In this paper, we introduce a parametric semantics for timed controllers called the Almost ASAP semantics. This semantics is a relaxation of the usual ASAP semantics (also called the maximal progress semantics) which is a mathematical idealization that can not be implemented by any physical device no matter how fast it is. On the contrary, any correct Almost ASAP controller can be implemented by a program on a hardware if this hardware is fast enough. We study the properties of this semantics, show how it can be analyzed using the tool HyTech, and illustrate its practical use on examples.

This paper considers finite-automata based algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this involves some difficult and delicate to implement algorithms. The contribution of this paper is to show, using topological arguments, that only a restricted class of automata on infinite words are necessary for handling real and integer linear arithmetic. This allows the use of substantially simpler algorithms, which have been successfully implemented.

We study the uniformity of Buchberger algorithms for computing Grobner bases with respect to a natural number parameter k in the exponents of the input polynomials. The problem is motivated by positive results of T. Takahashi on special exponential parameter series of polynomial sets in singularity theory. For arbitrary input sets uniformity is in general impossible. By way of contrast we show that the Buchberger algorithm is indeed uniform up to a finite case distinction on the exponential parameter k for inputs consisting of monomials and binomials only. Under this hypothesis the case distinction is algorithmic and partitions the parameter range into Presburger sets. In each case the Buchberger algorithm is uniform and can be described explicitly and algorithmically. In the course of the algorithm the exponential parameter k enters also the coe#cients as exponent. Thus the uniformity in k is established with respect to parametric exponents in both terms and coe#cients. These results are obtained as a consequence of a much more general theorem concerning Buchberger algorithms for sets of monomials and binomials with arbitrary parametric coe#cients and exponents, generalizing the construction of Grobner systems.

We present a formally verified quantifier elimination procedure for the first order theory over linear mixed real-integer arithmetics in higher-order logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for linear real arithmetics.

We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which “local ” decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let Tadd[Q] be the first-order theory of the real numbers in the language with symbols 0, 1, +, −, <,..., fa,... where for each a ∈ Q, fa denotes the function fa(x) = ax. Let Tmult[Q] be the analogous theory for the language with symbols 0, 1, ×, ÷, <,..., fa,.... We show that although T [Q] = Tadd[Q]∪Tmult[Q] is undecidable, the universal fragment of T [Q] is decidable. We also show that terms of T [Q] can fruitfully be put in a normal form. We prove analogous results for theories in which Q is replaced, more generally, by suitable subfields F of the reals. Finally, we consider practical methods of establishing quantifier-free validities that approximate our (impractical) decidability results. 1

This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in tactic style, i.e. by a proof-producing functional program, and once by reflection, i.e. by computations inside the logic rather than in the meta-language. Both formalizations are highly generic because they make only minimal assumptions w.r.t. the underlying logical system and theorem prover. An implementation in Isabelle/HOL shows that the reflective approach is between one and two orders of magnitude faster. 1

Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.

Abstract. Our goal is to identify families of relations that are useful for reasoning about software. We describe such families using decidable quantifier-free classes of logical constraints with a rich set of operations. A key challenge is to define such classes of constraints in a modular way, by combining multiple decidable classes. Working with quantifierfree combinations of constraints makes the combination agenda more realistic and the resulting logics more likely to be tractable than in the presence of quantifiers. Our approach to combination is based on reducing decidable fragments to a common class, Boolean Algebra with Presburger Arithmetic (BAPA). This logic was introduced by Feferman and Vaught in 1959 and can express properties of uninterpreted sets of elements, with set algebra operations and equicardinality relation (consequently, it can also express Presburger arithmetic constraints on cardinalities of sets). Combination by reduction to BAPA allows us to obtain decidable quantifier-free combinations

Abstract. We show how ellipticity of partial differential systems in the sense of Douglis and Nirenberg can be decided algorithmically by quantifier elimination on real closed fields. A concrete implementation based on MuPAD and Redlog is presented. 1

State Machines is described. Time can be continuous or discrete. Time constraints are defined by linear inequalities. Two semantics are considered: with and without non-deterministic bounded delays between actions. The simulator is easily configurable. Simulation tasks can be generated according to descriptions in a special language. The simulator is used for verification of formulas in an expressible timed predicate logic. Several features that facilitate the simulation are described: external functions definition, delays settings, constraints specification, and others. 1