Artificial Intelligence
Logic and Reasoning
Dr. Edjard Mota
Department of Computer Science
edjard.de-souza-mota@city.ac.uk
Class 1
Copyright 2016 Edjard Mota
Agenda
• Why Logic?
• Logic-based Languages
• Propositional Logic: Syntax and Semantics
• Models and Validity
• Rules of Inference
Class 1
Copyright 2016 Edjard Mota
Why Logic?
• How can we represent ordinary word-language
meaning so that a computer can process these
representations?
• How can we associate meaning representations
with an unlimited set of sentences?
• How can we use programs that connect the
meaning representations of sentences to stores of
knowledge?
Class 1
Copyright 2016 Edjard Mota
Why Logic?
Formal
Ambiguous Expressive
Inference
Natural languages
✖️
✔️✔️✔️
✔️✔️✔️
✔️✔️✔️
Programming
Languages
✔️✔️
✖️
✖️
➕/➖
Logic-based
Language
✔️✔️✔️
✖️
✔️
✔️
Formal language for expressing knowledge and
ways of reasoning, i.e. calculus for reasoning.
Class 1
Copyright 2016 Edjard Mota
Logic-based Languages
Definition 1. Logic is defined by:
• A set of sentences:
- a sentence is built from a set of primitives according to
syntax rules, i.e. the grammar.
• A set of interpretations:
- an interpretation gives a meaning to sentences.
• The valuation (meaning) function (V) is a mapping
• Inference rules to derive new sentences from existing ones
Class 1
Copyright 2016 Edjard Mota
Logic-based Languages
A, B and C below are symbols used to denote
propositions that are either true or false
Examples of atomic propositions/sentences
a : Alex goes to the party
b : Beki goes to the party
c : Chris goes to the party
Examples of compound propositions
If
then
If Alex goes to the party, then Beki also goes
a
b
a
Logical connectives
b
Other logical connectives: not, or, and if and only if
Class 1
Copyright 2016 Edjard Mota
Logic-based Languages
From
and other compound
If Chris goes to the party, then Alex also goes
It should be possible to conclude:
But, Beki does not go: ¬ b
It should be possible to conclude: ¬ a and ¬ c
What if Chris goes is true? Inconsistency found
c and ¬ c
Class 1
Copyright 2016 Edjard Mota
Propositional Logic - Syntax
Notation: elements of the logic written with this font
Definition 2. Syntax of the propositional language
• Alphabet: set of atoms {p, q, r, s, …} for propositions, T
(True), F(False), logical connectives ¬, ∧, ∨, →, ↔︎, variables
of formulas are in capital letter.
• Well-formed formula (wff) (sentence) in propositional logic
1. Any atom is a formula
2. T and F are formulas
3. If A is a formula, then ¬ A is a formula
4. If A and B are formula, then (A ∧ B), (A ∨ B), (A → B)
and (A ↔︎ B) are formulas
Class 1
Copyright 2016 Edjard Mota
Propositional Logic - Syntax
According to the definition, which of the following are wff?
1.
2.
3.
4.
5.
6.
Class 1
∨pq
(¬ (p → (q ∧ p)))
(¬ (p → (q = p))
(¬ (◇(q ∨ p))
(p ∧ ¬q) ∨ (q → r)
r ¬q
Copyright 2016 Edjard Mota
Propositional Logic Semantics
Truth-functionality of logical connectives.
Definition 3. The meaning of a compound sentence
is a function of the truth-value of its sub-sentences.
1. Negation: ¬A is false when A is true, and ¬A is true when A is
false
2. Conjunction A and B: (A ∧ B) is true only when both are true,
otherwise it is false.
3. Disjunction A and B: (A ∨ B) is false only when both are false,
otherwise it is true (inclusive disjunction).
4. Logical implication of A and B: (A → B) is false if A is true and B
is false, other otherwise it is true.
5. “If and only if”: A ↔︎ B, is true when A and B have the same
truth-value
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Semantics
Truth-tables for schema of sentences
Class 1
A
B
¬A
A∧B
A∨B
A→B
A ↔︎ B
T
T
F
T
T
T
T
T
F
F
F
T
F
T
F
T
T
F
T
T
T
F
F
T
F
F
T
T
Copyright 2016 Edjard Mota
Propositional Logic Semantics
Definition 4. An Interpretation is assignment of truthvalues (T or F) to the atoms of a sentence.
Example: consider the sentence A: (p → q) ∨ ¬ s
- The assignment I1 = {T, F,F} to {p, q, s} is an
interpretation that makes the sentence A true.
- We say that sentence A is satisfiable under I1 or that
A is satisfied in I1
Definition 5. An interpretation that satisfies a sentence is
called a Model for such a sentence
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Semantics
Build the truth-table for B: ((p → q) ∧ p) → q
p
q
p→q
(p → q)∧ p
((p → q)∧ p) → q
T
T
F
T
T
T
F
T
F
T
F
T
F
F
T
F
F
T
F
T
Sentence B is true under all interpretations! 🤔
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Semantics
Build the truth-table for C: ((p → q) ∧ (p ∧ ¬q)
p
q
¬q
p→q
(p ∧ ¬q)
((p → q)∧ (p ∧ ¬q)
T
T
F
T
F
F
T
F
T
F
T
F
F
T
F
T
F
F
F
F
T
T
F
F
Sentence C is false under all interpretations! 🤔
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Semantics
Complete the truth-table for A : (P → Q) ∨ ¬ S
p→q
¬s
(p → q) ∨¬s
F
F
T
T
F
T
F
F
F
T
T
F
T
T
T
p
q
s
F
F
F
F
F
T
F
T
F
F
T
T
T
F
T
Sentence A can be either true or false … 🤔
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Inference
When a given argument is valid?
Definition 6. An argument of the form “A1, A2, …, An Thus
B” is said to be valid if for any interpretation in which A1, A2,
…, An are all satisfied, B is also satisfied. We write
• We say that A1, A2, …, An entails B
• A1, A2, …, An is usually called a Knowledge Base (KB)
• ⊨ is called a logical consequence relation
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Inference
Examples of valid arguments
A⊨A
A∧B⊨A
A, A → B ⊨ B
A → B, ¬ B ⊨ ¬ A
Example of an invalid argument (⊭)
A⊨A∧B
Validity of an argument can be verified via truth-table!
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Inference
Example of validity check via truth-table
a → b, ¬ b ⊨ ¬ a ??
a
b
a→b
¬b
((a → b)∧ ¬ b)
¬a
F
F
T
T
T
T
F
T
T
F
F
T
T
F
F
F
F
F
T
T
T
F
F
F
Checking validity is computationally very expensive: a
sentence with n atoms will a truth-table with 2n lines! 💥
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Inference
Definition 7. Validity and Consistency of a formula.
• A sentence A is valid or a tautology (written ⊨ A) if
and only if it is true under all possible interpretations.
e.g. A ∨ ¬A
• A sentence A is inconsistent (or unsatisfiable) if and
only if it is false under all possible interpretations.
e.g. A ∧ ¬A
• A sentence is consistent (satisfiable) if and only if it
is not inconsistent.
Class 1
Copyright 2016 Edjard Mota
Propositional Logic Inference
Interesting observations … 😎
1. A formula is valid iff its negation is inconsistent.
2. A formula is inconsistent iff its negation is valid.
3. A formula is invalid iff there is at least one
interpretation under which the formula is false.
4. A formula is consistent (satisfiable) iff there is at least
one interpretation under which the formula is true.
5. If a formula is valid, then it is consistent, but not vice
versa.
6. If a formula is inconsistent, then it is invalid, but not
vice versa.
Class 1
Copyright 2016 Edjard Mota
Propositional Logic: what for?
Example: Modeling digital circuit of modern computers
• Does it meet its
specifications?
• Does it function
correctly for a given
instance of signals?
(p ∧ ¬ q) ∨ (¬ p ∧ q) ⟷ o
(r ∧ o) ⟷ a
(p ∧ q) ⟷ b
(o ∧ ¬ r) ∨ (¬ o ∧ r) ⟷ s
(a ∨ b) ⟷ c
Class 1
• 28 = 256 lines in the truth-table
• Can we reduce the number of
gates or components and keep
the same functionality?
Copyright 2016 Edjard Mota
Logical Equivalence as Reasoning
Definition 8. Two formulas A and B are logically
equivalent (A ≡ B) if and only if they have the same models.
