Strong Bimonoids Research Articles

Decidability Boundaries for the Finite-Image Property of Weighted Finite Automata

A weighted finite automaton has the finite-image property if the image of the weighted language associated with it is finite. We show two undecidability results concerning the finite-image property of weighted finite automata over semirings, respectively strong bimonoids. Firstly, we give a computable idempotent commutative past-finite ordered semiring such that it is undecidable, for an arbitrary deterministic weighted finite automaton [Formula: see text] over that semiring, whether [Formula: see text] has the finite-image property. Secondly, we give a computable commutative past-finite monotonic ordered strong bimonoid such that it is undecidable, for an arbitrary weighted finite automaton [Formula: see text] over that strong bimonoid, whether [Formula: see text] has the finite-image property. This shows that recent decidability results for suitable weighted finite automata over past-finite monotonic strong bimonoids cannot be extended to natural classes of ordered semirings and ordered strong bimonoids without further assumptions.

A Theorem on Supports of Weighted Tree Automata Over Strong Bimonoids

We investigate weighted tree automata over strong bimonoids. Intuitively, strong bimonoids are semirings which may lack the distributivity laws. For instance, each (not necessarily distributive) bounded lattice is a strong bimonoid. In general, for a weighted tree automaton over a strong bimonoid, its run semantics is different from its initial algebra semantics. We prove that for positive strong bimonoids the supports of these semantics are equal, and we also deal with a kind of reverse statement.

Finite-image property of weighted tree automata over past-finite monotonic strong bimonoids

We consider weighted tree automata over strong bimonoids (for short: wta). A wta A has the finite-image property if its recognized weighted tree language 〚A〛 has finite image; moreover, A has the preimage property if the preimage under 〚A〛 of each element of the underlying strong bimonoid is a recognizable tree language. For each wta A over a past-finite monotonic strong bimonoid we prove the following results. In terms of A's structural properties, we characterize whether it has the finite-image property. We characterize those past-finite monotonic strong bimonoids such that for each wta A it is decidable whether A has the finite-image property. In particular, the finite-image property is decidable for wta over past-finite monotonic semirings. Moreover, we prove that A has the preimage property. All our results also hold for weighted string automata.

Crisp-determinization of weighted tree automata over strong bimonoids

We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.

Chomsky-Schützenberger parsing for weighted multiple context-free languages

We prove a Chomsky-Schützenberger representation theorem for multiple context-free languages weighted over complete commutative strong bimonoids. Using this representation we devise a parsing algorithm for a restricted form of those devices.

Weak QMV algebras and some ring-like structures

In this work, we propose a new quantum structure--weak quantum MV algebras (wQMV algebras)--and define coupled bimonoids and strong coupled bimonoids. We find that the coupled bimonoids and strong coupled bimonoids are ring-like structures corresponding to lattice-ordered wQMV algebras and lattice-ordered QMV algebras, respectively. Using an automated reasoning tool, we give the smallest 4-element wQMV algebra but not a QMV algebra. We also show that lattice-ordered wQMV algebras are the real nondistributive generalization of MV algebras. Certainly, most important properties of quantum MV algebras (QMV algebras) are preserved by wQMV algebras. Furthermore, we can conclude that lattice-ordered wQMV algebras are the simplest unsharp quantum logical structures by far, based on which computation theory could be set up.

The relationships among several forms of weighted finite automata over strong bimonoids

Given a strong bimonoid P, we introduce three different behaviors of a weighted finite automaton over P (called a P−valued finite automaton), named the initial object semantics, final object semantics and run semantics. We define four forms for a P−valued nondeterministic finite automaton (P−NFA) and three forms for a P−valued deterministic finite automaton (P−DFA). Under the above-mentioned semantics, the equivalence and differences among the four forms of P−NFAs are discussed and the equivalence among the three forms of P−DFAs are given. Moreover, we show that some equivalence depends on right distributivity or left distributivity, or even requires P to degenerate into a semiring.

The algebraic characterizations for a formal power series over complete strong bimonoids.

On the basis of run semantics and breadth-first algebraic semantics, the algebraic characterizations for a classes of formal power series over complete strong bimonoids are investigated in this paper. As recognizers, weighted pushdown automata with final states (WPDAs for short) and empty stack (WPDAs^{emptyset }) are shown to be equivalent based on run semantics. Moreover, it is demonstrated that for every WPDA there is an equivalent crisp-simple weighted pushdown automaton with final states by run semantics if the underlying complete strong bimonoid satisfies multiplicatively local finiteness condition. As another type of generators, weighted context-free grammars over complete strong bimonoids are introduced, which are proven to be equivalent to WPDAs^{emptyset } based on each one of both run semantics and breadth-first algebraic semantics. Finally examples are presented to illuminate the proposed methods and results.

