Fragments Of First Order Logic

A sentence of first-order logic is satisfiable if it is true in some structure, and finitely satisfiable if it is true in some finite structure. For which fragments of first-order logic is there an effective method for determining satisfiability or finite satisfiability? Furthermore, if these problems are decidable for a particular fragment, what is their computational complexity? This book provides an up-to-date survey of the principal axes of research into these questions. Part I focusses on fragments defined by restricting the set of available formulas. Starting with the Aristotelian syllogistic and its relatives, we proceed to consider the two-variable fragment, the guarded fragment, the quantifier-prefix fragments and the fluted fragment. Part II investigates logics with counting quantifiers. We begin with De Morgan's numerical generalization of the Aristotelian syllogistic, before giving a detailed treatment of the two-variable fragment with counting quantifiers and its guarded subfragment, explaining the application of the latter to ontology-based query answering. Part III concerns logics characterized by semantic constraints, limiting the available interpretations of certain predicates. The origins of this idea lie in propositional modal logic, and therefore we start with a survey of modal and graded modal logics. We then investigate two-variable first-order logic in which certain distinguished binary predicates are interpreted as equivalence relations or transitive relations, extending these results to incorporate counting quantifiers. We finish, slightly breaching the bounds of first-order logic proper, with a chapter on logics interpreted over trees.

A sentence of first-order logic is satisfiable if it is true in some structure, and finitely satisfiable if it is true in some finite structure. The question arises as to whether there exists an algorithm for determining whether a given formula of first-order logic is satisfiable, or indeed finitely satisfiable. This question was answered negatively in 1936 by Church and Turing (for satisfiability) and in 1950 by Trakhtenbrot (for finite satisfiability).In contrast, the satisfiability and finite satisfiability problems are algorithmically solvable for restricted subsets—-or, as we say, fragments—-of first-order logic, a fact which is today of considerable interest in Computer Science. This book provides an up-to-date survey of the principal axes of research, charting the limits of decision in first-order logic and exploring the trade-off between expressive power and complexity of reasoning. Divided into three parts, the book considers for which fragments of first-order logic there is an effective method for determining satisfiability or finite satisfiability. Furthermore, if these problems are decidable for some fragment, what is their computational complexity? Part I focusses on fragments defined by restricting the set of available formulas. Topics covered include the Aristotelian syllogistic and its relatives, the two-variable fragment, the guarded fragment, the quantifier-prefix fragments and the fluted fragment. Part II investigates logics with counting quantifiers. Starting with De Morgan's numerical generalization of the Aristotelian syllogistic, we proceed to the two-variable fragment with counting quantifiers and its guarded subfragment, explaining the applications of the latter to the problem of query answering in structured data. Part III concerns logics characterized by semantic constraints, limiting the available interpretations of certain predicates. Taking propositional modal logic and graded modal logic as our cue, we return to the satisfiability problem for two-variable first-order logic and its relatives, but this time with certain distinguished binary predicates constrained to be interpreted as equivalence relations or transitive relations. The work finishes, slightly breaching the bounds of first-order logic proper, with a chapter on logics interpreted over trees.

This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Researchers in areas ranging from economics to computational linguistics have since realised its worth. The book is for novices and for more experienced readers, with two distinct tracks clearly signposted at the start of each chapter. The development is mathematical; prior acquaintance with first-order logic and its semantics is assumed, and familiarity with the basic mathematical notions of set theory is required. The authors focus on the use of modal languages as tools to analyze the properties of relational structures, including their algorithmic and algebraic aspects, and applications to issues in logic and computer science such as completeness, computability and complexity are considered. Three appendices supply basic background information and numerous exercises are provided. Ideal for anyone wanting to learn modern modal logic.

The refereed proceedings of the 14th International Conference on Rewriting Techniques and Applications, RTA 2003, held in Valencia, Spain in June 2003. The 26 revised regular papers and 6 system descriptions presented together with 3 invited contributions were carefully reviewed and selected from 61 submissions. All current aspects of rewriting are addressed.

The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both well-written expository articles, suitable for beginners approaching the subject for the first time, and advanced articles, which will help those already familiar with the field to deepen their expertise. Please visit: http://people.uleth.ca/~woods/RedSeriesPromo_WP/PubSLPR.html - Compact modal logic reference - Computational approaches fully discussed - Contemporary applications of modal logic covered in depth

This book offers a comprehensive treatment of the classical decision problem of mathematical logic and of the role of the classical decision problem in modern computer science. The text presents a revealing analysis of the natural order of decidable and undecidable cases and includes a number of simple proofs and exercises.

This book collects the refereed proceedings of the 8th Indian Conference on Logic and Its Applications, ICLA 2019, held in Delhi, India, in March 2019. The volume contains 13 full revised papers along with 6 invited talks presented at the conference. The aim of this conference series is to bring together researchers from a wide variety of fields in which formal logic plays a significant role. Areas of interest include mathematical and philosophical logic, computer science logic, foundations and philosophy of mathematics and the sciences, use of formal logic in areas of theoretical computer science and artificial intelligence, logic and linguistics, and the relationship between logic and other branches of knowledge. Of special interest are studies in systems of logic in the Indian tradition, and historical research on logic.

This volume contains the papers presented at the Sixth International Conference on Logic for Programming and Automated Reasoning (LPAR'99), held in Tbilisi, Georgia, September 6-10, 1999, and hosted by the University of Tbilisi. Forty-four papers were submitted to LPAR'99. Each of the submissions was reviewed by three program committee members and an electronic program com mittee meeting was held via the Internet. Twenty-three papers were accepted. We would like to thank the many people who have made LPAR'99 possible. We are grateful to the following groups and individuals: to the program committee and the additional referees for reviewing the papers in a very short time, to the organizing committee, and to the local organizers of the INTAS workshop in Tbilisi in April 1994 (Khimuri Rukhaia, Konstantin Pkhakadze, and Gela Chankvetadze). And last but not least, we would like to thank Konstantin - rovin, who maintained the program committee Web page; Uwe Waldmann, who supplied macros for these proceedings and helped us to install some programs for the electronic management of the program committee work; and Bill McCune, who implemented these programs.