Propositional Logic Intro,
Syntax
Jim Little
UBC CS 322 – CSP
October 17, 2014
Textbook § 5.1- 5.1.1 – 5.2
Slide 1
Lecture Overview
• Recap Planning
• Logic Intro
• Propositional Definite Clause Logic:
Syntax
Slide 2
Recap Planning
•
Represent possible actions with ….. STRIPS
•
Plan can be found by….. search
•
Or can be found by mapping planning problem
into… CSP
Slide 3
Solve planning as CSP: pseudo code
solved = false
for horizon h=0,1,2,…
map STRIPS into a CSP csp with horizon h
solve that csp
if solution exists then
return solution
else
horizon = horizon + 1
end
Slide 4
Planning as CSP
Slide 5
STRIPS to CSP applet
Allows you:
• to specify a planning problem in STRIPS
• to map it into a CSP for a given horizon
• the CSP translation is automatically loaded
into the CSP applet where it can be solved
Practice exercise using STRIPS to CSP is
available on AIspace
Slide 6
•
•
•
•
•
•
Now, do you know how to implement a
planner for….
•
Emergency Evacuation?
Robotics?
Space Exploration?
Manufacturing Analysis?
Games (e.g., Bridge)?
Generating Natural language
Product Recommendations ….
North Q 9 A A
J 7 K 9
6
5
5
3
West
6
Q
South
East
2
8
Slide 7
No , but you
(will) know the
key ideas !
• Ghallab, Nau, and Traverso
Automated Planning:
Theory and Practice
•
Morgan Kaufmann, May
2004
ISBN 1-55860-856-7
Web site:
 http://www.laas.fr/planning
Slide 8
Lecture Overview
• Recap Planning
• Logic Intro
• Propositional Definite Clause Logic:
Syntax
Slide 9
Lecture Overview
• Recap of Lecture 6
• Planning as CSP
• Logic
• Intro
• Propositional Definite Clause Logic (PDCL)
• Logical Consequences and Proof Procedures
• Bottom Up Proof Procedure
Slide 10
Where Are We?
Environment
Stochastic
Deterministic
Problem Type
Arc
Consistency
Constraint Vars +
Satisfaction ConstraintsSearch
Static
Belief Nets
Logics
Query
Variable
Search
Elimination
Sequential
Planning
First Part of
the Course
STRIPS
Search
Slide 11
Representation
Decision Nets
Variable
Elimination
Markov Processes
Value
Iteration
Reasoning
Technique
Where Are We?
Environment
Stochastic
Deterministic
Problem Type
Arc
Consistency
Constraint Vars +
Satisfaction ConstraintsSearch
Static
Belief Nets
Logics
Query
Variable
Search
Elimination
Sequential
Planning
Back to static
problems, but with
richer
representation
STRIPS
Search
Slide 12
Representation
Decision Nets
Variable
Elimination
Markov Processes
Value
Iteration
Reasoning
Technique
Logics in AI: Similar slide to the one for planning
Propositional Definite
Clause Logics
Propositional
Logics
Description
Logics
Ontologies
Semantic Web
Information
Extraction
Semantics and Proof
Theory
First-Order
Logics
Satisfiability Testing
(SAT)
Production Systems
Cognitive Architectures
Video Games
Summarization
Slide 13
Hardware Verification
Software Verification
Product Configuration
Applications
Tutoring Systems
Logics in AI: Similar slide to the one for planning
Propositional Definite
Clause Logics
Propositional
Logics
Description
Logics
Ontologies
Semantic Web
Information
Extraction
Semantics and Proof
Theory
First-Order
Logics
Satisfiability Testing
(SAT)
Production Systems
Cognitive Architectures
Hardware Verification
Software Verification
Product Configuration
You will know
You will know a little
Video Games
Summarization
Slide 14
Applications
Tutoring Systems
What you already know about logic...
• From programming: Some logical operators
• If ((amount > 0) && (amount < 1000)) || !(age < 30)
• ...
