LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

Full Citation in the ACM Digital Library

When Reachability Meets Grzegorczyk

Vector addition systems with states, or equivalently vector addition systems, or Petri nets are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem is reachability: whether from a given initial configuration there exists a sequence of valid execution steps that reaches a given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of computation.

In this paper, we survey results about the reachability problem focusing on the general problem. We also show how a recent paper about the reachability problem in fixed dimension combined with vector addition systems with states weakly computing Grzegorczyk hierarchy provides a logspace reduction of the general reachability problem to the bounded case. This result, not included in the original paper due to a lack of space shows that the reachability problem can obviously be decided by a deterministic brute-force exploration. We provide perspectives based on this observation.

A tale of intersection types

Intersection types have come a long way since their introduction in the Seventies. They have been exploited for characterising behaviours of λ-terms and π-calculus processes, building λ-models, verifying properties of higher-order programs, synthesising code, and enriching the expressivity of programming languages. This paper is a light overview of intersection types and some of their applications.

Automatic Structures: Twenty Years Later

Automatic structures made their appearance at LICS twenty years ago, at LICS 2000. However, their roots are much older. The idea of automata based decision procedures for logical theories can be traced back to the early days of automata theory and to the work of Büchi, Elgot, Trakhtenbrot and Rabin in the 1960s. The explicit notion of automatic structures has first been proposed in 1976 in the (unfortunately largely unnoticed) PhD thesis of Hodgson, and later been reinvented by Khoussainov and Nerode in 1995.

In this tutorial, we present an introduction into the history and basic definitions of automatic structures, and survey the achievements in the study of different variants of automatic structures. We discuss their most important mathematical and algorithmic properties, their characterisations in terms of logical interpretations, and we present some of the mathematical techniques that are used for the analysis of automatic structures and for proving limitations of these concepts.

Contextual Types, Explained: Invited Tutorial

Contextual objects characterize an object M together with the typing context Ψ in which it is meaningful. This idea is then also internalized within the type theory itself using the notion of a contextual type which pairs the type A of an object together with the context Ψ in which the object is well-typed. In this tutorial, we review the origins of this idea and show its power in characterizing partial programs, mechanizing meta-theory, and meta-programming. Starting from the simply typed setting, we give an overview of existing work which adopts contextual types to dependent type theories and touch on future research directions.

Logic Beyond Formulas: A Proof System on Graphs

In this paper we present a proof system that operates on graphs instead of formulas. We begin our quest with the well-known correspondence between formulas and cographs, which are undirected graphs that do not have P4 (the four-vertex path) as vertex-induced subgraph; and then we drop that condition and look at arbitrary (undirected) graphs. The consequence is that we lose the tree structure of the formulas corresponding to the cographs. Therefore we cannot use standard proof theoretical methods that depend on that tree structure. In order to overcome this difficulty, we use a modular decomposition of graphs and some techniques from deep inference where inference rules do not rely on the main connective of a formula. For our proof system we show the admissibility of cut and a generalization of the splitting property. Finally, we show that our system is a conservative extension of multiplicative linear logic (MLL) with mix, meaning that if a graph is a cograph and provable in our system, then it is also provable in MLL+mix.

A Higher Structure Identity Principle

The ordinary Structure Identity Principle states that any property of set-level structures (e.g., posets, groups, rings, fields) definable in Univalent Foundations is invariant under isomorphism: more specifically, identifications of structures coincide with isomorphisms. We prove a version of this principle for a wide range of higher-categorical structures, adapting FOLDS-signatures to specify a general class of structures, and using two-level type theory to treat all categorical dimensions uniformly. As in the previously known case of 1-categories (which is an instance of our theory), the structures themselves must satisfy a local univalence principle, stating that identifications coincide with "isomorphisms" between elements of the structure. Our main technical achievement is a definition of such isomorphisms, which we call "indiscernibilities," using only the dependency structure rather than any notion of composition.

The Integers as a Higher Inductive Type

We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.

Space-efficient Query Evaluation over Probabilistic Event Streams

Real-time decision making in IoT applications relies upon space-efficient evaluation of queries over streaming data. To model the uncertainty in the classification of data being processed, we consider the model of probabilistic strings --- sequences of discrete probability distributions over a finite set of events, and initiate the study of space complexity of streaming computation for different classes of queries over such probabilistic strings.

We first consider the problem of computing the probability that a word, sampled from the distribution defined by the probabilistic string read so far, is accepted by a given deterministic finite automaton. We show that this regular pattern matching problem can be solved using space that is only poly-logarithmic in the string length (and polynomial in the size of the DFA) if we are allowed a multiplicative approximation error. Then we show how to generalize this result to quantitative queries specified by additive cost register automata --- these are automata that map strings to numerical values using finite control and registers that get updated using linear transformations. Finally, we consider the case when updates in such an automaton involve tests, and in particular, when there is a counter variable that can be either incremented or decremented but decrements only apply when the counter value is non-zero. In this case, the desired answer depends on the probability distribution over the set of possible counter values that can range from 0 to n for a string of length n. Under a mild assumption, namely probabilities of the individual events are bounded away from 0 and 1, we show that there is an algorithm that can compute all n entries of this probability distribution vector to within additive 1/poly(n) error using space that is only Õ(n). In establishing these results, we introduce several new technical ideas that may prove useful for designing space-efficient algorithms for other query models over probabilistic strings.

Algebraic models of simple type theories: A polynomial approach

We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed λ-calculi, the computational λ-calculus, and predicate logic.

Simple type theories are given models in presheaf categories, with structure specified by algebras of polynomial endofunctors that correspond to natural deduction rules. Initial models, which we construct, abstractly describe the syntax of simple type theories. Taking substitution structure into consideration, we further provide sound and complete semantics in structured cartesian multicategories. This development generalises Lambek's correspondence between the simply-typed λ-calculus and cartesian-closed categories, to arbitrary simple type theories.

Approximating Values of Generalized-Reachability Stochastic Games

Simple stochastic games are turn-based 2½-player games with a reachability objective. The basic question asks whether one player can ensure reaching a given target with at least a given probability. A natural extension is games with a conjunction of such conditions as objective. Despite a plethora of recent results on the analysis of systems with multiple objectives, the decidability of this basic problem remains open. In this paper, we present an algorithm approximating the Pareto frontier of the achievable values to a given precision. Moreover, it is an anytime algorithm, meaning it can be stopped at any time returning the current approximation and its error bound.

Reconciling noninterference and gradual typing

One of the standard correctness criteria for gradual typing is the dynamic gradual guarantee, which ensures that loosening type annotations in a program does not affect its behavior in arbitrary ways. Though natural, prior work has pointed out that the guarantee does not hold of any gradual type system for information-flow control. Toro et al.'s GSLRef language, for example, had to abandon it to validate noninterference.

