159'302 LECTURE

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159.302 LECTURE
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Propositional Logic Syntax
Source MIT OpenCourseWare
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Propositional Logic
What is a logic?
A formal language
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Propositional Logic
What is a logic?
A formal language

• Syntax what expressions are legal

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Propositional Logic
What is a logic?
A formal language

• Syntax what expressions are legal

• Semantics what legal expressions mean

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Propositional Logic
What is a logic?
A formal language

• Syntax what expressions are legal

• Semantics what legal expressions mean

• Proof System a way of manipulating syntactic
  expressions to get other syntactic expressions
  (which will tell us something new)

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Propositional Logic
What is a logic?
A formal language

• Syntax what expressions are legal

• Semantics what legal expressions mean

• Proof System a way of manipulating syntactic
  expressions to get other syntactic expressions
  (which will tell us something new)

Why proofs?

• Two kinds of inferences an agent might want to
  make

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Propositional Logic
What is a logic?
A formal language

• Syntax what expressions are legal

• Semantics what legal expressions mean

• Proof System a way of manipulating syntactic
  expressions to get other sytactic expressions
  (which will tell us something new)

Why proofs?

• Two kinds of inferences an agent might want to
  make

• Multiple percepts gt conclusions about the world

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Propositional Logic
What is a logic?
A formal language

• Syntax what expressions are legal

• Semantics what legal expressions mean

• Proof System a way of manipulating syntactic
  expressions to get other sytactic expressions
  (which will tell us something new)

Why proofs?

• Two kinds of inferences an agent might want to
  make

• Multiple percepts gt conclusions about the world

• Current state operator gt properties of next
  state

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Propositional Logic
Syntax what youre allowed to write

• for(thing tfizz t fuzz t)

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Propositional Logic
Syntax what youre allowed to write

• for(thing tfizz t fuzz t)

• Colorless green ideas sleep furiously.

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Propositional Logic
Syntax what youre allowed to write

• for(thing tfizz t fuzz t)

• Colorless green ideas sleep furiously.

Sentences (wwfs well-formed formulas)
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Propositional Logic
Syntax what youre allowed to write

• for(thing tfizz t fuzz t)

• Colorless green ideas sleep furiously.

Sentences (wwfs well-formed formulas)

• true and false are sentences

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Propositional Logic
Syntax what youre allowed to write

• for(thing tfizz t fuzz t)

• Colorless green ideas sleep furiously.

Sentences (wwfs well-formed formulas)

• true and false are sentences

• Propositional variables are sentences P, Q, R, Z

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Propositional Logic
Syntax what youre allowed to write

• for(thing tfizz t fuzz t)

• Colorless green ideas sleep furiously.

Sentences (wwfs well-formed formulas)

• true and false are sentences

• Propositional variables are sentences P, Q, R, Z

• If f and ? are sentences, then so are

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Propositional Logic
Syntax what youre allowed to write

• for(thing tfizz t fuzz t)

• Colorless green ideas sleep furiously.

Sentences (wwfs well-formed formulas)

• true and false are sentences

• Propositional variables are sentences P, Q, R, Z

• If f and ? are sentences, then so are

• Nothing else is a sentence

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Precedence
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159.302 LECTURE
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Propositional Logic Semantics
Source MIT OpenCourseWare
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Propositional Logic
Semantics

• Meaning of a sentence is truth value t, f

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Propositional Logic
Semantics

• Meaning of a sentence is truth value t, f

• Interpretation is an assignment of truth values
  to the propositional variables

holds( , i) Sentence is t in
interpretation i
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Propositional Logic
Semantics

