The undecidability of arbitrary arrow update logic

Arbitrary Arrow Update Logic is a dynamic modal logic with a modality to quantify over arrow updates. Some properties of this logic have already been established, but until now it remained an open question whether the logic's satisfiability problem is decidable. Here, we show by a reduction of the tiling problem that the satisfiability problem of Arbitrary Arrow Update Logic is co-RE hard, and therefore undecidable

Temporal Stream Logic: Synthesis beyond the Bools

Reactive systems that operate in environments with complex data, such as mobile apps or embedded controllers with many sensors, are difficult to synthesize. Synthesis tools usually fail for such systems because the state space resulting from the discretization of the data is too large. We introduce TSL, a new temporal logic that separates control and data. We provide a CEGAR-based synthesis approach for the construction of implementations that are guaranteed to satisfy a TSL specification for all possible instantiations of the data processing functions. TSL provides an attractive trade-off for synthesis. On the one hand, synthesis from TSL, unlike synthesis from standard temporal logics, is undecidable in general. On the other hand, however, synthesis from TSL is scalable, because it is independent of the complexity of the handled data. Among other benchmarks, we have successfully synthesized a music player Android app and a controller for an autonomous vehicle in the Open Race Car Simulator (TORCS.

To Be Announced

In this survey we review dynamic epistemic logics with modalities for quantification over information change. Of such logics we present complete axiomatizations, focussing on axioms involving the interaction between knowledge and such quantifiers, we report on their relative expressivity, on decidability and on the complexity of model checking and satisfiability, and on applications. We focus on open problems and new directions for research

Satisfiability of Arbitrary Public Announcement Logic with Common Knowledge is Σ11\Sigma^1_1-hard

Arbitrary Public Announcement Logic with Common Knowledge (APALC) is an extension of Public Announcement Logic with common knowledge modality and quantifiers over announcements. We show that the satisfiability problem of APALC on S5-models, as well as that of two other related logics with quantification and common knowledge, is $\Sigma^1_1$-hard. This implies that neither the validities nor the satisfiable formulas of APALC are recursively enumerable. Which, in turn, implies that APALC is not finitely axiomatisable.Comment: In Proceedings TARK 2023, arXiv:2307.0400

Undecidability of the Spectral Gap (full version)

We show that the spectral gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.Comment: v1: 146 pages, 56 theorems etc., 15 figures. See shorter companion paper arXiv:1502.04135 (same title and authors) for a short version omitting technical details. v2: Small but important fix to wording of abstract. v3: Simplified and shortened some parts of the proof; minor fixes to other parts. Now only 127 pages, 55 theorems etc., 10 figures. v4: Minor updates to introductio

Arbitrary Arrow Update Logic with Common Knowledge is neither RE nor co-RE

Arbitrary Arrow Update Logic with Common Knowledge (AAULC) is a dynamic epistemic logic with (i) an arrow update operator, which represents a particular type of information change and (ii) an arbitrary arrow update operator, which quantifies over arrow updates. By encoding the execution of a Turing machine in AAULC, we show that neither the valid formulas nor the satisfiable formulas of AAULC are recursively enumerable. In particular, it follows that AAULC does not have a recursive axiomatization.Comment: In Proceedings TARK 2017, arXiv:1707.0825

Quantifying over information change with common knowledge

Public announcement logic (PAL) extends multi-agent epistemic logic with dynamic operators modelling the effects of public communication. Allowing quantification over public announcements lets us reason about the existence of an announcement that reaches a certain epistemic goal. Two notable examples of logics of quantified announcements are arbitrary public announcement logic (APAL) and group announcement logic (GAL). While the notion of common knowledge plays an important role in PAL, and in particular in characterisations of epistemic states that an agent or a group of agents might make come about by performing public announcements, extensions of APAL and GAL with common knowledge still haven’t been studied in detail. That is what we do in this paper. In particular, we consider both conservative extensions, where the semantics of the quantifiers is not changed, as well as extensions where the scope of quantification also includes common knowledge formulas. We compare the expressivity of these extensions relative to each other and other connected logics, and provide sound and complete axiomatisations. Finally, we show how the completeness results can be used for other logics with quantification over information change.publishedVersio