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18 votes
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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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  • 12.3k
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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8 votes

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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  • 5,261
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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  • 1,175
4 votes
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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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3 votes
Accepted

Are both safety and liveness properties closed under finite intersection?

Safety properties are closed under finite intersection. This can be seen by following Alpern and Schneider's characterisation which showed that safety properties are limit-closed when viewed ...
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  • 800
3 votes
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Is every countable, finite-branching LTS bisimilar to a tree?

Q1: Yes, every LTS is bisimilar to its unfolding, which is a tree. Q2: No, by a cardinality argument. For instance take infinite binary trees with $L=\{a,b\}$. Each tree has countable set of states ...
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  • 7,643
3 votes

Is modal $\mu$-calculus "equivalent" to bisimulation?

For Q1, the answer is yes if we consider image-finite systems: for all node $t$ and label $a$, the number of $a$-successors of $t$ must be finite. In this case you don't even need fixpoints of the $\...
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  • 7,643
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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  • 5,261
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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  • 1,112
2 votes
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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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  • 5,261
2 votes
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Regular safety properties and bad prefixes of $\omega$-regular properties

Your construction for bad prefixes is not correct on NBA's. For instance take the NBA on alphabet $A=\{a,b\}$ with two initial states $q_a$ and $q_b$ where for both $x\in A$, $q_x$ goes to an ...
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  • 7,643
2 votes
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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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  • 3,314
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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  • 7,643
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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1 vote
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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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  • 5,261
1 vote

Algorithms to synthesize optimal plans satisfying temporal logic constraints

Maybe take a look at http://www.syntcomp.org/ This is a competition of tools solving the LTL synthesis problem (and some related problems).
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  • 366
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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  • 366
1 vote

Reference for CTL* logic

See Venema, Yde. Temporal Logic. The blackwell guide to Philosophical Logic.
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1 vote
Accepted

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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  • 366

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