Limited expressivity of propositional
Physics of the Wumpus World: Modeling is difficult with
Propositional Logic
Schemas:
TDDC17
Seminar 6
First-Order Logic
Resolution
Nonmonotonic Reasoning
( Bx,y ⇔ (Px,y+1 ∨ Px,y-1 ∨ Px+1,y ∨ Px-1,y ))#
Def. of breeze in pos [x,y]
( Sx,y ⇔ (Wx,y+1 ∨ Wx,y-1 ∨ Wx+1,y ∨ Wx-1,y ))#
Def. of stench in pos [x,y]
(W1,1 ∨ W1,2 ∨ … ∨ W4,4 ))"
..., etc.
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Ontological Commitment of 1st-order logic
Higher-Order Logic
2nd-Order Logic
There is only one wumpus!
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Spectrum of Logics and Languages
Circumscription
There is at least one wumpus!
Facts
Objects
Relations
Default Logic
Modal Logics: Epistemic, Doxastic, Temporal,...
1st-Order Logic + Fixpoints
Inductive Definitions
Datalog
1st-Order Logic
Description Logic
Model with:
• 5 objects
• 2 binary relations
• brother
• on-head
• 3 unary relations
• person
• crown
• king
• 1 unary function
• left-leg()
Definite Clauses
Horn Clauses
Compile!
Propositional Logic
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Language and Syntax: Terms
FOL: Language and Syntax: Components
Name
individuals
in a
domain of
discourse
Name
relations between
objects
Refer to some or
all objects with
particular
constraints
• Object Constants - Infinite set of object constants.
• Convention: Begin with capital letters or numerals
• Examples: Aa, 125, Q, Battery1
•Variables – Infinite set of variables
• Convention: Use lower case letters
• Examples: p,q,r,s,t,…, p1,p2, and so on.
Terms
and separated by commas, is a term.
• Examples: fatherOf(John, p), onTopOf(B1), armOf(R1)
• Function Constants - Infinite set of function constants of all arities.
• Convention: Begin with lower case letters and have an arity
• Examples: presidentOf, onTopOf (arity 1), times, plus (arity 2)
•Relation Constants - Infinite set of relation constants of all arities.
• Convention: Begin with capital letters and have an arity.
• Examples: Large, Clear (arity 1), Parent (arity 2)
•Quantifier symbols -- ∀and ∃.
• ∀ is called the universal quantifier. ∃ is called the existential quantifier
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Connectives
and
Delimiters
Propositional connectives: ∧, ∨, ⊃, and, ¬$
Delimiters: [, ], (, ). Separator: ,
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Language and Syntax: wffs
• Atoms: a relation constant of arity n followed by n terms in parentheses
and separated by commas is an atom. An atom is a wff.
• Examples: P(A,B,c), IsBlock(onTopOf(B1))
• Propositional wffs: Any expression formed out of predicate calculus wffs
in the same way that the propositional calculus forms wffs out of other wffs
is a wff, called a propositional wff.
• Examples: P(A,B1,C) ⊃ (IsBlock(onTopOf(B1)) ∧ Heavy(C))
#
• An object constant is a term
• Examples: Aa, 125, Q, Battery1
• A variable symbol is a term.
• Examples: p,q,r,s,t,…, p1,p2, and so on.
• A function constant of arity n, followed by n terms in parentheses
Semantics: Restricted to propositional wffs
+ Boolean
combinations
atoms
Boolean
combinations
of
ground atoms
• If ω is a wff and ν is a variable symbol, then both (∀ν)ω
and (∃ν)ω are wffs.
The language restricted to propositional wffs can be used to refer
to objects in the world as well as propositions (with internal structure) about
objects (properties and relations). Using the language, we can refer to:
• an infinite number of objects (or individuals) in the world. (but not sets of objects!)
• an infinite number of functions on individuals
• an infinite number of relations on individuals
Language
IsBlock(onTopOf(B))
P(A,B,C)
Interpretation
• ν is the variable quantified over and ω is said to be within the scope of
the quantifier.
