Finite Model Theory

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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

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
• In this course we will learn FMT together
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

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
• Background in Model Theory
• A taste of what’s different in FMT
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

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

f ::= 9 R. f | 8 R.f

This is SYNTAX

Model or s-Structure

A = <A, R1A, …, RnA, c1A, …, cmA>

STRUCT[s] = all s-structures

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
• Godel’s completeness theorem
• Compactness theorem
• Lowenheim-Skolem theorem
• [Godel’s incompleteness theorem]

We discuss these in some detail next

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
• 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 Inference
• We write G²f if: 8 A, if A ²G then A ²f
• Note: ² is a semantic operation
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
• 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

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

Proof: [in class]

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
• 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

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
• 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
• 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
• Completeness, Compactness, LS
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 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 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; ?