Finite and Algorithmic Model Theory最新文献

Logical aspects of spatial databases 空间数据库的逻辑方面

Finite and Algorithmic Model Theory Pub Date : 1900-01-01 DOI: 10.1017/cbo9780511974960.003

B. Kuijpers, J. V. D. Bussche

引用次数: 1

Model theoretic methods for fragments of FO and special classes of (finite) structures FO碎片和特殊类型(有限)结构的模型理论方法

Finite and Algorithmic Model Theory Pub Date : 1900-01-01 DOI: 10.1017/cbo9780511974960.007

M. Otto

引用次数: 13

Some connections between finite and infinite model theory 有限和无限模型理论之间的一些联系

Finite and Algorithmic Model Theory Pub Date : 1900-01-01 DOI: 10.1017/cbo9780511974960.004

Vera Koponen

{"title":"Some connections between finite and infinite model theory","authors":"Vera Koponen","doi":"10.1017/cbo9780511974960.004","DOIUrl":"https://doi.org/10.1017/cbo9780511974960.004","url":null,"abstract":"Most of the work in model theory has, so far, considered in nite structures and the methods and results that have been worked out in this context can usually not be transferred to the study of nite structures in an obvious way. In addition, some basic results from in nite model theory fail within the context of nite models. The theory about nite structures has largely developed in connection with theoretical computer science, in particular complexity theory [12]. The question arises whether these two \"worlds\", the study of in nite structures and the study of nite structures, can be weaved together in some way and enrich each other. In particular, one may ask if it is possible to adapt notions and methods which have played an important role in in nite model theory to the context of nite structures, and in this way get a better understanding of fairly large and su ciently well-behaved classes of nite structures. If we are to study structures in relation to some formal language, then the question arises which one to choose. Most of in nite model theory considers rst-order logic. Within nite model theory various restrictions and extensions of rst-order logic have been considered, since rst-order logic may be considered as being both too strong and too weak (in different senses) for the study of nite structures. A reasonable candidate for studying nite structures, with a viewpoint from in nite model theory, is the language Ln, rst order logic L restricted to formulas in which at most n variables occur, whether free or bound. Theories consisting of only Ln-formulas, even those which are \"complete\" within Ln, may have both nite and in nite models, or only nite models, or only in nite models. The language Ln has the nice properties of being closed under subformulas, quanti cation and negation. Also, there is a pebble game which distinguishes whether two structures satisfy exactly the same Ln-sentences or not ([23] and implicitly in [29]). The notion of a type plays an important role in in nite model theory. In nite model theory the notion of an Ln-type, i.e. a type restricted to Ln-formulas, has been used; the number of di erent Ln-types of an Ln-theory can be seen as a measure of the complexity of the theory. Dawar observed [5] that for every Ln-theory T with nite models there is an upper bound, depending only on the number of Ln-types (in n free variables) of T , of the size of the smallest model of T . Later Grohe proved that this upper bound is not recursive [17]. The language Ln has also been considered in the context of (only) in nite models in the work of Hedman [19] where complete theories (within full rst-order logic) which are axiomatizable by Ln-sentences are studied. For a general overview about interactions (and di erences) between nite and in nite model theory, see [30]. For a survey about the use of nite variable logics in nite model theory, see [16]. Within in nite model theory the area of stability theory has had great in uence. It studi","PeriodicalId":423487,"journal":{"name":"Finite and Algorithmic Model Theory","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123545866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Automata-based presentations of infinite structures 无限结构的自动机表示

Finite and Algorithmic Model Theory Pub Date : 1900-01-01 DOI: 10.1017/CBO9780511974960.002

V. Bárány, E. Grädel, S. Rubin

{"title":"Automata-based presentations of infinite structures","authors":"V. Bárány, E. Grädel, S. Rubin","doi":"10.1017/CBO9780511974960.002","DOIUrl":"https://doi.org/10.1017/CBO9780511974960.002","url":null,"abstract":"The model theory of finite structures is intimately connected to various fields in computer science, including complexity theory, databases, and verification. In particular, there is a close relationship between complexity classes and the expressive power of logical languages, as witnessed by the fundamental theorems of descriptive complexity theory, such as Fagin’s Theorem and the ImmermanVardi Theorem (see [78, Chapter 3] for a survey). However, for many applications, the strict limitation to finite structures has turned out to be too restrictive, and there have been considerable efforts to extend the relevant logical and algorithmic methodologies from finite structures to suitable classes of infinite ones. In particular this is the case for databases and verification where infinite structures are of crucial importance [130]. Algorithmic model theory aims to extend in a systematic fashion the approach and methods of finite model theory, and its interactions with computer science, from finite structures to finitely-presentable infinite ones. There are many possibilities to present infinite structures in a finite manner. A classical approach in model theory concerns the class of computable structures; these are countable structures, on the domain of natural numbers, say, with a finite collection of computable functions and relations. Such structures can be finitely presented by a collection of algorithms, and they have been intensively studied in model theory since the 1960s. However, from the point of view of algorithmic model theory the class of computable structures is problematic. Indeed, one of the central issues in algorithmic model theory is the effective evaluation of logical formulae, from a suitable logic such as, for instance, first-order logic (FO), monadic second-order logic (MSO), or a fixed point logic like LFP or the modal μ-calculus. But on computable structures, only the quantifier-free formulae generally admit effective evaluation, and already the existential fragment of first-order logic is undecidable, for instance on the computable structure (N,+, · ). This leads us to the central requirement that for a suitable logic L (depending on the intended application) the model-checking problem for the class C of finitely presented structures should be algorithmically solvable. At the very","PeriodicalId":423487,"journal":{"name":"Finite and Algorithmic Model Theory","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115746150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 26