We show that we can solve this conflict by avoiding a feature of prior proposals: type-guided classification, or the use of type ascription to classify data. Gradual languages require run-time secrecy labels to enforce security dynamically; if type ascription merely checks these labels without modifying them (that is, without classifying data), it cannot violate the dynamic gradual guarantee. We demonstrate this idea with GLIO, a gradual type system based on the LIO library that enforces both the gradual guarantee and noninterference, featuring higher-order functions, general references, coarsegrained information-flow control, security subtyping and first-class labels. We give the language a domain-theoretic semantics, using Pitts' framework of relational structures to prove noninterference and the dynamic gradual guarantee.

Complexity of controlled bad sequences over finite sets of Nd

We provide upper and lower bounds for the length of controlled bad sequences over the majoring and the minoring orderings of finite sets of Nd. The results are obtained by bounding the length of such sequences by functions from the Cichon hierarchy. This allows us to translate these results to bounds over the fast-growing complexity classes.

The obtained bounds are proven to be tight for the majoring ordering, which solves a problem left open by Abriola, Figueira and Senno (Theor. Comp. Sci, Vol. 603). Finally, we use the results on controlled bad sequences to prove upper bounds for the emptiness problem of some classes of automata.

Deciding Differential Privacy for Programs with Finite Inputs and Outputs

Differential privacy is a de facto standard for statistical computations over databases that contain private data. Its main and rather surprising strength is to guarantee individual privacy and yet allow for accurate statistical results. Thanks to its mathematical definition, differential privacy is also a natural target for formal analysis. A broad line of work develops and uses logical methods for proving privacy. A more recent and complementary line of work uses statistical methods for finding privacy violations. Although both lines of work are practically successful, they elide the fundamental question of decidability.

This paper studies the decidability of differential privacy. We first establish that checking differential privacy is undecidable even if one restricts to programs having a single Boolean input and a single Boolean output. Then, we define a non-trivial class of programs and provide a decision procedure for checking the differential privacy of a program in this class. Our procedure takes as input a program P parametrized by a privacy budget ϵ and either establishes the differential privacy for all possible values of ϵ or generates a counter-example. In addition, our procedure works for both to ϵ-differential privacy and (ϵ, δ)-differential privacy. Technically, the decision procedure is based on a novel and judicious encoding of the semantics of programs in our class into a decidable fragment of the first-order theory of the reals with exponentiation. We implement our procedure and use it for (dis)proving privacy bounds for many well-known examples, including randomized response, histogram, report noisy max and sparse vector.

Universal equivalence and majority of probabilistic programs over finite fields

We study decidability problems for equivalence of probabilistic programs, for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication and conditionals and thus is sufficiently expressive to encode boolean and arithmetic circuits. We consider two variants of equivalence: the first one considers an interpretation over the finite field Fq, while the second one, which we call universal equivalence, verifies equivalence over all extensions Fqk of Fq. The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F2k for any k. While the first problem is obviously decidable, we establish its exact complexity which lies in the counting hierarchy. To show decidability, and a doubly exponential upper bound, of the universal variant we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. Finally we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy.

Modal Logics with Composition on Finite Forests: Expressivity and Complexity

We study the expressivity and complexity of two modal logics interpreted on finite forests and equipped with standard modalities to reason on submodels. The logic ML(|) extends the modal logic K with the composition operator | from ambient logic, whereas ML(*) features the separating conjunction * from separation logic. Both operators are second-order in nature. We show that ML(|) is as expressive as the graded modal logic GML (on trees) whereas ML(*) is strictly less expressive than GML. Moreover, we establish that the satisfiability problem is Tower-complete for ML(*), whereas it is (only) AExpPol-complete for ML(|), a result which is surprising given their relative expressivity. As by-products, we solve open problems related to sister logics such as static ambient logic and modal separation logic.

A Hennessy-Milner Theorem for ATL with Imperfect Information

We show that a history-based variant of alternating bisimulation with imperfect information allows it to be related to a variant of Alternating-time Temporal Logic (ATL) with imperfect information by a full Hennessy-Milner theorem. The variant of ATL we consider has a common knowledge semantics, which requires that the uniform strategy available for a coalition to accomplish some goal must be common knowledge inside the coalition, while other semantic variants of ATL with imperfect information do not accomodate a Hennessy-Milner theorem. We also show that the existence of a history-based alternating bisimulation between two finite Concurrent Game Structures with imperfect information (iCGS) is undecidable.

Mixing Probabilistic and non-Probabilistic Objectives in Markov Decision Processes

In this paper, we consider algorithms to decide the existence of strategies in MDPs for Boolean combinations of objectives. These objectives are omega-regular properties that need to be enforced either surely, almost surely, existentially, or with non-zero probability. In this setting, relevant strategies are randomized infinite memory strategies: both infinite memory and randomization may be needed to play optimally. We provide algorithms to solve the general case of Boolean combinations and we also investigate relevant subcases. We further report on complexity bounds for these problems.

Hardness Characterisations and Size-Width Lower Bounds for QBF Resolution

We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) by circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems (Beyersdorff & Pich, LICS 2016), but leaving open the most important case of QBF resolution. Different from the Frege case, our characterisation uses a new version of decision lists as its circuit model, which is stronger than the CNFs the system works with. Our decision list model is well suited to compute countermodels for QBFs.

Our characterisation works for both Q-Resolution and QU-Resolution, which we show to be polynomially equivalent for QBFs of bounded quantifier alternation.

Using our characterisation we obtain a size-width relation for QBF resolution in the spirit of the celebrated result for propositional resolution (Ben-Sasson & Wigderson, J. ACM 2001). However, our result is not just a replication of the propositional relation --- intriguingly ruled out for QBF in previous research (Beyersdorff et al., ACM ToCL 2018) ---but shows a different dependence between size, width, and quantifier complexity.

We demonstrate that our new technique elegantly reproves known QBF hardness results and unifies previous lower-bound techniques in the QBF domain.

The Complexity of Reachability in Affine Vector Addition Systems with States

Vector addition systems with states (VASS) are widely used for the formal verification of concurrent systems. Given their tremendous computational complexity, practical approaches have relied on techniques such as reachability relaxations, e.g., allowing for negative intermediate counter values. It is natural to question their feasibility for VASS enriched with primitives that typically translate into undecidability. Spurred by this concern, we pinpoint the complexity of integer relaxations w.r.t. arbitrary classes of affine operations.

More specifically, we provide a trichotomy on the complexity of integer reachability in VASS extended with affine operations (affine VASS). Namely, we show that it is NP-complete for VASS with resets, PSPACE-complete for VASS with (pseudo-)transfers and VASS with (pseudo-)copies, and undecidable for any other class. We further present a dichotomy for standard reachability in affine VASS: it is decidable for VASS with permutations, and undecidable for any other class. This yields a complete and unified complexity landscape of reachability in affine VASS.

Temporal Constraint Satisfaction Problems in Fixed-Point Logic

Finite-domain constraint satisfaction problems are either solvable by Datalog, or not even expressible in fixed-point logic with counting. The border between the two regimes can be described by a strong height-one Maltsev condition. For infinite-domain CSPs, the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (Q; <); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the Boolean rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.

