Questions tagged [linear-temporal-logic]
LTL (linear temporal logic or linear-time temporal logic) is a temporal logic that can encode assertions about the future of traces.
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How do I prove relations of two CTL formulas?
If I have two CTL equations, how do I prove they're equivalent or that one implies the other?
What's the general approach? Disproving is obvious, but I am unable to figure out how to prove the ...
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About next operator duality in LTL
I'm learning LTL (linear temporal logic) and we learned that a model $M$ and a starting position $q_0$ satisfies a LTL formula $\psi$ iff for every path $\pi$ starting at $q_0$ it is true that $\pi \...
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Errors in examples of Vardi's paper "Linear Temporal Logic and Linear Dynamic Logic on Finite Traces"
The paper Linear Temporal Logic and Linear Dynamic Logic on Finite Traces has the following examples on page 4:
Q1. (Update to Q1: solved. See the comment by DCTLib.) The first example says that the ...
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Is first order linear temporal logic a special case of first order modal logic?
Propositional linear temporal logic is a special case of propositional modal logic. Is first order linear temporal logic likewise a special case of first order modal logic?
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A proof of the completeness of PLTL (Propositional Linear Temporal Logic)
In what paper(s), textbook(s), and/or classnote(s) can I find a detailed proof of the completeness of a certain proof system for PLTL (Propositional Linear Temporal Logic)?
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What is the Kripke semantic for a linear temporal logic?
I've read that in general for a modal formula P, a world w and a Kripke frame ⟨W,R⟩
w⊨□P if and only if for every u∈W, if wRu then u⊢P
In case of LTL, being a modal logic, I assumed that the worlds ...
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Safety VS. Liveliness Property
I have to prove whether a certain property is safety or liveliness. The property represents the absence of deadlock so I expected it to be a safety property from what I read online.
The issue is that ...
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How can I prove that LTL formula is valid?
I do not know with which technique i can prove if a LTL forumula is valid.
Let's say we have for example this one: ¬q U(¬p ∧ ¬q) → ¬Gp. How can prove if this valid or not? (should be true in any state ...
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Switching between doing something and not doing it within $k$ steps in LTL
Imagine that we are designing a system in which we have the action to brake or not brake. How can we write an LTL specification that guarantees that we cannot switch between braking and not braking ...
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Understanding a proof from a paper (model checking game)
I'm reading the paper: "Model Checking Games for Branching Time Logics" by Martin Lange and Colin Stirling - https://carrick.fmv.informatik.uni-kassel.de/~mlange/papers/jlc2000.pdf. The ...
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Logic foundation for formal verification
What types of logic should one study as foundation before diving into the area of software verification? What I can think of are:
Hoare Logic (for proving correctness of imperative programs)
Linear ...
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Getting rid of "FG" in a LTL equation
i am currently struggling with a Linear temporal logic equation: $$\phi=FG( \lnot a\lor X \lnot a )$$
For my understanding, it means that starting at a certain point in the future, proposition a can ...
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Translating Natural Language to LTL Formulae
I'm brand new to LTL and working on becoming better with LTL formulae. I've got two examples where I am unsure whether my LTL formula is correct.
I'm given the sentences, and my assumption is that $l$ ...
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What does “linear” in Linear Temporal Logic refer to?
Consider the term linear temporal logic (in the meaning of linear-time temporal logic). In linear temporal logic, what does linear refer to:
to temporal or
to logic?
If I interpret http://en....
user127009
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Complexity of LTL realizability of safety games with Next operator only
It is known that the computational complexity of deciding whether an LTL specification is realizable in a safety game is 2EXP-complete (that is, you receive an LTL formula, where some variables belong ...
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Can an LTL formula uniquely be represented by an expression tree?
Just like we can represent a mathematical expression uniquely with a binary expression tree, I was wondering if we could do the same for LTL formulae? Such that no two different looking LTL formula ...
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Given LTL formulas $m$ and $p$, is there a tool that can check whether $m \models p$ does hold?
To the best of my understanding, $m \models p$ asks whether the LTL formula $p$ satisfies the LTL formula $m$. In other words, $m \to p$ is a tautology. Here are some examples of where $m \models p$ ...