You know what they mean in a “procedural” way
Logic is the language of Mathematics. To define formal structures
(e.g., sets, graphs) and to prove statements about those
"(x)triangle(x) ¾¾
®[A = B = C¬¾
®a = b = g ]
We use logic as a Representation and Reasoning System
that can be used to formalize a domain and to reason about it
Slide 15
Logic: a framework for representation & reasoning
• When we represent a domain about which we have only
partial (but certain) information, we need to represent….
• Objects, properties, sets, groups, actions, events, time, space, …
• All these can be represented as
• Objects
• Relationships between objects
• Logic is the language to express knowledge about the
world this way
• http://en.wikipedia.org/wiki/John_McCarthy (1927 - 2011)
Logic and AI “The Advice Taker”
Coined “Artificial Intelligence”. Dartmouth W’shop (1956)
Slide 16
Why Logics?
“Natural” to express knowledge about the world
(more natural than a “flat” set of variables &
constraints)
e.g. “Every 101 student will pass the course”
Course (c1)
Name-of (c1, 101)
"(z)student(z) & registered(z, c1) ¾¾
® will _ pass(z, c1)
• It is easy to incrementally add knowledge
• It is easy to check and debug knowledge
• Provides language for asking complex queries
17
• Well understood formalSlideproperties
Logic: A general framework for
reasoning
General problem: Query answering
• tell the computer how the world works
• tell the computer some facts about the world
• ask a yes/no question about whether other facts must be true
Solving it with Logic
1.
2.
3.
4.
5.
Begin with a task domain.
Distinguish those things you want to talk about (the ontology)
Choose symbols in the computer to denote elements of your ontology
Tell the system knowledge about the domain
Ask the system whether new statements about the domain are true or
false
live_w4?
lit_l2?
Slide 18
Example: Electrical Circuit
/down
/ up
Slide 19
/down
/ up
Syntax: are these sentences that a
reasoning procedure can process?
Semantics: what do these statements say
about the world I need to represent?
Slide 20
To Define a Logic We Need
• Syntax: specifies the symbols used, and how they
can be combined to form legal sentences
• Knowledge base is a set of sentences in the language
• Semantics: specifies the meaning of symbols and
sentences
• Reasoning theory or proof procedure: a
specification of how an answer can be produced.
• Sound: only generates correct answers with respect to
the semantics
• Complete: Guaranteed to find an answer if it exists
Slide 21
Propositional Definite Clauses
We will start with a simple logic
• Primitive elements are propositions: Boolean variables that can be
{true, false}
Two kinds of statements:
• that a proposition is true
• that a proposition is true if one or more other propositions are true
Why only propositions?
• We can exploit the Boolean nature for efficient reasoning
• Starting point for more complex logics
We need to specify: syntax, semantics, proof procedure
Slide 22
Lecture Overview
• Recap of Lecture 6
• Planning as CSP
• Logic
• Intro
• Propositional Definite Clause Logic (PDCL)
• Syntax and Semantics
• Logical Consequences and Proof Procedures
• Bottom Up Proof Procedure
Slide 23
Propositional Definite Clauses: Syntax
Definition (atom)
An atom is a symbol starting with a lower case letter
Examples: p1;
live_l1
Definition (body)
A body is an atom or is of the form b1 ∧ b2 where b1 and b2 are bodies.
Examples: p1 ∧ p2;
ok_w1 ∧ live_w0
Definition (definite clause)
Examples: p1 ← p2;
A definite clause is
live_w0 ← live_w1 ∧ up_s2
- an atom or
- a rule of the form h ← b where h is an
atom (“head”) and b is a body. (Read this as “h if b”.)
Definition (KB)
A knowledge base (KB) is a set of Slide
definite
clauses
24
atoms
definite
clauses,
KB
rules
Slide 25
PDCL Syntax: more examples
Definition (definite clause)
A definite clause is
- an atom or
- a rule of the form h ← b where h is an atom (‘head’) and b is a body.
(Read this as ‘h if b.’) A body is an atom or is of the form b1 ∧ b2 where
b1 and b2 are bodies
How many of the clauses below are legal PDCL clauses?
a) ai_is_fun
A.
