AI – Lecture No.4
Knowledge Representation: Symbolic Logic
Search and KR
Knowledge Base
Set of sentences
Three levels of a knowledge-based agent:
 knowledge level
 logical level
 implementation level
KR languages
Symbolic logic language
A logic involves:
 a language with a syntax
 semantics
 inference rules
Rule of inference – check you know the definition, we have already discussed about
Proof
Ex:  = {P, P  Q, T},
 R Q  T
Logic – is a formal system
Propositional logic
Syntax: atoms, logical connectives, wffs
Semantics
 Interpretation
 Satisfiability and models
 Properties of formulas: valid, satisfiable, inconsistent, equivalence
 Entailment
 logically entails  or  logically follows from  or  is a logical
consequence of 
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Model theory
Proof theory
 Sound and completeness
 Sound IR or set of IR
 Complete IR or set of IR
PSAT, CNF PSAT NP-complete problem
kSAT
2 important theorems
 The deduction theorem
 Reductio ad absurdum (refutation theorem)
First-order logic
Syntax: object constants, relational constants, function constants, variables, logical
connectives, logical quantifiers over variables, term, literal, clause, wffs
Equality symbol
Semantics
Interpretation, model
Domain
Conceptualisation
Examples:
 All objects (in the universe) are red apples
 All apples are red
 There is a red apple
 All purple mushrooms are poisonous
x (Purple(x)  Mushroom(x))  Poisoneous(x)
x Purple(x)  (Mushroom(x)  Poisoneous(x))
x Mushroom (x)  (Purple (x)  Poisoneous(x))
 All packages in room 27 are smaller than any of the packages in room 28.
 Every package in the room 27 is smaller than one of the packages in room 28.
Hold for FOPL:
Properties of formulas, entailement, model theory, proof theory, soundness and
completeness of IR, Deduction theorem
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Inference rules in FOPL
 Modus Ponens (MP)
 Modus Tolens (MT)
 And elimination (AE)
 And Introduction (AI)
 Universal Instantiation (UI)
 Existential Instantiation (EI)
 Resolution
Example: Harry and Ralph
Situation Calculus
Diachronic rules – rules that describe the way in which world changes or does not
change
Result: A x S  S
Result(action, situation) = new situation
Effect axioms
xys (OnTable(x,s)  OnTable(y,s)  Clear(x,s)  Clear(y,s)  x  y) 
On(x,y, Result(Stack(x,y),s))
Frame axioms
xys (On(x,y,s)  Clear(x,s)  Clear(x, Result(Unstack(x,y),s))
 Frame problem
 Qualification problem
 Ramification problem
Deducing hidden properties if the world
 Causal rules – infer effect from cause, reflect direction of causality in the world:
some hidden property in the world causes certain percepts to be generated.
l1l2s At(Wumpus,l1,s)  Adjacent(l1,l2)  Smelly(l2)
l1l2s At(Pit,l1,s)  Adjacent(l1,l2)  Breezy(l2)
 Diagnostic rules - Infer cause from effect, infer the presence of hidden properties
directly from perceipt-derived information
l s At(Agent,l, s)  Breeze(s)  Breezy(l)
l s At(Agent,l, s)  Stench(s)  Smelly(l)
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Problem in NL
Horses are faster than dogs and there is a greyhound that is faster than every rabbit. We
know that Harry is a horse and that Ralph is a rabbit. Derive that Harry is faster than
Ralph.
Problem translated in FOPL
x y Horse(x)  Dog(y)  Faster(x,y)
y Greyhound(y)  (z Rabbit(z)  Faster(y,z))
Horse(Harry)
Rabbit(Ralph)
Added axioms to represent commonsense knowledge
y Greyhound(y)  Dog(y)
x y z Faster(x,y)  Faster(y,z)  Faster(x,z)
Theorem: Faster(Harry, Ralph) ?
Theorem proving using Proof Theory and a set of inference rules
1. x y Horse(x)  Dog(y)  Faster(x,y)
2. y Greyhound(y)  (z Rabbit(z)  Faster(y,z))
3. y Greyhound(y)  Dog(y)
4. xyz Faster(x,y)  Faster(y,z)  Faster(x,z)
5. Horse(Harry)
6. Rabbit(Ralph)
7. Greyhound(Greg)  (z Rabbit(z)  Faster(Greg,z))
2, EI
8. Greyhound(Greg)
7, AE
9. z Rabbit(z)  Faster(Greg,z))
7, AE
10. Rabbit(Ralph)  Faster(Greg,Ralph)
9, UI
11. Faster(Greg,Ralph)
6,10, MP
12. Greyhound(Greg)  Dog(Greg)
3, UI
13. Dog(Greg)
12, 8, MP
14. Horse(Harry)  Dog(Greg)  Faster(Harry, Greg)
1, UI
15. Horse(Harry)  Dog(Greg)
5, 13, AI
16. Faster(Harry, Greg)
14, 15, MP
17. Faster(Harry, Greg)  Faster(Greg, Ralph)  Faster(Harry,Ralph)
4, UI
18. Faster(Harry, Greg)  Faster(Greg, Ralph)
16, 11, AI
19. Faster(Harry,Ralph)
17, 19, MP
QED
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Ontological commitments – connected to the nature of reality
Epistemological commitments – possible states of knowledge an agent can have using
various types of knowledge
Necessity and contingency
Necessary knowledge (analytical)
Contingency knowledge
Level of knowledge
Modal logic
A modal is an expression (like ‘necessarily’ or ‘possibly’) that is used to qualify
the truth of a judgement. Modal logic is, strictly speaking, the study of the deductive
behavior of the expressions ‘it is necessary that’ and ‘it is possible that’. However, the
term ‘modal logic’ may be used more broadly for a family of related systems.
These include logics for belief, for tense and other temporal expressions, for the
deontic (moral) expressions such as ‘it is obligatory that’ and ‘it is permitted that’, and
many others.
Narrowly construed, modal logic studies reasoning that involves the use of the
expressions ‘necessarily’ and ‘possibly’. However, the term ‘modal logic’ is used
more broadly to cover a family of logics with similar rules and a variety of different
symbols.
A list describing the best known of these logics follows.
Modal Logics


It is necessary that ..
It is possible that ..
p =  p
Deontic Logic
O It is obligatory that ..
P It is permitted that ..
F It is forbidden that ..
Temporal Logic
G It will always be the case that ..
F It will be the case that ..
H It has always been the case that ..
P It was the case that..
Doxastic Logic
Bx x believes that ..
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