Egon Borger Erich Gradel Yuri Gurevich

The Classical Decision Problem

SpringerVerlag

Berlin Heidelb erg New York

London Paris Tokyo

Hong Kong Barcelona Budap est

Preface

This b o ok is addressed to all those logicians computer scientists mathe

maticians philosophers of science as well as the students in all these disci

plines who may b e interested in the development and current status of one

of the ma jor themes of mathematical logic in the twentieth century namely

the classical decision problem known also as Hilb erts Entscheidungsproblem

The text provides a comprehensive mo dern treatment of the sub ject includ

ing complexity theoretic analysis

We have made an eort to combine the features of a research monograph

and a textb o ok Only the basic knowledge of the language of rstorder logic

is required for understanding of the main parts of the b o ok and we use

standard terminology The chapters are written in such a way that various

combinations of them can b e used for introductory or advanced courses on

undecidability decidability and complexity of logical decision problems This

explains a few intended redundancies and rep etitions in some of the chapters

The annotated bibliography the historical remarks at the end of the chap

ters and the index allow the reader to use the text also for quick reference

purp oses

The b o ok is the result of an eort which went over a decade Many p eo

ple help ed us in various ways with English with pictures and latex with

comments and information It is a great pleasure to thank David Basin

Bertil Brandin Martin Davis Anatoli Degtyarev Igor Durdanovic Dieter

Ebbinghaus Ron Fagin Christian FermullerPhokion Kolaitis Alex Leitsch

Janos Makowsky Karl Meinke Jim Huggins Silvia Mazzanti Vladimir

Orevkov Martin Otto Eric Rosen Rosario Salomone Wolfgang Thomas

Jurek Tyszkiewicz Moshe Vardi Stan Wainer and Suzanne Zeitman This

list is incomplete and we ap ologize to those whose names have b een inad

vertently omitted We are sp ecially thankful to Saharon Shelah for his help

with the Shelah case and to Cyril Allauzen and Bruno Durand for providing

an app endix with a new simplied pro of for the unsolvability of the domino

problem Also we use this opp ortunity to thank Springer Verlag the Omega

group and in particular Gert M ullerfor the patience and b elief in our long

standing promise to write this b o ok

August Egon Borger

Erich Gradel Yuri Gurevich VI I I

Table of Contents

Preface VI I

Introduction The Classical Decision Problem

The Original Problem

The Transformation of the Classical Decision Problem

What Is and What Isnt in this Bo ok

Part I Undecidable Classes

Reductions

Undecidability and Conservative Reduction

The ChurchTuring Theorem and Reduction Classes

Trakhtenbrots Theorem and Conservative Reductions

Inseparability and Mo del Complexity

Logic and Complexity

Prop ositional Satisability

The Sp ectrum Problem and Fagins Theorem

Capturing Complexity Classes

A Decidable PrexVocabulary Class

The Classiability Problem

The Problem

Well Partially Ordered Sets

The Well Quasi Ordering of Prex Sets

The Well Quasi Ordering of Arity Sequences

The Classiability of PrexVocabulary Sets

Historical Remarks

Undecidable Standard Classes for Pure Predicate Logic

The Kahr Class

Domino Problems

Formalization of Domino Problems by

Formulae

Graph Interpretation of Formulae

X Table of Contents



The Remaining Cases Without

3 

Existential Interpretation for

The Gurevich Class

The Pro of Strategy

Reduction to DiagonalFreeness

Reduction to ShiftReduced Form

Reduction to F Elimination Form

i

Elimination of Monadic F

i

The KostyrkoGenenz and Suranyi Classes

Historical Remarks

Undecidable Standard Classes with Functions or Equality

Classes with Functions and Equality

Classes with Functions but Without Equality

Classes with Equality but Without Functions the Goldfarb

Classes

2 

Formalization of Natural Numbers in

=

Using Only One Existential Quantier

Enco ding the NonAuxiliary Binary Predicates



Enco ding the Auxiliary Binary Predicates of NUM

Historical Remarks

Other Undecidable Cases

Krom and Horn Formulae

Krom Prex Classes Without Functions or Equality

Krom Prex Classes with Functions or Equality

Few Atomic Subformulae

Few Function and Equality Free Atoms

Few Equalities and Inequalities

Horn Clause Programs With One Krom Rule

Undecidable Logics with Two Variables

FirstOrder Logic with the Choice Op erator

TwoVariable Logic with Cardinality Comparison

Conjunctions of PrexVocabulary Classes

Reduction to the Case of Conjunctions

Another Classiability Theorem

Some Results and Op en Problems

Historical Remarks

Table of Contents XI

Part I I Decidable Classes and Their Complexity

Standard Classes with the Finite Mo del Prop erty

Techniques for Proving Complexity Results

Domino Problems Revisited

Succinct Descriptions of Inputs

The Classical Solvable Cases

Monadic Formulae

The BernaysSchonnkelRamsey Class

The GodelKalmarSchutteClass a Probabilistic Pro of

Formulae with One

 

