Editors: Mads Dam, Valentin Goranko

In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in univalent type theory, where the comparisons between them can be given more elementarily than in set-theoretic foundations. Specifically, we construct maps between the various types of structures, and show that assuming the Univalence axiom, some of the comparisons are equivalences. We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure. We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.

Given a graph $F$, let $I(F)$ be the class of graphs containing $F$ as an induced subgraph. Let $W[F]$ denote the minimum $k$ such that $I(F)$ is definable in $k$-variable first-order logic. The recognition problem of $I(F)$, known as Induced Subgraph Isomorphism (for the pattern graph $F$), is solvable in time $O(n^{W[F]})$. Motivated by this fact, we are interested in determining or estimating the value of $W[F]$. Using Olariu's characterization of paw-free graphs, we show that $I(K_3+e)$ is definable by a first-order sentence of quantifier depth 3, where $K_3+e$ denotes the paw graph. This provides an example of a graph $F$ with $W[F]$ strictly less than the number of vertices in $F$. On the other hand, we prove that $W[F]=4$ for all $F$ on 4 vertices except the paw graph and its complement. If $F$ is a graph on $t$ vertices, we prove a general lower bound $W[F]>(1/2-o(1))t$, where the function in the little-o notation approaches 0 as $t$ inreases. This bound holds true even for a related parameter $W^*[F]\le W[F]$, which is defined as the minimum $k$ such that $I(F)$ is definable in the infinitary logic $L^k_{\infty\omega}$. We show that $W^*[F]$ can be strictly less than $W[F]$. Specifically, $W^*[P_4]=3$ for $P_4$ being the path graph on 4 vertices. Using the lower bound for $W[F]$, we also obtain a succintness result for existential monadic second-order logic: A usage of just one monadic quantifier sometimes reduces the first-order quantifier depth at a […]

The computational properties of modal and propositional dependence logics have been extensively studied over the past few years, starting from a result by Sevenster showing NEXPTIME-completeness of the satisfiability problem for modal dependence logic. Thus far, however, the validity and entailment properties of these logics have remained mostly unaddressed. This paper provides a comprehensive classification of the complexity of validity and entailment in various modal and propositional dependence logics. The logics examined are obtained by extending the standard modal and propositional logics with notions of dependence, independence, and inclusion in the team semantics context. In particular, we address the question of the complexity of validity in modal dependence logic. By showing that it is NEXPTIME-complete we refute an earlier conjecture proposing a higher complexity for the problem.

We show that the class of chordal claw-free graphs admits LREC$_=$-definable canonization. LREC$_=$ is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm, and therefore a logarithmic-space isomorphism test, for the class of chordal claw-free graphs. As a further consequence, LREC$_=$ captures logarithmic space on this graph class. Since LREC$_=$ is contained in fixed-point logic with counting, we also obtain that fixed-point logic with counting captures polynomial time on the class of chordal claw-free graphs.

It has been known since Ehrhard and Regnier's seminal work on the Taylor expansion of $\lambda$-terms that this operation commutes with normalization: the expansion of a $\lambda$-term is always normalizable and its normal form is the expansion of the Böhm tree of the term. We generalize this result to the non-uniform setting of the algebraic $\lambda$-calculus, i.e. $\lambda$-calculus extended with linear combinations of terms. This requires us to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's techniques rely heavily on the uniform, deterministic nature of the ordinary $\lambda$-calculus, and thus cannot be adapted; second is the absence of any satisfactory generic extension of the notion of Böhm tree in presence of quantitative non-determinism, which is reflected by the fact that the Taylor expansion of an algebraic $\lambda$-term is not always normalizable. Our solution is to provide a fine grained study of the dynamics of $\beta$-reduction under Taylor expansion, by introducing a notion of reduction on resource vectors, i.e. infinite linear combinations of resource $\lambda$-terms. The latter form the multilinear fragment of the differential $\lambda$-calculus, and resource vectors are the target of the Taylor expansion of $\lambda$-terms. We show the reduction of resource vectors contains the image of any $\beta$-reduction step, from which we deduce that Taylor expansion and normalization commute on the nose. We moreover identify a […]

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of $\omega$-tree languages derived from weak regular $\omega$-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived $\omega$-tree languages to learning the underlying class of $\omega$-word languages, for any class of $\omega$-word languages recognized by a deterministic Büchi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [1995] for the class of weak regular $\omega$-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic Büchi $\omega$-word acceptors. A previous claim of an algorithm to learn regular $\omega$-trees due to Jayasrirani, Begam and Thomas [2008] is unfortunately incorrect, as shown in Angluin [2016].

For a regular cardinal $\kappa$, a formula of the modal $\mu$-calculus is $\kappa$-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of $\kappa$-directed sets. We define the fragment $C_{\aleph_1}(x)$ of the modal $\mu$-calculus and prove that all the formulas in this fragment are $\aleph_1$-continuous. For each formula $\phi(x)$ of the modal $\mu$-calculus, we construct a formula $\psi(x) \in C_{\aleph_1 }(x)$ such that $\phi(x)$ is $\kappa$-continuous, for some $\kappa$, if and only if $\phi(x)$ is equivalent to $\psi(x)$. Consequently, we prove that (i) the problem whether a formula is $\kappa$-continuous for some $\kappa$ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment $C_{\aleph_0}(x)$ studied by Fontaine and the fragment $C_{\aleph_1}(x)$. We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal $\mu$-calculus. An ordinal $\alpha$ is the closure ordinal of a formula $\phi(x)$ if its interpretation on every model converges to its least fixed-point in at most $\alpha$ steps and if there is a model where the convergence occurs exactly in $\alpha$ steps. We prove that $\omega_1$, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, $\omega$, $\omega_1$ by […]

We study an extension of modal $\mu$-calculus to sets with atoms and we study its basic properties. Model checking is decidable on orbit-finite structures, and a correspondence to parity games holds. On the other hand, satisfiability becomes undecidable. We also show expressive limitations of atom-enriched $\mu$-calculi, and explain how their expressive power depends on the structure of atoms used, and on the choice between basic or vectorial syntax.

In implementing evaluation strategies of the lambda-calculus, both correctness and efficiency of implementation are valid concerns. While the notion of correctness is determined by the evaluation strategy, regarding efficiency there is a larger design space that can be explored, in particular the trade-off between space versus time efficiency. Aiming at a unified framework that would enable the study of this trade-off, we introduce an abstract machine, inspired by Girard's Geometry of Interaction (GoI), a machine combining token passing and graph rewriting. We show soundness and completeness of our abstract machine, called the \emph{Dynamic GoI Machine} (DGoIM), with respect to three evaluations: call-by-need, left-to-right call-by-value, and right-to-left call-by-value. Analysing time cost of its execution classifies the machine as ``efficient'' in Accattoli's taxonomy of abstract machines.

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the monomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. We also study the bounded-degree polynomial calculus. Over the rationals, it captures fixed-point logic with counting if we restrict the bit-complexity of the coefficients. For unrestricted coefficients, we can only say that the bounded-degree polynomial calculus is at most as powerful as bounded variable infinitary counting logic, but a precise logical characterisation of its power remains an open problem. These connections between logics and proof systems allow us to establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and also over finite fields. This is a corrected version of the paper (arXiv:1802.09377) published originally on January 23, 2019.