• If all variable symbols besides ν in ω are quantified over, then (Qν)ω is
called a closed wff. (Q can be ∀or ∃. )
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3
World
P: {<A,B,C>,
<D,F,E>}
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A
C
1
B onTopOf: {<B>-->A
<C> -->B
<A> -->Floor}
Semantics of Quantification
Interpretations
An Interpretation of an expression in the predicate calculus is an assignment
that maps:
• object constants into constants (objects) in the world
• n-ary function constants into n-ary functions
• n-ary relation constants into n-ary relations
• The set of objects to which the object constants assignments are
made is called the domain of the interpretation.
(∀ν)ω has the value True (under a given interpretation), just in case ω has the
value True for all assignments of the variable symbol ν to objects in the domain.
(∃ν)ω has the value True (under a given interpretation), just in case ω has the
value True for at least one assignment of the variable symbol ν to
objects in the domain.
A Extended Interpretation is an interpretation that also
maps all variables into the domain (an assignment)
Every object on the floor is clear
There is at least one object on the floor
Some objects are big
Given an extended interpretation for its component parts, an atom has value
True just in case the denoted relation holds for those individuals
denoted by its terms, otherwise the atom has value False.
(∀ x) [On(x, Floor) ⊃ Clear(x)] #
The values of non-atomic wffs can be determined in the same way
as for the propositional formulas using truth tables.
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Artificial Intelligence & Integrated Computer Systems Division
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An Example
On(B,A)
On(A, Floor)
On(C,Floor)
Clear(B)
Clear(C)
KB1
On(B,A)
On(C,Floor)
Clear(B)
Clear(C)
Clear(B) ∧ Clear(C) ⊃ On(A,Floor)
KB2
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Another Example
Object Constants: A, B, C, Floor
Relation Constants: On, Clear
B
A
Floor
A --> A
B --> B
C --> C
Floor --> Floor
On --> {<A, Floor>,<B,A>,
<C,Floor>}
Clear --> {<B>, <C>}
Interpretation
C
On(B,A)
On(C,Floor)
Clear(B)
Clear(C)
Object Constants: A, B, C, Floor
Relation Constants: On, Clear
Clear(B) ∧ Clear(C) ⊃ On(A,Floor)
Are these sentences True
using the
interpretation to the right?
(∃x)On(x,Floor)
Find one extended interpretation where
On(x, Floor) is true.
(∀x) Clear(B) ∧ Clear(C) ⊃ On(x,Floor)
Make sure the non-quantified part of the formula
is true for all extended interpretations.
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B
A
Floor
A --> A
B --> B
C --> C
Floor --> Floor
On --> {<A, Floor>,<B,A>,
<C,Floor>}
Clear --> {<B>, <C>}
Interpretation
C
On Domains and Models
• Language with 2 constants, R,J and one binary relation
On Domains and Models
• Language with 2 constants, R,J and one binary relation
Some models
Some models
Naming individuals:
• More than one name
• No name
Database Semantics
Many applications assume
UNA, DC
Naming individuals:
•Unique Names Assumptions
•Domain Closure
Under-constrained in FOL!
Richard has two brothers?
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Artificial Intelligence & Integrated Computer Systems Division
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Substitutions
Resolution and Unification
Recall that terms of an expression can be object constants, variables or
functional expressions which include function constants and terms.
Propositional Case:
{R} {P} {¬P, ¬R}
Resolve on P
{¬R}
Resolve on R
{}
Matching a positive and
negative literal is more or
less trivial because the
atom in question has no
structure.
In First-Order Logic Case:
{P(f(y),A), Q(B,C)}
{¬P(x,A), R(x,C)}
{?}
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A substitution instance of an expression is obtained by substituting terms
for variables in that expression. We denote a substitution instance by ωs,
where ω is an expression and s a substitution.
P[z, f(w), B]
Alphabetic Variant
P[x, f(y), B]
P[g(z), f(A), B]
Matching the internal
structure of atoms is more
complex and involves
generating substitutions
P[C, f(A), B]
Ground instance
{x/z, y/w}
{x/g(z), y/A}
{x/C, y/A}
A substitution is represented by a set of ordered pairs s = {ψ1/τ1, ψ2/τ2, . . ., ψn/τn },
where the pair ψi/τi means that term τi is substituted for every occurrence of the
variable ψi throughout the scope of the substitution.