A Satisability Test for all all

The Ackermann Class

The Ackermann Class with Equality

Standard Classes of Mo dest Complexity

The Relational Classes in P NP and CoNP

Fragments of the Theory of One Unary Function

Other Functional Classes

Finite Mo del Prop erty vs Innity Axioms

Historical Remarks

Monadic Theories and Decidable Standard Classes with In

nity Axioms

Automata Games and Decidability of Monadic Theories

Monadic Theories

Automata on Innite Words and the Monadic Theory

of One Successor

Tree Automata Rabins Theorem and Forgetful De

terminacy

The Forgetful Determinacy Theorem for Graph Games

The Monadic SecondOrder Theory of One Unary Function

Decidability Results for One Unary Function

The Theory of One Unary Function is not Elementary

Recursive

The Shelah Class

Algebras with One Unary Op eration

Canonic Sentences

Terminology and Notation

Satisability

Satisability

Renements

Villages

Contraction

Towns

XI I Table of Contents

The Final Reduction

Historical Remarks

Other Decidable Cases

FirstOrder Logic with Two Variables

Unication and Applications to the Decision Problem

Unication

Herbrand Formulae

Positive FirstOrder Logic

Decidable Classes of Krom Formulae

The Chain Criterion

The AanderaaLewis Class

The Maslov Class

Historical Remarks

A App endix Tiling Problems

A Introduction

A The Origin Constrained Domino Problem

A Robinsons Ap erio dic Tile Set

A The Unconstrained Domino Problem

A The Periodic Problem and the Inseparability Result

Annotated Bibliography

Index

Introduction The Classical Decision

Problem

The Original Problem

The original classical decision problem can b e stated in several equivalent

ways

The satisability problem or the consistency problem for rstorder logic

given a rstorder formula decide if it is consistent

The validity problem for rstorder logic given a rstorder formula decide

if it is valid

The provability problem for a sound and complete formal pro of system for

rstorder logic given a rstorder formula decide if it is provable in the

system

Recall that a formula is satisable or consistent if it has a mo del It is

valid or logically true if it holds in all mo dels where it is dened A pro of

system is sound if every provable formula is valid it is complete if every valid

formula is provable

It was Hilb ert who drew attention of mathematicians to the classical

decision problem and made it into a central problem of mathematical logic

He called it das Entscheidungsproblem literally the decision problem In

the b eginning of this century he was developing the formalist programme for

the foundations of mathematics see and thus was interested

in axiomatizing various branches of mathematics by means of nitely many

rstorder axioms In principle such an axiomatization reduces proving a

mathematical statement to p erforming a mechanical derivation in a xed

formal logical system see b elow Obviously the Entscheidungsproblem is very

imp ortant in this context

stel lt sich die Frage der Widerspruchsfreiheit als ein Problem

der reinen PradikatenLogik dar Eine solche Frage fallt unter

1

das Entscheidungsproblem page

Hilb ert and Ackermann formulated a sound formal pro of system for rst

order logic and conjectured that the system is complete Later Godel

1

the question of consistency presents itself as a problem of the pure predicate

logic Such a question falls under the Entscheidungsprobl em

Introduction The Classical Decision Problem

proved the completeness The pro of is found in standard logic textb o oks

eg For our purp oses the details of a formal system

are of no imp ortance We will simply assume that some sound and complete

formal pro of system for rstorder logic has b een xed Notice that there

is a mechanical pro cedure that derives all valid rstorder formulae in some

