ABSTRACT. A formal system for expressing mathematical reasoning, or for a general interactive proof system, should address the issue of representation of collections  of mathematical objects and describe the laws of identifications of these objects. These two questions are closely connected: a collection is itself a mathematical object and the problem of identifying two collections can be quite subtle. The laws of identifications of algebraic, ordered and topological structures were analysed in details by Bourbaki (in his "theory of structures") in the setting of set theory. Category theory suggests new laws of identifications (equivalenceof categories) which may be quite complex to express in this setting.

 Voevoedsky discovered an elegant and uniform way to capture these laws of identifications in the setting of  dependent type theory, with the formulation of the univalence axiom (2009)which generalizes the axiom of extensionality as formulated by Church in simple type theory. The goal of this talk will be to explain this axiom and to survey some formalizations carried outusing it, as well as some metamathematical results (such as models and proof theoretic strength).

ABSTRACT. In a recent paper, Herbelin developed a calculus dPAω in which constructive proofs for the axioms of countable and dependent choices could be derived via the encoding of a proof of countable universal quantification as a stream of it components. However, the property of normalization (and therefore the one of soundness) was only conjectured. The difficulty for the proof of normalization is due to the simultaneous presence of dependent dependent types (for the constructive part of the choice), of control operators (for classical logic), of coinductive objects (to encode functions of type N → A into streams (a₀, a₁, ...)) and of lazy evaluation with sharing (for these coinductive objects). Building on previous works, we introduce in this paper a variant of dPAω presented as a sequent calculus. On the one hand, we take advantage of a variant of Krivine classical realizability we developed to prove the normalization of classical call-by-need. On the other hand, we benefit of dL, a classical sequent calculus with dependent types in which type safety is ensured using delimited continuations together with a syntactic restriction. By combining the techniques developed in these papers, we manage to define a realizability interpretation à la Krivine of our calculus that allows us to prove normalization and soundness.

ABSTRACT. Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, torus, suspensions, truncations and pushouts. All of these types are justified by the semantics and have judgmental computation rules for all constructors, including the higher dimensional ones, and the universes are closed under these type formers.

ABSTRACT. It was recently shown by van den Broeck at al. that the symmetric weighted first-order model counting problem (WFOMC) for sentences of two-variable logic FO2 is in polynomial time, while it is Sharp-P_1 complete for FO3-sentences. We extend the result for FO2 in two independent directions: to sentences of the form "phi and \forall\exists^=1 psi" with phi and psi in FO2, and to sentences formulated in the uniform one-dimensional fragment of FO, a recently introduced extension of two-variable logic with the capacity to deal with relation symbols of all arities. Note that the former generalizes the extension of FO2 with a functional relation symbol. We also identify a complete classification of first-order prefix classes according to whether WFOMC is in polynomial time or Sharp-P_1 complete.

ABSTRACT. The logic MMSNP is a restricted fragment of existential second-order logic which allows to express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and Vardi who showed that every MMSNP sentence is computationally equivalent to a finite-domain constraint satisfaction problem (CSP); the involved probabilistic reductions were derandomized by Kun using explicit constructions of expander structures. We present a new proof of the reduction to finite-domain CSPs which does not rely on the results of Kun. Our approach uses the fact that every MMSNP sentence describes a finite union of CSPs for countably infinite ω-categorical structures; moreover, by a recent result of Hubicka and Nesetril, these structures can be expanded to homogeneous structures with finite relational signature and the Ramsey property. This allows us to use the universal-algebraic approach to study the computational complexity of MMSNP. In particular, we verify the more general Bodirsky-Pinsker dichotomy conjecture for CSPs in MMSNP.

ABSTRACT. We consider an extension of the unary negation fragment of first-order logic in which arbitrarily many binary symbols may be required to be interpreted as equivalence relations. We show that this extension has the finite model property. More specifically, we show that every satisfiable formula has a model of at most doubly exponential size. We argue that the satisfiability (= finite satisfiability) problem for this logic is \TwoExpTime-complete. We also transfer our results to a restricted variant of the guarded negation fragment with equivalence relations.

ABSTRACT. Satisfiability of Boolean circuits is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is connected with solving equations over finite algebras. We want to characterize finite algebras A with polynomial time algorithm deciding if an equation over A has a solution. We are also looking for polynomial time algorithms deciding if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization (in terms of nice structural algebraic properties) for a very broad class of algebras from congruence modular varieties, including groups, rings, lattices and their extensions.

ABSTRACT. For a given set of queries (which are formulae in some query language) {Q_1, Q_2, ... Q_k} and for another query Q we say that {Q_1, Q_2, ... Q_k} determines Q if -- informally speaking -- for every database D, the information contained in the views Q_1(D), Q_2(D), ... Q_k(D) is sufficient to compute Q(D).

Query Determinacy Problem is the problem of deciding, for given {Q_1, Q_2, ... Q_k} and Q whether {Q_1, Q_2, ... Q_k} determines Q.

Many versions of this problem, for different query languages, were studied in database theory. In this paper we solve a problem stated in [CGLV02] showing that Query Determinacy Problem is undecidable for the Regular Path Queries -- the paradigmatic query language of graph databases.

