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Logic plays a fundamental role in computer science analogous to that played by calculus in physics and traditional engineering. In software engineering, systems with the ability to perform logical deduction are being applied to the synthesis, verification and transformation of computer programs. In artificial intelligence, such systems are applied to the understanding of natural language and the formation of commonsense and robotic plans. Expert systems, deductive databases, and logic-programming systems may be regarded as rudimentary applications of this new deductive technology.
This book provides a carefully paced introduction to the logical underpinnings of these applications. Exceptionally clear and laced with examples, this book requires no prior knowledge of logic or programming. It interleaves a basic treatment of logic with a discussion of its application in automated deduction. This work provides most readable introduction to the logical concepts and techniques underlying computer programming.
Propositional Logic.
Foundations.
Deductive Tableaux.
Introduction.
The Language.
The Meaning of a Sentence.
Semantic Rules.
Validity.
Universal and Existential Closure.
Valid Sentence Schemata.
Equivalence.
Safe Substitution.
Valid Schemata with Substitution.
Polarity.
Force of Quantifiers.
Quantifier Removal: Intuitive Preview.
Removing Both Forces.
Removing Strict Universal Force.
Removing Strict Existential Force.
Summary of the Skolemization Process.
Unification.
Deductive Tableaux: Notification and Meaning.
Basic Properties.
The Deductive Process.
Rewriting Rule.
Splitting Rules.
Resolution Rule.
Equivalence Rule.
Quantifier-Elimination Rules.
Examples of Complete Proofs.
Definition of a Theory.
Augmenting Theories.
Theory of Strict Partial Orderings.
Theory of Equivalence Relations.
Theory of Equality.
Theory of Weak Partial Orderings.
Theory of Groups.
Theory of Pairs.
Relativized Quantifiers.
Finite Theories.
Equality Rule.
Finite Theories with Equality.
Basic Properties.
The Addition Function.
Multiplication and Exponentiation.
Predecessor and Subtraction.
The Less-than Relation.
The Complete Induction Principle.
Quotient and Remainder.
Proof of Complete Induction.
The Divides Relation.
The Least-Number Principle.
The Theory.
Basic Functions and Relations.
The Decomposition Induction Principle.
The Reverse Function.
The Subtuple Relation.
The Complete Induction Principle.
Nonnegative Integers and Tuples.
The Permutation Relation.
The Ordered Relation.
The Sorting Function.
Quicksort.
Basic Properties.
The Left and Right Function.
The Subtree Relation.
Tuples and Trees.
The Nonnegative Integers.
The Tuples.
The Trees.
Well-Founded Relations.
The Well-Founded Induction Principle.
Use of a Well-Founded Induction.
Lexicographic Relations.
Use of Lexicographic Induction.
The Well-Founded Induction Rule.
Well-Founded Induction Over Pairs.
Deduction Procedures.
Proposition Logic.
Predicate Logic.
Special Theories. 0201548860T04062001