Logic Programming and Logarithmic Space

We present an algebraic view on logic programming, related to proof theory and more specifically linear logic and geometry of interaction. Within this construction, a characterization of logspace (deterministic and non-deterministic) computation is given via a synctactic restriction, using an encoding of words that derives from proof theory. We show that the acceptance of a word by an observation (the counterpart of a program in the encoding) can be decided within logarithmic space, by reducing this problem to the acyclicity of a graph. We show moreover that observations are as expressive as two-ways multi-heads finite automata, a kind of pointer machines that is a standard model of logarithmic space computation

A Unifying Survey on Weighted Logics and Weighted Automata: Core Weighted Logic: Minimal and Versatile Specification of Quantitative Properties

International audienceLogical formalisms equivalent to weighted automata have been the topic of numerous research papers in the recent years. It started with the seminal result by Droste and Gastin on weighted logics over semir-ings for words. It has been extended in two dimensions by many authors. First, the weight domain has been extended to valuation monoids, valuation structures, etc., to capture more quantitative properties. Along another dimension, different structures such as ranked or unranked trees, nested words, Mazurkiewiz traces, etc., have been considered. The long and involved proofs of equivalences in all these papers are implicitely based on the same core arguments. This article provides a meta-theorem which unifies these different approaches. Towards this, we first introduce a core weighted logic with a minimal number of features and a simplified syntax. Then, we define a new semantics for weighted automata and weighted logics in two phases—an abstract semantics based on multisets of weight structures (independent of particular weight domains) followed by a concrete semantics. We show at the level of the abstract semantics that weighted automata and core weighted logic have the same expressive power. We show how previous results can be recovered from our result by logical reasoning. In this paper, we prove the meta-theorem for words, ranked and unranked trees, showing the robustness of our approach

Aperiodic Weighted Automata and Weighted First-Order Logic

By fundamental results of Sch\"utzenberger, McNaughton and Papert from the 1970s, the classes of first-order definable and aperiodic languages coincide. Here, we extend this equivalence to a quantitative setting. For this, weighted automata form a general and widely studied model. We define a suitable notion of a weighted first-order logic. Then we show that this weighted first-order logic and aperiodic polynomially ambiguous weighted automata have the same expressive power. Moreover, we obtain such equivalence results for suitable weighted sublogics and finitely ambiguous or unambiguous aperiodic weighted automata. Our results hold for general weight structures, including all semirings, average computations of costs, bounded lattices, and others.Comment: An extended abstract of the paper appeared at MFCS'1

Weighted Pushdown Systems with Indexed Weight Domains

The reachability analysis of weighted pushdown systems is a very powerful technique in verification and analysis of recursive programs. Each transition rule of a weighted pushdown system is associated with an element of a bounded semiring representing the weight of the rule. However, we have realized that the restriction of the boundedness is too strict and the formulation of weighted pushdown systems is not general enough for some applications. To generalize weighted pushdown systems, we first introduce the notion of stack signatures that summarize the effect of a computation of a pushdown system and formulate pushdown systems as automata over the monoid of stack signatures. We then generalize weighted pushdown systems by introducing semirings indexed by the monoid and weaken the boundedness to local boundedness

Coalgebraic Trace Semantics for Continuous Probabilistic Transition Systems

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems. Finally we consider two example systems with uncountable state spaces and apply our theory to calculate their trace measures

Varieties of Languages in a Category

Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of languages in a category C, and prove that they correspond to pseudovarieties of monoids in a closed monoidal category D, provided that C and D are dual on the level of finite objects. By suitable choices of these categories our result uniformly covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer, respectively, and yields new Eilenberg-type correspondences

A Coalgebraic Approach to Reducing Finitary Automata

Compact representations of automata are important for efficiency. In this paper, we study methods to compute reduced automata, in which no two states accept the same language. We do this for finitary automata (FA), an abstract definition that encompasses probabilistic and weighted automata. Our procedure makes use of Milius' locally finite fixpoint. We present a reduction algorithm that instantiates to probabilistic and S-linear weighted automata (WA) for a large class of semirings. Moreover, we propose a potential connection between properness of a semiring and our provided reduction algorithm for WAs, paving the way for future work in connecting the reduction of automata to the properness of their associated coalgebras

Tropical time series, iterated-sum signatures and quasisymmetric functions

Driven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of (real-valued) time series that are not easily available using existing signature-type objects

Syntactic Monoids in a Category

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is an entropic variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in case the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids