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Finite Model Theory. Lecture 1: Overview and Background. 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

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finite model theory

Finite Model Theory

Lecture 1: Overview and Background

motivation
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
organization
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
resources
Resources

www.cs.washington.edu/599ds

Books

  • Leonid Libkin, Elements of Finite Model Theorymain 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)
today s outline
Today’s Outline
  • Background in Model Theory
  • A taste of what’s different in FMT
classical model theory
Classical Model Theory
  • Universal algebra + Logic = Model Theory
  • Note: the following slides are not representative of the rest of the course
first order logic fo
First Order Logic = FO

Vocabulary: s = {R1, …, Rn, c1, …, cm}

Variables: x1, x2, …

t ::= c | x

f ::= R(t, …, t) | t=t | fÆf | fÇf | :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

model or s structure
Model or s-Structure

A = <A, R1A, …, RnA, c1A, …, cmA>

STRUCT[s] = all s-structures

interpretation
Interpretation
  • Given:
    • a s-structure A
    • A formula f with free variables x1, …, xn
    • N constants a1, …, an2 A
  • Define A ²f(a1, …, an)
    • Inductively on f
classical results
Classical Results
  • Godel’s completeness theorem
  • Compactness theorem
  • Lowenheim-Skolem theorem
  • [Godel’s incompleteness theorem]

We discuss these in some detail next

satisfiability validity
Satisfiability/Validity
  • f is satisfiable if there exists a structure A s.t. A ²f
  • f is valid if for all structures A, A ²f
  • Note: f is valid iff :f is not satisfiable
logical inference
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

logical inference13
Logical Inference
  • We write G²f if: 8 A, if A ²G then A ²f
  • Note: ² is a semantic operation
godel s completeness result
Godel’s Completeness Result

Theorem (soundness) If G`f then G²f

Theorem (completeness) If G²f then G`f

Which one is easy / hard ?

It follows that G²f is r.e.

Note: we always assume that G is r.e.

godel s completness result
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 ²G

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 `]

compactness theorem
Compactness Theorem

Theorem If for any finite G0µG, G0 is satisfiable, then G is satisfiable

Proof: [in class]

completeness v s compactness
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
application
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 ²G
  • Then show that G has an infinite model A [in class]
lowenheim skolem theorem
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

decidability
Decidability
  • CN(G) = {f | G²f}
  • A theory T is a set s.t. CN(T) = T
  • T is complete if 8f either T²f or T²: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
some great theories
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
summary of classical results
Summary of Classical Results
  • Completeness, Compactness, LS
a taste of fmt
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

a taste of fmt24
A taste of FMT
  • Proof Suppose CONN is expressed by f, i.e. G ²f 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 ²f but there is no path from s to t, contradiction

THIS PROOF IS INSSUFFICIENT OF US. WHY ?

a taste of fmt25
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 s¹; ?