1. Chapter 2: Analysis and Verification of Non-Real-Time Systems
Albert M. K. Cheng

2. Specification, Analysis, and Verification of Untimed Systems
Many approaches for proving correctness are based on symbolic logic and/or languages and automata
Performance is often studied via mathematical modeling, simulation, and runtime measurements
Analysis and verification techniques for real-time systems are often based on or extensions of their untimed counterparts

3. Symbolic Logic
Propositional logic (zero-order logic): write declarative sentences called proposition that can be either true (T) or false (F). Example use: specifying and verifying digital logic circuits
Predicate logic (first-order logic): allows the use of quantifiers to indicate for which values the specified sentence is true. Example use: specifying and verifying computer programs

4. Propositional Logic Example: P “sensor detects intruder”
Q “alarm sounds”
R “police is alerted”
We can state P -> Q and Q -> R
We can show that P -> R

5. Resolution Principle for Propositional Logic
For any two clauses C_1 and C_2, if there is a literal L_1 in C_1 and there is a literal L_2 in C_2 such that “L_1 and L_2” is false, then the resolvent of C_1 and C_2 is the disjunction of the remaining clauses in C_1 and C_2 after removing L_1 and L_2 from C_1 and C_2, respectively.

6. Resolution Principle - Example
P or Q
~Q or R or ~S
resolvent is P or R or ~S

7. Verification using Automata
Given:
Implementation automaton A_I
Specification automaton A_S
Verify:
A_I satisfies A_S by showing that the language accepted by A_I is a subset of the language accepted by A_S, that is,
L(A_I) is subset of L(A_S)

8. Predicate Logic Function, terms, predicates
Atoms, bound and free variables
Interpretation
Closed formula
Satisfiable formula
Valid formula
Prenex normal form
Skolem standard form

9. Proving Unsatisfiability of a Clause Set using the Resolution Procedure
Substitution, variant
Unification, unifier
Resolvent
Resolution theorem

10. Languages and their Representations
Regular expressions
Deterministic finite automaton (DFA)
Nondeterministic finite automaton (NFA)

11. Example 1: Untimed automaton representing climate control unit
cold
turn_on_ac
turn_on_heater
hot
S_0
S_5
S_2
turn_off_heater
comfort
comfort
turn_off_ac
S_6
S_3

12. Untimed Process
Pair (E,S), where E is the event set and S is the set of possible traces
Example: Traffic light
event set = {green, yellow, red}
trace = {green}{yellow}{red}{green}{yellow}
{red}…

13. Untimed Trace Linear sequence of observable events of a process:
p bar = p_1 p_2 p_3 …
infinite word over nonempty subsets of event set