? B.

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Summary

• Inference is the process of deriving new
  sentences from old ones.
• Propositional logic is a very simple language
  consisting of proposition symbols and logical
  connectives. It can handle propositions that are
  known true, known false, or completely unknown.
• The set of possible models, given a fixed
  propositional vocabulary is finite, so entailment
  can be checked by enumerating models. Efficient
  model-checking inference algorithms for
  propositional logic include backtracking and
  local-search methods and can often solve large
  problems very quickly.

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Summary

• Inference rules are patterns of inference that
  can be used to find proofs. The resolution rule
  yields a complete inference algorithm for
  knowledge bases that are expressed in conjuctive
  normal form.

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Exercises

• Russell, page 236
• 7.3 b
• 7.5
• 7.7
• 7.8
• 7.12 a,b

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Assignment (3rd Jan)

• (1) Construct a truth table for the propositions
• (Q ? P ) ? P
• (P ? Q ) V R ? P

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• (2) Simplify the following propositional form
• (A ? B) V (A ? D) ? (B V D)
• (3) Prove the following logical implications
• P ? (P V Q)
• P ? (P V Q) ? Q
• (P ? Q) ? (Q ? R) ? (P ? R)

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• (4) Let P be the proposition It is snowing
• Let Q be the proposition I will go to town
• Let R be the proposition I have time
• Using logical connectives, write a proposition
  which symbolizes each of the following
• If it is snowing and I have time, then I will go
  to town.
• I will go to town only if I have time.
• It is not snowing.
• It is snowing, and I will not go to town.
• Write the following sentences in English
• Q ? (R ? P)
• (Q ? R) ? (R ? Q)

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• (5) Establish whether the following propositions
  are tautologies, contingencies or contradictions
• P V P
• P ? P
• P ? (P V Q)
• (P ? Q) ? P
• (P ? Q) ? P ? P ? Q
• (P ? Q) V (R ? S) ? (P V R) ? (Q V S)

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• (6) For each of the following expressions, use
  identities to find equivalent expressions which
  use only ? and and are as simple as possible.
• P V Q V R
• P V (Q ? R) ? P
• P ? (Q ? P)

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• (7) Establish the following tautologies by
  simplifying the left side to the form of the
  right side
• (P ? P) ? (P ? P) ? 0

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First-Order Logic

• We used propositional logic as our representation
  language because it sufficed to illustrate the
  basic concepts of logic and knowledge-based
  agents. Unfortunately, propositional logic is too
  puny a language to represent knowledge of complex
  environments in a concise way.
• First-order logic is sufficiently expressive to
  represent a good deal of our commonsense
  knowledge.

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Programming Languages

• Programming languages such as C or Java or Lisp
  are by far the largest class of formal languages
  in common use.
• Programmers themselves represent, in a direct
  sense, only computational processes.
• Data structures within programs can represent
  facts. For example, a 1 by 2 array may represent
  4 different cases.

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• What programming languages lack is any general
  mechanism for deriving facts from other facts
  each update to a data structure is done by a
  domain-specific procedure whose details are
  derived by the programmer from his or her own
  knowledge of the domain.
• This procedural approach can be contrasted with
  the declarative nature of propositional logic, in
  which knowledge and inference are separate, and
  inference is entirely domain-independent.

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• Another drawback of data structure in programs is
  the lack of any easy way to say There is a king
  in square 4,5 or if the queen is in square 5,2
  then he is not in 7,6. Programs can store a
  single value for each variable, and some systems
  allow the value to be unknown, but they lack the
  expressiveness required to handle partial
  information.

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• Propositional logic is declarative language
  because its semantics is based on a truth
  relation between sentences and possible worlds.
  It also has sufficient expressive power to deal
  with partial information, using disjunction and
  negation.
• Propositional logic has a third property that is
  desirable in representation languages, namely
  compositionality which means that the meaning of
  a sentence is a function of the meaning of its
  parts.
• Eg.
• A it is raining
• B I have no umbrella
• A ? B It is raining and I have no umbrella
• Impossible A ? B IIUM is a university ?

