ABSTRACT. We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P=NP, no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic (FP) or fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic (or the version with counting quantifiers) to separate instances with a high optimum from those with a low optimum for fixed-size instances.

ABSTRACT. We propose a fragment of many-sorted second order logic called EQSMT and show that checking satisfiability of sentences in this fragment is decidable. EQSMT formulae have an \exists\forall quantifier prefix (over variables, functions and relations) making EQSMT conducive for modeling synthesis problems. Moreover, EQSMT allows reasoning using a combination of background theories provided that they have a decidable satisfiability problem for the \exists\forall FO-fragment (e.g., linear arithmetic). Our decision procedure reduces the satisfiability of EQSMT~formulae to satisfiability queries of \exists\forall formulae of each individual background theory, allowing us to use existing efficient SMT solvers supporting \exists\forall reasoning for these theories; hence our procedure can be seen as effectively quantified SMT (EQSMT) reasoning.

ABSTRACT. Quantitative extensions of parity games have recently attracted significant interest. These extensions include parity games with energy and payoff conditions as well as finitary parity games and their generalization to parity games with costs. Finitary parity games enjoy a special status among these extensions, as they offer a native combination of the qualitative and quantitative aspects in infinite games: the quantitative aspect of finitary parity games is a quality measure for the qualitative aspect, as it measures the limit superior of the time it takes to overwrite an odd color by a larger even one. Finitary parity games have been extended to parity games with costs, where each transition is labelled with a non-negative weight that reflects the costs incurred by taking it. We lift this restriction and consider parity games with costs with arbitrary integer weights. We show that solving such games is in NP $\cap$ coNP, the signature complexity for games of this type. We also show that the protagonist has finite-state winning strategies, and provide tight exponential bounds for the memory he needs to win the game. Naturally, the antagonist may need need infinite memory to win. Finally, we present tight bounds on the quality of winning strategies for the protagonist.

ABSTRACT. Recently, Dallal, Neider, and Tabuada studied a generalization of the classical game-theoretic model used in program synthesis, which additionally accounts for unmodeled intermittent disturbances. In this extended framework, one is interested in computing optimally resilient strategies, i.e., strategies that are resilient against as many disturbances as possible. Dallal, Neider, and Tabuada showed how to compute such strategies for safety specifications.

In this work, we compute optimally resilient strategies for a much wider range of winning conditions and show that they do not require more memory than winning strategies in the classical model. Our algorithms only have a polynomial overhead in comparison to the ones computing winning strategies. In particular, for parity conditions optimally resilient strategies are positional and can be computed in quasipolynomial time.

ABSTRACT. Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. It is desirable to classify the computational complexity of VCSPs depending on a fixed set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified in this sense. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear homogeneous cost functions. We show that such VCSPs can be solved in polynomial time when the cost functions are additionally submodular, and that this is indeed a maximally tractable class: adding any cost function that is not submodular leads to an NP-hard VCSP.

ABSTRACT. A probabilistic finite automaton (PFA) A is said to be regular-approximable with respect to (x,y), if there is a regular language that contains all words accepted by A with probability at least x+y, but does not contain any word accepted with probability at most x. We show that the problem of determining if a PFA A is regular-approximable with respect to (x,y) is not recursively enumerable. We then show that many tractable sub-classes of PFAs identified in the literature --- hierarchical PFAs, polynomially ambiguous PFAs, and eventually weakly ergodic PFAs --- are regular-approximable with respect to all (x,y). Establishing the regular-approximability of a PFA has the nice consequence that its value can be effectively approximated, and the emptiness problem can be decided under the assumption of isolation.

ABSTRACT. We extend work of Lautemann, Schwentick and Stewart on characterisations of the 'positive' polynomial-time predicates (posP, also called mP by Grigni and Sipser) to function classes. They showed that restricting 'negation' in various characterisations of P yield the same class of predicates, inducing a robust definition of posP. We observe that many of those results scale to the functional setting too, giving a natural class of functions, 'posFP', capturing feasible monotone computation. Our main result is the obtention of a function algebra for posFP, by imposing a simple uniformity condition on the bounded recursion operator in Cobham's characterisation of FP.

ABSTRACT. We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment includes Voevodsky's univalence axiom and a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false.

ABSTRACT. We prove cut-elimination for a sequent-style proof system which is sound and complete for the equational theory of Kleene algebra, and where proofs are potentially non-wellfounded infinite trees. We extend these results to systems with meets and residuals, capturing `star-continuous' action lattices in a similar way. We recover the equational theory of all action lattices by restricting to regular proofs (with cut)---those proofs that are unfoldings of finite graphs.

ABSTRACT. The semantic notion of dependent safety is a common generalization of the notion of absoluteness used in set theory and the notion of domain independence used in database theory for characterizing safe queries. This notion has been used in previous works to provide a unified theory of constructions and operations as they are used in different branches of Mathematics and computer science, including set theory, computability theory, and database theory. In this paper we provide a complete syntactic characterization of general first-order dependent safety. We also show that this syntactic safety relation can be used for characterizing the set of strictly decidable relations on the natural numbers, as well as for characterizing rudimentary set theory and absoluteness of formulas within it.