--------- [CGLV02] D. Calvanese, G. De Giacomo, M. Lenzerini, and M.Y. Vardi; Lossless regular views; Proc. of the twenty-first ACM SIGMOD-SIGACT- SIGART symposium on Principles of Database Systems, PODS 2002

ABSTRACT. We present two Dialectica-like constructions for models of intensional Martin-Löf type theory based on Gödel's original Dialectica interpretation and the Diller-Nahm variant, bringing dependent types to categorical proof theory. We set both constructions within a logical predicates style theory for display map categories where we show that 'quasifibred' versions of dependent products and universes suffice to construct their standard counterparts. To support the logic required for dependent products in the first construction, we propose a new semantic notion of finite sum for dependent types, generalizing finitely-complete extensive categories. The second avoids extensivity assumptions using biproducts in a Kleisli category for a fibred additive monad.

ABSTRACT. We describe a dependent type theory, and a denotational model for it, that incorporates both intensional and extensional semantic universes. In the former, terms and types are interpreted as strategies on certain graph games, which are concrete data structures of a generalized form, and in the latter as stable functions on event domains.

The concrete data structures themselves form an event domain, with which we may interpret an (extensional) universe type of (intensional) types. A dependent game corresponds to a stable function into this domain; we use its trace to define dependent product and sum constructions as it captures precisely how unfolding moves combine with the dependency to shape the possible interaction in the game. Since each strategy computes a stable function on CDS states, we can lift typing judgements from the intensional to the extensional setting, giving an expressive type theory with recursively defined type families and type operators.

We define an operational semantics for intensional terms, giving a functional programming language based on our type theory, and prove that our semantics for it is computationally adequate. By extending it with a simple non-local control operator on intensional terms, we can precisely characterize behaviour in the intensional model. We demonstrate this by proving full abstraction and full completeness results.

ABSTRACT. Existing approaches to temporal verification of higher-order functional programs have either sacrificed compositionality in favor of achieving automation or vice-versa. In this paper we present a dependent-refinement type \& effect system to ensure that well-typed programs satisfy given temporal properties, and also give an algorithmic approach---based on deductive reasoning over a fixpoint logic---to typing in this system. The first contribution is a novel type-and-effect system capable of expressing \emph{dependent temporal} effects, which are fixpoint logic predicates on event sequences and program values, extending beyond the (non-dependent) temporal effects used in recent proposals. Temporal effects facilitate compositional reasoning whereby the temporal behavior of program parts are summarized as effects and combined to form those of the larger parts. As a second contribution, we show that type checking and typability for the type system can be reduced to solving first-order fixpoint logic constraints. Finally, we present a novel deductive system for solving such constraints. The deductive system consists of rules for reasoning via invariants and well-founded relations, and is able to reduce formulas containing both least and greatest fixpoints to predicate-based reasoning.

ABSTRACT. Church-Turing computability was extended by Brouwer who considered non-lawlike computability in the form of free choice sequences. Those are essentially unbounded sequences whose elements are chosen freely, i.e. not subject to any law. In this work we develop a new type theory BITT, which is an extension of the type theory of the Nuprl proof assistant, that embeds the notion of choice sequences. Supporting the evolving, non-deterministic nature of these objects required major modi cations to the under- lying type theory. Though the construction of a choice sequence is non-deterministic, once certain choices were made, they must remain consistent. To ensure this, BITT uses the underlying library as state and store choices as they are created. Another salient feature of BITT is that it uses a Beth-like semantics to account for the dynamic nature of choice sequences. We formally define BITT and use it to interpret and validate essential axioms governing choice sequences. These results provide a foundation for a fully intuitionistic version of Nuprl.

ABSTRACT. We present Quantitative Type Theory, a Type Theory that records usage information for each variable in a judgement, based on a previous system by McBride. The usage information is used to give a realizability semantics using a variant of Linear Combinatory Algebras, refining the usual realizability semantics of Type Theory by accurately tracking resource behaviour. We define the semantics in terms of Quantitative Categories with Families, a novel extension of Categories with Families for modelling resource sensitive type theories.

ABSTRACT. Probabilistic algorithms for both decision and search problems can offer significant complexity improvements over deterministic algorithms. One major difference, however, is that they may output different solutions for different choices of randomness. This makes correctness amplification impossible for search algorithms and is less than desirable in setting where uniqueness of output is important such as generation of system wide cryptographic parameters or distributed setting where different sources of randomness are used. Pseudo-deterministic algorithms are a class of randomized search algorithms, which output a unique answer with high

probability. Intuitively, they are indistinguishable from deterministic algorithms by a polynomial time observer of their input/output behavior. In this talk I will describe what is known about pseudo-deterministic algorithms in the sequential, sub-linear and parallel setting. We will also describe an extension of pseudo-deterministic algorithms to interactive proofs for search problems where the verifier is guaranteed with high probability to output the same output on different executions, regardless of the prover strategies. Based on joint work with Goldreich, Ron, Grossman and Holden.