Partially Punctual Metric Temporal Logic is Decidable

Abstract

TIFRMetric Temporal Logic MTL[ U I , S I ] is one of the most studied real time logics. It exhibits considerable diversity in expressiveness and decidability properties based on the permitted set of modalities and the nature of time interval constraints I. Henzinger et al., in their seminal paper showed that the non-punctual fragment of MTL called MITL is decidable. In this paper, we sharpen this decidability result by showing that the partially punctual fragment of MTL (denoted PMTL) is decidable over strictly monotonic finite point wise time. In this fragment, we allow either punctual future modalities, or punctual past modalities, but never both together. We give two satisfiability preserving reductions from PMTL to the decidable logic MTL[ U I ]. The first reduction uses simple projections, while the second reduction uses a novel technique of temporal projections with oversampling. We study the trade-off between the two reductions: while the second reduction allows the introduction of extra action points in the underlying model, the equisatisfiable MTL[ U I ] formula obtained is exponentially succinct than the one obtained via the first reduction, where no oversampling of the underlying model is needed. We also show that PMTL is strictly more expressive than the fragments MTL[ U I , S] and MTL[ U, S I ].

Generalizing Non-punctuality for Timed Temporal Logic with Freeze Quantifiers

“…Raskin as well as Wilke added automata modalities to MITL as well as an Event-Clock logic ECL [22,23] and showed the decidability of satisfaction. The current authors showed that MTL[U, S NP ] (where U can use punctual intervals but S is restricted to non-punctual intervals), when extended with counting as well as regular expression modalities preserves decidability of satisfaction [12,[14][15][16]. Recently, Ferrère showed the EXPSPACE decidability of MIDL which is LTL[U] extended with a fragment of timed regular expression modality [5].…”

“…We now describe the technicalities associated with our reduction. We use the technique of equisatisfiability modulo oversampling [12,16]. Let Σ and OVS be disjoint set of propositions.…”

“…Next, we flatten φ to eliminate the nested F k I1,...,I k and P k I1,...,I k modalities while preserving satisfiability. Flattening is well studied [11,12,16,19].…”

Metric Temporal Logic with Counting

“…Situation 2: Starting from (i 1 , j 1 ) with time stamps (0,0), if the Spoiler chooses a U (0,1)#a∼c move and lands up at some point between x 1 and y 1 , Duplicator will play copy-cat and achieve an identical configuration. Consider the case when Spoiler lands up at y 1 6 . In response, Duplicator moves to y ′ 1 .…”

“…A pebble is kept at the inbetween positions x 2 , x ′ 2 respectively. If Spoiler chooses to pick the pebble in Duplicator's word, then we obtain the configuration (i 3 , j 3 ) with time 6 The argument when Spoiler lands up at x1 or a point in between x1, y1 is exactly the same 2 ), with the lag of one segment (seg(i 3 ) = 1, seg(j 3 ) = 2).…”