Experiences with enumeration of integer projections of parametric polytopes

Many compiler optimization techniques depend on the ability to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric polytope) is a function of the symbolic parameters that may appear in the constraints. In an extended problem (the "integer projection" of a parametric polytope), some of the variables that appear in the constraints may be existentially quantified and then the enumerated set corresponds to the projection of the integer points in a parametric polytope. This paper shows how to reduce the enumeration of the integer projection of parametric polytopes to the enumeration of parametric polytopes. Two approaches are described and experimentally compared. Both can solve problems that were considered very difficult to solve analytically

Symbolic and analytic techniques for resource analysis of Java bytecode

Recent work in resource analysis has translated the idea of amortised resource analysis to imperative languages using a program logic that allows mixing of assertions about heap shapes, in the tradition of separation logic, and assertions about consumable resources. Separately, polyhedral methods have been used to calculate bounds on numbers of iterations in loop-based programs. We are attempting to combine these ideas to deal with Java programs involving both data structures and loops, focusing on the bytecode level rather than on source code

Fluted Logic with Counting

The fluted fragment is a fragment of first-order logic in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. It is known that the fluted fragment possesses the finite model property. In this paper, we extend the fluted fragment by the addition of counting quantifiers. We show that the resulting logic retains the finite model property, and that the satisfiability problem for its (m+1)-variable sub-fragment is in m-NExpTime for all positive m. We also consider the satisfiability and finite satisfiability problems for the extension of any of these fragments in which the fluting requirement applies only to sub-formulas having at least three free variables

Integer Affine Transformations of Parametric Z-polytopes and Applications to Loop Nest Optimization

The polyhedral model is a well-known compiler optimization framework for the analysis and transformation of affine loop nests. We present a new method concerning a difficult geometric operation that is raised by this model: the integer affine transformation of parametric Z-polytopes. The result of such a transformation is given by a worst-case exponential union of Z-polytopes. We also propose a polynomial algorithm (for fixed dimension), to count points in arbitrary unions of a fixed number of parametric Z-polytopes. We implemented these algorithms and compared them to other existing algorithms, for a set of applications to loop nest analysis and optimization

Mini-Workshop: Ehrhart-Quasipolynomials: Algebra, Combinatorics, and Geometry

[no abstract available

Geometric decision procedures and the VC dimension of linear arithmetic theories

This paper resolves two open problems on linear integer arithmetic (LIA), also known as Presburger arithmetic. First, we give a triply exponential geometric decision procedure for LIA, i.e., a procedure based on manipulating semilinear sets. This matches the running time of the best quantifier elimination and automata-based procedures. Second, building upon our first result, we give a doubly exponential upper bound on the Vapnik–Chervonenkis (VC) dimension of sets definable in LIA, proving a conjecture of D. Nguyen and I. Pak [Combinatorica 39, pp. 923–932, 2019]. These results partially rely on an analysis of sets definable in linear real arithmetic (LRA), and analogous results for LRA are also obtained. At the core of these developments are new decomposition results for semilinear and -semilinear sets, the latter being the sets definable in LRA. These results yield new algorithms to compute the complement of (-)semilinear sets that do not cause a non-elementary blowup when repeatedly combined with procedures for other Boolean operations and projection. The existence of such an algorithm for semilinear sets has been a long-standing open problem.</p

Master index

Pla general, del mural ceràmic que decora una de les parets del vestíbul de la Facultat de Química de la UB. El mural representa diversos símbols relacionats amb la química

Binary reachability of timed-register pushdown automata and branching vector addition systems

Timed-register pushdown automata constitute a very expressive class of automata, whose transitions may involve state, input, and top-of-stack timed registers with unbounded differences. They strictly subsume pushdown timed automata of Bouajjani et al., dense-timed pushdown automata of Abdulla et al., and orbit-finite timed-register pushdown automata of Clemente and Lasota. We give an effective logical characterisation of the reachability relation of timed-register pushdown automata. As a corollary, we obtain a doubly exponential time procedure for the non-emptiness problem. We show that the complexity reduces to singly exponential under the assumption of monotonic time. The proofs involve a novel model of one-dimensional integer branching vector addition systems with states. As a result interesting on its own, we show that reachability sets of the latter model are semilinear and computable in exponential time