An Introduction to the Technique of Formative Processes in Set Theory

161 Pages·2018·2.385 MB·English

by  Domenico Cantone,Pietro Ursino (auth.)

Domenico Cantone
Pietro Ursino
An Introduction
to the Technique
of Formative
Processes in Set
Theory
An Introduction to the Technique of Formative
Processes in Set Theory
Domenico Cantone Pietro Ursino
(cid:129)
An Introduction
to the Technique
of Formative Processes
in Set Theory
123
Domenico Cantone Pietro Ursino
Dipartimento di Matematica eInformatica Dipartimento di Fisica eMatematica
Universitàdegli Studi diCatania Universitàdegli Studi dell'Insubria
Catania Como
Italy Italy
ISBN978-3-319-74777-4 ISBN978-3-319-74778-1 (eBook)
https://doi.org/10.1007/978-3-319-74778-1
LibraryofCongressControlNumber:2018932530
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To Maria Pia and Riccardo,
and to the memory of my beloved parents
To Melita and Miranda
Preface
Thedecisionprobleminsettheoryhasbeenexploredintensivelyinthelastdecades,mainly
withthefollowinggoals:
- mechanical formalization of mathematics with a proof verifier based on the set-
theoreticformalism[FOS80,OS02,COSU03,OCPS06,SCO11],
- enhancement of the services offered by high-level specification languages including
setsandfunctionsamongtheirprimitives[SDDS86,KP95],
- handling set constraints [DPPR98, RB15, CRF15] in an advanced declarative pro-
grammingsetting.
Thisresearchhasgeneratedalargebodyofresults,givingrisetothefieldof Computable
Set Theory [CFO89,CF95,COP01,SCO11].
In the design of decision procedures in set theory, one needs inevitably to face the
problem of manipulating collections of sets, subject to certain relationships among them
thatmustbemaintainedinvariant. Infact,foragivenfragmentFofsettheoryofwhichone
isinvestigatingthesatisfiabilityproblem,atypicalapproachconsistsoffindinga uniform
method that simplifies the satisfying assignments of any given formula in F, in such a
way that some of their representatives—and, possibly, some additional finite structures
attached to them—can be confined into a finite collection of candidate assignments that
can be effectively generated and tested for satisfaction. As we shall see, this amounts to
provingthatFenjoysasmall model property orasmall witness-model property.
Thisbookprovidesanintroductiontothemostadvancedofthesesimplificationmethods
developedincomputablesettheory,namely,thetechniqueofformativeprocesses. Eachset
carriesinitsextensionacomplete‘constructionplan’,whichcanbeconvenientlyunraveled
and manipulated according to one’s needs. In the case of a finite number of sets ,
F
which collectively may form a satisfying assignment for a given set-theoretic formula, it
provesadvantageoustorepresent throughitsVennpartition,orsomeofitsextensions,
F
rather than by the mere collection of its members. The ‘construction plan’ of such
anextendedVennpartition(cid:6)of isaformativeprocess. ByappropriatelymHanFipulating
F
, one can produce other formative processes whose induced set collections enjoy
0
HthFesamepropertiesas ,pluspossiblyadditionalones(e.g.,rankboundednessoFrinfinite
F
cardinality).
In the mid-1980s, a similar technique began to emerge in the process of solving the
satisfiability problem for the fragments of quantifier-free formulae of set theory called
MLSP(viz.,Multi-Level Syllogistic with the Powerset operator;see [CFS85])andMLSSP
(see[Can87]and[Can91]). Specifically,MLSP-formulaearethepropositionalcombinations
vii
viii Preface
ofatomsofthefollowingtypes
x=y z; x=y z; x=y z; x y; x=P(y);
[ \ n 2
(wherex;y;z standforsetvariables,andP()isthepowersetoperator),whereasMLSSP-
(cid:1)
formulaeareobtainedbyextendingtheendowmentof MLSPwithatomsoftheformx=
y ,involvingthesingletonoperator. Thistechniquewasthenrestatedin[CU97,COU02],
f g
inmuchlikethecurrentterms,andfurtherrefinedin[Urs05,CU14],whereitwasappliedto
solvethesatisfiabilityproblemfortheextensionMLSSPFof MLSSPwithatomicformulae
oftheformFinite(x),intendedtoexpressthatxhasafinitecardinality. However,unlike
MLSSP-formulae, some formulae in MLSSPF are satisfied by infinite assignments only.
