Table Of ContentCognitive Technologies
Dov M. Gabbay
Reactive
Kripke
Semantics
Cognitive Technologies
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Dov M. Gabbay
Reactive Kripke Semantics
Dov M. Gabbay
Department of Computer Science
Bar-Ilan University
Israel
King’s College London
UK
University of Luxembourg
Luxembourg
ManagingEditors
Dov M. Gabbay Jörg Siekmann
Department of Computer Science Deutsches Forschungszentrum
Bar-Ilan University für Künstliche Intelligenz (DFKI)
Israel Univers ität desSaarla ndes
Saarbrücken
King’s College London Germany
UK
University of Luxembourg
Luxembourg
ISSN 1611-2482 Cognitive Technologies
ISBN 978-3-642-41388-9 ISBN 978-3-642-41389-6 (eBook)
DOI 10.1007/978-3-642-41389-6
Springer Heidelberg New York Dordrecht London
LibraryofCongressControlNumber:2013954970
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Contents
1 ATheoryofHypermodalLogics ................................. 1
1.1 Introduction ............................................... 1
1.2 Hypermodalities ........................................... 4
1.3 Casestudy:ShiftingbetweenKandT modalities ............... 11
1.4 Translationsofhypermodality ............................... 16
1.5 AxiomatisingHS ........................................... 18
1
1.6 Conclusion:hypermodalityincontext ......................... 27
2 IntroducingReactiveKripkeSemanticsandArcAccessibility....... 29
2.1 Motivationandbackground .................................. 29
2.2 Thereactiveparadigmingeneral ............................. 42
2.3 Connectionwithhyper-modalities ............................ 52
2.4 SwitchreactiveKripkemodels .............................. 54
2.5 Non-deterministicreactiveKripkemodels ..................... 60
2.6 Connectionwithfibringlogics .............................. 63
2.7 Dedicatedreactivityconnectives .............................. 67
3 IntroducingReactiveModalTableaux ............................ 77
3.1 Settingthescene ........................................... 77
3.2 ReactiveBethtableaux...................................... 95
3.3 Conclusion................................................ 99
4 ReactiveIntuitionisticTableaux.................................. 101
4.1 Introduction ............................................... 101
4.2 ReactiveKripkeframesforintuitionisticlogic . ................. 103
4.3 Foldingreactiveframes...................................... 110
4.4 Reactivetableaux .......................................... 115
V
VI Contents
5 CompletenessTheoremsforReactiveModalLogics ................ 119
5.1 Overview ................................................. 119
5.2 IntroducingthelogicKR,reactivemodalK .................... 131
5.3 CompletenesstheoremforKR ............................... 154
5.4 Concludingremarks ........................................ 164
6 ModalLogicsofReactiveFrames ................................ 169
6.1 Introduction ............................................... 169
6.2 Reactivemodels............................................ 173
6.3 Axiomatisations............................................ 177
6.4 Resultsandfinalcomments .................................. 200
7 GlobalViewonReactivity:SwitchGraphsandTheirLogics........ 203
7.1 Introduction ............................................... 203
7.2 Switchgraphs ............................................. 210
7.3 Reactivehybridswitchlogics ................................ 224
8 ReactiveAutomata ............................................. 237
8.1 Introductionandbackground................................. 237
8.2 Reactiveautomata.......................................... 241
8.3 Reactivityandnon-reactivity................................. 245
8.4 Savingstatesofnon-deterministicautomatausingreactivelinks ... 246
8.5 Reducingthenumberofstatesofdeterministicautomata.......... 249
8.6 Examplesofsizereduction................................... 251
