A Logical Characterization of Timed (non-)Regular Languages

CLTLoc (Constraint LTL over clocks) is a quantifier-free extension of LTL allowing variables behaving like clocks over real numbers. CLTLoc is in PSPACE [9] and its satisfiability can polynomially be reduced to a SMT problem, allowing a feasible implementation of a decision procedure. We used CLTLoc to capture the semantics of metric temporal logics over continuous time, such as Metric Interval Temporal Logic (MITL), resulting in the first successful implementation of a tool for checking MITL satisfiability [7]. In this paper, we assess the expressive power of CLTLoc, by comparing it with various temporal formalisms over dense time.When interpreted over timed words, CLTLoc is equivalent to Timed Automata. We also define a monadic theory of orders, extending the one introduced by Kamp, which is expressively equivalent to CLTLoc. We investigate a decidable extension with an arithmetical next operator, which allows the expression of timed non-ω-regular languages

Making Metric Temporal Logic Rational

We study an extension of MTL in pointwise time with regular expression guarded modality Reg_I(re) where re is a rational expression over subformulae. We study the decidability and expressiveness of this extension (MTL+Ureg+Reg), called RegMTL, as well as its fragment SfrMTL where only star-free rational expressions are allowed. Using the technique of temporal projections, we show that RegMTL has decidable satisfiability by giving an equisatisfiable reduction to MTL. We also identify a subclass MITL+UReg of RegMTL for which our equisatisfiable reduction gives rise to formulae of MITL, yielding elementary decidability. As our second main result, we show a tight automaton-logic connection between SfrMTL and partially ordered (or very weak) 1-clock alternating timed automata

LNCS

Imprecision in timing can sometimes be beneficial: Metric interval temporal logic (MITL), disabling the expression of punctuality constraints, was shown to translate to timed automata, yielding an elementary decision procedure. We show how this principle extends to other forms of dense-time specification using regular expressions. By providing a clean, automaton-based formal framework for non-punctual languages, we are able to recover and extend several results in timed systems. Metric interval regular expressions (MIRE) are introduced, providing regular expressions with non-singular duration constraints. We obtain that MIRE are expressively complete relative to a class of one-clock timed automata, which can be determinized using additional clocks. Metric interval dynamic logic (MIDL) is then defined using MIRE as temporal modalities. We show that MIDL generalizes known extensions of MITL, while translating to timed automata at comparable cost

Revisiting timed logics with automata modalities

© 2019 ACM. It is well known that (timed) ω-regular properties such as 'p holds at every even position' and 'p occurs at least three times within the next 10 time units' cannot be expressed in Metric Interval Temporal Logic (MITL) and Event Clock Logic (ECL). A standard remedy to this deficiency is to extend these with modalities defined in terms of automata. In this paper, we show that the logics EMITL0, ∞ (adding non-deterministic finite automata modalities into the fragment of MITL with only lower- and upper-bound constraints) and EECL (adding automata modalities into ECL) are already as expressive as EMITL (full MITL with automata modalities). In particular, the satisfiability and model-checking problems for EMITL0, ∞ and EECL are PSPACE-complete, whereas the same problems for EMITL are EXPSPACE-complete. We also provide a simple translation from EMITL0, ∞ to diagonal-free timed automata, which enables practical satisfiability and model checking based on off-the-shelf tools

Logics Meet 1-Clock Alternating Timed Automata

This paper investigates a decidable and highly expressive real time logic QkMSO which is obtained by extending MSO[<] with guarded quantification using block of less than k metric quantifiers. The resulting logic is shown to be expressively equivalent to 1-clock ATA where loops are without clock resets, as well as, RatMTL, a powerful extension of MTL[U_I] with regular expressions. We also establish 4-variable property for QkMSO and characterize the expressive power of its 2-variable fragment. Thus, the paper presents progress towards expressively complete logics for 1-clock ATA

Satisfiability Checking of Multi-Variable TPTL with Unilateral Intervals Is PSPACE-Complete

We investigate the decidability of the ${0,\infty}$ fragment of Timed Propositional Temporal Logic (TPTL). We show that the satisfiability checking of TPTL$^{0,\infty}$ is PSPACE-complete. Moreover, even its 1-variable fragment (1-TPTL$^{0,\infty}$) is strictly more expressive than Metric Interval Temporal Logic (MITL) for which satisfiability checking is EXPSPACE complete. Hence, we have a strictly more expressive logic with computationally easier satisfiability checking. To the best of our knowledge, TPTL$^{0,\infty}$ is the first multi-variable fragment of TPTL for which satisfiability checking is decidable without imposing any bounds/restrictions on the timed words (e.g. bounded variability, bounded time, etc.). The membership in PSPACE is obtained by a reduction to the emptiness checking problem for a new "non-punctual" subclass of Alternating Timed Automata with multiple clocks called Unilateral Very Weak Alternating Timed Automata (VWATA$^{0,\infty}$) which we prove to be in PSPACE. We show this by constructing a simulation equivalent non-deterministic timed automata whose number of clocks is polynomial in the size of the given VWATA$^{0,\infty}$.Comment: Accepted in Concur 202