Quantifier elimination (QE) is a key task in formal verification algorithms, and the ability to return partial results, such as under-approximations, is beneficial for many QE clients. In Linear Real Arithmetic (LRA), existing QE methods often fail to preserve syntactic convexity, that is, they return a disjunction even for a conjunctive input, or they return a large non-minimal representation. We define the novel concept of Bidirectional Model-Based Projection and a new QE algorithm for LRA (BMBP-QE) that (i) returns a conjunctive over-approximation and a disjunctive under-approximation when interrupted early, (ii) returns a minimal conjunction when the input is conjunctive, and (iii) applies to arbitrary LRA formulae. We show that BMBP-QE outperforms SMT-based QE algorithms, offering improvements in both runtime and result size.

Abstract The theory of arithmetic is integral to many uses of SMT solvers. Z3 has implemented native solvers for arithmetic reasoning since its first release. We present a full re-implementation of Z3’s original arithmetic solver. It is based on substantial experiences from user feedback, engineering and experimentation. While providing a comprehensive overview of the main components we emphasize selected new insights we arrived at while developing and testing the solver.

Monadic predicates play a prominent role in many decidable cases, including decision procedures for symbolic automata. We are here interested in discovering whether a formula can be rewritten into a Boolean combination of monadic predicates. Our setting is quantifier-free formulas whose satisfiability is decidable, such as linear arithmetic. Here we develop a semidecision procedure for extracting a monadic decomposition of a formula when it exists.

Consider a finite sequence of permutations of the elements $1,\ldots,n$ with the property that each element changes its position by at most 1 from any permutation to the next. We call such a sequence a tangle, and we define a move of element $i$ to be a maximal subsequence of at least two consecutive permutations during which its positions form an arithmetic progression of common difference +1 or -1. We prove that for any initial and final permutations, there is a tangle connecting them in which each element makes at most 5 moves, and another in which the total number of moves is at most 4n. On the other hand, there exist permutations that require at least 3 moves for some element, and at least 2n-2 moves in total. If we further require that every pair of elements exchange positions at most once, then any two permutations can be connected by a tangle with at most $O(\log n)$ moves per element, but we do not know whether this can be reduced to O(1) per element, or to O(n) in total. A key tool is the introduction of certain restricted classes of tangle that perform pattern-avoiding permutations.

UNLABELLED: Evolutionary biologists are often interested in finding correlations among biological traits across a number of species, as such correlations may lead to testable hypotheses about the underlying function. Because some species are more closely related than others, computing and visualizing these correlations must be done in the context of the evolutionary tree that relates species. In this note, we introduce PhyloDet (short for PhyloDetective), an evolutionary tree visualization tool that enables biologists to visualize multiple traits mapped to the tree. AVAILABILITY: http://research.microsoft.com/cue/phylodet/

This paper deals with testing of nondeterministic software systems. We assume that a model of the nondeterministic system is given by a directed graph with two kind of vertices: states and choice points. Choice points represent the nondeterministic behaviour of the implementation under test (IUT). Edges represent transitions. They have costs and probabilities. Test case generation in this setting amounts to generation of a game strategy. The two players are the testing tool (TT) and the IUT. The game explores the graph. The TT leads the IUT by selecting an edge at the state vertices. At the choice points the control goes to the IUT. A game strategy decides which edge should be taken by the TT in each state. This paper presents three novel algorithms 1) to determine an optimal strategy for the bounded reachability game, where optimality means maximizing the probability to reach any of the given final states from a given start state while at the same time minimizing the costs of traversal; 2) to determine a winning strategy for the bounded reachability game, which guarantees that given final vertices are reached, regardless how the IUT reacts; 3) to determine a fast converging edge covering strategy, which guarantees that the probability to cover all edges quickly converges to 1 if TT follows the strategy.