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• Propositional logic lacks the expressive power to
  describe an environment with many objects
  concisely.
• Example any piece in a square adjacent to the
  king can be killed.
• The language of first-order logic is built around
  objects and relations.
• First-order logic can also express facts about
  some or all of the objects in the universe. This
  represent general laws or rules.

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Models for First-Order Logic

• Models of FOL have objects in them. The domain of
  a model is the set of objects it contains these
  objects are sometimes called domain elements.
• Atomic Sentence is formed from a predicate symbol
  (relation) followed by a parenthesized list of
  terms (objects).
• Brother(Al-Hasan, Al-Husain)
• Married(Father(Al-Hasan), Mother(Al-Husain))
• An atomic sentence is true in a given model,
  under a given interpretation, if the relation
  referred to by the predicate symbol holds among
  the objects referred to by the arguments.

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• Logical connectives are used to construct more
  complex senteces.
• Brother(Ahmed, Father(Naim))
• Brother(Fadhil, Anees) ? Brother(Ali, Khalid)
• King(Zul) ? King(Khairul)

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Quantifiers

• Used to express properties of entire collections
  of objects instead of enumerating the objects by
  name.
• Universal (For all)
• Existential
• Universal
• All kings are persons
• Å x King(x) ? Person(x)
• For all x, if x is a king, then x is a person.
• x is a variable

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• Å x P, where P is any logical expression, means
  that P is true for every object x.
• If the assertion P(x) is true for every possible
  value of x, then Å x P(x) is true otherwise Å x
  P(x) is false.
• For any predicate P and any element c of the
  universe (domain) Å x P(x) ? P(c).

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Examples

• Å x x lt x 1 true
• Å x x 3 false
• If A is an integer array with 50 entries, A1,
  A2 A50 then we can assert that all entries
  are nonzero as follows
• Å i (1 i and i 50) ? Ai ? 0
• The entries of the array are sorted in
  non-decreasing order if the following assertion
  holds.
• Å i (1 i and i 50) ? Ai Ai1
• Å x Å y x y gt x is true if U I.

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Existential

• 3 x, P(x) is read as There exists a value of x
  for which the assertion P(x) is true.
• The symbol 3 is also read as for some or for
  at least one.
• P(c) ? 3 x, P(x)

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Examples

• 3 x, x lt x 1 is true in I.
• 3 x x 4 is true in I.
• 3 x x x 1 is false.

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There is one and only one x3!

• 3! x x lt 1 is true in I0.
• 3! x x 5 is true.
• 3! x x gt 1 is false.

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Example

• Let the universe consist of the integer 1, 2, and
  3. then
• Å x P(x)
• is equivalent to the conjunction
• P(1) ? P(2) ? P(3)
• The proposition
• 3xP(x)
• is equivalent to the disjunction
• P(1) V P(2) V P(3)
• The proposition
• 3!x P(x)
• is equivalent to
• P(1) ? P(2) ? P(3) V
• P(1) ? P(2) ? P(3) V
• P(1) ? P(2) ? P(3)

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Example

• In I0, Å x P(x) means P(1) ? P(2) ? P(3) ?
• The predicate P(x,y,z) represents x y z
  all of its 3 variables are free.
• If we assign x 2 then the result is the
  predicate P with a bound variable P(2,y,z) which
  can be considered as a new predicate of 2
  variables R(y,z)
• R(y,z) ?? 3xP(x,y,z) ?? 3xxyz ?? (y z)

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• Å x Å y P(x,y) denotes Å x Å y P(x,y)
• Å x 3 y P(x,y) can be read as no matter what
  value of x is chosen, a value of y can be found
  such that
• 3 y Å x P(x,y) is read as A value of y can be
  chosen so that no matter what value is chosen for
  x
• Å x Å y can be replaced by Å y Å x
• 3 x 3 y can be replaced by 3 y 3 x

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Examples

• Å x 3 y x is married to y
• 3 y Å x x is married to y
• In I, Å x 3 y x y 0
• In I, 3 y Å x x y 0
• In I, Å x Å y 3!z x y z
• In I, Å x 3!z Å yx y z
• In I, 3!xx 6 0
• In I, 3!x Å yx y 0
• In I, Å y 3!xx y lt 0
• In I, Å y 3!xx y lt 0

