Is "true U Φ" a valid Computation Tree Logic formula?

Question

I'm taking a course on computer-aided verification, and we are just starting to cover Computational Tree Logic after covering Linear Temporal Logic. My lecture has said that ◇Φ ≡ "true U Φ" is valid CTL, whereas □Φ is not valid CTL. I agree with the second part, because a CTL formula is a Φ and the rules for Φ do not include □Φ. But they also don't include ◇Φ or "true U Φ" - only the rules for Ψ include those, and the rules for Φ state that any Ψ must be preceded by ∃ or ∀, and neither ◇Φ or "true U Φ" are. Wikipedia seems to agree with me on this.

Has he just made a mistake, or am I missing something here? Image of the rules for CTL we have been given

Answer 1

I would dare say that none of those formulas belong to CTL. Perhaps there might have been a misunderstanding, it would be worth contacting your lecturer privately asking about this issue (in a polite manner).

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Let me define a mapping between your notation and the one I've been taught.

◇ := F ~ eventually
□ := G ~ globally
◯ := X ~ next
∀ := A ~ necessarily
∃ := E ~ possibly

The LTL formula ◇ϕ (Fϕ) states that ϕ is necessarily going to hold in the future. The corresponding CTL formulation would be ∀◇ϕ (AFϕ).

The LTL formula □ϕ (Gϕ) states that ϕ is globally holding forever. The corresponding CTL formulation would be ∀□ϕ (AGϕ).

The CTL until operator can appear in one of two forms:

• ∀(ϕ1 U ϕ2): in all possible execution paths, ϕ1 holds until ϕ2 holds (at least once).

• ∃(ϕ1 U ϕ2): in at least one execution path, ϕ1 holds until ϕ2 holds (at least once).

See also: CTL operators.