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Finite Model Theory

Lecture 1: Overview and Background

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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

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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

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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)

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Today’s Outline

• Background in Model Theory

• A taste of what’s different in FMT

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Classical Model Theory

• Universal algebra + Logic = Model Theory

• Note: the following slides are not representative of the rest of the course

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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

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Model or -Structure

A = <A, R1A, …, Rn

A, c1A, …, cm

A>A = <A, R1A, …, Rn

A, c1A, …, cm

A>

STRUCT[] = all -structures

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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

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Classical Results

• Godel’s completeness theorem

• Compactness theorem

• Lowenheim-Skolem theorem

• [Godel’s incompleteness theorem]

We discuss these in some detail next

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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

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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

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Logical Inference

• We write ² if: 8 A, if A ² then A ²

• Note: ² is a semantic operation

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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.

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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 `]

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Compactness Theorem

Theorem If for any finite 0 µ , 0 is satisfiable, then is satisfiable

Proof: [in class]

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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

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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]

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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

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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

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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

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Summary of Classical Results

• Completeness, Compactness, LS

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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

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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 ?

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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 ; ?