TDDC17
Seminar 6
First-Order Logic
Resolution
Nonmonotonic Reasoning
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Limited expressivity of propositional
Physics of the Wumpus World: Modeling is difficult with
Propositional Logic
Schemas:
( 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.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
There is at least one wumpus!
There is only one wumpus!
Spectrum of Logics and Languages
Higher-Order Logic
2nd-Order Logic
Circumscription
Default Logic
Modal Logics: Epistemic, Doxastic, Temporal,...
1st-Order Logic + Fixpoints
Inductive Definitions
Datalog
1st-Order Logic
Description Logic
Definite Clauses
Horn Clauses
Compile!
Propositional Logic
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Ontological Commitment of 1st-order logic
Facts
Objects
Relations
Model with:
• 5 objects
• 2 binary relations
• brother
• on-head
• 3 unary relations
• person
• crown
• king
1
• unary function
• left-leg()
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
FOL: Language and Syntax: Components
• 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.
Name
individuals
in a
domain of
discourse
• Function Constants - Infinite set of function constants of all arities.
Name
relations between
objects
Refer to some or
all objects with
particular
constraints
• 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
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Language and Syntax: Terms
Terms
• 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
and separated by commas, is a term.
• Examples: fatherOf(John, p), onTopOf(B1), armOf(R1)
Connectives
and
Delimiters
Propositional connectives: ∧, ∨, ⊃, and, ¬$
Delimiters: [, ], (, ). Separator: ,
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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))
#
• If ω is a wff and ν is a variable symbol, then both
and (∃ν)ω are wffs.
+ Boolean
combinations
atoms
Boolean
combinations
of
ground atoms
(∀ν)ω
• 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 ∃. )
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Semantics: Restricted to propositional 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
3
World
P: {<A,B,C>,
<D,F,E>}
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
A
C
1
B onTopOf: {<B>-->A
<C> -->B
<A> -->Floor}
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.
A Extended Interpretation is an interpretation that also
maps all variables into the domain (an assignment)
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.
The values of non-atomic wffs can be determined in the same way
as for the propositional formulas using truth tables.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Semantics of Quantification
(∀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.
Every object on the floor is clear
There is at least one object on the floor
Some objects are big
(∀ x) [On(x, Floor) ⊃ Clear(x)] #
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
An Example
Object Constants: A, B, C, Floor
Relation Constants: On, Clear
On(B,A)
On(A, Floor)
On(C,Floor)
Clear(B)
Clear(C)
B
A
Floor
KB1
On(B,A)
On(C,Floor)
Clear(B)
Clear(C)
Clear(B) ∧ Clear(C) ⊃ On(A,Floor)
KB2
C
A --> A
B --> B
C --> C
Floor --> Floor
On --> {<A, Floor>,<B,A>,
<C,Floor>}
Clear --> {<B>, <C>}
Interpretation
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Another Example
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.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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
Some models
Naming individuals:
• More than one name
• No name
Under-constrained in FOL!
Richard has two brothers?
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
On Domains and Models
• Language with 2 constants, R,J and one binary relation
Some models
Naming individuals:
•Unique Names Assumptions
•Domain Closure
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Database Semantics
Many applications assume
UNA, DC
Resolution and Unification
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)}
{?}
Matching the internal
structure of atoms is more
complex and involves
generating substitutions
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Substitutions
Recall that terms of an expression can be object constants, variables or
functional expressions which include function constants and terms.
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]
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.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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}
(ω s1)s2 = ω (s1s2)
s1(s2s3) = (s1s2)s3
Associative
s1s2 ≠ s2s1
Not Commutative
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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.
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´.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Unification Algorithm
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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.
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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.
3-6: Additions for FOL
Deal with quantifiers and variable names
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Example
Eliminate Implications:
Reduce the scope of negation:
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Department of Computer and Information Science
Linköping University, Sweden
Standardize/rename Variables
Skolemize:
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Department of Computer and Information Science
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Drop Universal Quantifiers:
Put into CNF form using DeMorgan’s rules, etc.:
Finished!
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Example
(∀h)[[Big(h) ∧ House(h) ⊃ Work(h)] ∨%
[(∃m) Cleans(m,h) ∧ ¬(∃g) Garden(g,h)]]%
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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}
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
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)
Is package A in room 27?
Package A is either in room 27 or room 28:
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)
Artificial Intelligence & Integrated Computer Systems Division
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∃u Inroom(A,u) ?"
Robot’s Situation
Room 27
Robot’s current
Location.
