{"title":"The Complexity of Proving the Discrete Jordan Curve Theorem","authors":"Phuong Nguyen, S. Cook","doi":"10.1145/2071368.2071377","DOIUrl":"https://doi.org/10.1145/2071368.2071377","url":null,"abstract":"The Jordan Curve Theorem (JCT) states that a simple closed curve divides the plane into exactly two connected regions. We formalize and prove the theorem in the context of grid graphs, under different input settings, in theories of bounded arithmetic that correspond to small complexity classes. The theory V0 (corresponding to AC0(2)) proves that any set of edges that form disjoint cycles divides the grid into at least two regions. The theory V0 (corresponding to AC0) proves that any sequence of edges that form a simple closed curve divides the grid into exactly two regions. As a consequence, the Hex tautologies and the st-Connectivity tautologies have polynomial size AC0(2)-Frege-proofs, which improves results of Buss which only apply to the stronger proof system TC0-Frege.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129442868","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":"Resource modalities in game semantics","authors":"Paul-André Melliès, Nicolas Tabareau","doi":"10.1109/LICS.2007.41","DOIUrl":"https://doi.org/10.1109/LICS.2007.41","url":null,"abstract":"The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of a misleading conception: the belief that linear logic is more primitive than game semantics. We advocate the contrary here: that game semantics is conceptually more primitive than linear logic. Starting from this revised point of view, we design a categorical model of resources in game semantics, and construct an arena game model where the usual notion of bracketing is extended to multi-bracketing in order to capture various resource policies: linear, affine and exponential.","PeriodicalId":137827,"journal":{"name":"22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114081109","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}