Finitary Semantics of Linear Logic and Higher-Order Model-Checking

• Published in: Mathematical Foundations of Computer Science 2015 pp. 256 - 268
• Main Authors: Grellois, Charles, Melliès, Paul-André
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2015
• Series: Lecture Notes in Computer Science
• Subjects: Finitary semantics; Higher-order model-checking; Linear logic; Parity games; Selection problem
• ISBN: 3662480565; 9783662480564
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-662-48057-1_20

In this paper, we explain how the connection between higher-order model-checking and linear logic recently exhibited by the authors leads to a new and conceptually enlightening proof of the selection problem originally established by Carayol and Serre using collapsible pushdown automata. The main idea is to start from an infinitary and colored relational semantics of the λY\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,Y$$\end{document}-calculus formulated in a companion paper, and to replace it by a finitary counterpart based on finite prime-algebraic lattices. Given a higher-order recursion scheme G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}$$\end{document}, the finiteness of its interpretation in the resulting model enables us to associate to any MSO formula φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} a higher-order recursion scheme Gφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{\varphi }$$\end{document} resolving the selection problem.