order

To explain how proving a mathematical statement reduces to p erforming a

mechanical derivation assume that T is a nitely axiomatizable mathemat

ical theory Without loss of generality the axioms have no free individual

variables that is are sentences indeed if an axiom has free individual vari

ables replace it with its universal closure Let b e the conjunction of the

axioms another rstorder sentence a mathematical claim in the termi

nology of Hilb ert and the implication Then is a theorem of T

if and only if is valid if and only if is provable in the xed formal pro of

system Thus the mathematical question whether is a theorem of T reduces

to the logical question whether is valid which in its turn reduces to the

question whether the mechanical pro cedure mentioned ab ove derives

Many imp ortant mathematical problems reduce to logic this way

Let us add another example

Example Reduction of the Riemann Hyp othesis to the validity problem for

some rstorder sentence Recall that a Diophantine equation is an equation

P x x where P is a p olynomial with integers co ecients and the

1 k

variables x range over integers In the authors exhibit a Diophantine

i

equation E that is solvable if and only if the Riemann Hyp othesis fails It

suces to nd a nitely axiomatizable theory T and a sentence such that

is provable in T if and only if E is solvable the desired is then the

implication where is the conjunction of the universal closures of

the axioms of T

Recall that the standard arithmetic A is the set of natural numbers with

distinguished element the successor function addition multiplication and

the order relation Let L b e the rstorder language of A Robinsons sys

tem Q is a nitely axiomatizable theory in L such that an arbitrary existential

Lsentence is provable in Q if and only if it holds in A A similar

theory is called N in

Cho ose T to b e Q It suces to construct an existential Lsentence in

such a way that E is solvable if and only if holds in A

In fact an arbitrary Diophantine equation D can b e expressed by an exis

tential formula in such a way Since a disjunction of existential sentences

D

is equivalent to an existential sentence it suces to check that an existential

Lsentence can express the given equation P x x together with an

1 k

atomic constraint x or x for every variable x But this is obvious

i i i

3 5

For example an equation x y with constraints x y is

3 5

equivalent to an equation x y with constraints x y

The Original Problem

5 3

which is equivalent to an equation y x with constraints x y

which is obviously expressible by an existential Lsentence

The classical decision problem is called the main problem of mathematical

logic by Hilb ert and Ackermann

Das Entscheidungsproblem ist gelostwenn man ein Verfahren kennt

das bei einem vorgelegten logischen Ausdruck durch end lich viele Op

erationen die Entscheidung uber die Al lgemeing ultigkeitbzw Erf ull

barkeit erlaubt Das Entscheidungsproblem muss als das Haupt

2

problem der mathematischen Logik bezeichnet werden pp

Hilb ert and Ackermann were not alone in their evaluation of the imp or

tance of the classical decision problem Their attitude has b een shared by

other leading logicians of the time Bernays and Schonnkel wrote

Das zentrale Problem der mathematischen Logik welches auch mit

den Fragen der Axiomatik im engsten Zusammenhang steht ist das

3

Entscheidungsproblem

Herbrands pap er starts with

We could consider the fundamental problem of mathematics to be the

fol lowing Problem A What is the necessary and sucient condition

for a theorem to be true in a given theory having only a nite number

of hypotheses

The pap er ends with

The solution of this problem would yield a general method in math