3
B.
4
C. 5
D. 6
b) ai_is_fun ∨ ai_is_boring
c) ai_is_fun ← learn_useful_techniques
d) ai_is_fun ← learn_useful_techniques ∧ notTooMuch_work
e) ai_is_fun ← learn_useful_techniques ∧  TooMuch_work
f)
ai_is_fun ← f(time_spent, material_learned)
g) srtsyj ← errt ∧ gffdgdgd
Slide 26
E. 7
PDC Syntax: more examples
Legal PDC clause
Not a legal PDC clause
a) ai_is_fun
b) ai_is_fun ∨ ai_is_boring
c) ai_is_fun ← learn_useful_techniques
d) ai_is_fun ← learn_useful_techniques ∧ notTooMuch_work
e) ai_is_fun ← learn_useful_techniques ∧  TooMuch_work
f)
ai_is_fun ← f(time_spent, material_learned)
g) srtsyj ← errt ∧ gffdgdgd
Do any of these statements mean anything?
Syntax doesn't answer this question!
Slide 27
To Define a Logic We Need
• Syntax: specifies the symbols used, and how they
can be combined to form legal sentences
• Knowledge base is a set of sentences in the language
• Semantics: specifies the meaning of symbols and
sentences
• Reasoning theory or proof procedure: a
specification of how an answer can be produced.
• Sound: only generates correct answers with respect to
the semantics
• Complete: Guaranteed to find an answer if it exists
Slide 28
Propositional Definite Clauses:
Semantics
• Semantics allows you to relate the symbols in the
logic to the domain you’re trying to model.
Definition (interpretation)
An interpretation I assigns a truth value to each atom.
Slide 29
Propositional Definite Clauses:
Semantics
• Semantics allows you to relate the symbols in the logic to
the domain you're trying to model.
Definition (interpretation)
An interpretation I assigns a truth value to each atom.
Slide 30
Propositional Definite Clauses:
Semantics
Semantics allows you to relate the symbols in the logic to the
domain you're trying to model.
Definition (interpretation)
An interpretation I assigns a truth value to each atom.
We can use the interpretation to determine the truth value of
clauses
Definition (truth values of statements)
• A body b1 ∧ b2 is true in I if and only if b1 is true in
I and b2 is true in I.
• A rule h ← b is false in I if and only if b is true in I
and h is false in I.
Slide 31
PDC Semantics: Example
Truth values under different interpretations
F=false, T=true
a1
a2
a1 ∧ a2
I1
F
F
F
I2
F
T
I3
T
I4
T
h
b
h←b
I1
F
F
T
F
I2
F
T
F
F
F
I3
T
F
T
T
T
I4
T
T
T
Slide 32
PDC Semantics: Example
Truth values under different interpretations
F=false, T=true
h
a1 a2 h ← a1 ∧ a2
I1
F
F
F
I2
F
F
T
I3
F
T
F
I4
F
T
T
I5
T
F
F
I6
T
F
T
I7
T
T
F
I8
T
T
T
Slide 33
PDC Semantics: Example for truth
values
Truth values under different interpretations
F=false, T=true
h
a1 a2 h ← a1 ∧ a2
h
b
h←b
I1
F
F
F
T
I1
F
F
T
I2
F
F
T
T
I2
F
T
F
I3
F
T
F
T
I3
T
F
T
I4
F
T
T
F
I4
T
T
T
I5
T
F
F
T
I6
T
F
T
T
I7
T
T
F
T
I8
T
T
T
T
h ← a1 ∧ a2
Body of the clause: a1 ∧ a2
Body is only true if both
a1 and a2 are true in I
Slide 34
Propositional Definite Clauses:
Semantics
Semantics allows you to relate the symbols in the logic to the
domain you're trying to model.
Definition (interpretation)
An interpretation I assigns a truth value to each atom.
We can use the interpretation to determine the truth value of
clauses
Definition (truth values of statements)
• A body b1 ∧ b2 is true in I if and only if b1 is true in
I and b2 is true in I.
