{"title":"Logical complexity of graphs: a survey","authors":"O. Pikhurko, O. Verbitsky","doi":"10.1090/conm/558/11050","DOIUrl":"https://doi.org/10.1090/conm/558/11050","url":null,"abstract":"We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth $D(G)$ of a graph $G$ is equal to the minimum quantifier depth of a sentence defining $G$ up to isomorphism. The logical width $W(G)$ is the minimum number of variables occurring in such a sentence. The logical length $L(G)$ is the length of a shortest defining sentence. We survey known estimates for these graph parameters and discuss their relations to other topics (such as the efficiency of the Weisfeiler-Lehman algorithm in isomorphism testing, the evolution of a random graph, quantitative characteristics of the zero-one law, or the contribution of Frank Ramsey to the research on Hilbert's Entscheidungsproblem). Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are allowed) or more expressible (after powering with counting quantifiers).","PeriodicalId":110641,"journal":{"name":"AMS-ASL Joint Special Session","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123020209","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}
{"title":"Spectra and Systems of Equations","authors":"J. Bell, S. Burris, K. Yeats","doi":"10.1090/conm/558/11048","DOIUrl":"https://doi.org/10.1090/conm/558/11048","url":null,"abstract":"In a previous work we introduced an elementary method to analyze the periodicity of a generating function defined by a single equation y=G(x,y). This was based on deriving a single set-equation Y = Gammma(Y) defining the spectrum of the generating function. This paper focuses on extending the analysis of periodicity to generating functions defined by a system of equations y = G(x,y). \u0000The final section looks at periodicity results for the spectra of monadic second-order classes whose spectrum is determined by an equational specification - an observation of Compton shows that monadic-second order classes of trees have this property. This section concludes with a substantial simplification of the proofs in the 2003 foundational paper on spectra by Gurevich and Shelah, namely new proofs are given of: \u0000(1) every monadic second-order class of $m$-colored functional digraphs is eventually periodic, and \u0000(2) the monadic second-order theory of finite trees is decidable.","PeriodicalId":110641,"journal":{"name":"AMS-ASL Joint Special Session","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122812615","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}
{"title":"Application of logic to combinatorial sequences and their recurrence relations","authors":"E. Fischer, Tomer Kotek, J. Makowsky","doi":"10.1090/conm/558/11047","DOIUrl":"https://doi.org/10.1090/conm/558/11047","url":null,"abstract":"2010 Mathematics Subject Classification. 03-02, 03C98, 05-02, 05A15, 11B50 . Partially supported by an ISF grant number 1101/06 and an ERC-2007-StG grant number 202405. Partially supported by the Fein Foundation and the Graduate School of the Technion Israel Institute of Technology. Partially supported by the Israel Science Foundation for the project “Model Theoretic Interpretations of Counting Functions” (2007-2010) and the Grant for Promotion of Research by the Technion–Israel Institute of Technology.","PeriodicalId":110641,"journal":{"name":"AMS-ASL Joint Special Session","volume":"7 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":"122794531","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}
{"title":"Methods for Algorithmic Meta Theorems","authors":"Martin Grohe, S. Kreutzer","doi":"10.1090/conm/558/11051","DOIUrl":"https://doi.org/10.1090/conm/558/11051","url":null,"abstract":"Algorithmic meta-theorems state that certain families of algorithmic problems, usually defined in terms of logic, can be solved efficiently. This is a survey of algorithmic meta-theorems, highlighting the general methods available to prove such theorems rather than specific results.","PeriodicalId":110641,"journal":{"name":"AMS-ASL Joint Special Session","volume":"53 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":"131823392","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}
{"title":"(Un)countable and (Non)effective Versions of Ramsey's Theorem","authors":"D. Kuske","doi":"10.1090/conm/558/11057","DOIUrl":"https://doi.org/10.1090/conm/558/11057","url":null,"abstract":"We review Ramsey’s theorem and its extensions by Jockusch for computable partitions, by Sierpiński and by Erdős and Rado for uncountable homogeneous sets, by Rubin for automatic partitions, and by the author for ω-automatic (in particular uncountable) partitions.","PeriodicalId":110641,"journal":{"name":"AMS-ASL Joint Special Session","volume":"47 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":"126830378","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}
{"title":"Compton's Method for Proving Logical Limit Laws","authors":"J. Bell, S. Burris","doi":"10.1090/conm/558/11049","DOIUrl":"https://doi.org/10.1090/conm/558/11049","url":null,"abstract":"Developments in the study of logical limit laws for both labelled and unlabelled structures, based on the methods of Compton (1987/1989), are surveyed, and a sandwich theorem is proved for multiplicative systems.","PeriodicalId":110641,"journal":{"name":"AMS-ASL Joint Special Session","volume":"50 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":"121200677","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}
{"title":"Partitions and Permutation Groups","authors":"A. Blass","doi":"10.1090/conm/558/11060","DOIUrl":"https://doi.org/10.1090/conm/558/11060","url":null,"abstract":"","PeriodicalId":110641,"journal":{"name":"AMS-ASL Joint Special Session","volume":"42 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":"126806052","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}
{"title":"Two Problems on Homogeneous Structures, Revisited","authors":"G. Cherlin","doi":"10.1090/conm/558/11055","DOIUrl":"https://doi.org/10.1090/conm/558/11055","url":null,"abstract":"We take up Peter Cameron's problem of the classification of count- ably infinite graphs which are homogeneous as metric spaces in the graph metric (Cam98). We give an explicit catalog of the known examples, together with results supporting the conjecture that the catalog may be complete, or nearly so. We begin in Part I with a presentation of Fra¨osse's theory of amalgamation classes and the classification of homogeneous structures, with emphasis on the case of homogeneous metric spaces, from the discovery of the Urysohn space to the connection with topological dynamics developed in (KPT05). We then turn to a discussion of the known metrically homogeneous graphs in Part II. This includes a 5-parameter family of homogeneous metric spaces whose connections with topological dynamics remain to be worked out. In the case of diameter 4, we find a variety of examples buried in the tables at the end of (Che98), which we decode and correlate with our catalog. In the final Part we revisit an old chestnut from the theory of homoge- neous structures, namely the problem of approximating the generic triangle free graph by finite graphs. Little is known about this, but we rephrase the problem more explicitly in terms of finite geometries. In that form it leads to questions that seem appropriate for design theorists, as well as some ques- tions that involve structures small enough to be explored computationally. We also show, following a suggestion of Peter Cameron (1996), that while strongly regular graphs provide some interesting examples, one must look beyond this class in general for the desired approximations.","PeriodicalId":110641,"journal":{"name":"AMS-ASL Joint Special Session","volume":"62 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":"131765586","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}
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