ematics and would enable mathematical logic to play with respect to

classical mathematics the role that analytic geometry plays with re

spect to ordinary geometry

In Herbrand adds

In a sense it the classical decision problem BGG is the most

general problem of mathematics

Ramsey wrote that his pap er was

concerned with a special case of one of the leading problems in mathe

matical logic the problem of nding a regular procedure to determine

the truth or falsity of any given logical formula p

2

The Entscheidungsproblem is solved when we know a pro cedure that allows for

any given logical expression to decide by nitely many op erations its validity

or satisability The Entscheidungsproblem must b e considered the main

problem of mathematical logic

3

The cental problem of mathematical logic which is also most closely related to

the questions of axiomatics is the Entscheidungsproblem

Introduction The Classical Decision Problem

The ro ots of the classical decision problem can b e traced while back

Philosophers were interested in a general problemsolving metho d The me

dieval thinker Raimundus Lullus called such a metho d ars magna Leibniz

was the rst to realize that a comprehensive and precise symbolic language

characteristica universalis is a prerequisite for any general problem solv

ing metho d He thought ab out a calculus calculus ratiocinator to resolve

mechanically questions formulated in the universal language A universal

symbolic language restricted to mathematics had to wait until when

Frege published the language allowed Russel and Whitehead to

embed virtually the whole b o dy of then known mathematics into a for

4

mal framework Leibniz distinguished b etween two dierent versions of ars

magna The rst version ars inveniendi nds all true scientic statements

The other ars iudicandi allows one to decide whether any given scientic

statement is true or not

In the framework of rstorder logic an ars inveniendi exists the collec

tion of valid rstorder formulae is recursively enumerable hence there is an

algorithm that lists all valid formulae The classical decision problem can b e

viewed as the ars iudicandi problem in the rstorder framework It can b e

sharp ened to a yesno question Do es there exist an algorithm that decides

the validity of any given rstorder formula Some logicians felt sceptical

ab out ever nding such an algorithm It wasnt clear however whether the

scepticism could b e justied by a theorem John von Neumann wrote

Es scheint also da es keinen Weg gibt um das al lgemeine Entschei

dungskriterium daf urob eine gegebene Normalformel a beweisbar ist

aufzunden Nachweisen konnenwir freilich gegenwartignichts Es

ist auch gar kein Anhaltspunkt daf urvorhanden wie ein solcher Un

entscheidbarkeitsbeweis zu f uhren ware Und die Unentscheid

barkeit ist sogar die Conditio sine qua non daf ur da es uberhaupt

einen Sinn habe mit den heutigen heuristischen Methoden Mathe

matik zu treiben An dem Tage an dem die Unentscheidbarkeit

aufhortew urde auch die Mathematik im heutigen Sinne aufhoren zu

existieren an ihre Stel le w urde eine absolut mechanische Vorschrift

treten mit deren Hilfe jedermann von jeder gegebenen Aussage ent

scheiden konnteob diese bewiesen werden kann oder nicht

Wir m ussen uns also auf den Standpunkt stel len Es ist al lgemein

unentscheidbar ob eine gegebene Normalformel beweisbar ist oder

nicht Das einzige was wir tun konnen ist beliebig viele be

weisbare Normalformeln aufzustel len Auf diese Art konnenwir

von vielen Normalformeln feststel len da sie beweisbar sind Aber

auf diesem Weg kann uns niemals die Feststellung gelingen da eine

5

Normalformel nicht beweisbar ist pp

4

See the forthcoming b o ok by M Davis in this connection

5

It app ears thus that there is no way of nding the general criterion for deciding