First-order tree-to-tree functions

We study tree-to-tree transformations that can be defined in first-order logic or monadic second-order logic. We prove a decomposition theorem, which shows that every transformation can be obtained from prime transformations, such as tree-to-tree homomorphisms or pre-order traversal, by using combinators such as function composition.

Extensions of ω-Regular Languages

We consider extensions of monadic second-order logic over ω-words, which are obtained by adding one language that is not ω-regular. We show that if the added language L has a neutral letter, then the resulting logic is necessarily undecidable. A corollary is that the ω-regular languages are the only decidable Boolean-closed full trio over ω-words.

A Cellular Howe Theorem

We introduce a categorical framework for operational semantics, in which we define substitution-closed bisimilarity, an abstract analogue of the open extension of Abramsky's applicative bisimilarity. We furthermore prove a congruence theorem for substitution-closed bisimilarity, following Howe's method. We finally demonstrate that the framework covers the call-by-name and call-by-value variants of λ-calculus in big-step style. As an intermediate result, we generalise the standard framework of Fiore et al. for syntax with variable binding to the skew-monoidal case.

On the Weisfeiler-Leman Dimension of Finite Groups

In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we define the Weisfeiler-Leman algorithm for groups. In fact we define three versions of the algorithm. In contrast to graphs, where the three analogous versions readily agree, for groups the situation is more intricate. For groups, we show that their expressive power is linearly related. We also give descriptions in terms of counting logics and bijective pebble games for each of the versions.

In order to construct examples of groups, we devise an isomorphism and non-isomorphism preserving transformation from graphs to groups. Using graphs of high Weisfeiler-Leman dimension, we construct similar but non-isomorphic groups with equal ™(log n)-subgroup-profiles, which nevertheless have Weisfeiler-Leman dimension 3. These groups are nilpotent groups of class 2 and exponent p, they agree in many combinatorial properties such as the combinatorics of their conjugacy classes and have highly similar commuting graphs.

The results indicate that the Weisfeiler-Leman algorithm can be more effective in distinguishing groups than in distinguishing graphs based on similar combinatorial constructions.

A Fixed Point Theorem on Lexicographic Lattice Structures

We introduce the notion of a lexicographic lattice structure, namely a lattice whose elements can be viewed as stratified entities and whose ordering relation compares elements in a lexicographic manner with respect to their strata. These lattices arise naturally in many non-monotonic formalisms, such as normal logic programs, higher-order logic programs with negation, and boolean grammars. We consider functions over such lattices that may overall be non-monotonic, but retain a restricted form of monotonicity inside each stratum. We demonstrate that such functions always have a least fixed point which is also their least pre-fixed point. Moreover, we prove that the sets of pre-fixed and post-fixed points of such functions, are complete lattices. For the special case of a trivial lexicographic lattice structure whose elements essentially consist of a unique stratum, our theorem gives as a special case the well-known Knaster-Tarski fixed point theorem. Moreover, our work considerably simplifies and extends recent results on non-monotonic fixed point theory, providing in this way a useful and convenient tool in the semantic investigation of non-monotonic formalisms.

Re-pairing brackets

Consider the following one-player game. Take a well-formed sequence of opening and closing brackets (a Dyck word). As a move, the player can pair any opening bracket with any closing bracket to its right, erasing them. The goal is to re-pair (erase) the entire sequence, and the cost of a strategy is measured by its width: the maximum number of nonempty segments of symbols (separated by blank space) seen during the play.

For various initial sequences, we prove upper and lower bounds on the minimum width sufficient for re-pairing. (In particular, the sequence associated with the complete binary tree of height n admits a strategy of width sub-exponential in log n.) Our two key contributions are (1) lower bounds on the width and (2) their application in automata theory: quasi-polynomial lower bounds on the translation from one-counter automata to Parikh-equivalent nondeterministic finite automata. The latter result answers a question by Atig et al. (2016).

The Benefit of Being Non-Lazy in Probabilistic λ-calculus: Applicative Bisimulation is Fully Abstract for Non-Lazy Probabilistic Call-by-Name

We consider the probabilistic applicative bisimilarity (PAB) --- a coinductive relation comparing the applicative behaviour of probabilistic untyped λ-terms according to a specific operational semantics. This notion has been studied by Dal Lago et al. with respect to the two standard parameter passing policies, call-by-value (cbv) and call-by-name (cbn), using a lazy reduction strategy not reducing within the body of a function. In particular, PAB has been proven to be fully abstract with respect to the contextual equivalence in cbv [6] but not in lazy cbn [16].

We overcome this issue of cbn by relaxing the laziness constraint: we prove that PAB is fully abstract with respect to the standard head reduction contextual equivalence. Our proof is based on Leventis' Separation Theorem [19], using probabilistic Nakajima trees as a tree-like representation of the contextual equivalence classes.

Finally, we prove also that the inequality full abstraction fails, showing that the probabilistic applicative similarity is strictly contained in the contextual preorder.

An Approach to Regular Separability in Vector Addition Systems

We study the problem of regular separability of languages of vector addition systems with states (VASS). It asks whether for two given VASS languages K and L, there exists a regular language R that includes K and is disjoint from L. While decidability of the problem in full generality remains an open question, there are several subclasses for which decidability has been shown: It is decidable for (i) one-dimensional VASS, (ii) VASS coverability languages, (iii) languages of integer VASS, and (iv) commutative VASS languages.

We propose a general approach to deciding regular separability. We use it to decide regular separability of an arbitrary VASS language from any language in the classes (i), (ii), and (iii). This generalizes all previous results, including (iv).

Modal Intuitionistic Logics as Dialgebraic Logics

Duality is one of the key techniques in the categorical treatment of modal logics. From the duality between (modal) algebras and (descriptive) frames one derives e.g. completeness (via a syntactic characterisation of algebras) or definability (using a suitable version of the Goldblatt-Thomason theorem). This is by now well understood for classical modal logics and modal logics based on distributive lattices, via extensions of Stone and Priestley duality, respectively. What is conspicuously absent is a comprehensive treatment of modal intuitionistic logic. This is the gap we are closing in this paper. Our main conceptual insight is that modal intuitionistic logics do not appear as algebra/coalgebra dualities, but instead arise naturally as dialgebras. Our technical contribution is the development of dualities for dialgebras, together with their logics, that instantiate to large class of modal intuitionistic logics and their frames as special cases. We derive completeness and expressiveness results in this general case. For modal intuitionistic logic, this systematises the existing treatment in the literature.

Cones as a model of intuitionistic linear logic

For overcoming the limitations of probabilistic coherence spaces which do not seem to provide natural interpretations of continuous data types such as the real line, we introduced with Pagani and Tasson a model of probabilistic higher order computation based on (positive) cones, and a class of totally monotone functions that we called "stable". Then Crubillé proved that this model is a conservative extension of the earlier probabilistic coherence space model. We continue these investigations by showing that the category of cones and linear and Scott-continuous functions is a model of intuitionistic linear logic. To define the tensor product, we use the special adjoint functor theorem, and we prove that this operation is an extension of the standard tensor product of probabilistic coherence spaces. We also show that these latter are dense in cones, thus allowing to lift the main properties of the tensor product of probabilistic coherence spaces to general cones. Finally we define in the same way an exponential of cones and extend measurability to these new operations.