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Substitution Properites
The composition of two substitutions s1 and s2, denoted by s1s2, is obtained
by first applying s2 to the terms of s1 and then adding any pairs of s2 having
variables not occurring among the variables of s1.
s1= { z/g(x,y) }
s2= { x/A, y/B, w/C, z/D}
s1s2= { z/g(A,B) , x/A, y/B, w/C}
Most General Unifiers (MGU)
Denote the set of substitution instances of a set {ωi} of expressions by {ωi}s.
A set of expressions is unifiable if there exists a substitution s such that
ω1 s = ω2 s = ω3 s . . . .
In such a case , s is said to be a unifier of {ωi} since its use collapses
the set to a singleton.
s= {x/A, y/B} unifies {P[x, f(y), B], P[x, f(B), B]} yielding {P[A, f(B), B]}
s binds too much! We do not need to bind x to A to unify the 2 expressions.
(ω s1)s2 = ω (s1s2)
s1(s2s3) = (s1s2)s3
Associative
s1s2 ≠ s2s1
Not Commutative
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Unification Algorithm
The most general (or simplest) unifier ( mgu), g of {ωi} , has the property
that if s is any unifier of {ωi} yielding {ωi}s, then there exists a substitution s´
such that {ωi}s = {ωi}gs´.
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Skolemization: Required for CNF Form
Let ω be a quantified sentence of the form (∀ν1) . . . (∀νn)(∃y)ω(νi, y),
where the variables ν1 . . . νn are universally quantified, the variable y
is existentially quantified, and these variables appear in the component
sentence ω. The Skolemization of (∀ν1) . . . (∀νn)(∃ y)ω(νi, y) is the
sentence (∀ν1) . . . (∀νn)ω(νi,f(ν1, . . , νn)), where y has been replaced with
f(ν1, . . , νn), in the subexpression ω. f must be a new function symbol that
does not already appear in the theory.
(∃x)(∀y)[ Mother(x) ∧ Mother-of(x,y)]
(∀y) (∃x) [Mother(x) ∧ Mother-of(x,y)]
Skolemization
x is the mother of everyone
any y has a mother
(∀y) [Mother(f(y)) ∧ Mother-of(f(y),y)]
(∃x) Mother(x)
Mother(M1)
Skolem constant
The point is to remove all existential quantifiers leaving only universal quantifiers.
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Artificial Intelligence & Integrated Computer Systems Division
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Example
CNF Form for First-Order Logic
1. Eliminate implication signs (using the ∨ form).
2. Reduce the scopes of negation signs. (using DeMorgan Laws, double negation)
3. Standardize the variables. Rename quantified variables so that each
quantifier has its own variable symbol.
4. Eliminate existential quantifiers (replace with Skolem functions or constants)
5. Convert to prenex form. (move all ∀ quantifiers to front of the formula).
6. Eliminate universal quantifiers (all variables are universally quantified).
7. Put the matrix (quantifier free part of 5) into conjunctive normal form using
DeMorgan’s rules, etc.
Eliminate Implications:
Reduce the scope of negation:
3-6: Additions for FOL
Deal with quantifiers and variable names
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Standardize/rename Variables
Skolemize:
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Drop Universal Quantifiers:
Put into CNF form using DeMorgan’s rules, etc.:
Finished!
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Example
(∀h)[[Big(h) ∧ House(h) ⊃ Work(h)] ∨%
[(∃m) Cleans(m,h) ∧ ¬(∃g) Garden(g,h)]]%
Resolution for First-Order Logic
Suppose γ1 ∪{φ} and γ2 ∪ {¬ψ} are two clauses and φ and ¬ψ are
positive and negative literals, respectively. Then, %
γ1 ∪{φ}
γ2 ∪ {¬ψ}%
where
φµ = ψµ%
[γ1 ∪ γ2]µ%
and µ is the mgu
of φ and ψ .
