1. First-Order Logic
CSE 573
A handful of GENERAL SEARCH TECHNIQUES lie at the heart of practically all work in AI
We will encounter the SAME PRINCIPLES again and again in this course, whether we are talking about
GAMES
LOGICAL REASONING
MACHINE LEARNING
These are principles for SEARCHING THROUGH A SPACE OF POSSIBLE SOLUTIONS to a problems.

2. 573 Core Topics Reinforcement Learning Supervised Learning Inference
Planning
Knowledge
Representation
Logic-Based
Probabilistic
Search
Problem Spaces
Agency
© Daniel S. Weld

3. Immediate Schedule Today Thurs 10/16 Tues 10/21 Thurs 10/23 Tues 10/28
First-Order Logic
Class  3:30pm
“Intelligence in Wikipedia”
Jesse Davis – Machine Learning
Probabilistic Reasoning
Jesse Davis
Learning Probabilistic Representations
CSPs will come later
© Daniel S. Weld

4. Logic-Based KR     Propositional logic First-order logic
Syntax (CNF, Horn clauses, …)
Semantics (Truth Tables)
Inference (FC, Resolution, DPLL, WalkSAT)
Restricted Subsets
First-order logic
Syntax (quantifiers, skolem functions, …
Semantics (Interpretations)
Inference (FC, Resolution, Compilation)
Restricted Subsets (e.g. Frame Systems)
Representing events, action & change   
© Daniel S. Weld

5. Propositional. Logic vs. First Order
Objects,
Properties,
Relations
Facts (P, Q)
Ontology
Syntax
Semantics
Inference
Algorithm
Complexity
Atomic sentences
Connectives
Variables & quantification
Sentences have structure: terms
father-of(mother-of(X)))
Interpretations
(Much more complicated)
Truth Tables
Unification
Forward, Backward chaining
Prolog, theorem proving
DPLL, GSAT
Fast in practice
Semi-decidable
NP-Complete
© Daniel S. Weld

6. FOL Definitions Constants: a,b, dog33. Variables: X, Y.
Name a specific object.
Variables: X, Y.
Refer to an object without naming it.
Functions: dad-of
Mapping from objects to objects.
Terms: dad-of(dog33)
Refer to objects
Atomic Sentences: in(dad-of(dog33), food6)
Can be true or false
Correspond to propositional symbols P, Q
© Daniel S. Weld

7. More Definitions Logical connectives: and, or, not, => Quantifiers:
 Forall
 There exists
Examples
Dumbo is grey
Elephants are grey
There is a grey elephant
© Daniel S. Weld

8. Quantifier / Connective Interaction
E(x) == “x is an elephant”
G(x) == “x has the color grey”
x E(x)  G(x)
x E(x) G(x)
x E(x)  G(x)
x E(x) G(x)
© Daniel S. Weld

9. Nested Quantifiers: Order matters!
x y P(x,y)  y x P(x,y)
Examples
Every dog has a tail
Every dog shares a tail! ?
d t has(d,t)
t d has(d,t)
Someone is loved by everyone
x y loves(y, x)
© Daniel S. Weld

10. Semantics Syntax: a description of the legal arrangements of symbols
(Def “sentences”)
Semantics: what the arrangement of symbols means in the world
Inference
Sentences
Sentences
Representation
Semantics
Semantics
World
Models
Models
© Daniel S. Weld

11. Propositional Logic: SEMANTICS
“Interpretation” (or “possible world”)
Specifically, TRUTH TABLES
Assignment to each variable either T or F
Assignment of T or F to each connective P T F Q T F
P  Q
© Daniel S. Weld

12. Models
Depiction of one possible “real-world” model
© Daniel S. Weld

13. Interpretations=Mappings syntactic tokens  model elements
Depiction of one possible interpretation, assuming
Constants: Functions: Relations:
Richard John
Leg(p,l)
On(x,y) King(p)
© Daniel S. Weld

14. Interpretations=Mappings syntactic tokens  model elements
Another interpretation, same assumptions
Constants: Functions: Relations:
Richard John
Leg(p,l)
On(x,y) King(p)
© Daniel S. Weld

15. Satisfiability, Validity, & Entailment
S is valid if it is true in all interpretations
S is satisfiable if it is true in some interp
S is unsatisfiable if it is false all interps
S1 entails S2 if
forall interps where S1 is true,
S2 is also true |=
© Daniel S. Weld

16. Skolemization Existential quantifiers aren’t necessary!
Existential variables can be replaced by
Skolem functions (or constants)
Args to function are all surrounding  vars
d t has(d, t)
x y loves(y, x)
d has(d, f(d) )
y loves(y, f() )
y loves(y, f97 )
© Daniel S. Weld

17. FOL Reasoning FO Forward & Backward Chaining FO Resolution
Many other types of theorem proving
Restricted representations
Description logics
Horn Clauses
Compilation to SAT
© Daniel S. Weld