Uniformisations of Regular Relations Over Bi-Infinite Words

We consider the problem of deciding whether a given mso-definable relation over bi-infinite words contains an mso-definable function with the same domain. We prove that this problem is decidable. There are two obstacles to the existence of such uniformisations: the first is related to the existence of non-trivial automorphisms of bi-infinite words, whereas the second, more subtle obstacle, is related to the existence of finite, discrete dynamical systems, where no trajectory can be selected by an mso formula.

One-Clock Priced Timed Games are PSPACE-hard

The main result of this paper is that computing the value of a one-clock priced timed game (OCPTG) is PSPACE-hard. Along the way, we provide a family of OCPTGs that have an exponential number of event points. Both results hold even in very restricted classes of games such as DAGs with treewidth three. Finally, we provide a number of positive results, including polynomial-time algorithms for even more restricted classes of OCPTGs such as trees.

Lower Bounds for QBFs of Bounded Treewidth

The problem of deciding the validity (QSat) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a successful parameter. A well-known result by Chen [9] is that QSat when parameterized by the treewidth of the primal graph and the quantifier rank of the input formula is fixed-parameter tractable. More precisely, the runtime of such an algorithm is polynomial in the formula size and exponential in the treewidth, where the exponential function in the treewidth is a tower, whose height is the quantifier rank. A natural question is whether one can significantly improve these results and decrease the tower while assuming the Exponential Time Hypothesis (ETH). In the last years, there has been a growing interest in the quest of establishing lower bounds under ETH, showing mostly problem-specific lower bounds up to the third level of the polynomial hierarchy. Still, an important question is to settle this as general as possible and to cover the whole polynomial hierarchy. In this work, we show lower bounds based on the ETH for arbitrary QBFs parameterized by treewidth and quantifier rank. More formally, we establish lower bounds for QSat and treewidth, namely, that under ETH there cannot be an algorithm that solves QSat of quantifier rank i in runtime significantly better than i-fold exponential in the treewidth and polynomial in the input size. In doing so, we provide a reduction technique to compress treewidth that encodes dynamic programming on arbitrary tree decompositions. Further, we describe a general methodology for a more finegrained analysis of problems parameterized by treewidth that are at higher levels of the polynomial hierarchy. Finally, we illustrate the usefulness of our results by discussing various applications of our results to problems that are located higher on the polynomial hierarchy, in particular, various problems from the literature such as projected model counting problems.

Coherence and normalisation-by-evaluation for bicategorical cartesian closed structure

We present two proofs of coherence for cartesian closed bicategories. Precisely, we show that in the free cartesian closed bicategory on a set of objects there is at most one structural 2-cell between any parallel pair of 1-cells. We thereby reduce the difficulty of constructing structure in arbitrary cartesian closed bicategories to the level of 1-dimensional category theory. Our first proof follows a traditional approach using the Yoneda lemma. For the second proof, we adapt Fiore's categorical analysis of normalisation-by-evaluation for the simply-typed lambda calculus. Modulo the construction of suitable bicategorical structures, the argument is not significantly more complex than its 1-categorical counterpart. It also opens the way for further proofs of coherence using (adaptations of) tools from categorical semantics.

Linear Dependent Type Theory for Quantum Programming Languages: Extended Abstract

Modern quantum programming languages integrate quantum resources and classical control. They must, on the one hand, be linearly typed to reflect the no-cloning property of quantum resources. On the other hand, high-level and practical languages should also support quantum circuits as first-class citizens, as well as families of circuits that are indexed by some classical parameters. Quantum programming languages thus need linear dependent type theory. This paper defines a general semantic structure for such a type theory via certain fibrations of monoidal categories. The categorical model of the quantum circuit description language Proto-Quipper-M in [28] constitutes an example of such a fibration, which means that the language can readily be integrated with dependent types. We then devise both a general linear dependent type system and a dependently typed extension of Proto-Quipper-M, and provide them with operational semantics as well as a prototype implementation.

Combining probabilistic and non-deterministic choice via weak distributive laws

Combining probabilistic choice and non-determinism is a long standing problem in denotational semantics. From a category theory perspective, the problem stems from the absence of a distributive law of the powerset monad over the distribution monad. In this paper we prove the existence of a weak distributive law of the powerset monad over the finite distribution monad. As a consequence, we retrieve the well-known convex powerset monad as a weak lifting of the powerset monad to the category of convex algebras. We provide applications to the study of trace semantics and behavioral equivalences of systems with an interplay between probability and non-determinism.

A Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity

Robin Milner (1984) gave a sound proof system for bisimilarity of regular expressions interpreted as processes: Basic Process Algebra with unary Kleene star iteration, deadlock 0, successful termination 1, and a fixed-point rule. He asked whether this system is complete. Despite intensive research over the last 35 years, the problem is still open.

This paper gives a partial positive answer to Milner's problem. We prove that the adaptation of Milner's system over the subclass of regular expressions that arises by dropping the constant 1, and by changing to binary Kleene star iteration is complete. The crucial tool we use is a graph structure property that guarantees expressibility of a process graph by a regular expression, and that is preserved when going over from a process graph to its bisimulation collapse.

Successor-Invariant First-Order Logic on Classes of Bounded Degree

We study the expressive power of successor-invariant first-order logic, which is an extension of first-order logic where the usage of an additional successor relation on the structure is allowed, as long as the validity of formulas is independent on the choice of a particular successor.

We show that when the degree is bounded, successor-invariant first-order logic is no more expressive than firstorder logic.

Multimodal Dependent Type Theory

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion --- demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

Counting Bounded Tree Depth Homomorphisms

We prove that graphs G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at most k. Here G, G' are homomorphism-indistinguishable over a class F of graphs if for each graph F ϵ F, the number of homomorphisms from F to G equals the number of homomorphisms from F to G'.

Bisimulation Finiteness of Pushdown Systems Is Elementary

We show that in case a pushdown system is bisimulation equivalent to a finite system, there is already a bisimulation equivalent finite system whose size is elementarily bounded in the description size of the pushdown system. As a consequence we obtain that it is elementarily decidable if a given pushdown system is bisimulation equivalent to some finite system. This improves a previously best-known ACKERMANN upper bound for this problem.

A tier-based typed programming language characterizing Feasible Functionals

The class of Basic Feasible Functionals BFF2 is the type-2 counterpart of the class FP of type-1 functions computable in polynomial time. Several characterizations have been suggested in the literature, but none of these present a programming language with a type system guaranteeing this complexity bound. We give a characterization of BFF2 based on an imperative language with oracle calls using a tier-based type system whose inference is decidable. Such a characterization should make it possible to link higher-order complexity with programming theory. The low complexity (cubic in the size of the program) of the type inference algorithm contrasts with the intractability of the aforementioned methods and does not restrain strongly the expressive power of the language.