Hence,thefragmentMLSSPFcannotenjoytheclassical smallmodelproperty. Instead,the
decidabilityof MLSSPFcanbeobtainedbyprovingasmall witness-model property,using
thetechniqueofformativeprocesses. Specifically, givenan MLSSPF-formula (cid:8)admitting
onlyinfinitesatisfyingassignments,onecanshowthattheassociatedformula(cid:8) ,obtained
(cid:0)
bydroppingfrom(cid:8)allliteralsinvolvingthepredicateFinite(),admitsasmallsatisfying
(cid:1)
assignmentwhoseformativeprocesscanbe‘pumped’intoanewinfiniteformativeprocess.
Anapproachstrictlyrelatedtothetechniqueofformativeprocesseswasalsoundertaken
by Omodeo and Policriti in [OP10, OP12] for the satisfaction problem of the Bernays-
Schönfinkel-Ramsey (BSR) class of set-theoretic statements (namely, the class of prenex
statements, with a prefix of the form , involving the predicate symbols of equality
(cid:3) (cid:3)
9 8
‘=’ and membership ‘ ’ only). Since BSR-statements with a prefix of the form can
2 9988
express infinite sets, as proved in [PP88], the decidability result for the BSR-class has
beenestablishedbyprovingforitasmallwitness-modelproperty. Specifically,ithasbeen
shown that any satisfiable BSR-formula (cid:8) admits a non-necessarily finite model that can
berepresentedbyafinitegraphofboundedsize withspecialnodes,calledrotors,standing
fordenumerablyinfinitefamiliesofwell-behaved sets.
In the form in which they were originally conceived, several decision algorithms in
computablesettheoryareoftenhardtoextendwithnewset-theoreticconstructsthatwere
notbuiltintothemfromtheoutset. Weexpectthatafullunderstandingofthetechnique
of formative processes can enhance its applicability, simplifying combinations of decision
procedures,andpossiblygivinganewresearchboosttothefascinatingareaofcomputable
settheory. Infact,theconsolidationoftheknownpartofcomputablesettheoryisessential
not only to promote new discoveries on decidability, but also to convert the theoretical
resultsintotechnologicaladvancesinthefieldofautomatedreasoning. Forinstance,even
themostbasiclayerofautomatedsetreasoning,theso-calledmulti-levelsyllogistic (MLS),
has benefited from being revisited under a tableaux-based approach, which renders its
implementationfarmoreefficient(see[CZ00,CZ99]and[COP01,Chapter14]).
We hope that the present book may contribute to a renewed interest in computable
set theory and, specifically, in the related technique of formative processes. We expect
that the technique of formative processes might be an essential tool in the solution of
some long-standing open problems in the area, such as the satisfiability problem for the
extension MLSC of multi-level syllogistic with the Cartesian product. This appears to be
particularly relevant insofar as the satisfiability problem for MLSC can be seen, in some
sense,astheset-theoreticcounterpartofthewell-knownHilbert’stenthproblem.
Preface ix
Content: Thebookisdividedintotwoparts. Thefirstpartispreliminaryandconsists
ofthreechapters. Chapter1introducesthebasicset-theoreticterminologyandproperties
used throughout the rest of the book. Chapter 2 is an introduction to the decision prob-
lem in set theory. After presenting the syntax and semantics of a comprehensive theory
that encompasses all the fragments of set theory considered in the book, the novel no-
tionofsatisfiability by partitions isintroduced. Therelationsofsimulation andimitation
among partitions, which maintain invariant the collection of formulae satisfied by them,
are then examined, and a simple case study—the theory BST—is explored in detail. It
is proved, in particular, that BST has an NP-complete decision problem. Subsequently,
various expressiveness results are obtained and then used to prove two negative results
about decidability and rank dichotomicity. The latter is a novel concept, strictly related
to the small model property. Chapter 3 is devoted to formative processes. Preliminarily,
formative processes, their supports—syllogistic boards—and their applications to the de-
cision problem of MLSP-like fragments are illustrated by means of worked-out examples
and pictures. Then all these notions are rigorously presented along with some of their
properties. Finally, the so-called shadowing relationship among formative processes and
its properties are discussed, proving that the final partitions of two shadowing formative
processesimitateeachother.