If A and B are logically equivalent, then A → B and B → A
(E1)
a → b ≡ ¬b → ¬a
(E2)
a → b ≡ ¬a ∨ b
(E3) (a ∧ b) ∧ c ≡ a ∧ (b ∧ c)
(E4) a ∧ (b ∨ c) ≡ (a ∧ b) ∨ (b ∧ c)
(E5)
¬(a ∨ b) ≡ ¬a ∧ ¬b
(E6)
¬(a ∧ b) ≡ ¬a ∨ ¬b
(E7)
a∨b ≡ b∨a
(E8)
a∧b ≡ b∧a
(E9)
¬a ∧ a ≡ F
(E10)
¬a ∨ a ≡ T
Class 1
Contrapositive
Conditional
Associativity
Distributivity
DeMorgan Laws
Commutativity
Contradiction
Tautology
Copyright 2016 Edjard Mota
Logical Equivalence as Reasoning
“Suppose the stocks go down if the prime interest rate goes up.
Suppose also that most people are unhappy when stock prices go
down. Assume that the prime interest rate goes up. Show that you
can conclude that most people are unhappy”.
(1) If the prime interest goes up, stock prices go down.
(2) If stock prices go down, most people are unhappy.
(3) The prime interest goes up.
(4) Most people are unhappy.
[given]
[given]
[given]
goal
Considering the following atoms
(1) p → s.
p - Prime interest rate goes up.
(2) s → u.
(3) p.
s - Stock prices go down.
(4) u??
u - Most people are unhappy.
We have to prove that (4) is true whenever (1)∧(2)∧(3) is
true.
Class 1
Copyright 2016 Edjard Mota
≡ (p ∧ (¬p ∨ s) ∧ (¬s ∨ u))
(E8)
≡ ((p ∧ ¬p)as
∨(p ∧Reasoning
s)) ∧ (¬s ∨ u)) (E4)
Logical Equivalence
≡ (F ∨(p ∧ s) ∧ (¬s ∨ u))
(E9)
≡ ((p ∧ s) ∧ (¬s ∨ u))
≡ ((p ∧ s ∧ ¬s) ∨ (p ∧ s ∧ u))
(E4)
≡ ((p ∧ F) ∨ (p ∧ s ∧ u))
(E9)
≡ (F ∨ (p ∧ s ∧ u))
≡ (p ∧ s ∧ u)
Thus, if ((p → s) ∧ (s → u)) ∧ p) is true, then (p ∧ s ∧ u) is true.
But (p ∧ s ∧ u) is true only if p, s and u are all true. Therefore, u is
true. We say u is a logical consequence of (p → s), (s → u) and p
Class 1
Copyright 2016 Edjard Mota
Logical Consequence
Theorem of Entailment. Given formulas 𝓕={F1, F2, …, Fn}
and a formula G. We say that G is a logical consequence of
𝓕(𝓕 ⊨ G) iff the formula (F1 ∧ F2 ∧ … ∧ Fn → G) is valid (i.e.
a tautology), we say that 𝓕 entails G (𝓕 ⊢ G)
Definition 7 says ¬ (F1 ∧ F2 ∧ … ∧ Fn → G) is inconsistent
¬(F1 ∧ F2 ∧ … ∧ Fn → G) ≡ ¬(¬(F1 ∧ F2 ∧ … ∧ Fn) ∨ G)
(E2)
≡ ¬(¬(F1 ∧ F2 ∧ … ∧ Fn) ∧ ¬G) (E5)
≡ (F1 ∧ F2 ∧ … ∧ Fn) ∧ ¬G
≡ F1 ∧ F2 ∧ … ∧ Fn ∧ ¬G
(E3)
In other words 𝓕 ∪ {¬G} is unsatisfiable! 𝓕 ∪ {¬G} ⊢ F
Class 1
Copyright 2016 Edjard Mota
Example of Reasoning
(1) If the prime interest goes up (p), stock prices go down (s).
(2) If stock prices go down, most people are unhappy (u).
(3) The prime interest goes up.
(4) Most people are unhappy.
[given]
[given]
[given]
goal
Is ((p → s) ∧ (s → u) ∧ p) → u a valid argument?