Descriptive complexity of real computation and probabilistic independence logic

We introduce a novel variant of BSS machines called Separate Branching BSS machines (S-BSS in short) and develop a Fagin-type logical characterisation for languages decidable in nondeterministic polynomial time by S-BSS machines. We show that NP on S-BSS machines is strictly included in NP on BSS machines and that every NP language on S-BSS machines is a countable disjoint union of closed sets in the usual topology of Rn. Moreover, we establish that on Boolean inputs NP on S-BSS machines without real constants characterises a natural fragment of the complexity class ∃R (a class of problems polynomial time reducible to the true existential theory of the reals) and hence lies between NP and PSPACE. Finally we apply our results to determine the data complexity of probabilistic independence logic.

A calculus of expandable stores: Continuation-and-environment-passing style translations

The call-by-need evaluation strategy for the λ-calculus is an evaluation strategy that lazily evaluates arguments only if needed, and if so, shares computations across all places where it is needed. To implement this evaluation strategy, abstract machines require some form of global environment. While abstract machines usually lead to a better understanding of the flow of control during the execution, facilitating in particular the definition of continuation-passing style translations, the case of machines with global environments turns out to be much more subtle.

The main purpose of this paper is to understand how to type a continuation-and-environment-passing style translation, that is to say how to soundly translate in continuation-passing style a calculus with global environment. To this end, we introduce Fγ, a generic calculus to define the target of such translations. In particular, Fγ features a data type for typed stores and a mechanism of explicit coercions witnessing store extensions along environment-passing style translations. On the logical side, this broadly amounts to a Kripke forcing-like translation mixed with a negative translation (for the continuation-passing part). Since Fγ allows for the definition of such translations for different source calculi (call-by-need, call-by-name, call-by-value) with different type systems (simple types, system F), we claim that it precisely captures the computational content of continuation-and-environment-passing style translations.

Intermediate problems in modular circuits satisfiability

In [15] a generalization of Boolean circuits to arbitrary finite algebras had been introduced and applied to sketch P versus NP-complete borderline for circuits satisfiability over algebras from congruence modular varieties. However the problem for nilpotent (which had not been shown to be NP-hard) but not supernilpotent algebras (which had been shown to be polynomial time) remained open.

In this paper we provide a broad class of examples, lying in this grey area, and show that, under the Exponential Time Hypothesis and Strong Exponential Size Hypothesis (saying that Boolean circuits need exponentially many modular counting gates to produce boolean conjunctions of any arity), satisfiability over these algebras have intermediate complexity between ω(2c logh-1 n) and O(2c logh n), where h measures how much a nilpotent algebra fails to be supernilpotent. We also sketch how these examples could be used as paradigms to fill the nilpotent versus supernilpotent gap in general.

Our examples are striking in view of the natural strong connections between circuits satisfiability and Constraint Satisfaction Problem for which the dichotomy had been shown by Bulatov [4] and Zhuk [28].

The Surprising Power of Constant Depth Algebraic Proofs

A major open problem in proof complexity is to prove superpolynomial lower bounds for AC0[p]-Frege proofs. This system is the analog of AC0 [p], the class of bounded depth circuits with prime modular counting gates. Despite strong lower bounds for this class dating back thirty years ([28, 30]), there are no significant lower bounds for AC0 [p]-Frege. Significant and extensive degree lower bounds have been obtained for a variety of subsystems of AC0[p]-Frege, including Nullstellensatz ([3]), Polynomial Calculus ([9]), and SOS ([14]). However to date there has been no progress on AC0 [p]-Frege lower bounds.

In this paper we study constant-depth extensions of the Polynomial Calculus [13]. We show that these extensions are much more powerful than was previously known. Our main result is that small depth (≤ 43) Polynomial Calculus (over a sufficiently large field) can polynomially effectively simulate all of the well-studied semialgebraic proof systems: Cutting Planes, Sherali-Adams, Sum-of-Squares (SOS), and Positivstellensatz Calculus (Dynamic SOS). Additionally, they can also quasi-polynomially effectively simulate AC0[q]-Frege for any prime q independent of the characteristic of the underlying field. They can also effectively simulate TC0-Frege if the depth is allowed to grow proportionally. Thus, proving strong lower bounds for constant-depth extensions of Polynomial Calculus would not only give lower bounds for AC0 [p]-Frege, but also for systems as strong as TC0-Frege.

Interaction Laws of Monads and Comonads

We introduce and study functor-functor and monad-comonad interaction laws as mathematical objects to describe interaction of effectful computations with behaviors of effect-performing machines. Monad-comonad interaction laws are monoid objects of the monoidal category of functor-functor interaction laws. We show that, for suitable generalizations of the concepts of dual and Sweedler dual, the greatest functor resp. monad interacting with a given functor or comonad is its dual while the greatest comonad interacting with a given monad is its Sweedler dual. We relate monad-comonad interaction laws to stateful runners. We show that functor-functor interaction laws are Chu spaces over the category of endofunctors taken with the Day convolution monoidal structure. Hasegawa's glueing endows the category of these Chu spaces with a monoidal structure whose monoid objects are monad-comonad interaction laws.

Consuming and Persistent Types for Classical Logic

We prove that type systems are able to capture exact measures related to dynamic properties of functional programs with control operators, which allow implementing intricate continuations and backtracking. Our type systems give the number of evaluation steps to normal form as well as the size of this normal form without any evaluation being needed.

We focus on Parigot's λμ-calculus, a computational interpretation of classical natural deduction, and our type systems are based upon non-idempotent intersection and union types. We introduce two kinds of arrows: consuming ones, which type function applications that are evaluated/destroyed during reduction, and persistent ones, which type those that are never consumed. These two forms of arrows are the essential tool of our typing systems to deal with control operators, as we are going to show.

The main contribution of this paper is a type framework capturing exact measures within the λμ-calculus for 3 different evaluation strategies, namely, head, leftmost-outermost and maximal, associated to the well-known notions of head, weak and strong normalization respectively. Moreover, this is done in a parametric way: we do not provide one type system for each reduction strategy, as it is usually done in the literature, but we give a unique typing system parametrizing key concepts and factorizing common proofs.

Refinement-Based Game Semantics for Certified Abstraction Layers

Formal methods have advanced to the point where the functional correctness of various large system components has been mechanically verified. However, the diversity of semantic models used across projects makes it difficult to connect these component to build larger certified systems. Given this, we seek to embed these models and proofs into a generalpurpose framework where they could interact. We believe that a synthesis of game semantics, the refinement calculus, and algebraic effects can provide such a framework.

To combine game semantics and refinement, we replace the downset completion typically used to construct strategies from posets of plays. Using the free completely distributive completion, we construct strategy specifications equipped with arbitrary angelic and demonic choices and ordered by a generalization of alternating refinement. This provides a novel approach to nondeterminism in game semantics.

