Transcript lecture01
Finite Model Theory
Lecture 1: Overview and Background
1
Motivation
• Applications:
– DB, PL, KR, complexity theory, verification
• Results in FMT often claimed to be known
– Sometimes people confuse them
• Hard to learn independently
– Yet intellectually beautiful
• In this course we will learn FMT together
2
Organization
•
•
•
•
Powerpoint lectures in class
Some proofs on the whiteboard
No exams
Most likely no homeworks
– But problems to “think about”
• Come to class, participate
3
Resources
www.cs.washington.edu/599ds
Books
• Leonid Libkin, Elements of Finite Model Theory main text
• H.D. Ebbinghaus, J. Flum, Finite Model Theory
• Herbert Enderton A mathematical Introduction to Logic
• Barwise et al. Model Theory (reference model theory book;
won't really use it)
4
Today’s Outline
• Background in Model Theory
• A taste of what’s different in FMT
5
Classical Model Theory
• Universal algebra + Logic = Model Theory
• Note: the following slides are not
representative of the rest of the course
6
First Order Logic = FO
Vocabulary: s = {R1, …, Rn, c1, …, cm}
Variables: x1, x2, …
t ::= c | x
f ::= R(t, …, t) | t=t | f Æ f | f Ç f | : f | 9 x. f | 8 x.f
In the future:
Second Order Logic = SO
Add:
f ::= 9 R. f | 8 R.f
This is SYNTAX
7
Model or s-Structure
A = <A, R1A, …, RnA, c1A, …, cmA>
STRUCT[s] = all s-structures
8
Interpretation
• Given:
– a s-structure A
– A formula f with free variables x1, …, xn
– N constants a1, …, an 2 A
• Define A ² f(a1, …, an)
– Inductively on f
9
Classical Results
•
•
•
•
Godel’s completeness theorem
Compactness theorem
Lowenheim-Skolem theorem
[Godel’s incompleteness theorem]
We discuss these in some detail next
10
Satisfiability/Validity
• f is satisfiable if there exists a structure A
s.t. A ² f
• f is valid if for all structures A, A ² f
• Note: f is valid iff : f is not satisfiable
11
Logical Inference
• Let G be a set of formulas
• There exists a set of inference rules that
define G ` f [white board…]
Proposition Checking G ` f is recursively
enumerable.
Note: ` is a syntactic operation
12
Logical Inference
• We write G ² f if: 8 A, if A ² G then A ² f
• Note: ² is a semantic operation
13
Godel’s Completeness Result
Theorem (soundness) If G ` f then G ² f
Theorem (completeness) If G ² f then G ` f
Which one is easy / hard ?
It follows that G ² f is r.e.
Note: we always assume that G is r.e.
14
Godel’s Completness Result
• G is inconsistent if G ` false
• Otherwise it is called consistent
• G has a model if there exists A s.t. A ² G
Theorem (Godel’s extended theorem) G is consistent
iff it has a model
This formulation is equivalent to the previous one
[why ? Note: when proving it we need certain
properties of `]
15
Compactness Theorem
Theorem If for any finite G0 µ G, G0 is
satisfiable, then G is satisfiable
Proof: [in class]
16
Completeness v.s. Compactness
• We can prove the compactness theorem
directly, but it will be hard.
• The completeness theorem follows from the
compactness theorem [in class]
• Both are about constructing a certain
model, which almost always is infinite
17
Application
• Suppose G has “arbitrarily large finite
models”
– This means that 8 n, there exists a finite model
A with |A| ¸ n s.t. A ² G
• Then show that G has an infinite model A
[in class]
18
Lowenheim-Skolem Theorem
Theorem If G has a model, then G has an
enumerable model
Upwards-downwards theorem:
Theorem [Lowenheim-Skolem-Tarski] Let l
be an infinite cardinal. If G has a model
then it has a model of cardinality l
19
Decidability
• CN(G) = {f | G ² f}
• A theory T is a set s.t. CN(T) = T
• T is complete if 8 f either T ² f or T ² : f
• If T is finitely axiomatizable and complete
then it is decidable.
• Los-Vaught test: if T has no finite models
and is l-categorical then T is complete
20
Some Great Theories
• Dense linear orders with no endpoints [in class]
• (N, 0, S) [in class]
• (N, 0, S, +) Pressburger Arithmetic
• (N, +, £) : Godel’s incompleteness theorem
21
Summary of Classical Results
• Completeness, Compactness, LS
22
A Taste of FMT
Example 1
• Let s = {R}; a s-structure A is a graph
• CONN is the property that the graph is
connected
Theorem CONN is not expressible in FO
23
A taste of FMT
• Proof Suppose CONN is expressed by f, i.e. G ²
f iff G is connected
• Let s’=s [ {s,t}
yk = : 9 x1, …, xk R(s,x1) Æ … Æ R(xk,t)
• The set G = {f} [ {y1, y2, …} is satisfiable (by
compactness)
• Let G be a model: G ² f but there is no path from
s to t, contradiction
THIS PROOF IS INSSUFFICIENT OF US. WHY ?
24
A taste of FMT
Example 2
• EVEN is the property that |A| = even
Theorem If s = ; then EVEN is not in FO
• Proof [in class]
But what do we do if s ; ?
25