Robot’s Goal:
Fetch Package A and bring it
to the current location.
Room 28
Where is package A?
Go there, pick it up, go back (generate a plan)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
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Resolution example
Is package A in room 27?
(7)
Negate the query:
(7’)
Put 1-6 into CNF form
2-6 are already in CNF form. Call them 2’-6’
Put 1 into CNF form:
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CNF form
Eliminate implications:
Reduce scope of negation sign:
Simplify:
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Department of Computer and Information Science
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CNF form
Standardize/rename variables:
Drop universals:
1’
Finished!
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Resolution
Theory is in CNF form:
1’
2’
3’
4’
5’
6’
7’
Negated query:
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
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Resolution Proof
Resolve 7’ and 4’
8:
Resolve 8 and 1’
9:
Resolve 9 and 2’
10:
Resolve 10 and 3’
Continued next slide......
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
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Resolution Proof
10:
Resolve 10 and 3’
11:
Resolve 11 and 6’
12:
Resolve 12 and 5’
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Reasoning About Action
and Change
Nonmonotonic Logics
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
Reasoning about action and change
•
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)
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
KPLAB
Knowledge Processing Lab
Reasoning about Action and Change
How do we represent and
develop efficient inference
mechanisms for dynamic
behaviors of agents in
incompletely specified
environments?
Actions & Effects
Epistemics & Causality
Temporal & Spatial Reasoning
Sensing & Observation
Planning & Plan Execution
Prediction and Explanation
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
KPLAB
Knowledge Processing Lab
Some Problems
[t1 ]at(gold, 2, 3)
??
Is the gold still at 2,3 after
the agent moves to 1,2 ?
Are the pits in the same place after
the agent moves to 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.
How can this rule of thumb be represented
succinctly in logic?
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
KPLAB
Knowledge Processing Lab
Some More Problems
[t1 ]inpocket(agent, bubblegum) ??
The Ramification Problem
[t1 ]at(agent, 1, 2)
[t,t1 ]Goto(agent, 1, 1, 1, 2)
[t]at(agent, 1, 1)
[t]at(gold, 2, 3)
[t]inpocket(agent, bubblegum)
Artificial Intelligence & Integrated Computer Systems Division
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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
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?
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
KPLAB
Knowledge Processing Lab
Some Formalisms
•
•
•
•
•
•
Situation Calculus -- McCarthy (1959-63)
•
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
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
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KPLAB
Knowledge Processing Lab
Using TAL Narratives for Projection
#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
Given that I execute these actions
and sense these percepts, what can
I say about the wumpus world?
3
P
2
1
1
2
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3
4
KPLAB
Knowledge Processing Lab
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))
….
…..
#obs [10] Wumpus(1,3) & !Alive(agent1)
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
Artificial Intelligence & Integrated Computer Systems Division
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2
3
KPLAB
Knowledge Processing Lab
4
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
Artificial Intelligence & Integrated Computer Systems Division
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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
Multi-Agent Iphone Game
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
KPLAB
Knowledge Processing Lab
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
Artificial Intelligence & Integrated Computer Systems Division
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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
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
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KPLAB
Knowledge Processing Lab
Sufficient and Necessary Conditions
P is a sufficient condition
for something being Q
P is a necessary condition
for something being Q
The only things that are Q are P. P
is both necessary and sufficient
Artificial Intelligence & Integrated Computer Systems Division
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KPLAB
Knowledge Processing Lab
Circumscription Informally
Narrative
Theory
in L(FL)
Narrative
Theory
in L(FL)
Make those sufficient conditions
for change the only conditions
+ Circumscription
Axiom
No spurious change!
Models for the narrative
Sufficient conditions for change
Intended set of preferred
Models for the narrative
By Minimizing Occludes we minimize unnecessary change.
If a feature does not have to change value relative to the axioms
it won’t!
Artificial Intelligence & Integrated Computer Systems Division
Department of Computer and Information Science
Linköping University, Sweden
KPLAB
Knowledge Processing Lab
λxy. (P (x) → Q(y)) to t̄ = ⟨a, z⟩ is the formula P (a) → Q(z).
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
Circumscription
i=1
for (Ū ≤ V̄ ) ∧ (V̄ ≤ Ū ), and Ū < V̄ for (Ū ≤ V̄ ) ∧ ¬(V̄ ≤ Ū ).