Binary Resolution: resolves on 2 literals
clause 1
clause 2
{Q(x), R(x)} ∪{P(x,x)}
{¬Q(B), S(y)} ∪{¬P(A, z)}
{Q(A), R(A), ¬Q(B), S(y)}
resolvant
{z/A, x/A}
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Soundness and Completeness
First-Order Logic Resolution is Sound!
First-Order Logic Resolution is not refutation
complete using the current resolution rule.
It can be made complete with the addition of
Factoring: the removal of redundant literals
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An Example: Package Delivery Robot
All of the packages in room 27 are smaller than any of the packages in room
28:
1. (∀x,y){[Package(x)∧ Package(y) ∧ Inroom(x,27) ∧ Inroom(y,28) ]#
⊃ Smaller(x,y)}#
A is a package and B is a package:
2. Package(A)#
3. Package(B)
Inroom(A,27) ?
4. Inroom(A,27) ∨ Inroom(A,28)
In which room is package A?
Package B is in room 27:
5. Inroom(B,27) #
Package B is not smaller than package A:
6. ¬Smaller(B,A)
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Is package A in room 27?
Package A is either in room 27 or room 28:
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∃u Inroom(A,u) ?"
Resolution example
Robot’s Situation
Room 27
Robot’s current
Location.
(7)
Negate the query:
(7’)
Put 1-6 into CNF form
Robot’s Goal:
Fetch Package A and bring it
to the current location.
2-6 are already in CNF form. Call them 2’-6’
Room 28
Where is package A?
Is package A in room 27?
Put 1 into CNF form:
Go there, pick it up, go back (generate a plan)
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Artificial Intelligence & Integrated Computer Systems Division
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CNF form
CNF form
Standardize/rename variables:
Eliminate implications:
Drop universals:
1’
Reduce scope of negation sign:
Simplify:
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Finished!
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Resolution
Resolution Proof
Theory is in CNF form:
1’
2’
Resolve 7’ and 4’
3’
4’
5’
6’
7’
8:
Resolve 8 and 1’
9:
Resolve 9 and 2’
10:
Resolve 10 and 3’
Negated query:
Continued next slide......
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Artificial Intelligence & Integrated Computer Systems Division
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Resolution Proof
10:
Resolve 10 and 3’
11:
Resolve 11 and 6’
12:
Resolve 12 and 5’
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Reasoning About Action
and Change
Nonmonotonic Logics
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Reasoning about action and change
•
Reasoning about Action and Change
How do we represent and
develop efficient inference
mechanisms for dynamic
behaviors of agents in
incompletely specified
environments?
One of the most difficult problems in formal
knowledge representation!
•
•
•
•
Late 60’s and 70’s: difficulties in modeling
Invention of nonmonotonic logic in 70’s
Great progress in 80’s and 90’s
Need for scalable, efficient solutions for
incomplete environments in the 00’s. (WWW,
Robotics)
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KPLAB
Knowledge Processing Lab
Actions & Effects
Epistemics & Causality
Temporal & Spatial Reasoning
Sensing & Observation
Planning & Plan Execution
Prediction and Explanation
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KPLAB
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Some More Problems
Some Problems
[t1 ]inpocket(agent, bubblegum) ??
[t1 ]at(gold, 2, 3)
??
Is the gold still at 2,3 after
the agent moves to 1,2 ?
The Ramification Problem
Are the pits in the same place after
the agent moves to 1,2 ?
[t1 ]at(agent, 1, 2)
[t1 ]at(agent, 1, 2)
[t,t1 ]Goto(agent, 1, 1, 1, 2)
[t]at(agent, 1, 1)
[t]at(gold, 2, 3)
The Frame Problem
The world tends to remain inert. Most actions
are local and do not disturb the larger frame.
Most features in the world do not change.
[t,t1 ]Goto(agent, 1, 1, 1, 2)
[t]at(agent, 1, 1)
[t]at(gold, 2, 3)
[t]inpocket(agent, bubblegum)
How can this rule of thumb be represented
succinctly in logic?
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KPLAB
Knowledge Processing Lab
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There are many ramifications to an
action, causal dependencies that
become true when an action
is executed
How can one succinctly represent these
ramified effects without making
action specifications overly detailed?