• A rule h ← b is false in I if and only if b is true in I
and h is false in I.
• A knowledge base KB is true in I if and only if
Slide
35 in I.
every clause in KB is
true
Propositional Definite Clauses:
Semantics
Definition (interpretation)
An interpretation I assigns a truth value to each atom.
Definition (truth values of statements)
• A body b1 ∧ b2 is true in I if and only if b1 is true in
I and b2 is true in I.
• A rule h ← b is false in I if and only if b is true in I
and h is false in I.
• A knowledge base KB is true in I if and only if
every clause in KB is true in I.
Definition (model)
A model of a knowledge base KB is an interpretation in
which KB is true.
Similar to CSPs: a model
Slide 36of a set of clauses is
an interpretation that makes all of the clauses true
PDC Semantics: Knowledge Base (KB)
• A knowledge base KB is true in I if and only if
every clause in KB is true in I.
I1
p
q
r
s
true
true
false
false
Slide 37
PDC Semantics: Knowledge Base (KB)
• A knowledge base KB is true in I if and only if
every clause in KB is true in I.
I1
p
q
r
s
true
true
false
false
Slide 38
PDC Semantics: Example for models
Definition (model)
A model of a knowledge base KB is an interpretation in
which every clause in KB is true.
p←q
KB =
Which of the interpretations below are models of KB?
q
r ←s
A.
p
q
r
s
I1
T
T
T
T
I2
F
F
F
F
I3
T
T
F
F
I4
T
T
T
F
I5
F
T
F
Slide 39
T
I3
B.
I 1 , I3
C.
I 1, I3, I4
D.
All of them
PDC Semantics: Example for models
Definition (model)
A model of a knowledge base KB is an interpretation in
which every clause in KB is true.
p←q
KB =
Which of the interpretations below are models of KB?
All interpretations where KB is true: I1, I3, and I4
q
r ←s
p
q
r
s
I1
T
T
T
T
I2
F
F
F
F
I3
T
T
F
F
I4
T
T
T
F
I5
F
T
F
Slide 40
T
p←q
q
r←s
KB
PDC Semantics: Example for models
Definition (model)
A model of a knowledge base KB is an interpretation in
which every clause in KB is true.
p←q
KB =
Which of the interpretations below are models of KB?
All interpretations where KB is true: I1, I3, and I4
q
r ←s
p
q
r
s
p←q
q
r←s
KB
I1
T
T
T
T
T
T
T
T
I2
F
F
F
F
T
F
T
F
I3
T
T
F
F
T
T
T
T
I4
T
T
T
F
T
T
T
T
I5
F
T
F
Slide 41 F
T
T
F
F
OK but….
…. Who cares? Where are we going with this?
Remember what we want to do with Logic
1)
Tell the system knowledge about a task domain.
•
•
2)
This is your KB
which expresses true statements about the world
Ask the system whether new statements about the
domain are true or false.
•
We want the system responses to be
– Sound: only generates correct answers with respect to the semantics
– Complete: Guaranteed to find an answer if it exists
Slide 42
For Instance…
1)
Tell the system knowledge about a task domain.
 p  q.

KB   q.
 r  s.

2)
Ask the system whether new statements about the
domain are true or false
p?
r?
s?
Slide 43
Or, More Interestingly
1)
2)
•
Tell the system knowledge about a task domain.
Ask the system whether new statements about the
domain are true or false
live_w4?
Slide 44
lit_l2?
To Obtain This We Need One More Definition
We want a reasoning procedure that can find all and
only the logical consequences of a knowledge base
Slide 45
Study for midterm (Mon Oct 27)
Midterm: ~6 short questions (10pts each) + 2 problems (20pts each)
• Study: textbook and slides
• Work on all practice exercises and revise assignments!
• While you revise the learning goals, work on practice questions
• Will post a couple of problems from previous offering (maybe
slightly more difficult) … but I’ll give you the solutions 
Slide 46
Next class
• Definite clauses Semantics and Proofs
(textbook 5.1.2, 5.2.2)
Slide 47