Publication: Tableaux approaches for region based theories of space
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Date
2016-09
Authors
Özdemir, Zafer
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Publisher
İstanbul Kültür Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Anabilim Dalı / Matematiğin Temelleri ve Matematiksel Lojik Bilim Dalı
Abstract
Uzay n b olgeye dayal teorilerini i cin tablo yakla s m n inceledi gimiz bu tez be s b ol umden olu smaktad r. Tezin ilk k sm nda problemin tan m ve uygulanan y ontemler verilmi stir. Tezin ikinci ve u c unc u k sm nda uzay n b olgeye dayal teorisinin sentaks, semantik ve aksiyomatik ozellikleri tan t larak uzay n b olgeye dayal teorisi i cin tablo kurallar verilmi s ve tablo y onteminin temel kavramlar olan; ba slang c tablosu, a c k ve kapal tablo, dal, d u g um kavramlar ifade edilerek tablo kurallar n n uygulamalar orneklerle ayr nt l olarak a c klanm st r. Tablo kurallar n n sonlanma ve sa glaml k teoremleri ve kan tlar verilmi stir. Ayr ca tablo kurallar i cin taml k teoreminin kan t nda kullan lan sistematik tablo in sa y ontemi, do gruluk lemmas ve taml k teoremleri ve ispat lar verilmi stir. Ek olarak, uzay n b olgeye dayal teorisinin modellerini geni sleterek simetrik, yans mal ve ge ci smeli olmas durumunda tablo kurallar tan mlanarak, sonlanma, sa glaml k ve taml k teoremlerinin kan t nda kullan lan yard mc teoremler ve bu teoremlerin kan tlar ayr nt l olarak verilmi stir. Tezin d ord unc u k sm nda, ba g nt l mant klar n bir geni slemesi olan evrensel modalite i ceren ba g nt l mant klar n; sentaks, semantik ve aksiyomatik ozellikleri tan t larak, tablo kurallar verilmi s ve uygulamalar orneklerle ayr nt l olarak a c klanm st r. Ard ndan tablo kurallar n n sonlanma ve sa glaml k teoremleri ve kan tlar verilmi stir. Ayr ca tablo kurallar i cin do gruluk lemmas ve taml k teoreminin kan tlar verilmi stir. Be sinci b ol umde, ba g nt l mant klar n farkl bir semanti gi olan reel say aral klar uzerindeki yorumu uzerine cal s lm s ve bu ba glamda; sentaks, semantik ve aksiyomatik ozellikleri tan t larak, tablo kurallar verilmi s ve tablo kurallar n n uygulamalar orneklerle ayr nt l olarak a c klanm st r. Ek olarak, tablo kurallar n n sonlanma ve sa glaml k teoremleri kan tlar ile verilmi stir. Ayr ca tablo kurallar i cin do gruluk lemmas ve taml k teoreminin kan t verilmi stir.
Description
In this study, we examined tableaux approaches for region based theories of space. This thesis consists of ve chapters. In the rst chapter is devoted to statement of the problem, review of contents, methods applied. In Chapter 2 and Chapter 3, we introduce region based theories of space. In Chapter 2, we introduce historical background, syntax and semantics, axiomatization of the theory. In semantics subsection, we present relational semantics and topological semantics. Following sections continue with applications and de nability result in region based theories of space. In Chapter 3, we study tableau approaches for region based theories of space, we give basic de nitions about tableau approaches, we prove termination, soundness and completeness theorems. In last section of Chapter 3, we give tableaux rules for variants of region based theories of space. In its subsections, we prove soundness and completeness theorem for variants. In Chapter 4, we presents generalized contact logics. The rst section of Chapter 4 consists of syntax-semantics, de nability and axiomatizationn. The chapter continue with general tableau approaches for generalized contact logics. Section 4.4 and 4.3, consist of soundness-completeness theorems and proofs. In the end of Chapter 4, we give tableau rules for variants. And then we give soundness-completeness theorem and proofs. Chapter 6 is devoted to the study of interval semantics of contact logics and its tableau approaches. In particular, we give syntax and semantics. After that we present tableau rules for contact logics interpreted over intervals. The last two sections are about soundness and completeness theorems. We give proofs of these theorems.
Keywords
Matematik, Mathematics