(p → s) ∧ (s → u) ∧ p ∧ ¬u ≡ (p → s) ∧ p ∧ ((¬s ∨ u) ∧ ¬u)
Class 1
≡
≡
≡
≡
≡
≡
≡
≡
(p → s) ∧ p ∧ ((¬s ∧ ¬u) ∨ (u ∧ ¬u)
(p → s) ∧ p ∧ ((¬s ∧ ¬u) ∨ F
((¬p ∨ s) ∧ p) ∧ (¬s ∧ ¬u)
((¬p ∧ p)∨ (s ∧ p)) ∧ (¬s ∧ ¬u)
(F ∨ (s ∧ p)) ∧ (¬s ∧ ¬u)
(s ∧ p ∧ ¬s ∧ ¬u)
(s ∧ ¬s ∧ p ∧ ¬u)
F
Copyright 2016 Edjard Mota
Propositional Logical Reasoning
Definition 9. Modus Ponens: given an implication rule and
fact we can infer the conclusion.
Forward Chaining Reasoning. (computationally linear)
1.given a query or goal G.
2.until the goal is found: fires any rules whose premises are
satisfied, adding their conclusion to the set of sentences
Example of deducing u from {(p → s), (s → u), p}
True
fact
p
s
Class 1
May draw conclusions not
Rule fired Conclusion Goal? “relevant” to the goal, e.g.
if p then prices goes up
no
p→s
s
if p then banks profit is higher
…
yes
s→u
u
Copyright 2016 Edjard Mota
Propositional Logical Reasoning
Definition 10. Resolution: from a disjunction, if one of the
disjuncts is false, then you can infer the rest is true.
It “looks like” Modus Ponens. Based on logical the equivalences
then we may say
part of the KB
Goal
Theorem of entailment, Resolution and thinking backwards allows
a more efficient reasoning technique: Backward Chaining.
Class 1
Copyright 2016 Edjard Mota
Propositional Logical Reasoning
Definition 10. Propositional Horn Clause: restricted form of
clause which is either
1. a fact (called unit clause)
A. (also written A ←.)
2. an implication with only one consequent (called the head), i.e.
A ← B1 ∧ B2 ∧… ∧ Bn. (the premisses is called the body)
3. a propositional goal is a clause with no consequent, i.e.
← B1 ∧ B2 ∧… ∧ Bn. (or equivalent to ¬B1 ∨ ¬B2 ∨… ∨ ¬Bn.)
are called sub-goals.
A definite program clause or definite logic program (DLP) is
a set of Horn Clauses only of types (1) and (2).
Remember: for a set 𝓕 of formulas and goal Backward Chaining
Reasoning ∪ {¬G} ⊢ F, means G is a logical consequence of 𝓕.
Class 1
Copyright 2016 Edjard Mota
Propositional Logical Reasoning
Backward Chaining Reasoning (BCR) For a given DLP 𝓕 and a goal G
• If 𝓕 has a clause Z = A ← B1 ∧ B2 ∧… ∧ Bn , s.t A matches G, then apply
resolution to Z and A,
if n=0, then Z is a fact and so G is entailed from 𝓕
else for i = 1 to n, apply BCR on 𝓕 and Bi as a goal
if every Bi a has been entailed from 𝓕,then so is G
• If for G or some sub-goal B during BCR no clause was found with head
matching it, then G is not entailed from 𝓕
𝓕 B11 ← C11 ∧… ∧ Ckk
B2 ← …
…
A ← B11 ∧ B2 ∧… ∧ Bn
A ← B’1 ∧ B’2 ∧… ∧ B’n
…
Class 1
G
A
1st resolution step
BB11 BB22 … BBnn
…
…
…
Copyright 2016 Edjard Mota
Propositional Logical Reasoning
Example of deducing u from 𝓕 = {(s ← p), (u ← s), p}
New goal
Matching rule
Sub-goal
u
u
u
s
u←s
s←p
p←
s
p
s
s
p
empty
PROLOG: Logic Programming Language
Inverse implication is written as “:-”
Logic program
Query to the system
Class 1
p
p
p.
s :- p.
u :- s.
?- u.
true.
Copyright 2016 Edjard Mota
Summary
Main concepts of (Propositional) Logic
Syntax: specifies the formal structure of sentences
Semantics: defines the truth of sentences relate to facts in the world
specifying the.
Models are interpretations that make the value of a sentence to be
true.
Entailment: necessary truth of one sentence given another
Inference: calculus for deriving sentences from other sentences
Validity of a sentence can be checked via truth-tables or inference
rules based on logical equivalence.
Inference mechanisms:
Forward chaining: validates a goal by deriving it from facts and
rules via Modus Ponens
Backward chaining: validates a goal by applying resolution only
on rules whose consequent matches the goal, and turn
the premisses of such rule in sub-goals to be validated
Class 1
Copyright 2016 Edjard Mota