A Decidable Spatial Logic with Cone-Shaped Cardinal Directions

What's Decidable about Halpern and Shoham's Interval Logic? The Maximal Fragment ABBL

“…Not surprisingly, a few such results can be found in the literature. In [12], Montanari et al improve the result given in [8] by proving PSPACEcompleteness and maximality with respect to decidability of the fragment BBDDLL over the rational line. In [13], the satisfiability problem for ABBĀ (resp., AEĒĀ), interpreted over finite linear orders, has been shown to be decidable, but not primitive recursive.…”

“…PROBLEM FOR ABBL In [8], Montanari et al give an automaton-based algorithm to check satisfiability of formulas of a spatial modal logic based on an encoding of the problem into a suitable fragment of CTL. The very same technique can be used to check the satisfiability of an ABBL-formula ϕ.…”

“…Among them, we mention the fragments DD and AĀ. In [8], Montanari et al introduce a spatial modal logic based on cone-shaped cardinal directions over the rational plane (cone logic for short) and they prove PSPACEcompleteness of its satisfiability problem. Moreover, they show that the decidability of DD, interpreted over the rational line, can be easily derived from that of cone logic.…”

Reasoning with Time Intervals: A Logical and Computational Perspective

“…The fragment AABBDD is the minimal undecidable fragment that includes all decidable ones on dense linear orders, and it contains two, incomparable, maximal decidable fragments, namely, BBDD and ABBL. The fragment AA is still NEXPTIME-complete [54], and NEXPTIME-hardness already holds, again, for A and A; ABBL, is in EXPSPACE, and EXPSPACE-hardness already holds for the fragments ABB and AB [55]; the fragments ABBA and ABB are already undecidable [48]; the fragment BBDD (which includes L and L as definable operators) is in PSPACE, and PSPACE-hardness holds for D and D alone [56]; finally, BB is NP-complete [43], and, obviously, NP-hardness holds for B and B alone too. Probably, we could extend the NP-completeness (in particular, NP-membership) of BB can be extended to BBLL and each one of its fragments, as we did in the strongly discrete case, and the EXPSPACE-completeness (in particular, EXPSPACEhardness) of AB might be possibly adapted to the fragment AB, but, still, this is currently ongoing research.…”

Maximal Decidable Fragments of Halpern and Shoham’s Modal Logic of Intervals

“…The most significant ones are the logic BB (resp., EĒ) of Allen's "begun by/begins" (resp., "ended by/ends") relations [9], the logic AĀ of temporal neighborhood, whose modalities correspond to Allen's "meets/met by" relations (it can be easily shown that Allen's "before/after" relations can be expressed in AĀ) [8], and the logic DD of the subinterval/superinterval relations, whose modalities correspond to Allen's "contains/during" relations [14]. In this paper, we focus our attention on the logic AĀBB that joins BB and AĀ (the case of AĀEĒ is fully symmetric).…”