Second-order logic

Abstract: This monograph on classical logic presents fundamental concepts and results in a rigorous mathematical style. Applications to automated theorem proving are considered and usable programs in Prolog are provided. This material can be used both as a first text in formal logic and as an introduction to automation issues, and is intended for those interested in computer science and mathematics at the beginning graduate level. The book begins with propositional logic, then treats first-order logic, and finally, first-order logic with equality. In each case the initial presentation is semantic: Boolean valuations for propositional logic, models for first-order logic, and normal models when equality is added. This defines the intended subjects independently of a particular choice of proof mechanism. Then many kinds of proof procedures are introduced: tableau, resolution, natural deduction, Gentzen sequent and axiom systems. Completeness issues are centered in a model existence theorem, which permits the coverage of a variety of proof procedures without repetition of detail. In addition, results such as compactness, interpolation, and the Beth definability theorem are easily established. Implementations of tableau theorem provers are given in Prolog, and resolution is left as a project for the student.

Abstract: Contains a precise and complete description of the computational logic develo by the authors; will serve also as a reference guide to the associated mechanical theorem proving system. Annotation copyright Book News, Inc. Portland, Or.

Abstract: The model-checking problem for a logic L on a class C of structures asks whether a given L-sentence holds in a given structure in C. In this paper, we give super-exponential lower bounds for fixed-parameter tractable model-checking problems for first-order and monadic second-order logic. We show that unless PTIME=NP, the model-checking problem for monadic second-order logic on finite words is not solvable in time f(k)/spl middot/p(n), for any elementary function f and any polynomial p. Here k denotes the size of the input sentence and n the size of the input word. We prove the same result for first-order logic under a stronger complexity theoretic assumption from parameterized complexity theory. Furthermore, we prove that the model-checking problems for first-order logic on structures of degree 2 and of bounded degree d/spl ges/3 are not solvable in time 2(2/sup o(k)/)/spl middot/p(n) (for degree 2) and 2(2/sup 2o(k)/)/spl middot/p(n) (for degree d), for any polynomial p, again under an assumption from parameterized complexity theory. We match these lower bounds by corresponding upper bounds.

Abstract: One: Logic and Set Theory.- 1.1. First Order Logic.- 1.1.1. Basic Concepts.- 1.1.2. Metalogic.- 1.2. Second Order Logic.- 1.2.1. Basic Concepts.- 1.2.2. The Expressive Power of Second Order Logic.- 1.3. First Order Theories.- 1.3.1. Some Examples of First Order Theories.- 1.3.2. Peano Arithmetics (PA).- 1.4. Zermelo-Fraenkel Set Theory.- 1.4.1. Basic Set Theory.- 1.4.2. The Set Theoretic Universe.- Two: Partial Orders.- 2.1. Universal Algebra.- 2.2. Partial Orders and Equivalence Relations.- 2.3. Chains and Linear Orders.- Three: Semantics with Partial Orders.- 3.1. Instant Tense Logic.- 3.2. Algebraic Semantics, Functional Completeness and Expressibility.- 3.3. Some Linguistic Considerations Concerning Instants.- 3.4. Information Structures.- 3.5. Partial Information and Vagueness.- Four: Constructions with Partial Orders.- 4.1. Period Structures.- 4.2. Event Structures.- Five: Intervals, Events and Change.- 5.1. Interval Semantics.- 5.2. The Logic of Change in Interval Semantics.- 5.3. The Moment of Change.- 5.4. Supervaluations.- 5.5. Kamp's Logic of Change.- Six: Lattices.- 6.1. Basic Concepts.- 6.2. Universal Algebra.- 6.3. Filters and Ideals.- Seven: Semantics with Lattices.- 7.1. Boolean Types.- 7.2. Plurals.- 7.3. Mass Nouns.- Answers To Exercises.- References.