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Does Rice's theorem apply to sequential logic circuits?
I am wondering if Rice's theorem (or something similar to that) applies also to sequential circuits. I.e. given any finite sequential circuit, can there be an algorithm that can formally verify any ...
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Test two LTL expression trees for equivalence
Is there an algorithm on how to check if two LTL expressions (represented as binary trees) are semantically equivalent? Since there are many smaller equivalences such as
$a\Rightarrow b \equiv \neg a \...
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A way to express LTL (varient) to enforce a stream of data to satisfy some linear time logic
Linear Time Logic (LTL) is used for system verification. In my case, I am investing some time, to see the feasibility of using LTL this time to enforce a constraint on a stream of data. Enough of ...
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is (infinitely often p) ∨ (infinitely often ¬p) valid?
i'm trying to prove every trace over PROP = {p} is a model of the formula.
I am very stuck in figuring out a model pi that satisfies this formula, can anyone point me in the right direction?
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LTL Model of (infinitely often p) ∧ ( infinitely often q) ∧ (¬ Eventually (p ∧ q))?
Can anyone give a model of the following LTL formula?
$$
\Box\Diamond p \land \Box\Diamond q \land \lnot \Diamond (p \land q).
$$
That is, we want each of $p$ and $q$ to hold infinitely often, but $p \...
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Automaton-based model checking on finite traces
I want to check whether a formula in finite LTL is valid on a finite, linear trace.
For infite traces I would create a Kripke structure of the trace and a Büchi automaton for the negated formula, ...
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Model for formula $\Box (\psi \vee \phi) \Rightarrow (\Box \psi) \vee (\Box \phi)$
I am new to LTL and I am trying to understand how it works. My question is: is there such $\sigma$ that:
$ \sigma \models [\Box (\psi \vee \phi) \Rightarrow (\Box \psi) \vee (\Box \phi)]$
I know ...
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What is the LTL expression for "there is a value of y whose next value is 8"?
Basically i have a program which increases a variable by 1 in each iteration and resets it to 0 as soon as y becomes 8 (i.e. mod 8).
It is a quite simple example but it still bugs me out because i ...
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Validity of self refering state with linear temporal logic 'X' connective
Lets say we have model like the one above or a similar one where a node refers back to itself.
Now let's say if I want to know the validity of the formula:
$M, s_2 \models Xr$
Will this be valid ...
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Prove that $\text{EF p}$ can't be written in LTL
Why can't we somehow represent it's negation in LTL and go from there? I think maybe because it has (effectively) two existential quantifiers, so negating it does not work. But how do I prove it?
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LTL Logic Finally, Globally and Until to irreflexive Version
my professor said that we can transform the reflexive Finally, Globally and Until into irreflexive Finally, Globally, Until.
Can someone explain me this?
For irreflexive Finally we have $w \models F^...
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Büchi automaton to Linear Temporal Logic
Given a Büchi automaton what is the procedure to build an equivalent LTL formula? And what is its size?
I'm looking for references but I haven't found them so far.
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Are the two LTL properties $GF(\psi_1 \land F\psi_2 )$ and $GF(\psi_2 \land F\psi_1 )$ equivalent?
Is $GF(\psi_1 \land F\psi_2 )$ equivalent to the property $GF(\psi_2 \land F\psi_1 )$?
Attempt:
In the first property each state must eventually see $\psi_1$ and $\psi_2$, in the second property as ...
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Is model checking PSpace-hard *in formula size*?
Sistla/Clarke proved [SC82] that the LTL model-checking problem is PSpace-complete.
Sometimes people write that this problem is "PSpace-hard in $|\phi|$" (e.g. [LP85]). What does this mean formally?
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Negation of the semantics of the Until operator in LTL
I have been looking at the Until operator and the release operator and when introduced to the release operator it was suggested that it is equivalent to:
$\phi R \psi \equiv \neg(\neg\phi U \neg \psi)...
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LTL to Büchi automaton, deterministic?
I know it is possible to convert LTL formulas to Büchi automatons. But is it possible to convert a LTL formula to a deterministic Büchi automaton? Are there formulas that can't be converted to a ...