Thesecondpartofthebookconsistsoftwochapters,whicharedevotedtoapplications
of the technique of formative processes to the decision problems for the theories MLSSP
andMLSSPF,respectively. Specifically,givenapartition(cid:6)andaformativeprocess for
H
it, Chapter 4 presents a technique for extracting the subsequence S of the salient steps
of . The subsequence S enjoys the following two properties: (i)Hits length is bounded
byHa computable functionHof the size of the partition (cid:6), and (ii) there exists a formative
process whose sequence of steps is S and is such that is a shadow process of .
This amHounts to showing that MLSSP eHnjoys the small modHel property, thereby proviHng
itsdecidabbility. Inaddition,theshadowprocess preservesbcertainstructuralproperties
H
of . ThisfactisexploitedinChapter5toprovetheexistenceofsuitableinfiniteshadow
H
processes of (pumping mechanism). Let (cid:8) beban MLSSPF-formula and, again, let (cid:8)
(cid:0)
H
be the MLSSP-formula obtained by dropping from (cid:8) all finiteness literals. Then it turns
out that one can generate effectively all (descriptions of) infinite processes that are in
the shadowing relationship with the formative process of some satisfying assignment for
(cid:8) (small witness-model property). As proved in Chapter 5, the latter fact yields the
(cid:0)
decidabilityof MLSSPF.
All chapters (especially in the first part) contain a number of exercises to help the
readerbecomefamiliarwiththenewconceptsandtechniques. Someexercisessimplyask
the reader to provide or complete omitted proofs; others are devoted to deepening the
comprehensionofthetopicsdiscussedinthemaintext. Thereaderisstronglyencouraged
toworkthemoutforafullunderstandingoftheconceptsandtechniquesofthebook.
Besides some “mathematical maturity,” the reader is assumed to have familiarity with
the basic set-theoretic apparatus, as well as with some elementary notions from mathe-
maticallogicandcomputability.
Finally, to enhance readability, each proof is terminated by the empty square symbol
,whereasallexamplesandremarksareclosedbythefullsquaresymbol .
x Acknowledgments
Acknowledgments
WearegratefultoEugenioOmodeoandAlbertoPolicritiforsomeinitialandveryenjoyable
conversations in the late 1990s, when the technique of formative processes began to take
shapeinitscurrentform. WearealsothankfultoAlfioGiarlotta,whoprovidedveryhelpful
comments on the first part of the book and was kind enough to draw one of the figures,
and to Ronan Nugent, Senior Editor at Springer Computer Science, for his continuous
assistance and support. The second author is also indebted to Josie Nacci for her kind
helpinEnglishproofreadingChapters4and5.
Contents
I Theoretical Apparatus 1
1 Basics of Set Theory 3
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 The Decision Problem in Set Theory 21
2.1 TheTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
S
2.1.1 SyntaxandSemanticsof . . . . . . . . . . . . . . . . . . . . . . . 22
S
2.1.2 TheDecisionProblemforSubtheoriesof . . . . . . . . . . . . . . 26
S
Decidablefragmentsofsettheory . . . . . . . . . . . . . . . . . . . . 28
Smallmodelpropertyforsubtheoriesof . . . . . . . . . . . . . . . 30
S
Vennpartitionsofsetassignments . . . . . . . . . . . . . . . . . . . 30
2.1.3 SatisfiabilitybyPartitions . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 RelatingPartitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.2 Imitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 ASimpleCaseStudy: TheTheory BST . . . . . . . . . . . . . . . . . . . . 39
2.3.1 ComplexityIssues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 ExpressivenessResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1 ExpressingtheSingletonOperator . . . . . . . . . . . . . . . . . . . 46
2.4.2 ExpressingHereditarilyFiniteSets . . . . . . . . . . . . . . . . . . . 46
2.4.3 ExpressingTermsoftheFormP ( x ;:::;x ) . . . . . . . . . . . . 49
(cid:3) f 1 kg
2.4.4 ExpressingTwoRelationsaboutCardinalities . . . . . . . . . . . . . 50
2.5 TwoApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5.1 UndecidabilityofMLSuC+ . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.2 RankDichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Formative Processes 59
3.1 AGentleIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1.1 ConstructingaFormativeProcess. . . . . . . . . . . . . . . . . . . . 62
3.1.2 SyllogisticBoards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.3 SpecialEventsandShadowProcesses . . . . . . . . . . . . . . . . . 67
3.1.4 ApplicationstotheDecisionProblemof MLSP-likeTheories . . . . . 68
3.2 BasicsonSyllogisticBoards . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 FormativeProcessesandBasicEvents . . . . . . . . . . . . . . . . . . . . . 76
3.4 ShadowProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
xi