22c:188 Logic in Computer Science

Course introduction and syllabus overview. Discussion of the basic role of logic in Computer Science: logic as the formal foundation of CS. Applications of logic in CS.

Basic components of a logic: syntax, semantics and inference rules. Logic as a tool for stating and proving arguments formally. Deductive reasoning vs. other forms of reasoning (inductive, abductive, ...).
Introduction to propositional logic. Scope of the logic. The language: propositional symbols and connectives. A natural deduction-style calculus. Some rules of the calculus. Examples of derivations in the calculus.

More on the rules of the natural deduction calculus. Proofs and nested subproofs. More examples.

The complete set of natural deduction rules. Derived rules. Derivation exercises. Formal definition of the language of propositional logic.  

Parse trees for PL formulas. Formal semantics of PL formulas. Truth tables and interpretations. Compositional nature of the PL semantics. The logical entailment relation |= and its properties. Examples. Satisfiable, unsatisfiable, valid and invalid formulas. Relationship between logical entailment and provability in the natural deduction calculus. Soundness and completeness of the calculus. Significance of the calculus' soundness and completeness. Decidability of logical entailment in propositional logic.  

Sketch of soundness proof for natural deduction.
Alternative methods to check for logical entailment. Logical equivalence. Equivalence preserving transformations into a formal form.
A general, equivalence preserving, CNF conversion procedure. Checking the validity of PL formulas by conversion into CNF.  

Examples CNF conversion by means of equivalence preserving rewrite rules.
A general, satisfiability preserving, Definitional Clausal Form conversion procedure. Motivation and examples. Discussion on the complexity of the various transformations.
Introduction to DPLL: a decision procedure for the satisfiability of CNF formulas. Main idea and examples.

Sublanguages of propositional logic: Horn clauses. A linear satisfiability procedure for Horn clauses.
Examples.
Discussion of Hw1 solution.
Abstract description of DPLL by mean of a transition system. The rules of the systems. Examples of applications and strategies.  

More on the abstract DPLL system. Examples of executions.
Improvements to the original DPPL procedure: backjumping, learning and restarts. Motivations, discussion and examples. Discussion of termination and correctness of the DPLL system with backjumping and of the DPLL system extended with learning and restarts with backjumping.

Introduction to First-Order Logic. Domain of discourse, individuals, properties of individuals and relations over them. Quantification over individuals. The basic vocabulary of FOL: variables, constant, function and predicate symbols.

Syntax of FOL. Terms and formulas. Free and bound variables. Substitutions.
A natural deduction calculus for FOL.
Discussion of Homework 2.  

Midterm I

Free and bound variables. Substitutions. Proof rules for universal quantification. Derivation examples.
Discussion of Midterm I.

Proof rules for existential quantification and for equality. Derivation examples.
Quantifier equivalences and examples.

Introduction to FOL semantics. First-order models. Examples of models and formulas satisfied or falsified in them.

Satisfiability relation and logical entailment. The soundness and completeness of natural deduction wrt to logical entailment, and the undecidability of logical entailment.

The deduction and compactness theorems for FOL. Representational power of FOL. Inability of FOL to represent reachability in a graph.

Introduction to temporal logic. Motivation and applications. Syntax and informal semantics of Linear Temporal Logic. LTL Models. Examples of transition systems. Examples of LTL formulas and paths that falsify/satisfy them.  

Regular expressions for denoting infinite paths. Definition and discussion of the satisfiability relation between paths and LTL formulas and between states and LTL formulas. Semantical equivalence in LTL. Examples of equivalent LTL formulas.
Defining temporal operators in terms of others. Minimal subsets of temporal operators. Practice with formalizing properties in LTL.

Semantical equivalence in LTL. Logical properties of the various temporal operators. Defining operators in terms of others. Informal proofs of equivalence.
Applications of LTL: a mutual exclusion protocol. Modeling a two-process system; specifying properties such as safety, liveness and so on.

Discussion of Hw3 solutions.
More on the mutual exclusion protocol example; verifying properties manually against the model. Limits on the expressive power of LTL.  

Midterm II

Introduction to branching time logics and to CTL. Syntax and informal semantics of CTL.
Equivalence of CTL formulas. Significant examples.
Discussion of Midterm II's solutions.

More on Midterm II.
Brief discussion of applications of LTL and CTL in the hardware and software industry, and of model checking tooks like NuSMV.
Formal semantics of CTL.
Modeling English specifications into CTL. Examples of satisfiable/unsatisfiable CTL formulas in a given state of a model.

Informal introduction to CTL* and brief discussion and comparisons with LTL and CTL.
A rule based procedure for checking the satisfiability of CTL formulas in a given model.

Thanksgiving break

Thanksgiving break

An example of applying the rule-based procedure for model checking in CTL.
Brief introduction to LTL model checking with automata.
Discussion of Hw4's solutions.

An automata-based method for checking the satisfiability of LTL formulas in a given model. High-level description of the method and example.  

Modal logics, introduction and motivation. Modalities and their use to reason about necessity, time, knowledge, belief, etc. Connections to temporal logics.
The basic modal logic K. Syntax and semantics. Accessibility relations and Kripke models. Validity and valid formulas in K. Examples of models and satisfied/falsified formulas.

Logic engineering. Modeling various modalities. Kripke frames. Correspondence theorems for modal logics. Examples and informal arguments.
Defining modal logics axiomatically. A natural deduction proof system for K.  

Final exam at 2:15pm