Upload
donald-merritt
View
218
Download
0
Tags:
Embed Size (px)
Citation preview
1
Finite Model Theory
Lecture 1: Overview and Background
2
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
3
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
4
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)
5
Today’s Outline
• Background in Model Theory
• A taste of what’s different in FMT
6
Classical Model Theory
• Universal algebra + Logic = Model Theory
• Note: the following slides are not representative of the rest of the course
7
First Order Logic = FO
t ::= c | x::= R(t, …, t) | t=t | Æ | Ç | : | 9 x. | 8 x.
t ::= c | x::= R(t, …, t) | t=t | Æ | Ç | : | 9 x. | 8 x.
Vocabulary: = {R1, …, Rn, c1, …, cm}Variables: x1, x2, …
In the future:Second Order Logic = SOAdd: ::= 9 R. | 8 R. ::= 9 R. | 8 R.
This is SYNTAX
8
Model or -Structure
A = <A, R1A, …, Rn
A, c1A, …, cm
A>A = <A, R1A, …, Rn
A, c1A, …, cm
A>
STRUCT[] = all -structures
9
Interpretation
• Given:– a -structure A
– A formula with free variables x1, …, xn
– N constants a1, …, an 2 A
• Define A ² (a1, …, an)
– Inductively on
10
Classical Results
• Godel’s completeness theorem
• Compactness theorem
• Lowenheim-Skolem theorem
• [Godel’s incompleteness theorem]
We discuss these in some detail next
11
Satisfiability/Validity
• is satisfiable if there exists a structure A s.t. A ²
• is valid if for all structures A, A ²
• Note: is valid iff : is not satisfiable
12
Logical Inference
• Let be a set of formulas• There exists a set of inference rules that
define ` [white board…]
Proposition Checking ` is recursively enumerable.
Note: ` is a syntactic operation
13
Logical Inference
• We write ² if: 8 A, if A ² then A ²
• Note: ² is a semantic operation
14
Godel’s Completeness Result
Theorem (soundness) If ` then ²
Theorem (completeness) If ² then `
Which one is easy / hard ?
It follows that ² is r.e.
Note: we always assume that is r.e.
15
Godel’s Completness Result
• is inconsistent if ` false• Otherwise it is called consistent
• has a model if there exists A s.t. A ² Theorem (Godel’s extended theorem) 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 `]
16
Compactness Theorem
Theorem If for any finite 0 µ , 0 is satisfiable, then is satisfiable
Proof: [in class]
17
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
18
Application
• Suppose has “arbitrarily large finite models”– This means that 8 n, there exists a finite model
A with |A| ¸ n s.t. A ²
• Then show that has an infinite model A [in class]
19
Lowenheim-Skolem Theorem
Theorem If has a model, then has an enumerable model
Upwards-downwards theorem:
Theorem [Lowenheim-Skolem-Tarski] Let be an infinite cardinal. If has a model then it has a model of cardinality
20
Decidability
• CN() = { | ² }• A theory T is a set s.t. CN(T) = T• is complete if 8 either ² or ² :
• If T is finitely axiomatizable and complete then it is decidable.
• Los-Vaught test: if T has no finite models and is -categorical then T is complete
21
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
22
Summary of Classical Results
• Completeness, Compactness, LS
23
A Taste of FMT
Example 1
• Let = {R}; a -structure A is a graph
• CONN is the property that the graph is connected
Theorem CONN is not expressible in FO
24
A taste of FMT
• Proof Suppose CONN is expressed by , i.e. G ² iff G is connected
• Let ’= [ {s,t}k = : 9 x1, …, xk R(s,x1) Æ … Æ R(xk,t)
• The set = {} [ {1, 2, …} is satisfiable (by compactness)
• Let G be a model: G ² but there is no path from s to t, contradiction
THIS PROOF IS INSSUFFICIENT OF US. WHY ?
25
A taste of FMT
Example 2
• EVEN is the property that |A| = even
Theorem If = ; then EVEN is not in FO
• Proof [in class]
But what do we do if ; ?