Lecture 4 First-Order Theories - PowerPoint Presentation

Slide1

Lecture 4 First-Order Theories

Xiaokang QiuSlide2

x=1;y=1;while (*) {

x=x+2;

y=y+1;}

Q: is x+y>=2 always true?

Q: Are

these formulae valid in arithmetic?Slide3

First-Order Theories

Q: Which statements are true in arithmetic/set-theory/groups/fields?

A

theory is a set of FOL sentences in a FO languageFix a language for arithmetic:

(why no ?)

How to define a theory?Fix a standard model: (or ?)Peano Arithmetic: Theory of PA: Another way to define a theoryFix a set of axioms , then

 Slide4

First-Order Theories

Other theories?

Presburger

Arithmetic:

Integers:

Reals: Rationals: Arrays:  Slide5

First-Order Theories

Definition:

A theory

is computably

axiomatizable (recursively enumerable) if:There is a computable set of axioms

Theorem: is computably axiomatizable iff. is decidable.Why? Slide6

Which theories are decidable?

Decidable theories

:

double exponential

:

triple exponential: double exponential: double exponential (P if quantifier-free)Quantifier-free : NP-completeQuantifier-free equality (plain FOL): NP-completeUndecidable theories

(

Gödel’s

Incompleteness Theorem, 1931)

(Tarski-

Mostowski

, 1949)

Theory of Rings

(

Mal'cev

, 1961)

Set Theory

(Tarski, 1949)

 Slide7

Peano Arithmetic

Gödel’s Incompleteness Theorem (1931):

is not computably

axiomatizable

. Proof: Intuitively, is expressive enough to say “I am a liar.” (Russel’s Paradox)

Gödel number: Encode every formula to a number Encode every proof to is a proof of iff.

iff

. “the formula encoded by

is not provable”

There is a sentence

 Slide8

Decision Procedures for Various TheoriesSlide9

Quantifier Elimination

Definition:

A set of formulas

admits quantifier elimination if for any formula

, there is a quantifier free

such that .Theorem: admits quantifier elimination.Theorem: admits quantifier elimination.E.g.,

 Slide10

Rational Arithmetic QE

Step 1: Normalization

Convert

to Negation Normal Form (NNF)Step 2: Remove Negation

Step 3: Solve for

in Collect all terms compared to , e.g.,

Instantiate

in

with all possible

,

and

 Slide11

Examples:

 Slide12

Solving QF Rational Arithmetic

Solve satisfiability of

Each conjunction is

Just linear programming!

LP is solvable in

(weakly) polynomial timeTheorem: is decidable in double exponential time.  Slide13

Theory of Equality

is interpreted (reflexivity, symmetry, transitivity, congruence)

Other functions/predicates are

uninterpreted

Congruence:

Theory of Equality is undecidable

Why?

 Slide14

Theory of Equality

Theorem:

The theory of equality is QF-decidable and NP-complete.

Idea: build the set of sub-terms and guess the equality between them.

Example:

Guess equivalence classes, e.g.,

Check congruence and

 Slide15

Satisfiability Modulo TheoriesSlide16

How to combine decidable theories?

is a decidable theory over

is a decision procedure for

is a decidable theory over

is a decision procedure for

Can we build a decision procedure for

from

and

?

 Slide17

Example

is decidable

is QF-decidable

Is satisfiable in ? The combined theory is undecidable in general!Slide18

Nelson-Oppen Combination

Theorem (1979):

If

is a QF-decidable theory over

is a QF-decidable theory over

Both and are stably infinite (intuitively, both theories have infinite models)then is QF-decidable!Combinable theories:

+ Equality +

 Slide19

Nelson-Oppen Combination

Step 1: Purification

Split an

-formula

into an

-formula and an -formula such that and are equisatisfiableExample:  

 Slide20

Nelson-Oppen Combination

Step 2: Guess and Check

and

should agree on the equality

between shared variables!

Guess an equality:

Solve the two theories separately!

(if both theories are in NP, so is the combined procedure)Slide21

Nelson-Oppen Combination

More efficient procedure exists if the theory is

convex

.Definition:

A theory is convex if for any in the theory, if

, then for some .The procedure becomes deterministic – no need to guess on equality! and are convex; and are not!E.g.,