Chapter 2: Analysis and Verification of Non-Real-Time Systems

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Chapter 2 Analysis and Verification of
Non-Real-Time Systems

• Albert M. K. Cheng

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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

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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

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Propositional Logic

• Example P sensor detects intruder
• Q alarm sounds
• R police is alerted
• We can state P -gt Q and Q -gt R
• We can show that P -gt R

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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.

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Resolution Principle - Example

• P or Q
• Q or R or S
• resolvent is P or R or S

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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)

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Predicate Logic

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

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Proving Unsatisfiability of a Clause Set using
the Resolution Procedure

• Substitution, variant
• Unification, unifier
• Resolvent
• Resolution theorem

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Languages and their Representations

• Regular expressions
• Deterministic finite automaton (DFA)
• Nondeterministic finite automaton (NFA)

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Example 1 Untimed automaton representing climate
control unit
S_4
S_1
cold
turn_on_ac
hot
turn_on_heater
S_0
S_5
S_2
turn_off_heater
comfort
comfort
turn_off_ac
S_6
S_3
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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 greenyellowredgreenyellow
• red

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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