8.7 Conclusion................................................ 254
9 ReactivityandGrammars:AnExploration ....................... 257
9.1 Backgroundandmotivation .................................. 257
9.2 Switchinggrammars........................................ 267
9.3 Switchreactivegrammars ................................... 270
9.4 Areductionresult .......................................... 274
9.5 Stringtransformergrammars................................. 279
9.6 Reactivityandderivationstrategies............................ 289
9.7 Embeddedreactivity ........................................ 298
9.8 Reactivityandcontrary-to-duty............................... 302
9.9 Conclusions ............................................... 307
10 ReactiveFlowProducts......................................... 309
10.1 Introductionandorientation.................................. 309
10.2 Formalmodelsofreactiveflowproductsystems................. 328
10.3 Comparisonwithmulti-modallogicsformulti-agentsystems...... 338
Contents VII
11 ReactiveStandardDeonticLogic ................................ 345
11.1 Standarddeonticlogicanditsproblems ....................... 345
11.2 FormalpropertiesofSDLR1................................. 349
11.3 Contrary-to-dutyinSDLR1.................................. 353
11.4 Checkingforparadoxes ..................................... 364
11.5 GeneraltheoryofCTDsinSDLR1............................ 366
11.6 Towardsmoregenerality .................................... 371
11.7 ComparisonwiththeJones–Po¨rnsystemDL2................... 383
11.8 Conclusionandcomparison.................................. 385
12 ReactivePreferentialStructuresandNon-monotonicConsequence .. 389
12.1 Introduction .............................................. 389
12.2 AsemanticsforIBRS ...................................... 399
12.3 IBRSasgeneralizedpreferentialstructures .................... 403
References......................................................... 431
Index ............................................................. 439
Preface
Theconceptofreactivityhasitsrootsintheideasofmybookonfibringlogics[57].
Whenwefibrethetwologics,sayintuitionistic→andmodallogic(cid:2),wecanform
formulasusingbothconnectives,→and(cid:2).Sowecanwritesomethinglike
(cid:2)(A→(cid:2)B).
→ should behave like intuitionistic implication and (cid:2) like a modality, say an S4
modality.Theareaoffibringlogicsdevelopsmethodologiesofhowtogetproperties
of the fibred modal (S4) intuitionistic logic from the properties of the individual
¯
logics involved. We can also fibre two modal logics, say an S4 modality and a K
modality.Wecanwrite
(cid:2) (cid:2) A.
S4 K
Thefirst(cid:2)isanS4modalityandthesecondisaKmodality.Wecanalsowrite
(cid:2)(cid:2)A
providedweagreethatthefirstmodalityisS4andthesecondisK.
Howabout(cid:2)(cid:2)(cid:2)A?Well, weneedanagreementaboutthethirdmodality.Once
wehavesuchanagreement,weknowwhat(cid:2)(cid:2)(cid:2)Ameans.
Imagineforamomentthatwewritearbitraryformulaswithonemodalityonly,
andthatwehaveanagreementthatif(cid:2)isnestedintheformulatoadepthkitmust
behavelikea modality(cid:2) . Say(cid:2) isa Kmodality.(cid:2) isanS4modality,(cid:2) isaT
k 1 2 3
modalityandthenwerepeatinacycle,i.e.(cid:2)3m+n isa(cid:2)n modality,0 ≤ n ≤ 2.Thus
(cid:2)(A → (cid:2)(cid:2)B) is (cid:2) (A → (cid:2) (cid:2) B). We can write axioms for this (cid:2) modality and
K S4 T
show the axiom system to a third party without revealing our interpretation for (cid:2).
Weaskthequestion:whatkindofsemanticsweshouldseekfor(cid:2)?
OurthirdpartywillprobablyproceedinatraditionalmannerandlookforKripke
models(S,R),R⊆S×S andseekconditionsonR.Hedoesnotknowthecomplexity
ofthe interpretationwe hadin mindandwill probablyfailto findsuch conditions.
(Actually,therearelogicsforwhichtherearenocorrespondingconditionsonR.)
So whatkind of semantical interpretationshould we adoptfor (cid:2)? The idea we
cameupwithistothinkof(cid:2)Aastellingustowhichworldstogoinordertocheck
IX