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
T (P̄ , S̄) ∧ ∀X̄∀Ȳ . ¬[T (X̄, Ȳ ) ∧ X̄ < P̄ ]
(5.3)
where X̄ = ⟨X1 . . . , Xn ⟩ and Ȳ = ⟨Y1 , . . . , Ym ⟩ are tuples of relation variables
5.3 Circumscription
!
similar to P̄ and S̄, respectively.11
10
11
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:
which, in turn, is an abbreviation for
Minimizes
T (P̄ , S̄)∧
$
%
!"
Extensions
5.3
91
# Circumscription
#
∀x̄.(X (x̄) → P (x̄)) →
∀x̄.(P (x̄) → X (x̄)) .
∀X̄∀Ȳ . T (X̄, Ȳ ) ∧
of
selected
Observe
that (5.3) can be rewritten as
T (P̄ , S̄) ∧ ∀X̄∀Ȳ . {[T (X̄, Ȳ ) Definition
∧ [X̄ ≤ P̄ ]] →5.3.2.
[P̄ ≤ X̄]}
A formula A is said to be a consequence of the circumPredicates
n
n
i
i
i
i=1
i
i=1
scription of P̄ in T (P̄ , S̄) with variable S̄ if and only if Circ(T ; P̄ ; S̄) |= A.12
which, in turn, is an abbreviation for
!
T (P̄ , S̄)∧
Artificial
Computer Systems Division $
%
!"Intelligence & Integrated
Example 5.3.3. Let#
KPLAB
n
nT consist of the following formulas:
Department of Computer and#
Information Science
Knowledge
Linköping
University,
Sweden
∀x̄.(Xi (x̄) → Pi (x̄)) →
∀x̄.(Pi (x̄) → Xi (x̄)) .
∀X̄∀Ȳ . T (X̄, Ȳ ) ∧
i=1
Processing Lab
Bird(Tweety)
i=1
∀x.[(Bird(x) ∧ ¬Ab(x)) → F lies(x)].
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
Let P̄ = ⟨Ab⟩ and S̄ = ⟨F lies⟩.
!
Circ(T ; P̄ ; S̄) = T (P̄ , S̄) ∧
Example 5.3.3. Let T consist of the following formulas:
∀X∀Y. { [Bird(Tweety) ∧ ∀x.[(Bird(x) ∧ ¬X(x)) → Y (x)]∧
Bird(Tweety)
∀x.[(Bird(x) ∧ ¬Ab(x)) → F lies(x)].
∀x.[X(x) → Ab(x)] ] → ∀x.[Ab(x) → X(x)] } .
A Famous Example!
In its basic form, the idea is to find relational expressions for X and Y that
Let P̄ = ⟨Ab⟩ and S̄ = ⟨F lies⟩.
when substituted into the theory T , will result in strengthening the theory so
Circ(T ; P̄ ; S̄) = T (P̄ , S̄) ∧ additional inferences can be made. For example, substituting λx.False for X
and λx.Bird(x) for Y , one can conclude that
∀X∀Y. { [Bird(Tweety) ∧ ∀x.[(Bird(x) ∧ ¬X(x)) → Y (x)]∧
∀x.[X(x)
→ X(x)] } .
Circ(T ;→
P̄ ;Ab(x)]
S̄) |= ]T→∧∀x.[Ab(x)
A
(5.4)
In its basic form, the idea iswhere
to findArelational
expressions for X and Y that
is
when substituted into the theory T , will result in strengthening the theory so
additional inferences can be {∀x.[(Bird(x)
made. For example,
substituting
λx.False
X
∧ ¬False)
→ Bird(x)]
∧ for
∀x.[False
→ Ab(x)]} →
and λx.Bird(x) for Y , one can conclude that
Circ(T ; P̄ ; S̄) |= T ∧ A
where A is
(5.4)
∀x.[Ab(x) → False].
Since A can be simplified to the logically equivalent sentence
∀x.[Ab(x) → False],
{∀x.[(Bird(x) ∧ ¬False) → Bird(x)] ∧ ∀x.[False → Ab(x)]} →
which in turn is equivalent
to →
∀x.¬Ab(x),
∀x.[Ab(x)
False]. one can infer by (5.4) that
Since A can be simplified toCirc(T
the logically
equivalent
sentence
; P̄ ; S̄)
|= F lies(Tweety).
∀x.[Ab(x) → False],
!
12
Here |= denotes the entailment relation of the second-order logic.
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
KPLAB
Linköping University, Sweden
Knowledge Processing Lab
Here
|= denotes
the
relation of the second-order logic.
Department
of Computer
andentailment
Information Science
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