COS3761
Formal Logic III

Logic in Computer Science: Modelling and Reasoning about Systems

(2th edition)

by Michael Huth and Mark Ryan

1. Propositional logic
1.1. Declarative sentences
1.2. Natural deduction
1.2.1. Rules for natural deduction
1.2.2. Derived rules
1.2.3. Natural deduction in summary
1.2.5. An aside: proof by contradiction
1.3. Propositional logic as a formal language
1.4. Semantics of propositional logic
1.4.1. The meaning of logical connectives
1.4.2. Mathematical induction
1.4.3. Soundness of propositional logic
1.4.4. Completeness of propositional logic
1.5. Normal forms
1.5.1. Semantic equivalence, satisfiability and validity
1.5.2. Conjunctive normal forms and validity
1.5.3. Horn clauses and satisfiability
1.6. SAT solvers
1.6.1. A linear solver
1.6.2. A cubic solver
1.7. Exercises
1.8. Bibliographic notes
2. Predicate Logic
2.1. The need for a richer language
2.2. Predicate logic as a formal language
2.2.1. Terms
2.2.2. Formulas
2.2.3. Free and bound variables
2.2.4. Substitution
2.3. Natural deduction for predicate logic
2.3.1. Natural deduction rules
2.3.2. Quantifier equivalences
2.4. Semantics of predicate logic
2.4.1. Models
2.4.2. Semantic entailment
2.4.3. The semantics of equality
2.5. Undecidability of predicate logic
2.6. Expressiveness of predicate logic
2.6.1. Existential second-order logic
2.6.2. Universal second-order logic
2.7. Micromodels of software
2.7.1. State machines
2.7.2. Alma – re-visited
2.7.3. A software micromodel
2.8. Exercises
2.9. Bibliographic notes
3. Verification by Model Checking
3.1. Motivation for verification
3.2. Linear-time temporal logic
3.2.1. Syntax of LTL
3.2.2. Semantics of LTL
3.2.3. Practical patterns of specifications
3.2.4. Important equivalences between LTL formulas
3.2.5. Adequate sets of connectives for LTL
3.3. Model checking: systems, tools, properties
3.3.1. Example: mutual exclusion
3.3.2. The NuSMV model checker
3.3.3. Running NuSMV
3.3.4. Mutual exclusion revisited
3.3.5. The ferryman
3.3.6. The alternating bit protocol
3.4. Branching-time logic
3.4.1. Syntax of CTL
3.4.2. Semantics of computation tree logic
3.4.4. Important equivalences between CTL formulas
3.4.5. Adequate sets of CTL connectives
3.5. CTL and the expressive powers of LTL and CTL
3.5.1. Boolean combinations of temporal formulas in CTL
3.6. Model-checking algorithms
3.6.1. The CTL model-checking algorithm
3.6.2. CTL model checking with fairness
3.6.3. The LTL model-checking algorithm
3.7. The fixed-point characterisation of CTL
3.7.1. Monotone functions
3.7.2. The correctness of SATEG
3.7.3. The correctness of SATEU
3.8. Exercises
3.9. Bibliographic notes
4. Program verification
4.1. Why should we specify and verify code?
4.2. A framework for software verification
4.2.1. A core programming language
4.2.2. Hoare triples
4.2.3. Partial and total correctness
4.2.4. Program variables and logical variables
4.3. Proof calculus for partial correctness
4.3.1. Proof rules
4.3.2. Proof tableaux
4.3.3. A case study: minimal-sum section
4.4. Proof calculus for total correctness
4.5. Programming by contract
4.6. Exercises
4.7. Bibliographic notes
5. Modal logics and agents
5.1. Modes of truth
5.2. Basic modal logic
5.2.1. Syntax
5.2.2. Semantics
5.3. Logic engineering
5.3.1. The stock of valid formulas
5.3.2. Important properties of the accessibility relation
5.3.3. Correspondence theory
5.3.4. Some modal logics
5.4. Natural deduction
5.5. Reasoning about knowledge in a multi-agent system
5.5.1. Some examples
5.5.2. The modal logic KT45n
5.5.3. Natural deduction for KT45n
5.5.4. Formalising the examples
5.6. Exercises
5.7. Bibliographic notes
6. Binary decision diagrams
6.1. Representing boolean functions
6.1.1. Propositional formulas and truth tables
6.1.2. Binary decision diagrams
6.1.3. Ordered BDDs
6.2. Algorithms for reduced OBDDs
6.2.1. The algorithm reduce
6.2.2. The algorithm apply
6.2.4. The algorithm exists
6.2.5. Assessment of OBDDs
6.3. Symbolic model checking
6.3.1. Representing subsets of the set of states
6.3.2. Representing the transition relation
6.3.4. Synthesising OBDDs
6.4. A relational mu-calculus
6.4.1. Syntax and semantics
6.5. Exercises
6.6. Bibliographic notes