Dual tableaux: foundations, methodology, case studies

540 Pages·2011·2.7 MB·English

by  Orlowska E., Golinska-Pilarek J.

Dual Tableaux: Foundations, Methodology,
Case Studies
TRENDS IN LOGIC
StudiaLogicaLibrary
VOLUME 33
ManagingEditor
RyszardWo´jcicki,InstituteofPhilosophyandSociology,
PolishAcademyofSciences,Warsaw,Poland
Editors
VincentF.Hendricks,DepartmentofPhilosophyandScienceStudies,
RoskildeUniversity,Denmark
DanieleMundici,DepartmentofMathematics“UlisseDini”,
UniversityofFlorence,Italy
EwaOrłowska,NationalInstituteofTelecommunications,
Warsaw,Poland
KristerSegerberg,DepartmentofPhilosophy,UppsalaUniversity,
Sweden
HeinrichWansing,InstituteofPhilosophy,DresdenUniversityofTechnology,
Germany
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ever,thislistisnotexhaustive,moreover,therangeofapplications,comparisons
andsourcesofinspirationisopenandevolvesovertime.
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RyszardWo´jcicki
Forfurthervolumes:
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Ewa Orłowska Joanna Golin´ska-Pilarek
(cid:2)
Dual Tableaux: Foundations,
Methodology, Case Studies
123
Prof.EwaOrłowska JoannaGolin´ska-Pilarek
NationalInstituteof UniversityofWarsaw
Telecommunications andNationalInstitute
Szachowa1 ofTelecommunications
04-894,Warszawa,Poland KrakowskiePrzedmies´cie3
[email protected] 00-927,Warszawa,Poland
[email protected]
ISBN978-94-007-0004-8 e-ISBN978-94-007-0005-5
DOI10.1007/978-94-007-0005-5
SpringerDordrechtHeidelbergLondonNewYork
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To theMemoryof HelenaRasiowa