The Supports of Weighted Unranked Tree Automata

We investigate the supports of weighted unranked tree automata. Our main result states that the support of a weighted unranked tree automaton over a zero-sum free, commutative strong bimonoid is recognizable. For this, we use methods of Kirsten (DLT 2009), in particular, his construction of finite automata recognizing the supports of weighted automata on strings over zero-sum free, commutative semirings. We also get an effective construction of a finite tree automaton recognizing the support of a given weighted unranked tree automaton for zero-sum free, commutative strong bimonoids where Kirsten's zero generation problem is decidable. In addition, we give a translation of nested weighted automata into weighted unranked tree automata for arbitrary commutative strong bimonoids. As a consequence, we derive analogous results for the supports of nested weighted automata. Finally, we give similar results for the supports of weighted pushdown automata.

WEIGHTED NESTED WORD AUTOMATA AND LOGICS OVER STRONG BIMONOIDS

Nested words have been introduced by Alur and Madhusudan as a model for e.g. recursive programs or XML documents and have received much recent interest. In this paper, we investigate a quantitative automaton model and a quantitative logic for nested words. The behavior resp. the semantics map nested words to weights which are taken from a strong bimonoid. Strong bimonoids can be viewed as semirings without requiring the distributivity assumption which was essential in the classical theory of formal power series; strong bimonoids include e.g. all bounded lattices and many other structures from multi-valued logics. Our main results show that weighted nested word automata and suitable weighted MSO logics are expressively equivalent. This extends the classical Büchi-Elgot result from words to a weighted setting for nested words.

Weighted automata and multi-valued logics over arbitrary bounded lattices

We show that L -weighted automata, L -rational series, and L -valued monadic second order logic have the same expressive power, for any bounded lattice L and for finite and infinite words. We also prove that aperiodicity, star-freeness, and L -valued first-order and LTL-definability coincide. This extends classical results of Kleene, Büchi–Elgot–Trakhtenbrot, and others to arbitrary bounded lattices, without any distributivity assumption that is fundamental in the theory of weighted automata over semirings. In fact, we obtain these results for large classes of strong bimonoids which properly contain all bounded lattices.

A Kleene-Schützenberger Theorem for Trace Series over Bounded Lattices

We study weighted trace automata with weights in strong bimonoids. Traces form a generalization of words that allow to model concurrency; strong bimonoids are algebraic structures that can be regarded as “semirings without distributivity”. A very important example for the latter are bounded lattices, especially non-distributive ones. We show that if both operations of the bimonoid are locally finite, then the classes of recognizable and mc-rational trace series coincide and, in general, are properly contained in the class of c-rational series. Moreover, if, in addition, in the bimonoid the addition is idempotent and the multiplication is commutative, then all three classes coincide.

An improved algorithm for determinization of weighted and fuzzy automata

For a given weighted finite automaton over a strong bimonoid we construct its reduced Nerode automaton, which is crisp-deterministic and equivalent to the original weighted automaton with respect to the initial algebra semantics. We show that the reduced Nerode automaton is even smaller than the Nerode automaton, which was previously used in determinization related to this semantics. We determine necessary and sufficient conditions under which the reduced Nerode automaton is finite and provide an efficient algorithm which computes the reduced Nerode automaton whenever it is finite. In determinization of weighted finite automata over semirings and fuzzy finite automata over lattice-ordered monoids this algorithm gives smaller crisp-deterministic automata than any other known determinization algorithm.

Determinization of weighted finite automata over strong bimonoids

We consider weighted finite automata over strong bimonoids, where these weight structures can be considered as semirings which might lack distributivity. Then, in general, the well-known run semantics, initial algebra semantics, and transition semantics of an automaton are different. We prove an algebraic characterization for the initial algebra semantics in terms of stable finitely generated submonoids. Moreover, for a given weighted finite automaton we construct the Nerode automaton and Myhill automaton, both being crisp-deterministic, which are equivalent to the original automaton with respect to the initial algebra semantics, respectively, the transition semantics. We prove necessary and sufficient conditions under which the Nerode automaton and the Myhill automaton are finite, and we provide efficient algorithms for their construction. Also, for a given weighted finite automaton, we show sufficient conditions under which a given weighted finite automaton can be determinized preserving its run semantics.

Weighted finite automata over strong bimonoids

We investigate weighted finite automata over strings and strong bimonoids. Such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold. We define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions. Moreover, we show that in this case the given weighted automata can be determinized.