whether or not a wellformed formula a is provable We cannot however at

The Transformation of the Classical Decision Problem

GodelsIncompleteness Theorem was a breakthrough in logic Can

one use a similar metho d to prove the nonexistence of a decision algorithm for

the classical decision problem In an app endix to his pap er The fundamental

problem of mathematical logic Herbrand wrote

Note nal ly that although at present it seems unlikely that the de

cision problem can be solved it has not yet been proved that it is

impossible to do so

Herbrand Godeland Kleene developed a very general notion of recursive

functions In Church put forward a b old thesis Every computable

function from natural numbers to natural numbers is recursive in the sense of

HerbrandGodelKleeneHe showed that no recursive function could decide

the validity of rstorder sentences and concluded that that there was no

decision algorithm for the classical decision problem

Indep endently Alan Turing introduced computing devices which are

called now Turing machines He put forward a similar thesis a function from

strings to strings is computable if and only if it is computable by a Turing

machine He showed that no Turing machine could decide the validity of

rstorder sentences and also concluded that there is no decision algorithm for

the classical decision problem The equivalence of Churchs and Turings the

ses was quickly established The ChurchTuring thesis was largely accepted

and thus it was accepted that the yesno version of the classical decision

problem was solved negatively by Church and Turing

The Transformation of the Classical Decision

Problem

By the time of Churchs and Turings theses the area of the classical decision

problem had already a rich and fruitful history Numerous fragments of rst

order logic were proved decidable for validity and numerous fragments were

shown to b e as hard as the whole problem What do es it mean that a fragment

F is as hard for validity as the whole problem This means that there exists

the moment demonstrate this Indeed we have no clue as to how such a pro of

of undecidabil i ty would go The undecidabil ity is even the conditio sine

qua non for the contemporary practice of mathematics using as it do es heuristic

metho ds to make any sense The very day on which the undecidabi li ty would

cease to exist so would mathematics as we now understand it it would b e

replaced by an absolutely mechanical prescription by means of which anyone

could decide the provability or unprovability of any given sentence

Thus we have to take the p osition it is generally undecidable whether a

given wellformed formula is provable or not The only thing we can do is

to construct an arbitrary number of provable formulae In this way we can

establish for many wellformed formulae that they are provable But in this way we never succeed to establish that a wellformed formula is not provable

Introduction The Classical Decision Problem

an algorithm A that transforms an arbitrary formula into a formula in F in

such a way that A is valid if and only if is so such a fragment is called

a reduction class for validity Actually it had b een more common to sp eak

ab out satisability and nite satisability that is satisability in a nite

structure Reduction classes for satisability resp ectively nite satisability

are dened similarly

To convey a feeling of the eld let us quote some early results on frag

ments of pure rstorder predicate logic rstorder logic without function

symbols or equality But rst let us recall that a prenex formula is a formula

with all its quantiers up front View a string in the fourletter alphab et

 

f g as a regular expression denoting a collection of strings in the

3 

twoletter alphab et f g For example denotes the collection of strings

3 j  2 

of the form where j is an arbitrary natural number and denotes

i 2 j

the collection of strings of the form where i and j are arbitrary natural

numbers

In Lowenheim gave a decision pro cedure for the satisability

of predicate formulae with only unary predicates He proved also that formu

lae with binary predicates form a reduction class for satisability In

Herbrand sharp ened the latter result showing that just three binary

predicates suce In Kalmar showed that one binary predicate

suces

 

In Skolem showed that sentences form a reduction class

for satisability In Bernays and Schonnkel gave a decision pro

 

cedure for the satisability of sentences In Ackermann gave

 