KPLAB
Knowledge Processing Lab
Some Formalisms
More Problems
Goto
Action
[at(agent, x, y) ^ ad jacent(x, y, x1 , y1 ) ^ Sa f e(x1 , y1 )]
! [at(agent, x1 , y1 ) ^ ¬at(agent, x, y)]
But what if ......
A giant bird flies by and drops a turd and blocks the path?
The Wumpus breaks the rules and goes on the move?
The Gold falls off a shelf and hits the agent in the head knocking
it unconscious?
The agent is wearing red-white and blue and a virtual
Usama Bin Ladin enters the game as a virus and exterminates
anything with those colors?
The Qualification Problem
All actions have exceptions (possibly infinite). How can we
succinctly represent that actions work most of the time but
there are exceptions, and we can add them to the theory in
a manner that doesn’t force us to go into the action rules and
change them all the time?
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Knowledge Processing Lab
P
1
2
3
Situation Calculus (Revised) -- Reiter et al
Event Calculus -- Kowalski (logic programming)
Fluent Calculus -- Thielscher
The A Family - Lifschitz, Gelfond, Baral
Features and Fluents -- Sandewall
Temporal Action Logics (TAL) -- Doherty et al
KPLAB
Knowledge Processing Lab
#obs [0] Safe(1,1) & !Wumpus(1,1) & !Pit(1,1) & Alive(agent1)
#occ [0,1] goto(agent1,1,1,1,2)
#dep I([1] sense(agent1, breeze))
….
…..
#obs [10] Wumpus(1,3) & !Alive(agent1)
4
2
1
•
Using Narratives for Explanation
#obs [0] Safe(1,1) & !Wumpus(1,1) & !Pit(1,1) & Alive(agent1)
#occ [0,1] goto(agent1,1,1,1,2)
#dep I([1] sense(agent1, breeze))
#occ [1,2] goto(agent1, 1,2,1,1)
#occ [2,3] goto(agent1, 1,1,2,1)
#dep I([3] sense(agent1, stench))
4
3
•
•
•
•
•
Situation Calculus -- McCarthy (1959-63)
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KPLAB
Using TAL Narratives for Projection
Given that I execute these actions
and sense these percepts, what can
I say about the wumpus world?
•
4
Given that the agent executed these
actions and sensed these percepts and
given the following observations at
time 10, how can we explain them?
3
P
2
1 1
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2
3
KPLAB
Knowledge Processing Lab
4
Multi-Agent Iphone Game
Using TAL Narratives to Plan
#obs [0] Safe(1,1) & !Wumpus(1,1)
& !Pit(1,1) & Alive(agent1)
Initial State Specification
?
#obs [t] Winner(agent1) & Gold(1,1)
TALPlanner
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Goal
Generate a sequence of actions
which make the observations at t
true given the initial state observations
at time 0.
KPLAB
Knowledge Processing Lab
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Techniques for solving A&C Problems
Open & Closed World Assumptions
Predicate Completion
Negation as Failure to Prove
Circumscription
Default Logic
Common thread:
• Make assumptions about
incomplete information.
• Do this by refering to
meta-theoretic concepts.
• The entailment relation becomes
nonmonotonic.
If an atom is not in a database, assume it is false.
The sufficient conditions for a predicate in a theory are also the
necessary conditions.
If I can’t prove a literal is true, assume it is false.
The objects or tuples that can be shown to satisfy a relation are the only
ones that do.
If a formula is consistent with a theory then assume it is true
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KPLAB
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KPLAB
Knowledge Processing Lab
Monotonicity
Classical Logic:
IF
THEN
Practical Reasoning is often not as conservative. We often “jump” to
conclusions or assume something is true if there is no reason to
believe otherwise. (ceteris paribus in law)
XX
Nonmonotonic Logics do not have the property of monotonicity
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KPLAB
Knowledge Processing Lab
90
5 Non-Monotonic Reasoning
Sufficient and Necessary Conditions
the sequel, we will not distinguish between a theory T and the sentence denoting the conjunction of all members of T . We write T (P1 , . . . , Pn ) to indicate
that some (but not necessarily all) of the relation symbols occurring in T are
among P1 , . . . , Pn .
