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We study the question of which visibly pushdown languages (VPLs) are in the complexity class $\mathsf{AC}^0$ and how to effectively decide this question. Our contribution is to introduce a particular subclass of one-turn VPLs, called intermediate VPLs, for which the raised question is entirely unclear: to the best of our knowledge our research community is unaware of containment or non-containment in $\mathsf{AC}^0$ for any language in our newly introduced class. Our main result states that there is an algorithm that, given a visibly pushdown automaton, correctly outputs exactly one of the following: that its language $L$ is in $\mathsf{AC}^0$, some $m\geq 2$ such that $L$ is $\mathsf{ACC}^0(m)$-hard (implying that $L$ is not in $\mathsf{AC}^0$), or a finite disjoint union of intermediate VPLs that $L$ is constant-depth equivalent to. In the latter of the three cases one can moreover effectively compute $k,l\in\mathbb{N}_{>0}$ with $k\not=l$ such that the concrete intermediate VPL $L(S\rightarrow \varepsilon\mid a c^{k-1} S b_1\mid ac^{l-1}Sb_2)$ is constant-depth reducible to the language $L$. Due to their particular nature we conjecture that either all intermediate VPLs are in $\mathsf{AC}^0$ or all are not. As a corollary of our main result we obtain that in case the input language is a visibly counter language our algorithm can effectively determine if it is in $\mathsf{AC}^0$ - hence our main result generalizes a result by Krebs et al. stating that it is decidable if […]

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Call-by-need evaluation for the lambda-calculus can be seen as merging the best of call-by-name and call-by-value, namely the wise erasing behaviour of the former and the wise duplicating behaviour of the latter. To better understand how duplication and erasure can be combined, we design a degenerated calculus, dubbed call-by-silly, that is symmetric to call-by-need in that it merges the worst of call-by-name and call-by-value, namely silly duplications by-name and silly erasures by-value. We validate the design of the call-by-silly calculus via rewriting properties and multi types. In particular, we mirror the main theorem about call-by-need -- that is, its operational equivalence with call-by-name -- showing that call-by-silly and call-by-value induce the same contextual equivalence. This fact shows the blindness with respect to efficiency of call-by-value contextual equivalence. We also define a call-by-silly strategy and a call-by-silly abstract machine implementing the strategy. Moreover, we measure the number of steps taken by the strategy via tight multi types. Lastly, we prove that the call-by-silly strategy computes evaluation sequences of maximal length in the calculus.

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A system $\boldsymbolλ_θ$ is developed that combines modal logic and simply-typed lambda calculus, and that generalizes the system studied by Montague and Gallin. Whereas Montague and Gallin worked with Church's simple theory of types, the system $\boldsymbolλ_θ$ is developed in the typed base theory most commonly used today, namely the simply-typed lambda calculus. Further, the system $\boldsymbolλ_θ$ is controlled by a parameter $θ$ which allows more options for state types and state variables than is present in Montague and Gallin. A main goal of the paper is to establish some basic metatheory of $\boldsymbolλ_θ$: (i) an Andrews-like characterization of its models in terms of combinatory logic is given, and this combinatory logic involves a $\mathsf{BCKW}$-like basis rather than an $\mathsf{SKI}$-like basis and (ii) semantic conservation and expressibility results relating $\boldsymbolλ_θ$ to the maximal system $\boldsymbolλ_ω$ are proven. Similar results are proven for the relation between $\boldsymbolλ_ω$ and $\boldsymbolλ$, the corresponding ordinary simply-typed lambda calculus. This answers a question of Zimmermann in the semantics of the simply typed setting. In a companion paper this is extended to Church's simple theory of types. We further develop a partial correspondence between a pure combinatory logic centered on the $\mathsf{BCKW}$-like basis and the weak deductive system for $\boldsymbolλ_ω$ wherein $β$-reduction is not allowed under a lambda abstract, and we […]

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People increasingly use digital platforms to exchange resources in accordance with some policies stating what resources users offer and what they require in return. In this paper, we propose a formal model of these environments, focussing on how users' policies are defined and enforced, so ensuring that malicious users cannot take advantage of honest ones. To that end, we introduce the declarative policy language MuAC and equip it with a formal semantics. To determine if a resource exchange is fair, i.e., if it respects the MuAC policies in force, we introduce the non-standard logic MuACL that combines non-linear, linear and contractual aspects, and prove it decidable. Notably, the operator for contractual implication of MuACL is not expressible in linear logic. We define a semantics preserving compilation of MuAC policies into MuACL, thus establishing that exchange fairness is reduced to finding a proof in MuACL. Finally, we show how this approach can be put to work on a blockchain to exchange non-fungible tokens.

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A finite relational structure A is called compact if for any infinite relational structure B of the same type, the existence of a homomorphism from B to A is equivalent to the existence of homomorphisms from all finite substructures of B to A. We show that if A has width one, then the compactness of A can be proved in the axiom system of Zermelo and Fraenkel, but otherwise, the compactness of A implies the existence of non-measurable sets in 3-space.

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