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

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

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

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

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.

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

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)

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

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

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

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

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}…

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