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Do past time LTL and future time LTL have the same expressiveness?

The quick summary is that LTL with only past and no future modalities defines properties expressed over finite-words and these are the star-free subset of the regular languages. Standard LTL when ...
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9 votes
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CTL and LTL logic difference

There are already some rather good related answers regarding LTL versus CTL. In a nutshell, LTL is first and foremost a logic of traces, and an LTL formula is true for a transition system $S$ if and ...
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Do past time LTL and future time LTL have the same expressiveness?

The logics are expressively the same, though past operators make LTL exponentially more succinct. You can start here, from which there are references.
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5 votes

LTL property - safety or liveness?

Your first question is answered in this paper: https://www.cs.cornell.edu/fbs/publications/RecSafeLive.pdf Given an LTL formula, translate it into a Büchi automaton, and remove states that have ...
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Translation of Counter-free automata into Linear Temporal Logic

As mentioned in the comments, the translation is shown in: Volker Diekert and Paul Gastin. "First-order definable languages." (2008) http://www.lsv.fr/Publis/PAPERS/PDF/DG-WT08.pdf And it ...
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4 votes

Translation of Counter-free automata into Linear Temporal Logic

Sorry to provide the same answer as for this question, but this was proved in [1, Theorem 3.1]. [1] J. Cohen, D. Perrin and J.-É. Pin, On the expressive power of temporal logic for finite words, J. ...
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Reference request: An algebraic characterisation of LTL[XF]-definable word languages

An algebraic characterisation of the restricted temporal logic (fragment using only next and eventually, but not until) is given in [1]. The expressive power of this fragment is the set of regular ...
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Calculating least fixed points in equations

In general, we look at fixed-points of monotone functions over lattices, i.e. with some partial ordering over your elements. If your lattice is complete (it has a least and greatest element, called a ...
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Decomposition of safety and liveness properties

Yes. The automata in question are often called "Looping" automata (so you have a keyword to start from). A possible starting point is the following paper: https://faculty.idc.ac.il/udiboker/...
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3 votes
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Difference between CTMC, DTMC, and MDP

The "probabilistic" element in probabilistic model checking is that the system being checked is probabilistic, not that we add probabilities to an existing deterministic or non-deterministic system. ...
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3 votes
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Complexity of propositional LTL with past operators and freeze quantifier?

The answer was buried in a small section of the same paper that I was citing. Adding past operators to TPTL, in contrast of what happens with LTL, causes a huge increase in complexity as the ...
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Global satisfiability in LTL

The statement $\langle M,i\rangle\models \varphi$ for all $i\in \mathbb{N}$ is equivalent to $\langle M,0\rangle\models G\varphi$. Thus, you can check the latter.
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Are there temporal logics linear time properties that only have counterexamples that are more complex than a lasso?

Pierre Wolper defined in 1983 extended temporal logic (ETL, in Information and Computation 56, 72–99, doi:10.1016/S0019-9958(83)80051-5), where a temporal operator $\mathcal A(\varphi_1,\dots,\...
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2 votes

resolution based theorem prover for temporal logic

Your translation goes into Presburger arithmetic, which is decidable. You could take your translated formula, do quantifier elimination on it, and then hand it over to a proof-producing SMT solver. ...
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2 votes

Equivalent formula for LTL with and without past operators

The "equally expressive" statement means that if a formula of PLTL is a statement about the future, i.e. if it's evaluated at the first instant $0$ of the time domain $\mathbb N$, then there exists an ...
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2 votes
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Reference request: An algebraic characterisation of LTL[XF]-definable word languages

I've just found an answer to my question. It is inside Bojańczyk's notes titled "Languages recognised by finite semigroups and their generalisations to objects such as trees and graphs with an ...
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Why does LTL equivalence (with next oeprator) implie trace equivalence in finite transition machines?

The important thing to notice is that in finite transition systems, two systems $K_1,K_2$ over an alphabet $\Sigma$ are not trace equivalent (for infinite traces) if and only if there exists a finite ...
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2 votes
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Proving that a given formula in LTL is the smallest way to express it

I'm not sure whether it would work in your case, but in order to show succinctness results in modal/temporal logic (e.g. the fact the two-variable logic over words in exponentially more succinct than ...
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Proving that a given formula in LTL is the smallest way to express it

Your question is formally the same as this one: how many symbols $X$ are needed to write the polynomial $XX + XXXX + XXXXXX = XX(1 + XX(1 + XX))$? Well, you obtained $6$ as an upper bound. It is also ...
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Linear temporal logic in computational trees

This question should (and will) probably be migrated to cs.se. In the meantime, consider the computation tree of the depicted structure: in almost all paths, $p$ is seen only finitely often, making ...
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1 vote

Syntactic characterisation of the intersection of CTL and LTL

Note that by "equivalent", we actually need to interpret LTL over trees. Thus, we say that a CTL formula $\phi$ is equivalent to an LTL formula $\psi$, if $\phi$ is equivalent to the CTL* formula $A\...
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1 vote

Are there temporal logics linear time properties that only have counterexamples that are more complex than a lasso?

I think it depends on what you mean by linear-time temporal logics. If you mean temporal logics that have linear time semantics (i.e. cannot distinguish more than trace equivalence, a la van Glabbeek) ...
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LTL property - safety or liveness?

To answer your second question: there is one property that is both safety and liveness: True. With this exception, however, it is fair to say that a property is either safety or liveness or neither. "...
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1 vote

Temporal Logic and Access Control Models

There were a lot of publication on extending and or formalizing RBAC/SOD with LTL or other flavors of temporal logics, not sure of other AC models. Google Scholar (or simple google) search with yield ...
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