Download Lecture Slides on Branching- Time Temporal and Syntax of CTL | CMSC 630 and more Study notes Computer Science in PDF only on Docsity! CMSC 630 February 27, 2007 1' & $ % Branching-Time Temporal Logic Recall: • Two kinds of temporal logic: linear-time and branching-time • Linear-time: models are sequences • Branching-time: models are “trees” (alternatively, states in Kripke structures) How do we define a branching-time temporal logic? c©2007 Rance Cleaveland. All rights reserved. CMSC 630 February 27, 2007 2' & $ % CTL∗ = LTL + Path Quantifiers The CTL∗ approach: add “path quantifiers” to LTL! Eφ: satisfied by a state if there exists a path from the state and satisfying φ. E is a path quantifier. c©2007 Rance Cleaveland. All rights reserved. CMSC 630 February 27, 2007 5' & $ % Defining Semantics of CTL∗ ... given wrt Kripke structure M = 〈S,A, R, ℓ, sI〉 as relation |=M ⊆ (S × ΣCTL∗) ∪ (S ω × ΦCTL∗) ... i.e. states related to state formulas and paths to path formulas. Fix M in what follows. Notation E(M, s) ⊆ Sω : execution paths emanating from s. π ∈ E(M, s) if: • π[0] = s • For all i ≥ 0, 〈π[i], π[i + 1]〉 ∈ R or for all s′ ∈ S, 〈π[i], s′〉 6∈ R and π[i + 1] = π[i]. c©2007 Rance Cleaveland. All rights reserved. CMSC 630 February 27, 2007 6' & $ % Semantics of State Formulas s |=M a if a ∈ ℓ(s). s |=M ¬σ if s 6|=M σ. s |=M σ1 ∨ σ2 if s |=M σ1 or s |=M σ2. s |= Eφ if there exists π ∈ E(M, s) such that ℓ(π) |=M φ. c©2007 Rance Cleaveland. All rights reserved. CMSC 630 February 27, 2007 7' & $ % Semantics of Path Formulas π |=M σ if π[0] |=M σ. π |=M ¬φ if π 6|=M φ. π |=M φ1 ∨ φ2 if π |=M φ1 or π |=M φ2. π |=M Xφ if π[1..] |=M φ. π |=M φ1 U φ2 if ∃ i ≥ 0. σ[i..] |= φ2 and ∀ j < i. σ[j..] |= φ1 Note: ¬,∨, X, U as in LTL! Derived operators: usual LTL derived operators, A ≡ ¬E¬. c©2007 Rance Cleaveland. All rights reserved. CMSC 630 February 27, 2007 10' & $ % CTL ... ... a sublogic of CTL∗ ... development preceded that of CTL∗ ... stands for “Computation Tree Logic” ... first temporal logic used in model checking CTL formulas: state formulas in CTL∗ in which every path modality (U, X, G, etc.) is immediately preceded by a path quantifier. CTL AG(sent ⇒ AF received) Not CTL AG(sent ⇒ F received) c©2007 Rance Cleaveland. All rights reserved. CMSC 630 February 27, 2007 11' & $ % Formal CTL Syntax Note All CTL formulas are state formulas. σ ::= a | ¬σ | σ ∨ σ | EX σ | E(σ U σ) | E(σ R σ) ΣCTL: set of all CTL formulas. Recall that R is dual of U. Derived operators: AX, AU, AR, EG, EF, AG, AF, etc. c©2007 Rance Cleaveland. All rights reserved. CMSC 630 February 27, 2007 12' & $ % Expressiveness of CTL System Specifications • No more expressive than CTL∗, since ΣCTL ⊆ ΣCTL∗ . • Incomparable to LTL: – CTL can express possibility properties. – LTL can express “fairness” properties. AFG a expressible in LTL, not in CTL. • Consequently, strictly less expressive than CTL∗! (Why?) c©2007 Rance Cleaveland. All rights reserved.