a decision pro cedure for the satisability of sentences Godel

Kalmar and Schutte separately in and resp ec

 2 

tively discovered decision pro cedures for the satisability of pure sen

 2 

tences In another pap er Godelproved that every satisable sentence

3 

has a nite mo del and that sentences form a reduction class for satis

ability See for a p opular introduction to the classical decision

problem

The reaction of the logicians to the discoveries of Church and Turing was

that the classical decision problem was wider than the yesno version of it

Here is one of the earliest reactions

Solche Reduktionen des Entscheidungsproblems werden hoentlich

vorteilhaft sein f ursystematische Untersuchungen uber die Zahlaus

dr uckezB wenn man versuchen wil l eine Ubersicht zu bekommen

f urwelche Klassen von solchen man das Entscheidungsproblem wirk

lich losenkann Bekanntlich hat A Church bewiesen dass eine al l

6

gemeine Losungdieses Problems nicht moglichist

6

Such reductions a reference to the reductions prop osed by Skolem in the pap er

cited BGG will hop efully b e advantageous for systematic investigations of

rstorder formulae for example if one would like to try to arrive at a complete

picture for which classes of such formulae one can really solve the Entschei

The Transformation of the Classical Decision Problem

The logicians started to think ab out the classical decision problem as a

classication problem

Which fragments are decidable for satisability and which are undecidable

Which fragments are decidable for nite satisability and which are unde

cidable

Which fragments have the nite mo del prop erty and which contain axioms

of innity that is satisable formulae without nite mo dels

For a long time the classical decision problem remained a central problem

of mathematical logic With the development of computational complexity

theory the problem has b een rened If a fragment of rstorder logic is de

cidable for satisability then indeed there is an absolutely mechanical pro ce

dure that is an algorithm for deciding the satisability or unsatisability of

any given sentence But what is the computational complexity of determining

satisability Similarly if a given fragment is decidable for nite satisability

what is the computational complexity of determining nite satisability

Of course the unrestricted classiability problem is hop eless There are

just to o many fragments Some of them are of no interest to anybo dy Some

of them involve particular branches of mathematics Consider for example

the satisability problem for sentences where is the universal closure

of the conjunction of the axioms of elds and is an arbitrary formula in

the vocabulary of elds this problem rightfully b elongs to eld theory rather

than logic

Eventually the classical decision problem b ecame to mean the restriction

of the classication problem describ ed ab ove to traditional fragments This

description is admittedly not precise but it gives a go o d guidance which we

will follow One can argue that the complexity issue do es not really b elong

to the traditional classical decision problem This is true to o but it is imp os

sible to ignore the complexity issue these days in particular b ecause of the

relevance of the logical decision pro cedures to theorem proving and mo del

checking metho ds We will try to cover the known complexity results

As we have mentioned ab ove for a long time the classical decision problem

remained a central problem of mathematical logic The literature on the

sub ject is huge and contains a great wealth of material The classical decision

7

problem served as a lab oratory for various logic metho ds and esp ecially

reduction metho ds The classication results have b een used not only in logic

but also in theoretical computer science In particular they have b een used

as a guide to the study of zeroone laws for fragments of secondorder logic

Classical techniques inspired some pro ofs on the zeroone laws and some of

classical techniques have b een further extended See

in this connection

dungsproblem As it is known A Church has proved that a general solution of

this problem is not p ossible

7

By the way Ramsey proved his famous combinatorial lemma in a pap er on the classical decision problem

Introduction The Classical Decision Problem

There is a number of b o oks devoted to the classical decision problem In

the s Ackermann gave a comprehensive treatment of the solvable cases

known at the time and Suranyi gave a complementary comprehensive

treatment of reduction classes known at the time The b o ok of

Dreb en and Goldfarb illustrates the p otential of the socalled Herbrand ex

pansion technique in establishing solvability The complementary b o ok

of Lewis covers many reduction results on classical fragments of pure pred

icate logic Together the two b o oks give a systematic treatment of decision

problems for predicate logic without functions or equality

Nevertheless much of the wealth has never app eared in a b o ok form

Moreover by now the work on the classical decision problem is by and large

completed though some op en problems remain of course and most of the

ma jor classications have not b een ever covered in b o ok form That is exactly

what we intend to do in this b o ok

What Is and What Isnt in this Bo ok

We give most attention to the most traditional fragments of rstorder logic

namely to collections of prenex formulae given by restrictions on the quan

tier prex andor vocabulary Recall that there is a simple algorithm for

transforming an arbitrary rstorder formula to an equivalent one in the

prenex form

Strings in the twoletter alphab et f g will b e called prexes A prex set

is a set of prexes An arity sequence is a function p from the set of p ositive

integers to the set of nonnegative integers augmented with the rst innite

ordinal

Denition PrexVocabulary Classes For any prex set and

any arity sequences p and f p f resp ectively p f is the collection

=

of all prenex formulae of rstorder logic without equality resp ectively with

equality such that

the prex of b elongs to

the number of nary predicate symbols in is pn and

the number of nary function symbols in is f n

has no nullary predicate symbols with the exception of the logic constants

true and false no nullary function symbols and no free variables

Let us explain the last clause We will sp eak ab out logic without equality

but the same applies to logic with equality It is easy to see that the status

decidable or undecidable of the nite satisability problem for a prex

vocabulary class do es not change if nullary predicate symbols are allowed

Now let us consider the roleof nullary function symbols that is individual

0

constants Let C p f and C b e the version of C when one is allowed to use say individual constants It is easy to see that the status of the nite