Connecting algebraic effects and game semantics, we interpret effect signatures as games and define two categories of effect signatures and strategy specifications. The resulting models are sufficient to represent the behaviors of a variety of low-level components, including the certified abstraction layers used to verify the operating system kernel CertiKOS.

Large and Infinitary Quotient Inductive-Inductive Types

Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of type theories and constructive definitions of real, ordinal and surreal numbers. We develop new metatheory for large QIITs, large elimination, recursive equations and infinitary constructors. As in prior work, we describe QIITs using a type theory where each context represents a QIIT signature. However, in our case the theory of signatures can also describe its own signature, modulo universe sizes. We bootstrap the model theory of signatures using self-description and a Church-coded notion of signature, without using complicated raw syntax or assuming an existing internal QIIT of signatures. We give semantics to described QIITs by modeling each signature as a finitely complete CwF (category with families) of algebras. Compared to the case of finitary QIITs, we additionally need to show invariance under algebra isomorphisms in the semantics. We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures.

Coherence via Well-Foundedness: Taming Set-Quotients in Homotopy Type Theory

Suppose we are given a graph and want to show a property for all its cycles (closed chains). Induction on the length of cycles does not work since sub-chains of a cycle are not necessarily closed. This paper derives a principle reminiscent of induction for cycles for the case that the graph is given as the symmetric closure of a locally confluent and (co-)well-founded relation. We show that, assuming the property in question is sufficiently nice, it is enough to prove it for the empty cycle and for cycles given by local confluence.

Our motivation and application is in the field of homotopy type theory, which allows us to work with the higher-dimensional structures that appear in homotopy theory and in higher category theory, making coherence a central issue. This is in particular true for quotienting - a natural operation which gives a new type for any binary relation on a type and, in order to be well-behaved, cuts off higher structure (set-truncates). The latter makes it hard to characterise the type of maps from a quotient into a higher type, and several open problems stem from this difficulty.

We prove our theorem on cycles in a type-theoretic setting and use it to show coherence conditions necessary to eliminate from set-quotients into 1-types, deriving approximations to open problems on free groups and pushouts. We have formalised the main result in the proof assistant Lean.

Efficient Analysis of VASS Termination Complexity

The termination complexity of a given VASS is a function L assigning to every n the length of the longest non-terminating computation initiated in a configuration with all counters bounded by n. We show that for every VASS with demonic nondeterminism and every fixed k, the problem whether L ϵ Gk, where Gk is the k-th level in the Grzegorczyk hierarchy, is decidable in polynomial time. Furthermore, we show that if L ϵ G, then L grows at least as fast as the generator Fk+1 of Gk+1. Hence, for every terminating VASS, the growth of L can be reasonably characterized by the least k such that L ϵ Gk.

Furthermore, we consider VASS with both angelic and demonic nondeterminism, i.e., VASS games where the players aim at lowering/raising the termination time. We prove that for every fixed k, the problem whether L ϵ Gk for a given VASS game is NP-complete. Furthermore, if L ϵ Gk, then L grows at least as fast as Fk+1.

Good-for-games ω-Pushdown Automata

We introduce good-for-games ω-pushdown automata (ω-GFG-PDA). These are automata whose nondeterminism can be resolved based on the run constructed thus far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata.

Our main results are that ω-GFG-PDA are more expressive than deterministic ω-pushdown automata and that solving infinite games with winning conditions specified by ω-GFG-PDA is EXPTIME-complete. Thus, we have identified a new class of ω-contextfree winning conditions for which solving games is decidable. It follows that the universality problem for ω-GFG-PDA is in EXPTIME as well.

Moreover, we study closure properties of the class of languages recognized by ω-GFG-PDA and decidability of good-for-gameness of ω-pushdown automata and languages.

Pebble Minimization of Polyregular Functions

We show that a polyregular word-to-word function is regular if and only its output size is at most linear in its input size. Moreover a polyregular function can be realized by: a transducer with two pebbles if and only if its output has quadratic size in its input, a transducer with three pebbles if and only if its output has cubic size in its input, etc.

Moreover the characterization is decidable and, given a polyregular function, one can compute a transducer realizing it with the minimal number of pebbles.

We apply the result to mso interpretations from words to words. We show that mso interpretations of dimension k exactly coincide with k-pebble transductions.

The Complexity of Dynamic Data Race Prediction

Writing concurrent programs is notoriously hard due to scheduling non-determinism. The most common concurrency bugs are data races, which are accesses to a shared resource that can be executed concurrently. Dynamic data-race prediction is the most standard technique for detecting data races: given an observed, data-race-free trace t, the task is to determine whether t can be reordered to a trace t* that exposes a data-race. Although the problem has received significant practical attention for over three decades, its complexity has remained elusive. In this work, we address this lacuna, identifying sources of intractability and conditions under which the problem is efficiently solvable. Given a trace t of size n over k threads, our main results are as follows.

First, we establish a general O(k · n2·(k-1) upper-bound, as well as an O(nk) upper-bound when certain parameters of t are constant. In addition, we show that the problem is NP-hard and even W[1]-hard parameterized by k, and thus unlikely to be fixed-parameter tractable. Second, we study the problem over acyclic communication topologies, such as server-clients hierarchies. We establish an O(k2 · d · n2 · log n) upper-bound, where d is the number of shared variables accessed in t. In addition, we show that even for traces with k = 2 threads, the problem has no O(n2-ϵ) algorithm under the Orthogonal Vectors conjecture. Since any trace with 2 threads defines an acyclic topology, our upper-bound for this case is optimal up to polynomial improvements for up to moderate values of k and d. Finally, motivated by existing heuristics, we study a distance-bounded version of the problem, where the task is to expose a data race by a witness trace that is similar to t. We develop an algorithm that works in O(n) time when certain parameters of t are constant.

Sparse Hashing for Scalable Approximate Model Counting: Theory and Practice

Given a CNF formula F on n variables, the problem of model counting, also referred to as #SAT, is to compute the number of models or satisfying assignments of F. Recent years have witnessed a surge of effort towards developing efficient algorithmic techniques that combine the classical strongly 2-universal hash functions (from [Stockmeyer 1983]) with the remarkable progress in SAT solving over the past decade. These techniques augment the CNF formula F with random XOR constraints and invoke an NP oracle repeatedly on the resultant CNF-XOR formulas. In practice, the NP oracle calls are replaced by calls to a SAT solver and it is observed that runtime performance of modern SAT solvers (based on conflict-driven clause learning) on CNF-XOR formulas is adversely affected by the size of XOR constraints. The standard construction of 2-universal hash functions chooses every variable with probability p = 1/2 leading to XOR constraints of size n/2 in expectation. Consequently, the main challenge is to design sparse hash functions, where variables can be chosen with smaller probability and lead to smaller sized XOR constraints, which can then replace strongly 2-universal hash functions.

