Algorithmics–The spirit of computing

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique, which builds upon the techniques of Ambainis et al. [STOC 2016]. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al. [STOC 2016], to be shown in a unified manner.

https://dl.acm.org/doi/pdf/10.1145/2897518.2897644

Separations in query complexity using cheat sheets

We introduce EHS+, a novel temporal-epistemic logic defined on temporal intervals characterised by regular expressions. We investigate the complexity of verifying multi-agent systems against EHS+ specifications for a number of fragments of EHS+ with results ranging from PSPACE-completeness to non-elementary time. The findings show that, at least for the fragments under analysis, the increase in expressiveness obtained by using regular expressions rather than end-points as standard, can be achieved without increasing the complexity of the problem. We show that the expressiveness of regular expressions can also be adopted at the level of specifications without severe computational cost. To do so we introduce a further temporal-epistemic logic, called EHSRE, in which regular expressions are used within propositions, and give a polynomial time reduction of the model checking problem from EHSRE to EHS+.

/pdf/model-checking-multi-agent-systems-against-epistemic-hs-gpz88ttnhv.pdf

Model checking multi-agent systems against epistemic HS specifications with regular expressions

The model checking problem has thoroughly been explored in the context of standard point-based temporal logics, such as LTL, CTL, and CTL$$^{*}$$, whereas model checking for interval temporal logics has been brought to the attention only very recently.

In this paper, we prove that the model checking problem for the logic of Allen's relations started-by and finished-by is highly intractable, as it can be proved to be $${{\mathrm{\mathbf {EXPSPACE}}}}$$-hard. Such a lower bound immediately propagates to the full Halpern and Shoham's modal logic of time intervals HS. In contrast, we show that other noteworthy HS fragments, namely, Propositional Neighbourhood Logic extended with modalities for the Allen relation starts resp., finishes and its inverse started-by resp., finished-by, turn out to have--maybe unexpectedly--the same complexity as LTL i.e., they are $${{\mathrm{\mathbf {PSPACE}}}}$$-complete, thus joining the group of other already studied, well-behaved albeit less expressive, HS fragments.

Interval Temporal Logic Model Checking: The Border Between Good and Bad HS Fragments

Model checking has been successfully used in many computer science fields, including artificial intelligence, theoretical computer science, and databases. Most of the proposed solutions make use of classical, point-based temporal logics, while little work has been done in the interval temporal logic setting. Recently, a non-elementary model checking algorithm for Halpern and Shoham's modal logic of time intervals HS over finite Kripke structures (under the homogeneity assumption) and an EXPSPACE model checking procedure for two meaningful fragments of it have been proposed. In this paper, we show that more efficient model checking procedures can be developed for some expressive enough fragments of HS.

Complexity of ITL Model Checking: Some Well-Behaved Fragments of the Interval Logic HS

Model checking is a powerful method widely explored in formal verification. Given a model of a system, e.g., a Kripke structure, and a formula specifying its expected behaviour, one can verify whether the system meets the behaviour by checking the formula against the model. Classically, system behaviour is expressed by a formula of a temporal logic, such as LTL and the like. These logics are "point-wise" interpreted, as they describe how the system evolves state-by-state. However, there are relevant properties, such as those constraining the temporal relations between pairs of temporally extended events or involving temporal aggregations, which are inherently "interval-based", and thus asking for an interval temporal logic. In this paper, we give a formalization of the model checking problem in an interval logic setting. First, we provide an interpretation of formulas of Halpern and Shoham's interval temporal logic HS over finite Kripke structures, which allows one to check interval properties of computations. Then, we prove that the model checking problem for HS against finite Kripke structures is decidable by a suitable small model theorem, and we provide a lower bound to its computational complexity.

Checking interval properties of computations

Model checking is commonly recognized as one of the most effective tool in system verification. While it has been systematically investigated in the context of classical, point-based temporal logics, it is still largely unexplored in the interval logic setting. Recently, a non-elementary model checking algorithm for Halpern and Shoham's modal logic of time intervals HS, interpreted over finite Kripke structures, has been proposed, together with a proof of the EXPSPACE-hardness of the problem. In this paper, we devise an EXPSPACE model checking procedure for two meaningful HS fragments. It exploits a suitable contraction technique, that allows one to replace long enough tracks of a Kripke structure by equivalent shorter ones.

/pdf/a-model-checking-procedure-for-interval-temporal-logics-2k06aown8v.pdf

A Model Checking Procedure for Interval Temporal Logics based on Track Representatives

Model checking is one of the most powerful and widespread tools for system verification with applications in many areas of computer science and artificial intelligence. The large majority of model checkers deal with properties expressed in point-based temporal logics, such as LTL and CTL. However, there exist relevant properties of systems which are inherently interval-based. Model checking algorithms for interval temporal logics (ITLs) have recently been proposed to check interval properties of computations. As the model checking problem for full Halpern and Shoham's ITL (HS for short) turns out to be decidable, but computationally heavy, research has focused on its well-behaved fragments. In this paper, we provide an almost final picture of the computational complexity of model checking for HS fragments with modalities for (a subset of) Allen's relations meets, met by, starts, and ends.

Alberto Molinari

Papers

Checking interval properties of computations

Interval Temporal Logic Model Checking: The Border Between Good and Bad HS Fragments

Complexity of ITL Model Checking: Some Well-Behaved Fragments of the Interval Logic HS

A Model Checking Procedure for Interval Temporal Logics based on Track Representatives

Model checking well-behaved fragments of HS: the (almost) final picture