What Is and What Isnt in this Bo ok

0

satisability problem for C is that of the nite satisability problem for

0 0 7

p f where f g Instead of individual constants we could

sp eak ab out free individual variables Thus allowing individual constants or

free individual variables do es not give us more classes either

The denition of prexvocabulary classes ab ove seems to b e excessively

general Call a prex set closed if it contains all substrings even not contigu

ous substrings of its prexes Clearly one can restrict attention to closed

prex sets Further call a prex set standard if either it is the set of

all prexes or else it can b e given by a string w in the fourletter alpha

 

b et f g In the rst case is denoted all Thus every standard

prex set has a succinct notation Furthermore we can require without loss



of generality that w is reduced in the following sense cannot have as a



neighbor and similarly cannot have as a neighbor For example a string

   

reduces to clearly the two strings dene the same prex set

Call an arity sequence p standard if it satises the following condition

pn whenever the sum pn pn is innite Every standard

sequence can b e given a succinct notation The standard arity sequence that

assigns to each n will b e denoted all Any other standard sequence p has a

tail of zero es pm pm and will b e denoted by the sequence

p p pm In case m for readability we denote p with

rather than Similar notation can b e used for nonstandard sequences

with a tail of zero es Notice that every arity sequence reduces in a sense

made more precise in Sect to a standard arity sequence For example

all all and every sentence all can

b e easily rewritten as an equivalent sentence in all just replace

0 0

formulae Rx with formulae R x x where R is a binary predicate symbol

that do es not o ccur in

Denition A prexvocabulary class p f or p f is standard

=

if p and f are standard

The classication problem for the prexvocabulary fragments admits a

complete solution in a form of a nite table In particular there are only

nitely many minimal undecidable fragments with closed prex sets and all

these minimal fragments are standard This follows from the Classiability

Theorem of Gurevich proved in Sect Accordingly in the main b o dy of

the b o ok the prexvocabulary classes of interest will b e almost exclusively

standard classes The Classiability Theorem has provided guidance for re

search and it provides guidance for this b o ok

Let us review briey the contents of the b o ok The main part of Chapter

is devoted to the reduction theory which we explain from scratch and develop

to a certain depth The reduction theory helps us to give simpler pro ofs and

prop er lower complexity b ounds The rest of Chapter is devoted to the Classiability Theorem

Introduction The Classical Decision Problem

In Chapters and we give a complete treatment of the undecidable

prexvocabulary fragments of rstorder logic with or without function

symbols with or without equality In Chapter we present various other

undecidable fragments mainly dened in terms of additional restrictions on

the prop ositional structure of the formulae we study in particular Krom and

Horn formulae which have played an imp ortant role in the theory of logic

programming

In Chapters and we treat the decidable prexvocabulary fragments

of rstorder logic with or without function symbols with or without equal

ity Together with the results of Chapters and this gives a complete

classication of the decidable and undecidable prexvocabulary classes Ta

bles and summarize the decidabilityundecidability results on prex

vocabulary fragments

Undecidable Cases

A Pure predicate logic without functions without

898 Kahr

3

8 9 Suranyi



8 9 KalmarSuranyi



898 Denton



8989 Gurevich

3 

8 9 KalmarSuranyi



89 8 KostyrkoGenenz



9 898 Suranyi

 3

9 8 9 Suranyi

B Classes with functions or equality

8 Gurevich

=

8 Gurevich

=

2

8 Gurevich

2

8 Gurevich

2

8 9 Goldfarb

=

 2

9 8 9 Goldfarb

=

2 

8 9 Goldfarb

=

Table Minimal Undecidable Standard Classes

What Is and What Isnt in this Bo ok

Decidable Cases

A Classes with the nite mo del prop erty

 

9 8 all Ramsey

=

 2 

9 8 9 all Godel Kalmar Schutte

al l Lob Gurevich

 

9 89 all all MaslovOrevkov Gurevich



9 all all Gurevich

=

B Classes with innity axioms

all Rabin

=

 