In this paper, our goal is to address this challenge both from a theoretical and a practical perspective. First, we formalize a relaxation of universal hashing, called concentrated hashing, a notion implicit in prior works to design sparse hash functions. We then establish a novel and beautiful connection between concentration measures of these hash functions and isoperimetric inequalities on boolean hypercubes. This allows us to obtain tight bounds on variance as well as the dispersion index and show that p = O(log2 m/m) suffices for the design of sparse hash functions from {0, 1}n to {0, 1}m belonging to the concentrated hash family. Finally, we use sparse hash functions belonging to this concentrated hash family to develop new approximate counting algorithms. A comprehensive experimental evaluation of our algorithm on 1893 benchmarks demonstrates that the usage of sparse hash functions can lead to significant speedups. To the best of our knowledge, this work is the first study to demonstrate runtime improvement of approximate model counting algorithms through the usage of sparse hash functions, while still retaining strong theoretical guarantees (à la 2-universal hash functions).

Concurrent Separation Logic Meets Template Games

An old dream of concurrency theory and programming language semantics has been to uncover the fundamental synchronization mechanisms which regulate situations as different as game semantics for higher-order programs, and Hoare logic for concurrent programs with shared memory and locks. We establish a deep and unexpected connection between two recent lines of work on concurrent separation logic (CSL) and on template game semantics for differential linear logic (DiLL). Thanks to this connection, we reformulate in the purely conceptual style of template games for DiLL the asynchronous and interactive interpretation of CSL designed by Melliès and Stefanesco in a recent work. We believe that the analysis reveals something important about the secret anatomy of CSL, and more specifically about the subtle interplay, of a categorical nature, between sequential composition, parallel product, errors and locks.

The Hidden Subgroup Problem for Universal Algebras

The Hidden Subgroup Problem (HSP) is a computational problem which includes as special cases integer factorization, the discrete logarithm problem, graph isomorphism, and the shortest vector problem. The celebrated polynomial-time quantum algorithms for factorization and the discrete logarithm are restricted versions of a generic polynomial-time quantum solution to the HSP for abelian groups, but despite focused research no polynomial-time solution for general groups has yet been found. We propose a generalization of the HSP to include arbitrary algebraic structures and analyze this new problem on powers of 2-element algebras. We prove a complete classification of every such power as quantum tractable (i.e. polynomial-time), classically tractable, quantum intractable, or classically intractable. In particular, we identify a class of algebras for which the generalized HSP exhibits super-polynomial speedup on a quantum computer compared to a classical one.

On the computational content of Zorn's lemma

We give a computational interpretation to an abstract instance of Zorn's lemma formulated as a wellfoundedness principle in the language of arithmetic in all finite types. This is achieved through Gödel's functional interpretation, and requires the introduction of a novel form of recursion over non-wellfounded partial orders whose existence in the model of total continuous functionals is proven using domain theoretic techniques. We show that a realizer for the functional interpretation of open induction over the lexicographic ordering on sequences follows as a simple application of our main results.

Russian Constructivism in a Prefascist Theory

The results from this paper are twofold. First, we give a purely syntactic presheaf model of CIC. Contrarily to similar endeavours, this variant both preserves conversion and interprets full dependent elimination.

Using a particular instance of this model, we show how to extend CIC with Markov's principle, while preserving all good meta-theoretical properties like canonicity and decidability of type-checking. The resulting construction can be seen as a synthetic presentation of Coquand-Hofmann's syntactic model of PRAω + MP as the composition of Pédrot-Tabareau's exceptional model with our presheaf interpretation.

Extended Kripke lemma and decidability for hypersequent substructural logics

We establish the decidability of every axiomatic extension of the commutative Full Lambek calculus with contraction FLec that has a cut-free hypersequent calculus. The axioms include familiar properties such as linearity (fuzzy logics) and the substructural versions of bounded width and weak excluded middle. Kripke famously proved the decidability of FLec by combining structural proof theory and combinatorics. This work significantly extends both ingredients: height-preserving admissibility of contraction by internalising a fixed amount of contraction (a Curry's lemma for hypersequent calculi) and an extended Kripke lemma for hypersequents that relies on the componentwise partial order on n-tuples being an ω2-well-quasi-order.

Partial Univalence in n-truncated Type Theory

It is well known that univalence is incompatible with uniqueness of identity proofs (UIP), the axiom that all types are h-sets. This is due to finite h-sets having non-trivial automorphisms as soon as they are not h-propositions.

A natural question is then whether univalence restricted to h-propositions is compatible with UIP. We answer this affirmatively by constructing a model where types are elements of a closed universe defined as a higher inductive type in homotopy type theory. This universe has a path constructor for simultaneous "partial" univalent completion, i.e., restricted to h-propositions.

More generally, we show that univalence restricted to (n -- 1)-types is consistent with the assumption that all types are n-truncated. Moreover we parametrize our construction by a suitably well-behaved container, to abstract from a concrete choice of type formers for the universe.

Resolving finite indeterminacy: A definitive constructive universal prime ideal theorem

Dynamical methods were designed to eliminate the ideal objects abstract algebra abounds with. Typically granted by an incarnation of Zorn's Lemma, those ideal objects often serve for proving the semantic conservation of additional non-deterministic sequents, that is, with finite but not necessarily singleton succedents. Eliminating ideal objects dynamically was possible also because (finitary) coherent or geometric logic predominates in that area: the use of a non-deterministic axiom can be captured by a finite branching of the proof tree.

Incidentally, a paradigmatic case has widely been neglected in dynamical algebra: Krull's Lemma for prime ideals. Digging deeper just about that case, which we have dealt with only recently (with Yengui), has now brought us to unearth the general phenomenon underlying dynamical algebra: Given a claim of computational nature that usually is proved by the semantic conservation of certain additional non-deterministic axioms, there is a finite labelled tree belonging to a suitable inductively generated class which tree encodes the desired computation. Our characterisation works in the fairly universal setting of a consequence relation enriched with non-deterministic axioms; uniformises many of the known instances of the dynamical method; generalises the proof-theoretic conservation criterion we have offered before (with Rinaldi); and last but not least links the syntactical with the semantic approach: every ideal object used for the customary proof of a concrete claim can be approximated by one of the corresponding tree's branches.

An Efficient Normalisation Procedure for Linear Temporal Logic and Very Weak Alternating Automata

In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form Λni =1 GFφi ∨FGψi, where φi and ψi contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalises the formula, translates it into a special very weak alternating automaton, and applies a simple determinisation procedure, valid only for these special automata.

Sequential Colimits in Homotopy Type Theory

Sequential colimits are an important class of higher inductive types. We present a self-contained and fully formalized proof of the conjecture that in homotopy type theory sequential colimits appropriately commute with Σ-types. This result allows us to give short proofs of a number of useful corollaries, some of which were conjectured in other works: the commutativity of sequential colimits with identity types, with homotopy fibers, loop spaces, and truncations, and the preservation of the properties of truncatedness and connectedness under sequential colimits. Our entire development carries over to (∞, 1)-toposes using Shulman's recent interpretation of homotopy type theory into these structures.

Making Streett Determinization Tight

Optimal determinization construction of Streett automata is an important research problem because it is indispensable in numerous applications such as decision problems for tree temporal logics, logic games and system synthesis. This paper presents a transformation from nondeterministic Streett automata (NSA) with n states and k Streett pairs to equivalent deterministic Rabin transition automata (DRTA) with n5n(n!)n states, O(nn2) Rabin pairs for k = ω(n) and n5nknk states, O(knk) Rabin pairs for k = O(n). This improves the state of the art Streett determinization construction with n5n(n!)n+1 states, O(n2) Rabin pairs and n5nknkn! states, O(nk) Rabin pairs, respectively. Moreover, deterministic parity transition automata (DPTA) are obtained with 3(n(n + 1) -- 1)!(n!)n+1 states, 2n(n +1) priorities for k = ω(n) and 3(n(k +1) -- 1)!n!knk states, 2n(k + 1) priorities for k = O(n), which improves the best construction with nn(k + 1)n(k+1)(n(k + 1) -- 1)! states, 2n(k + 1) priorities. Further, we prove a lower bound state complexity for determinization construction from N-SA to deterministic Rabin (transition) automata i.e. n5n(n!)n for k = ω(n) and n5nknk for k = O(n), which matches the state complexity of the proposed determinization construction. Besides, we put forward a lower bound state complexity for determinization construction from NSA to deterministic parity (transition) automata i.e. 2ω(n2 log n) for k = ω(n) and 2ω(nk log nk) for k = O(n), which is the same as the state complexity of the proposed determinization construction in the exponent.

Register Automata with Extrema Constraints, and an Application to Two-Variable Logic

We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register values in subtrees. We show that the emptiness problem for these automata is decidable.

As an application, we prove decidability of the countable satisfiability problem for two-variable logic in the presence of a tree order, a linear order, and arbitrary atoms that are MSO definable from the tree order. As a consequence, the satisfiability problem for two-variable logic with arbitrary predicates, two of them interpreted by linear orders, is decidable.

On Computability of Logical Approaches to Branching-Time Property Verification of Programs

This paper studies the hardness of branching-time property verification of Turing-complete programming languages, as well as logical approaches to the verification problem. As these approaches reduce the verification problem to logical problems, e.g. the satisfiability problem of Horn clauses with certain extensions, it is natural to ask whether the logical problems are as hard as the verification problem or strictly harder. This paper reveals that logical problems used in most approaches are far more difficult than the verification problem; the only exception is the validity problem of first-order arithmetic with fixed-point operators. We also answers some other natural questions, for example, whether the extensions of Horn clauses are necessarily.

Automata Learning: An Algebraic Approach

We propose a generic categorical framework for learning unknown formal languages of various types (e.g. finite or infinite words, weighted and nominal languages). Our approach is parametric in a monad T that represents the given type of languages and their recognizing algebraic structures. Using the concept of an automata presentation of T-algebras, we demonstrate that the task of learning a T-recognizable language can be reduced to learning an abstract form of algebraic automaton whose transitions are modeled by a functor. For the important case of adjoint automata, we devise a learning algorithm generalizing Angluin's L*. The algorithm is phrased in terms of categorically described extension steps; we provide for a termination and complexity analysis based on a dedicated notion of finiteness. Our framework applies to structures like ω-regular languages that were not within the scope of existing categorical accounts of automata learning. In addition, it yields new learning algorithms for several types of languages for which no such algorithms were previously known at all, including sorted languages, nominal languages with name binding, and cost functions.

A Constructive Model of Directed Univalence in Bicubical Sets

Directed type theory is an analogue of homotopy type theory where types represent categories, generalizing groupoids. A bisimplicial approach to directed type theory, developed by Riehl and Shulman, is based on equipping each type with both a notion of path and a separate notion of directed morphism. In this setting, a directed analogue of the univalence axiom asserts that there is a universe of covariant discrete fibrations whose directed morphisms correspond to functions---a higher-categorical analogue of the category of sets and functions. In this paper, we give a constructive model of a directed type theory with directed univalence in bicubical, rather than bisimplicial, sets. We formalize much of this model using Agda as the internal language of a 1-topos, following Orton and Pitts. First, building on the cubical techniques used to give computational models of homotopy type theory, we show that there is a universe of covariant discrete fibrations, with a partial directed univalence principle asserting that functions are a retract of morphisms in this universe. To complete this retraction into an equivalence, we refine the universe of covariant fibrations using the constructive sheaf models by Coquand and Ruch.

Constructing Higher Inductive Types as Groupoid Quotients

In this paper, we show that all finitary 1-truncated higher inductive types (HITs) can be constructed from the groupoid quotient. We start by defining internally a notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. We finish by constructing a biinitial object in the bicategory of algebras in groupoids. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

A characterisation of ordered abstract probabilities

In computer science, especially when dealing with quantum computing or other non-standard models of computation, basic notions in probability theory like "a predicate" vary wildly. There seems to be one constant: the only useful example of an algebra of probabilities is the real unit interval. In this paper we try to explain this phenomenon. We will show that the structure of the real unit interval naturally arises from a few reasonable assumptions. We do this by studying effect monoids, an abstraction of the algebraic structure of the real unit interval: it has an addition x + y which is only defined when x + y ≤ 1 and an involution x~1 -- x which make it an effect algebra, in combination with an associative (possibly non-commutative) multiplication. Examples include the unit intervals of ordered rings and Boolean algebras.

We present a structure theory for effect monoids that are ω-complete, i.e. where every increasing sequence has a supremum.

We show that any ω-complete effect monoid embeds into the direct sum of a Boolean algebra and the unit interval of a commutative unital C*-algebra. This gives us from first principles a dichotomy between sharp logic, represented by the Boolean algebra part of the effect monoid, and probabilistic logic, represented by the commutative C*-algebra. Some consequences of this characterisation are that the multiplication must always be commutative, and that the unique ω-complete effect monoid without zero divisors and more than 2 elements must be the real unit interval. Our results give an algebraic characterisation and motivation for why any physical or logical theory would represent probabilities by real numbers.

On The Relational Width of First-Order Expansions of Finitely Bounded Homogeneous Binary Cores with Bounded Strict Width

The relational width of a finite structure, if bounded, is always (1, 1) or (2, 3). In this paper we study the relational width of first-order expansions of finitely bounded homogeneous binary cores where binary cores are structures with equality and some anti-reflexive binary relations such that for any two different elements a, b in the domain there is exactly one binary relation R with (a, b) ϵ R.

Our main result is that first-order expansions of liberal finitely bounded homogeneous binary cores with bounded strict width have relational width (2, MaxBound) where MaxBound is the size of the largest forbidden substructure, but is not less than 3, and liberal stands for structures that do not forbid certain finite structures of small size. This result is built on a new approach and concerns a broad class of structures including reducts of homogeneous digraphs for which the CSP complexity classification has not yet been obtained.