• T
• F
• T
• F
• T
• F
• T
• T
• F
• F

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Exercises

• Let S(x,y,z) denote the predicate xyz
  P(x,y,z) denote x.yz
  L(x,y) denote xlty
  
  Let the universe of discourse be the natural
  numbers N. Using
  the above predicates, express the following
  assertions.
• For every x and y, there is a z such that xyz
• No x is less than 0
• For all x, x0x
• For all x,y x.yy
• There is an x for all y such that x.yy

Correct syntax but the meaning is wrong
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Example

• Consider these statements, of which the first
  three are premises and the fourth is a valid
  conclusion.
• All hummingbirds are richly colored
• No large birds live on honey
• Birds that do not live on honey are dull in
  color
• Hummingbirds are small
• P(x) denotes x is a hummingbird
• Q(x) denotes x is large
• R(x) denotes x lives on honey
• S(x) denotes x is richly colored

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Exercises

• Determine which of the following propositions are
  true if the universe is the set of integers I and
  . denotes the operation of multiplication.
• True
• False
• False
• True

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Exercises

• Specify a universe of discourse for which the
  following propositions are true. Try to choose
  the universe to be as large a subset of the
  integers as possible.
• All integers greater than 10
• The universe contains only 3
• x belongs to I-, y 436-x
• y belong to I-, xlt-y

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Exercises

• Consider the universe of integers and let
  P(x,y,z) denote x-yz. Transcribe the following
  assertions into logical notation.
• For every x and y, there is some z such that
  x-yz.
• For every x and y, there is some z such that
  x-zy.
• There is an x such that for all y, y-xy.
• When 0 is subtracted from any integer, the result
  is the original integer.
• 3 subtracted from 5 gives 2.

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Quantifiers and Logical Operators

• The the universe be the integers and let
  
  N(x) denote x is a nonnegative integer,
  E(x) denote
  x is even
  O(x) denote x
  is odd and
  P(x) denote x is
  prime
• There exists an even integer
• Every integer is even or odd
• All prime integers are nonnegative
• The only even prime is two
• There is one and only one ven prime
• Not all integers are odd
• Not all primes are odd
• If an integer is not odd, then it is even

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Examples

• Consider the universe of integers and let
  P(x,y,z) denote xyz
• If x 0 then xy0 for all values of y
• If xy 0 for every y, then x 0

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Logical Relationships Involving Quantifiers
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Example
64
Example

• Let P(x,y,z) denote xy z E(x,y) denote x y
  and G(x,y) denote x gt y.
• Let the universe of discourse be the integers.
  Transcribe the following into logical notation
• If y 1, then xy x for any x.
• 3x 6 if and only if x 2
• xlty and y ltx is a sufficient condition for yx
• If x lty and zlt0, then xzgtyz

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Example

• Let the universe of discourse be the set of
  arithmetic assertions with predicates defined as
  follows
• P(x) denotes x is provable
• T(x) denotes x is true
• S(x) denotes x is satisfiable
• D(x,y,z) denotes z is the discjunction x V y
• Translate the following assertions into English
  statements. Make your transcription as natural as
  possible
• If y is the assertion w V x, z is the assertion x
  V w and y is provable, then z is provable
• Every arithmetic assertion which is provable is
  true
• If z x V y and z is provable, then x is
  provable or y is provable

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Example

• Put the following into logical notation. Choose
  predicates so that each assertion requires at
  least one quantifier.
• There is one and only one even prime.
• Every train is faster than some cars
• If it rains tomorrow, then somebody will get wet

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Example

• Find an assertion which is logically equivalent
  to but uses only the quantifier
  and the logical operator . Similarly, express
  in terms of and .

68
Example

• Show that the following is valid for the universe
  of natural numbers N either by expanding the
  statement or by applying identities.

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Example

• Show that the following is valid for the universe
  of natural numbers N either by expanding the
  statement or by applying identities.