9 89 all Shelah

=

Table Maximal Decidable Standard Classes

We give also a fairly complete complexity analysis of the decidable cases

One op en problem is to nd the exact complexities of the satisability and

nite satisability problems for the Shelah class For most of the maximal de

cidable standard fragments the satisability problem has a very high compu

tational complexity typically deterministic or nondeterministic exp onential

time the complexity is even nonelementary in the case of the Rabin class At

the end of Chapter we also present a classication of the standard classes

that have the nite mo del prop erty and of those having innity axioms The

decidability results in Chapter rely in our exp osition on a reduction to

SS the monadic secondorder theory of the innite binary tree The de

cidability of SS proved by Rabin is one of the most imp ortant and

dicult decidability theorems for mathematical theories We give a complete

pro of of this result in Sect In Chapter we present some other decidable

cases of the decision problem In addition the b o ok contains a quite exten

sive annotated bibliography and an app endix written by Cyril Allauzen and

Bruno Durand containing a new simplied pro of for the unsolvability of the

unconstrained domino problem which is used at many places in this b o ok

Some classications app ear for the rst time in a b o ok For example

the classications of prexvocabulary fragments in the cases of logic with

equality functions or b oth All complexity results app ear for the rst time in

a b o ok There are many new pro ofs eg those assisted by Shelah related

to the Shelah class There are also many new results

On the other hand there are many closely related topics that we do not cover in this b o ok We are concerned here with fragments of rstorder logic

Introduction The Classical Decision Problem

and do not deal with decision problems for secondorder logic higherorder

logic intuitionistic logic see linear logic see the forthcoming

b o ok or any other logic We do not deal with decision problems for

mathematical theories formalized in rstorder or any other logic in this

connection see

Furthermore even though the classical decision problem is more or less

nished in its most classical form there are various other natural versions and

extensions of it that we do not deal with here systematically For example

we do not deal with classications based on the resolution calculus in this

connection see But we do discuss various extensions of the classical

decision problem and various op en problems on our way Let us mention some

extensions and op en problems here

Extend the classiability theorem in various directions This is very im

p ortant without a prop er direction it is hard even to remember a myriad of

sp ecic results

Extend the prexvocabulary classication to imp ortant undecidable

mathematical theories see in this connection Find the computational

complexity of decidable prexvocabulary classes of imp ortant mathematical

theories see in many cases even the computational complexity

of the theory itself is unknown It would also b e interesting to extend the

classication to dierent logics

We were interested whether a given fragment contains a formula with

out nite mo dels Do es a given fragment contain a formula without recur

sive mo dels This direction is still covered by the Classiability Theorem

in particular there are nitely many minimal prexvocabulary classes with

formulae without recursive mo dels and each of them is standard Instead

of recursivity one can sp eak ab out other kinds of descriptive or computa

tional complexity Similarly do es a given fragment contain an axiom of an

essentially undecidable theory Since the fragment may b e not closed under

conjunction it is meaningful to ask if the fragment includes a nite set of sen

tences that form an axiomatization of an essentially undecidable theory Also

one may restrict attention to innite mo dels of certain complexity primitive

recursive mo dels recursive mo dels mo dels of such and such Turing degrees

Borel mo dels etc

In cases of fragments of reasonably low complexity b ound develop prac

tical solutions of the decision problem This problem is well recognized as a

ma jor b ottleneck for eg mo del checking an imp ortant current metho d

for computer verication of hardware and software correctness claims

One extension of the classical decision problem is related to the strictness

of reductions If one cares only ab out satisability it suces to require that a

0

reduction transforms a given formula into a formula which is satisable

if and only if is so We usually care ab out satisability and nite satisa

bility and thus consider so called conservative reductions when it is required

0 0

that i is satisable if and only if is satisable and ii is nitely sat

What Is and What Isnt in this Bo ok

isable if and only if is nitely satisable One may b e interested in even

0

stricter reductions For example one may require that and have the

same sp ectra or more generally that there is a simple connection b etween

the sp ectra On several o ccasions Suranyi insisted that there should exist

0

a general metho d that transforms a given mo del of to a mo del of On

the other hand one may consider not only recursive but also arithmetical

Borel etc transformations

There are many more sp ecic problems One is to examine Bo olean com

binations of prexvocabulary classes see Section in this connection

The b o ok is addressed to a wide audience and not only to professional

logicians There are scattered remarks and exercises addressing more sp ecial

audiences logicians p eople familiar with logic programming etc but the

main b o dy requires only the familiarity with basic notions of mathematical

logic This do es not mean of course that all parts are easy to read some

pro ofs are quite involved even after much simplication Finally let us note

that sometimes we will omit the adjective rstorder formulae languages and theories are by default rstorder in this b o ok