Circumscription Informally
Narrative
Theory
in L(FL)
An n-ary relation expression is an expression of the form
λx1 . . . xn . A(x1 , . . . , xn )
(n ≥ 0)
P is a sufficient condition
Q
wherefor
x1 ,something
. . . , xn arebeing
individual
variables and A(x1 , . . . , xn ) is any formula
Make those sufficient conditions
for change the only conditions
Narrative
Theory
in L(FL)
+ Circumscription
Axiom
No spurious change!
of first- or second-order logic. We identify an n-ary relation symbol P with
the relation expression λx1 . . . xn . P (x1 , . . . , xn ). Similarly, an n-ary relation
variable X is identified with the relation expression λx1 . . . , xn .X(x1 , . . . , xn ).
The following are relation expressions (below X is a relation variable):
P is a necessary
condition
λxy. (X(x) ∨ P (y));
for something being Q
λx. P (x);
Models for the narrative
Sufficient conditions for change
λx. False.
An n-ary relation expression U is intended to represent an n-ary relation
which is usually referred to as the extension of U .
By Minimizing Occludes we minimize unnecessary change.
5.3 Circumscription
91
If a feature does not have to change value relative to the axioms
Observe that (5.3) can be rewritten as
it won’t!
In the sequel, an n-ary relation expression λx1 . . . , xn . A(x1 , . . . , xn ) will often
be written as λx̄. A(x̄), where x̄ stands for a tuple ⟨x1 , . . . , xn ⟩.
The only things that are Q are P. P
Let U be a relation expression of the form λx̄. A(x̄), where x̄ = ⟨x1 , . . . , xn ⟩,
is both necessary and sufficient
and suppose that t̄ = ⟨t1 , . . . , tn ⟩ is an n-tuple of terms. The application of
U toArtificial
t̄, written
U (t̄),Computer
is the
A(t̄). For instance, the application
of
Intelligence & Integrated
Systemsformula
Division
KPLAB
of Computer and Information Science
λxy. Department
(P (x)University,
→ Q(y))
to t̄ = ⟨a, z⟩ is the formula P (a) → Q(z). Knowledge Processing
Linköping
Sweden
T (P̄ , S̄) ∧ ∀X̄∀Ȳ . {[T (X̄, Ȳ ) ∧ [X̄ ≤ P̄ ]] → [P̄ ≤ X̄]}
which, in turn, is an abbreviation for
Artificial Intelligence & Integrated Computer Systems Division
i=1
i=1
!
Example 5.3.3. Let T consist of the following formulas:
Definition 5.3.1. Let P̄ = ⟨P1 . . . , Pn ⟩ be a tuple of distinct relation symbols,
S̄ = ⟨S1 , . . . , Sm ⟩ be a tuple of distinct relation symbols disjoint with P̄ , and
let T (P̄ , S̄) be a theory. The circumscription of P̄ in T (P̄ , S̄) with varied S̄,
written Circ(T ; P̄ ; S̄), is the sentence
Let P̄ = ⟨Ab⟩ and S̄ = ⟨F lies⟩.
i=1
Bird(Tweety)
∀x.[(Bird(x) ∧ ¬Ab(x)) → F lies(x)].
(5.3)
∀x.[X(x) → Ab(x)] ] → ∀x.[Ab(x) → X(x)] } .
91
Note that U ≤ V means
that the
extension
is a subset
Observe
that
(5.3) can of
beUrewritten
as of the extension of V .
T (X̄, Ȳ ) is the sentence obtained from T (P̄ , S̄) by replacing all occurrences
P̄ ,nS̄)
∧ ∀X̄∀Ȳ . {[T (X̄,
∧ [X̄
≤ P̄ ]] → [P̄ of
≤ X̄]}
, respectively,
andȲ )all
occurrences
S1 . . . , Sm by
of P1 . . . , Pn by X1 . . T. ,(X
Y1 . . . , Ym , respectively.
Intuition:
Minimizes
Extensions
of selected
Predicates
∀X̄∀Ȳ .