• Meaning of a sentence is truth value t, f

• Interpretation is an assignment of truth values
  to the propositional variables

holds( , i) Sentece is t in
interpretation i
fails( , i) Sentence is f in
interpretation i
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Propositional Logic
Semantic Rules
holds(true, i) for all i
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Propositional Logic
Semantic Rules
holds(true, i) for all i
fails(false, i) for all i
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Propositional Logic
Semantic Rules
holds(true, i) for all i
fails(false, i) for all i
holds( , i) if and only if fails(
, i) (negation)
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Propositional Logic
Semantic Rules
holds(true, i) for all i
fails(false, i) for all i
holds( , i) if and only if fails(
, i) (negation)
holds( , i) iff holds( ,i) and
holds( , i) (conjunction)
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Propositional Logic
Semantic Rules
holds(true, i) for all i
fails(false, i) for all i
holds( , i) if and only if fails(
, i) (negation)
holds( , i) iff holds( ,i) and
holds( , i) (conjunction)
holds( , i) iff holds( ,i) or holds(
, i) (disjunction)
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Propositional Logic
Semantic Rules
holds(true, i) for all i
fails(false, i) for all i
holds( , i) if and only if fails(
, i) (negation)
holds( , i) iff holds( ,i) and
holds( , i) (conjunction)
holds( , i) iff holds( ,i) or holds(
, i) (disjunction)
holds( P, i) iff i(P) t
fails( P, i) iff i(P) f
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Propositional Logic
Some important shorthand

(antecedent consequent)
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Propositional Logic
Some important shorthand

(antecedent consequent)


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Propositional Logic
Some important shorthand

(antecedent consequent)


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Propositional Logic
Some important shorthand

(antecedent consequent)


Note that implication is not causality, if P is
f then is t
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Propositional Logic
Terminology
A sentence is valid iff its truth value is t in
all interpretations
A sentence is satisfiable iff its truth value is
t in at least one interpretation
A sentence is unsatisfiable iff its truth value
is f in all interpretations
All are finitely decidable.
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Examples
contrapositive
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Examples
contrapositive
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Examples
contrapositive
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Examples
contrapositive
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Examples
contrapositive
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Satisfiability

• Related to constraint satisfaction

• Given a sentence S, try to find an
  interpretation i such that holds(S,i)

• Analogous to finding an assignment of values to
  variables such that the constraints hold

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Satisfiability

• Related to constraint satisfaction

• Given a sentence S, try to find an
  interpretation i such that holds(S,i)

• Analogous to finding an assignment of values to
  variables such that the constraints hold

• Brute force method enumerate all
  interpretations and check

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Satisfiability

• Related to constraint satisfaction

• Given a sentence S, try to find an
  interpretation i such that holds(S,i)

• Analogous to finding an assignment of values to
  variables such that the constraints hold

• Brute force method enumerate all
  interpretations and check

• Better methods

Heuristic search Constraint propagation Stochastic
search
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Satisfiability Problems

• Scheduling nurses to work in a hospital

• propositional variables represent for example,
  that Pat is working on Tuesday at 2pm

• constraints on the schedule are represented
  using logical expressions over the variables

• Finding bugs in software

• propositional variables represent state of the
  program

• use logic to describe how the program works and
  to assert there is a bug

• if the sentence is satisfiable, youve found the
  bug!

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159.302 LECTURE
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Propositional Logic Proof
Source MIT OpenCourseWare
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A Good lecture?

• Imagine we knew that

• If today is sunny, then Tomas will be happy

• If Tomas is happy, the lecture will be good

• Today is sunny

Should we conclude that the lecture will be good?
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Checking Interpretations
Good lecture!
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Adding a variable
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Entailment

• A knowledge base (KB) entails a sentence S iff
  every interpretation that makes KB true also
  makes S true

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Computing Entailment

• enumerate all interpretations

• select those in which all elements of KB are true

• check to see if S is true in all of those
  interpretations

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Computing Entailment

• enumerate all interpretations

• select those in which all elements of KB are true

• check to see if S is true in all of those
  interpretations

• Way too many interpretations, in general!!

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Entailment and Proof

• A proof is a way to test whether a KB entails a
  sentence, without enumerating all possible
  interpretations

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Proof

• A proof is a sequence of sentences

• First ones are premises (KB)

• Then, you can write down on the next line the
  result of applying an inference rule to previous
  lines

• When S is on a line, you have proved S from KB

• If inference rules are sound, then any S you can
  prove from KB is entailed by KB

• If inference rules are complete, then any S that
  is entailed KB can be proven from KB

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Natural Deduction

• Some inference rules

Modus ponens
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Natural Deduction

• Some inference rules

Modus ponens
Modus tolens
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Natural Deduction

• Some inference rules

Modus ponens
Modus tolens
And-Introduction
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Natural Deduction

