For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatization of rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy-Milner logic, graded modal logic and probabilistic modal logic.

Probabilistic I/O automata (PIOA) constitute a model for distributed or concurrent systems that incorporates a notion of probabilistic choice. The PIOA model provides a notion of composition, for constructing a PIOA for a composite system from a collection of PIOAs representing the components. We present a method for computing completion probability and expected completion time for PIOAs. Our method is compositional, in the sense that it can be applied to a system of PIOAs, one component at a time, without ever calculating the global state space of the system (i.e. the composite PIOA). The method is based on symbolic calculations with vectors and matrices of rational functions, and it draws upon a theory of observables, which are mappings from delayed traces to real numbers that generalize the classical "formal power series " from algebra and combinatorics. Central to the theory is a notion of representation for an observable, which generalizes the clasical notion "linear representation " for formal power series. As in the classical case, the representable observables coincide with an abstractly defined class of "rational" observables; this fact forms the foundation of our method. 1

Abstract: We present a polynomial time algorithm for learning a large class of algebraic models of computation. We show that any arithmetic circuit whose partial derivatives induce a low-dimensional vector space is exactly learnable from membership and equivalence queries. As a consequence, we obtain polynomial-time algorithms for learning restricted algebraic branching programs as well as noncommutative set-multilinear arithmetic formulae. In addition, we observe that the algorithms of Bergadano et al. (1996) and Beimel et al. (2000) can be used to learn depth-3 set-multilinear arithmetic circuits. Previously only versions of depth-2 arithmetic circuits were known to be learnable in polynomial time. Our learning algorithms can be viewed as solving a generalization of the well known polynomial interpolation problem where the unknown polynomial has a succinct representation. We can learn representations of polynomials encoding exponentially many monomials. Our techniques combine a careful algebraic analysis of the partial derivatives of arithmetic circuits with “multiplicity automata ” learning algorithms due to Bergadano et al. (1997) and Beimel et al. (2000).

We introduce a lower bound technique that applies to a broad spectrum of functional representations including Binary Decision Diagrams (BDDs), Binary Moment Diagrams (*-BMDs), Hybrid Decision Diagrams (HDDs), and their variants. These representations have been widely used for formal verification of hardware systems, particularly symbolic model checking and digital-system design, testing and verification. We define a representation called the Binary Linear Diagram (BLD) that generalizes all these representations and then apply our lower bound technique to show exponential size bounds for a wide range of functions. We also give the first examples of integer functions including integer division, remainder, high/low-order words of multiplication, square root and reciprocal that require exponential size in all these representations. Finally, we show that there is a simple regular language that requires exponential size to be represented by any *-BMD, even though BDDs can represent any regul...

Abstract. Stochastic deterministic finite automata have been introduced and are used in a variety of settings. We report here a number of results concerning the learnability of these finite state machines. In the setting of identification in the limit with probability one, we prove that stochastic deterministic finite automata cannot be identified from only a polynomial quantity of data. If concerned with approximation results, they become Pac-learnable if the L ∞ norm is used. We also investigate queries that are sufficient for the class to be learnable. 1

In the early days of high-speed computers there was a hope that it would be possible to program a computer to translate from one language to another, and this led to an intensive study of language structure. The result has been disappointing in that we are still far from making translations by computer, but that is no cause for

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

Abstract not found