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Why is LTL Model Checking in PSPACE
Given a LTL formula $\phi$ and a transition system $T$ we have to do following steps:
Build a (non deterministic) Büchi automaton for $T$
Build a (non deterministic) Büchi automaton for $\phi$
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How to prove a LTL formula correct in a specific model?
I have been learning Verification by model checking recently and I get the following question:
$Whether\ the\ LTL\ formula\ M, q_3\ \models (X\ \lnot a) \rightarrow (F\ G\ \lnot a)\ is\ established\ ...
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Validity of □(P→ Q) → (◊P → ◊Q) in linear temporal logic
How can I prove that $\Box(P\rightarrow Q)\rightarrow (\Diamond P\rightarrow\Diamond Q)$ is valid in linear temporal logic (LTL)?
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distinguishing between CTL* formulas $A[FG p]$ and $AFAG p$ using transition system
It is to show, using a transition system, that the two formulas $A[FG p]$ and $AFAG p$ are not equivalent.
For me, it seems strange that they are not equivalent.
As the first one says that any ...
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Linear Temporal Logic (LTL) Syntax Infinitely Often
I'm a little confused about some LTL syntax.
When the Global and Future operator (GFx) or []<>x is used, what does it mean. In the lecture slides it is given as infinitely often. But I don't ...
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How to express the existence of winning strategy of the starter of a game in temporal logic?
Consider a two-player game. A winning strategy of a player is a strategy following which the player can always beat his opponent, no matter how his opponent responds.
A game can be unfolded to a ...
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How to graph search a LTL-generated Buchi automaton to generate valid execution paths
I have a set of tasks, and a LTL specification that describes which orders of the tasks are legal. I want to find a way to enumerate all permutations of the tasks that meet the specification.
For ...
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Proving the equivalence of an LTL and a CTL formula
For a lecture I am attending, we have to prove that
$$\forall \big(a \textsf{U} (b \land \forall \square a)\big) \equiv \big(a \textsf{U} (b \land \square a)\big).$$
That is, we need to prove that ...
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Linear Temporal Logic, Idempotent law
in many lecture notes, I have seen the LTL idempotent law for until and its equivalence is described as $\phi U(\phi U \psi) \equiv \phi U\psi$ and also $(\phi U \...
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LTL globally implies
I have got confused in undressing the informal definition of LTL below: $$G(\phi \Longrightarrow\psi)\Longrightarrow(G\phi \Longrightarrow G\psi)$$
in many literatures I have seen implies is said as ...
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Equivalence preserving operator from CTL* to LTL
The question is about an operator that transforms any CTL* formula ${\psi}$ into a (not necessarily equivalent) LTL formula ${A\psi^d}$, where $d$ means syntactically removing all $A,E$ quantifiers ...
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Acceptance conditions when translating LTL to Büchi automaton?
As an exercise in better understanding, I have been implementing the LTL to Generalized Büchi Automaton translation algorithm of Gerth, et al. (which is also discussed in Clarke, et al., Model ...
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LTL to automaton translation that can only read disjunctions of labels per transition
Is there any automaton (or automaton translation) as expressive as LTL that can only read disjunctions of atomic propositions per each transition? Standard LTL-to-automaton algorithms allow the ...
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How to draw a Non Deterministic Buchi Automaton (NBA) from the given property?
I'm new to LTL and Buchi automaton, and I have a hard time in constructing NBA from the given formula or the property. Could somebody please help me?
The following property is given, P= "Whenever ...
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LTL Until Tautology
Today I had to prove some tautologies as an exercise, one of them was:
(Ga)->(Fb) equivalent aU(b v ¬a)
It is clear that I have to prove the implication in both ways. After putting into negation ...
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LTL Formula Until with same atoms (aUa)
I am trying to solve some past papers in preparations for my next exam and I've stumbled across the following exercise:
While I managed to solve most parts of the exercise, I am still unsure about ...
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Minimal Deterministic Buchi Automata Product
Problem:
Let $\varphi = \varphi_1 \land \varphi_2$ be Deterministic Buchi Automata (DBA) expressible LTL formulas.
Let $A$, $A_1$ and $A_2$ be translated DBAs such that ${\cal{L}}(A) = {\cal{L}}(A_1)...
