Complexity of short Presburger arithmetic

We study complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integers involved in the inequalities. We prove that assuming Kannan's partition can be found in polynomial time, the satisfiability of Short-PA sentences can be decided in polynomial time. Furthermore, under the same assumption, we show that the numbers of satisfying assignments of short Presburger sentences can also be computed in polynomial time

On rationality of nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n × m matrix M into a product of a nonnegative n × d matrix W and a nonnegative d × m matrix H. NMF has a wide variety of applications, including bioinformatics, chemometrics, communication complexity, machine learning, polyhedral combinatorics, among many others. A longstanding open question, posed by Cohen and Rothblum in 1993, is whether every rational matrix M has an NMF with minimal d whose factors W and H are also rational. We answer this question negatively, by exhibiting a matrix M for which W and H require irrational entries. As an application of this result, we show that state minimization of labeled Markov chains can require the introduction of irrational transition probabilities. We complement these irrationality results with an NP- complete version of NMF for which rational numbers suffice

Parikh Image of Pushdown Automata

We compare pushdown automata (PDAs for short) against other representations. First, we show that there is a family of PDAs over a unary alphabet with $n$ states and $p \geq 2n + 4$ stack symbols that accepts one single long word for which every equivalent context-free grammar needs $\Omega(n^2(p-2n-4))$ variables. This family shows that the classical algorithm for converting a PDA to an equivalent context-free grammar is optimal even when the alphabet is unary. Moreover, we observe that language equivalence and Parikh equivalence, which ignores the ordering between symbols, coincide for this family. We conclude that, when assuming this weaker equivalence, the conversion algorithm is also optimal. Second, Parikh's theorem motivates the comparison of PDAs against finite state automata. In particular, the same family of unary PDAs gives a lower bound on the number of states of every Parikh-equivalent finite state automaton. Finally, we look into the case of unary deterministic PDAs. We show a new construction converting a unary deterministic PDA into an equivalent context-free grammar that achieves best known bounds.Comment: 17 pages, 2 figure

Model counting for complex data structures

We extend recent approaches for calculating the probability of program behaviors, to allow model counting for complex data structures with numeric fields. We use symbolic execution with lazy initialization to compute the input structures leading to the occurrence of a target event, while keeping a symbolic representation of the constraints on the numeric data. Off-the-shelf model counting tools are used to count the solutions for numerical constraints and field bounds encoding data structure invariants are used to reduce the search space. The technique is implemented in the Symbolic PathFinder tool and evaluated on several complex data structures. Results show that the technique is much faster than an enumeration-based method that uses the Korat tool and also highlight the benefits of using the field bounds to speed up the analysis

Symbolic Logic meets Machine Learning: A Brief Survey in Infinite Domains

The tension between deduction and induction is perhaps the most fundamental issue in areas such as philosophy, cognition and artificial intelligence (AI). The deduction camp concerns itself with questions about the expressiveness of formal languages for capturing knowledge about the world, together with proof systems for reasoning from such knowledge bases. The learning camp attempts to generalize from examples about partial descriptions about the world. In AI, historically, these camps have loosely divided the development of the field, but advances in cross-over areas such as statistical relational learning, neuro-symbolic systems, and high-level control have illustrated that the dichotomy is not very constructive, and perhaps even ill-formed. In this article, we survey work that provides further evidence for the connections between logic and learning. Our narrative is structured in terms of three strands: logic versus learning, machine learning for logic, and logic for machine learning, but naturally, there is considerable overlap. We place an emphasis on the following "sore" point: there is a common misconception that logic is for discrete properties, whereas probability theory and machine learning, more generally, is for continuous properties. We report on results that challenge this view on the limitations of logic, and expose the role that logic can play for learning in infinite domains

Towards concolic testing for hybrid systems

Hybrid systems exhibit both continuous and discrete behavior. Analyzing hybrid systems is known to be hard. Inspired by the idea of concolic testing (of programs), we investigate whether we can combine random sampling and symbolic execution in order to effectively verify hybrid systems. We identify a sufficient condition under which such a combination is more effective than random sampling. Furthermore, we analyze different strategies of combining random sampling and symbolic execution and propose an algorithm which allows us to dynamically switch between them so as to reduce the overall cost. Our method has been implemented as a web-based checker named HYCHECKER. HYCHECKER has been evaluated with benchmark hybrid systems and a water treatment system in order to test its effectiveness.CPCI-S(ISTP)[email protected]; [email protected]

Notes on Counting with Finite Machines

We determine the descriptional complexity (smallest number of states, up to constant factors) of recognizing languages {1^n} and {1^{t n} : t = 0, 1, 2, ...} with state-based finite machines of various kinds. This task is understood as counting to n and modulo n, respectively, and was previously studied for classes of finite-state automata by Kupferman, Ta-Shma, and Vardi (2001). We show that for Turing machines it requires log(n)/log(log(n)) states in the worst case, and individual values are related to Kolmogorov complexity of the binary encoding of n. For deterministic pushdown and counter automata, the complexity is log(n) and sqrt(n), respectively; for alternating counter automata, we show an upper bound of log(n). For visibly pushdown automata, i.e., if the stack movements are determined by input symbols, we consider languages {a^n b^n} and {a^{t n} b^{t n} : n t = 0, 1, 2, ...} and determine their complexity, of sqrt(n) and min(n_1 + n_2), respectively, with minimum over all factorizations n = n_1 n_2

The taming of the semi-linear set

Semi-linear sets, which are rational subsets of the monoid (Z^d,+), have numerous applications in theoretical computer science. Although semi-linear sets are usually given implicitly, by formulas in Presburger arithmetic or by other means, the effect of Boolean operations on semi-linear sets in terms of the size of description has primarily been studied for explicit representations. In this paper, we develop a framework suitable for implicitly presented semi-linear sets, in which the size of a semi-linear set is characterized by its norm—the maximal magnitude of a generator. We put together a toolbox of operations and decompositions for semi-linear sets which gives bounds in terms of the norm (as opposed to just the bit-size of the description), a unified presentation, and simplified proofs. This toolbox, in particular, provides exponentially better bounds for the complement and set-theoretic difference. We also obtain bounds on unambiguous decompositions and, as an application of the toolbox, settle the complexity of the equivalence problem for exponent-sensitive commutative grammars