• Some inference rules

Modus ponens
Modus tolens
And-Introduction
And-Elimination
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Natural Deduction Example
Prove S
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Natural Deduction Example
Prove S
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Natural Deduction Example
Prove S
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Natural Deduction Example
Prove S
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Natural Deduction Example
Prove S
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Natural Deduction Example
Prove S
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Proof Systems

• There are many natural deduction systems they
  are typically proof checkers, sound but not
  complete

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Proof Systems

• There are many natural deduction systems they
  are typically proof checkers, sound but not
  complete

• Natural deduction uses lots of inference rules
  which introduces a large branching factor in the
  search for a proof.

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Proof Systems

• There are many natural deduction systems they
  are typically proof checkers, sound but not
  complete

• Natural deduction uses lots of inference rules
  which introduces a large branching factor in the
  search for a proof.

• In general, you need to do proof by cases
  which introduces even more branching.

Prove R
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Propositional Resolution
Resolution rule

• Single inference rule is a sound and complete
  proof system

• Requires all sentences to be converted to
  conjunctive normal form

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159.302 LECTURE
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Propositional Logic Conjunctive Normal Form (CNF)
Source MIT OpenCourseWare
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Conjunctive Normal Form
CNF Formulas
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Conjunctive Normal Form
CNF Formulas
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Conjunctive Normal Form
CNF Formulas
, which is a disjunction of literals
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Conjunctive Normal Form
CNF Formulas
, which is a disjunction of literals
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Conjunctive Normal Form
CNF Formulas
, which is a disjunction of literals
, each of which is a variable or the negation of
a variable.
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Conjunctive Normal Form
CNF Formulas
, which is a disjunction of literals
, each of which is a variable or the negation of
a variable.

• Each clause is a requirement that must be
  satisfied and can be satisfied in multiple ways

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Conjunctive Normal Form
CNF Formulas
, which is a disjunction of literals
, each of which is a variable or the negation of
a variable.

• Each clause is a requirement that must be
  satisfied and can be satisfied in multiple ways

• Every sentence in propositional logic can be
  written in CNF

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Converting to CNF

• Eliminate arrows using definitions

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Converting to CNF

• Eliminate arrows using definitions

• Drive in negations using De Morgans Laws

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Converting to CNF

• Eliminate arrows using definitions

• Drive in negations using De Morgans Laws

• Distribute or over and

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Converting to CNF

• Eliminate arrows using definitions

• Drive in negations using De Morgans Laws

• Distribute or over and

• Every sentence can be converted to CNF, but it
  may grow exponentially in size

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CNF Conversion Example
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CNF Conversion Example

• Eliminate arrows using definitions

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CNF Conversion Example

• Eliminate arrows using definitions

• Drive in negations using De Morgans Laws

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CNF Conversion Example

• Eliminate arrows using definitions

• Drive in negations using De Morgans Laws

• Distribute or over and

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159.302 LECTURE
5
Propositional Resolution
Source MIT OpenCourseWare
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Propositional Resolution
Resolution rule
Resolution refutation

• Convert all sentences to CNF

• Negate the desired conclusion (converted to CNF)

• Apply resolution rule until either

• Derive false (a contradiction)

• Cant apply anymore

Resolution refutation is sound and complete

• If we derive a contradiction, then the
  conclusion follows from the axioms

• If we cant apply anymore, then the conclusion
  cannot be proven from the axioms

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Propositional Resolution
Prove R
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Propositional Resolution
Prove R
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Propositional Resolution
Prove R
unnecessary
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The Power of False
Prove Z

• Any conclusion follows from a contradiction
  and so strict logic systems are very brittle.

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Example Problem
Convert to CNF
Prove R
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Example Problem
Prove R
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Example Problem
Prove R
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Proof Strategies
Unit Preference prefer a resolution step
involving a unit clause (clause with one literal).

• Produces a shorter clause which is good since
  we are trying to produce a zero-length clause,
  that is, a contradiction.

Set of support choose a resolution involving the
negated goal or any clause derived from the
negated goal.

• Were trying to produce a contradiction that
  follows from the negated goal, so these are
  relevant clauses

• If a contradiction exists, one can find one
  using the set-of-support strategy.