T (X̄, Ȳ ) ∧
n
#
i=1
$
∀x̄.(Xi (x̄) → Pi (x̄)) →
n
#
i=1
{∀x.[(Bird(x) ∧ ¬False) → Bird(x)] ∧ ∀x.[False → Ab(x)]} →
∀x.[Ab(x) → False].
%
Since A can be simplified to the logically equivalent sentence
∀x̄.(Pi (x̄) → Xi (x̄)) .
!
Example 5.3.3. Let T consist of the following formulas:
KPLAB
Bird(Tweety)
∀x.[(Bird(x) ∧ ¬Ab(x)) → F lies(x)].
Let P̄ = ⟨Ab⟩ and S̄ = ⟨F lies⟩.
Knowledge Processing Lab
(5.4)
where A is
Definition 5.3.2. A formula A is said to be a consequence of the circumscription of P̄ in T (P̄ , S̄) with variable S̄ if and only if Circ(T ; P̄ ; S̄) |= A.12
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
In its basic form, the idea is to find relational expressions for X and Y that
when substituted into the theory T , will result in strengthening the theory so
additional inferences can be made. For example, substituting λx.False for X
and λx.Bird(x) for Y , one can conclude that
Circ(T ; P̄ ; S̄) |= T ∧ A
which, in turn, is an abbreviation for
T (P̄ , S̄)∧
!"
A Famous Example!
Circ(T ; P̄ ; S̄) = T (P̄ , S̄) ∧
∀X∀Y. { [Bird(Tweety) ∧ ∀x.[(Bird(x) ∧ ¬X(x)) → Y (x)]∧
where X̄ = ⟨X1 . . . , Xn ⟩ and Ȳ = ⟨Y1 , . . . , Ym ⟩ are tuples of relation variables
5.3 Circumscription
!
similar to P̄ and S̄, respectively.11
11
Knowledge Processing Lab
Definition 5.3.2. A formula A is said to be a consequence of the circumscription of P̄ in T (P̄ , S̄) with variable S̄ if and only if Circ(T ; P̄ ; S̄) |= A.12
Circumscription
for (Ū ≤ V̄ ) ∧ (V̄ ≤ Ū ), and Ū < V̄ for (Ū ≤ V̄ ) ∧ ¬(V̄ ≤ Ū ).
10
KPLAB
T (P̄ Department
, S̄)∧ of Computer and Information Science
Linköping
$
%
!" University, Swedenn
n
#
#
∀x̄.(Xi (x̄) → Pi (x̄)) →
∀x̄.(Pi (x̄) → Xi (x̄)) .
∀X̄∀Ȳ . T (X̄, Ȳ ) ∧
Lab
If U and V are relation expressions of the same arity, then U ≤ V stands for
∀x̄. (U (x̄) → V (x̄)).10 Similarly, if Ū = ⟨U1 , . . . , Un ⟩ and V̄ = ⟨V1 , . . . , Vn ⟩ are
similar tuples of relation expressions, i.e., for 1 ≤ i ≤ n, Ui and Vi are of the
n
!
[Ui ≤ Vi ]. We write Ū = V̄
same arity, then Ū ≤ V̄ is an abbreviation for
T (P̄ , S̄) ∧ ∀X̄∀Ȳ . ¬[T (X̄, Ȳ ) ∧ X̄ < P̄ ]
Intended set of preferred
Models for the narrative
∀x.[Ab(x) → False],
which in turn is equivalent to ∀x.¬Ab(x), one can infer by (5.4) that
Circ(T ; P̄ ; S̄) |= F lies(Tweety).
12
Artificial Intelligence & Integrated Computer Systems Division
Here
|= denotes
the
relation of the
Department
of Computer
andentailment
Information Science
Linköping University, Sweden
!
second-order logic.
KPLAB
Knowledge Processing Lab
What to Know!
•
•
Propositional and First-Order Logic
Resolution Theorem Proving for both.
•
You should be able to model and solve a
problem using resolution
•
This implies knowing transformation to CNF
forms
•
You should understand problems associated with
reasoning about action and change and intuitions
behind circumscription.
•
Look at the Situation Calculus in the book
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden