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I will survey a body of work, developed over the past decade or so, on algorithms for, and the computational complexity of, analyzing and model checking some important families of countably infinite-state Markov chains, Markov decision processes (MDPs), and stochastic games. These models arise by adding natural forms of recursion, branching, or a counter, to finite-state models, and they correspond...
We give a model of dependent type theory with one univalent universe and propositional truncation interpreting a type as a stack, generalizing the groupoid model of type theory. As an application, we show that countable choice cannot be proved in dependent type theory with one univalent universe and propositional truncation.
We study the descriptive complexity of summation problems in Abelian groups and semigroups. In general, an input to the summation problem consists of an Abelian semigroup G, explicitly represented by its multiplication table, and a subset X of G. The task is to determine the sum over all elements of X.
Data vectors generalise finite multisets: they are finitely supported functions into a commutative monoid. We study the question whether a given data vector can be expressed as a finite sum of others, only assuming that 1) the domain is countable and 2) the given set of base vectors is finite up to permutations of the domain.
Brunet and Pous showed at LICS 2015 that the equational theory of identity-free relational Kleene lattices (a fragment of Kleene allegories) is decidable in EXPSPACE. In this paper, we show that the equational theory of Kleene allegories is decidable, and is EXPSPACE-complete, answering the first open question posed by their work. The proof proceeds by designing partial derivatives on graphs, which...
This paper studies sets of rational numbers definable by continuous Petri nets and extensions thereof. First, we identify a polynomial-time decidable fragment of existential FO(ℚ,+,<) and show that the sets of rationals definable in this fragment coincide with reachability sets of continuous Petri nets. Next, we introduce and study continuous vector addition systems with states (CVASS), which are...
Compute the coarsest simulation preorder included in an initial preorder is used to reduce the resources needed to analyze a given transition system. This technique is applied on many models like Kripke structures, labeled graphs, labeled transition systems or even word and tree automata. Let (Q,→) be a given transition system and ℛinit be an initial preorder over Q. Until now, algorithms to compute...
We prove that the solvability of systems of linear equations and related linear algebraic properties are definable in a fragment of fixed-point logic with counting that only allows polylogarithmically many iterations of the fixed-point operators. This enables us to separate the descriptive complexity of solving linear equations from full fixed-point logic with counting by logical means. As an application...
It has been shown that for a general-valued constraint language Γ the following statements are equivalent: (1) any instance of VCSP(Γ) can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of VCSP(Γ) can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of Γ satisfies the “bounded width condition”, i.e...
We introduce fibred type-theoretic fibration categories which are fibred categories between categorical models of Martin-Löf type theory. Fibred type-theoretic fibration categories give a categorical description of logical predicates for identity types. As an application, we show a relational parametricity result for homotopy type theory. As a corollary, it follows that every closed term of type of...
In monadic programming, datatypes are presented as free algebras, generated by data values, and by the algebraic operations and equations capturing some computational effects. These algebras are free in the sense that they satisfy just the equations imposed by their algebraic theory, and remain free of any additional equations. The consequence is that they do not admit quotient types. This is, of...
We define a monadic translation of type theory, called the weaning translation, that allows for a large range of effects in dependent type theory—such as exceptions, non-termination, non-determinism or writing operations. Through the light of a call-by-push-value decomposition, we explain why the traditional approach fails with type dependency and justify our approach. Crucially, the construction...
Quantitative algebras (QAs) are algebras over metric spaces defined by quantitative equational theories as introduced by us in 2016. They provide the mathematical foundation for metric semantics of probabilistic, stochastic and other quantitative systems. This paper considers the issue of axiomatizability of QAs. We investigate the entire spectrum of types of quantitative equations that can be used...
This paper contributes to the techniques of topoalgebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a related Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction yields,...
A structure enjoys the Herbrand property if, whenever it satisfies an equality between some terms, these terms are unifiable. On such structures the expressive power of equalities becomes trivial, as their semantic satisfiability is reduced to a purely syntactic check.
We prove that non-emptiness of timed register pushdown automata is decidable in doubly exponential time. This is a very expressive class of automata, whose transitions may involve state and top-of-stack clocks with unbounded differences. It strictly subsumes pushdown timed automata of Bouajjani et al., dense-timed pushdown automata of Abdulla et al., and orbit-finite timed register pushdown automata...
We present new data structures for quasistrict higher categories, in which associativity and unit laws hold strictly. Our approach has low axiomatic complexity compared to traditional algebraic approaches, and gives a practical method for performing calculations in quasistrict 4-categories. It is amenable to computer implementation, and we exploit this to give a machine-verified algebraic proof that...
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