Preface
TheoriginofdualtableauxgoesbacktothepaperbyHelenaRasiowaandRoman
Sikorski‘OntheGentzentheorem’publishedinFundamentaMathematicaein1960.
Theauthorspresentedacutfreedeductionsystemfortheclassicalfirst-orderlogic
without identity. Since then the deduction systems in the Rasiowa–Sikorski style
havebeenconstructedforagreatvarietyoftheories,rangingfromwellestablished
non-classicallogicssuchasintuitionistic,modal,relevant,andmultiple-valuedlog-
ics, to important applied theories such as, among others, temporal, in particular
intervaltemporallogics, variouslogics of programs,fuzzy logics, logics of rough
sets,theoriesofspatialreasoningincludingregionconnectioncalculus,theoriesof
orderofmagnitudereasoning,andformalconceptanalysis.
Specificmethodologicalprinciplesofconstructionofdualtableauxwhichmake
possiblesuchabroadapplicabilityofthesesystemsare:
(cid:2) First,givenatheory,atruthpreservingtranslationisdefinedofthelanguageof
thetheoryintoanappropriatelanguageofrelations(mostoftenbinary);
(cid:2) Second,a dualtableauis constructedforthisrelationallanguageso thatitpro-
videsadeductionsystemfortheoriginaltheory.
Thismethodology,reflectingtheparadigm‘FormulasareRelations’,enablesusto
represent within a uniform formalism the three basic components of formal sys-
tems:syntax,semantics,anddeductionapparatus.Theessentialobservation,leading
toa relationalformalizationoftheories,isthatastandardrelationalstructure(i.e.,
a Boolean algebra together with a monoid) constitutes a common core of a great
variety of theories. Exhibiting this commoncore on all the three levels of syntax,
semanticsanddeduction,enablesusto createa generalframeworkforrepresenta-
tion,investigationandimplementationoftheories.
Therelationalapproachenablesustobuilddualtableauxinasystematic,mod-
ular way. First, deduction rules are defined for the common relational core of the
theories.These rulesconstitutea basis ofall the relationaldualtableau proofsys-
tems.Next,foranyparticulartheoryspecificrulesareaddedtothebasicsetofrules.
Theyreflectthesemanticconstraintsassumedinthemodelsofthetheory.Asacon-
sequence,we neednotimplementeach deductionsystem fromscratch,we should
onlyextendthebasicsystemwithamodulecorrespondingtothespecificpartofa
theoryunderconsideration.
vii
viii Preface
Relationaldualtableauxarepowerfultoolswhichperformnotonlyverification
ofvalidity(i.e.,verificationoftruthofthestatementsinallthemodelsofatheory)
butoftentheycanalsobeusedforprovingentailment(i.e.,verificationthattruthofa
finitenumberofstatementsimpliestruthofsomeotherstatement),modelchecking
(i.e.,verificationoftruthofastatementinaparticularfixedmodel),andsatisfaction
(i.e.,verificationthatastatementissatisfiedbysomefixedobjectsofamodel).
PartIofthebookisconcernedwiththetwosystemswhichprovideafoundation
forallofthedualtableausystemspresentedin thisbook.InChap.1we recallthe
originalRasiowa–Sikorskisystemandweextendittothesystemforfirst-orderlogic
with identity. We discuss relationshipsof dual tableaux with other deduction sys-
tems, namely, tableau systems, Hilbert-style systems, Gentzen-style systems, and
resolution. In Chaps.2 and 3 classical theories of binary relations and their dual
tableauxarepresented.Itisshownhowdualtableauxofthesetheoriesperformthe
abovementionedtasksofverificationof validity,entailment,modelchecking,and
verification of satisfaction. Some decidable classes of relational formulasare pre-
sentedinthisparttogetherwithdualtableaudecisionprocedures.
Part II is concerned with some non-classical theories of relations. In Chap.4
we present a theory of Peirce algebras and its dual tableau. Peirce algebras pro-
vide a means for representation of interactions between binary relations and sets.
In Chap.5 a theory of fork algebras and its dual tableau are presented. Fork al-
gebras are the algebras of binary relations which, together with all the classical
relationaloperations,haveaspecialoperation,referredtoasforkofrelations.While
therelationaltheoriesofChap.2serveasmeansofrepresentationforpropositional
languages,theforkoperationenablesusatranslationoffirst-orderlanguagesintoa
languageofbinaryrelations.InChap.6wepresentatheoryoftypedrelationsand
itsdualtableau.Thetheoryenablesustorepresentrelationsastheyareunderstood
inrelationaldatabases.Thetheorydealswithrelationsofvariousfinitearitiesand,
moreover,eachrelationhasitstypewhichismeanttobearepresentationofasubset
ofattributesonwhichtherelationisdefined.
In Parts III–V relational formalizations of various theories are presented. In
Part III relational dual tableaux are constructed for modal (Chap.7), intuitionistic
(Chap.8),relevant(Chap.9),andfinitelymany-valued(Chap.10)logics.
PartIVisconcernedwiththemajortheoriesofreasoningwithincompleteinfor-
mation.In Chaps.11and 12we dealwith logicsof roughsets and their relational
dualtableaux.Chapter13presentsa relationaltreatmentof formalconceptanaly-
sis. In Chap.14a monoidalt-normfuzzylogic is consideredand a relationaldual
tableauforthislogicisconstructed.Inthissystemternaryrelationsareneededfor
representationofthemonoidproductoperation.Next,inChap.15theoriesoforder
ofmagnitudereasoningareconsideredandtheirdualtableauxarepresented.
Part V is concerned with dual tableaux for temporal reasoning, spatial reason-
ing,andforlogicsofprograms.Thefirsttwochaptersofthatpartrefertotemporal
logics. In Chap.16 some classical temporal logics are dealt with and in Chap.17
relational dual tableaux for a class of interval temporal logics are presented. In
Chap.18 dualtableauxfor theoriesof spatialreasoningare constructed,including
Preface ix
asystemfortheregionconnectioncalculus.Chapter19includesdualtableauxfor
variousversionsofpropositionaldynamiclogicandforaneventstructurelogic.
In Part VI we consider some theories for which dual tableau systems are con-
structed directly within the theory, without translation into any relational theory.
InChap.20wepresentaclassofthresholdlogicswherebothweightsofformulas
andthresholdsareelementsofacommutativegroup.InChap.21wepresentacon-
structionof a signed dualtableau which is a decisionprocedurefor a well known
intermediatelogic.Chapter22includesdualtableauxforaclassoffirst-orderPost
logics. The reduct of this dual tableau for the propositional part of the logic is a
decision procedure.Chapter 23 presentsa propositionallogic endowedwith iden-
titytreatedasapropositionaloperationandsometheoriesbasedonthislogic.Dual
tableauxforall of these theoriesare presented.In Chap.24logicsandalgebrasof
conditionaldecisionsareconsideredtogetherwith theirdualtableaudecisionpro-
cedures.
The bookconcludeswith PartVII. In the single Chap.25 of this partwe make
a synthesis of what we learned in the process of developing dual tableaux in the
precedingchapters.Wecollectobservationsonhowthedualtableauxrulesshould
bedesignedoncetheconstraintsonthemodelsofthetheoriesordefinitionsofsome
specificconstantsaregiven.Wealsodiscusssomeusefulstrategiesforconstruction
ofdualtableauxproofs.
Allthedualtableausystemsconsideredinthebookareprovedtobesoundand
complete. We presenta generalmethod of provingcompletenessof dualtableaux
whichisshowntobebroadlyapplicabletomanytheories.
Researchersworkinginanyofthetheoriesmentionedinthetitlesofthechapters
willreceiveinthebookaformaltoolofspecificationandverificationofthoseprob-
lems in their theories which involve checking validity, satisfaction, or entailment.
Everytheorywhosedualtableauispresentedinachapterofthebookisbrieflyin-
troducedat the beginningof the chapter and a bibliographyis indicated where an
interested reader could trace developments, major results, and applications of the
theory.
Togetanideaofwhatdualtableauxareandhowtheyarerelatedtotheotherma-
jortypesofdeductionsystems,readingChap.1isrecommended.Afterreadingthe
introductorymaterialfromSects.1.1,...,1.4,andSects.2.1,...,2.8,eachchapter
inPartsIII,IV,andVmaybereadindependently.ThematerialofChap.7maybe
helpfulin reading Chapters 11, 12, 16, 17, and 19, since they are concernedwith
modal-stylelogics.
Readers interested in the formal methods of deduction and their application to
specificationandverificationwillfindinthebookanexhaustiveexpositionanddis-
cussionofdualtableauxandtheirmethodologyillustratedwithseveralcasestudies.