18. ? Forward Chaining Given Prove ?x dog(?x) => mammal(?x) dog(fido)
?x lifeform(?x) => mortal(?x)
?x mammal(?x) => lifeform(?x)
?x dog(?x) => mammal(?x)
dog(fido)
Prove
mortal(fido)
?x dog(?x) => mammal(?x)
dog(fido)
mammal(fido) ?
© Daniel S. Weld

19. Unification Emphasize variables with ?
Useful for FO inference (modus ponens, …)
Also for compilation of FOPC -> propositional
Unify(, ) returns “mgu”
Unify(city(?a), city(kent)) returns ?a/kent
Substitute(expr, mapping) returns new expr
Substitute(connected(?a, ?b), {?a/kent}) returns connected(kent, ?b)
© Daniel S. Weld

20. Unification Examples Unify( road(?a, kent), road(seattle, ?b) )
Unify( road(?a, ?a), road(seattle, kent) )
Unify( f(g(?x, dog), ?y), f(g(cat, ?y), dog) )
Unify( f(g(?x)), f(?y) )
Unify( f(g(?x)), f(?x) )
© Daniel S. Weld

21. Properties of Resolution
Sound
Complete
But satisifability of FO Logic is only semi-decidable
?x human(?x) => male(father(?x))
?y male(?y) => human(?y)
human(Fred)
Prove: male(Bob)
© Daniel S. Weld

22. Properties of Resolution
Sound
Complete
But satisifability of FO Logic is only semi-decidable
h(?x)  m(f(?x))
m(?y)  h(?y)
h(F)
Prove: m(B)
© Daniel S. Weld

23. Properties of Resolution
Sound
Complete
But satisifability of FO Logic is only semi-decidable
h(?x)  m(f(?x))
m(?y)  h(?y)
h(F)
m(B)
© Daniel S. Weld

24. Resolution Proof
{[m(?y), h(?y)], [h(?x), m(f(?x))], [h(F)], [m(B)]}
?x=F
[m(f(F))]
?y=f(F)
[h(f(F))]
?x=f(F)
[m(f(f(F)))]
. . .
© Daniel S. Weld

25. Compilation to Prop. Logic I
Typed Logic
city a,b connected(a,b)
Universe
Cities: seattle, tacoma, enumclaw
Equivalent propositional formula:
© Daniel S. Weld

26. Compilation to Prop. Logic II
Universe
Cities: seattle, tacoma, enumclaw
Firms: IBM, Microsoft, Boeing
First-Order formula
city c firm f hasHQ(c, f)
Equivalent propositional formula
© Daniel S. Weld

27. Hey! You said FO Inference is semi-decidable
But you compiled it to SAT
Which is NP Complete
So now we can always do the inference?!?
Tho it might take exponential time…
Something seems wrong here….????
© Daniel S. Weld

28. Revisiting Planning as SATQ0 Q2 R0 B1 R2 S0 S2 C1 T0 T2
Initial
State
Actions
Time …
© Daniel S. Weld

29. Medic SAT Compiler Planner Init state Actions Goal Plan Logic
Simplification
SAT
Solver
Compiler
Decoder
© Daniel S. Weld

30. Axioms Axiom Description / Example Init The initial state holds at t=0
Goal
The goal holds at t=2n
A  P, E
Paint(A,Red,t)  Block(A, t-1)
Paint(A,Red,t)  Color(A, Red, t+1)
Frame
Classical / Explanatory
At-least-one
Act1(…, t)  Act2(…, t)  …
Exclude
Act1(…, t)  Act2(…, t)
© Daniel S. Weld

31. Revisiting Planning as SATQ0 Q2 R0 B1 R2 S0 S2 C1 T0 T2
Initial
State
Actions
Time …
© Daniel S. Weld

32. Restricted Forms of FO Logic
Known, Finite Universes
Compile to SAT
Frame Systems
Ban certain types of expressions
Horn Clauses
Aka Prolog
Function-Free Horn Clauses
Aka Datalog
© Daniel S. Weld

33. KR with Description Logics
Abox
mother(jane)
child-of(jane, bob) …
Assertions
Tbox
person
Term Defs
father
mother
grandmother
© Daniel S. Weld

34. Tbox
Term definitions
FO Language + inference organized into a taxonomy, e.g:
father(x) = person(x) male(x)  y childof(y,x)
parent(x) = person(x)  y childof(y,x)
Complexity of classifying new terms
subsumption
person
Subsumption hierarchy 
father
mother
grandmother
© Daniel S. Weld

35. Abox
Assertions
Abox – separate language + inference for “propositional” assertions using Tbox terms
e.g. person(kelley)
© Daniel S. Weld

36. Debate Restricted language thesis Restricted classification thesis
Disjunction, negation, particularization, order…
Natural kinds
Restricted classification thesis
Concepts using contingent information:
Treatable disease, democratic country, illegal act
Counterargument
Constructs: Omit vs limit
Completeness
Efficiency
© Daniel S. Weld