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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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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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Counterexample Construction and Translation in CEGAR

As in this slide, M2 is an abstraction of M1 using the abstraction function f1 which essentially groups based on labels. Then we see that the property is not satisfied by the abstraction so we find ...
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When does Simulation Equivalent imply Bisimulation

Here, Given 2 Kripke structures M1 and M2, if there exists a simulation from M1 to M2 and a simulation from M2 to M1, is there a bisimulation from M1 to M2?, it is shown that for two LTSs M1 and M2 ...
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LTL Model Checking Worst Case size for NBA

I have a slide that gives the worst case time complexity of LTL model checking which uses a product construction of an LTS along with the NBA for the negation of the LTL formula. My question is, when ...
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Irregular Safety Properties

A regular safety property is a safety property whose bad prefixes form a regular language, https://moves.rwth-aachen.de/wp-content/uploads/WS1920/MC/mc2019_handout_lec4.pdf. My questions is, are there ...
2 votes
1 answer
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Can CTL* express every model's behaviour

CTL* as in https://en.wikipedia.org/wiki/CTL*, is a logic that combines CTL and LTL. I know that CTL* can express everything expressible in these two languages and more. My question is whether we can ...
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1 answer
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Linear Time properties classification

I know LT Properties can be classified with classes Invariants, Safety Properties, Liveness Properties as in https://en.wikipedia.org/wiki/Linear_time_property. My question is, does this cover all ...
14 votes
1 answer
832 views

The difference between dynamic logic and temporal logic

To find the difference, I'd just encountered with assertions below about temporal logic in Wikipedia: another variant of modal logic sharing many common features with dynamic logic, differs from ...
7 votes
1 answer
373 views

Non-deterministic Büchi vs Rabin: Automaton size for LTL->automaton

Is there any general result to show that which automaton is more succinct? I have a set of LTL properties and I would like to know (show) which automaton is more efficient in term of state number and ...
2 votes
1 answer
79 views

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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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 ...
1 vote
1 answer
130 views

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....
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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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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 ...
2 votes
1 answer
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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 ...
2 votes
1 answer
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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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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 ...
4 votes
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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 ...
1 vote
1 answer
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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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1 answer
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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 ...
1 vote
2 answers
365 views

Find equivalent LTL formula, without Y (Yesterday) operator. How can I handle first state?

The task is to find an equivalent LTL formula for $G(a \Rightarrow Yb)$, which doesn't contain the Y operator. My idea is to search for invalid path patterns with 2 $a$'s in a row, e.g. bbbbaab. ...
2 votes
2 answers
1k views

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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1 answer
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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 \...
1 vote
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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 ...
2 votes
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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 ...
5 votes
1 answer
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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 \...
1 vote
3 answers
135 views

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?
2 votes
1 answer
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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 ...
1 vote
1 answer
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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, ...
2 votes
2 answers
126 views

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 ...
3 votes
1 answer
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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 ...
0 votes
1 answer
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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 ...
1 vote
1 answer
180 views

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^...
2 votes
1 answer
186 views

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.
2 votes
1 answer
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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\ ...
1 vote
1 answer
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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 ...
4 votes
1 answer
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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? ...
1 vote
3 answers
351 views

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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1 answer
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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 ...
2 votes
2 answers
327 views

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 $\phi$ ...
3 votes
3 answers
172 views

Linear Temporal Logic with non-Boolean propositions (e.g. Integers)?

LTL works with Boolean propositions. People probably studied extensions to non-Boolean propositions... Do you know a good starting reference? (I am aware of STL, but it also seems to talk about ...
2 votes
1 answer
864 views

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 ...
4 votes
1 answer
1k views

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 ...
2 votes
2 answers
125 views

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 ...
2 votes
2 answers
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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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1 answer
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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 ...