Multi-linear iterative K-Sigma-semialgebras

We consider $K$-semialgebras for a commutative semiring $K$ that are at the same time $\Sigma$-algebras and satisfy certain linearity conditions. When each finite system of guarded polynomial fixed point equations has a unique solution over such an algebra, then we call it an iterative multi-linear $K$-$\Sigma$-semialgebra. Examples of such algebras include the algebras of $\Sigma$-tree series over an alphabet $A$ with coefficients in $K$, and the algebra of all rational tree series. We show that for many commutative semirings $K$, the rational $\Sigma$-tree series over $A$ with coefficients in $K$ form the free multi-linear iterative $K$-$\Sigma$-semialgebra on $A$

Munchausen Iteration

We present a method for solving polynomial equations over idempotent omega-continuous semirings. The idea is to iterate over the semiring of functions rather than the semiring of interest, and only evaluate when needed. The key operation is substitution. In the initial step, we compute a linear completion of the system of equations that exhaustively inserts the equations into one another. With functions as approximants, the following steps insert the current approximant into itself. Since the iteration improves its precision by substitution rather than computation we named it Munchausen, after the fictional baron that pulled himself out of a swamp by his own hair. The first result shows that an evaluation of the n-th Munchausen approximant coincides with the 2^n-th Newton approximant. Second, we show how to compute linear completions with standard techniques from automata theory. In particular, we are not bound to (but can use) the notion of differentials prominent in Newton iteration

Universal numerical algorithms and their software implementation

The concept of a universal algorithm is discussed. Examples of this kind of algorithms are presented. Software implementations of such algorithms in C++ type languages are discussed together with means that provide for computations with an arbitrary accuracy. Particular emphasis is placed on universal algorithms of linear algebra over semirings.Comment: 16 pages, no figure

Free iterative and iteration K-semialgebras

We consider algebras of rational power series over an alphabet $\Sigma$ with coefficients in a commutative semiring $K$ and characterize them as the free algebras in various classes of algebraic structures

Free inductive K-semialgebras

We consider rational power series over an alphabet $\Sigma$ with coefficients in a ordered commutative semiring $K$ and characterize them as the free ordered $K$-semialgebras in various classes of ordered $K$-semialgebras equipped with a star operation satisfying the least pre-fixed point rule and/or its dual. The results are generalizations of Kozen's axiomatization of regular languages

Revisiting Algebra and Complexity of Inference in Graphical Models

This paper studies the form and complexity of inference in graphical models using the abstraction offered by algebraic structures. In particular, we broadly formalize inference problems in graphical models by viewing them as a sequence of operations based on commutative semigroups. We then study the computational complexity of inference by organizing various problems into an "inference hierarchy". When the underlying structure of an inference problem is a commutative semiring -- i.e. a combination of two commutative semigroups with the distributive law -- a message passing procedure called belief propagation can leverage this distributive law to perform polynomial-time inference for certain problems. After establishing the NP-hardness of inference in any commutative semiring, we investigate the relation between algebraic properties in this setting and further show that polynomial-time inference using distributive law does not (trivially) extend to inference problems that are expressed using more than two commutative semigroups. We then extend the algebraic treatment of message passing procedures to survey propagation, providing a novel perspective using a combination of two commutative semirings. This formulation generalizes the application of survey propagation to new settings

A unifying approach to software and hardware design for scientific calculations and idempotent mathematics

A unifying approach to software and hardware design generated by ideas of Idempotent Mathematics is discussed. The so-called idempotent correspondence principle for algorithms, programs and hardware units is described. A software project based on this approach is presented.Comment: 16 pages, no figures; sibmitted to Reliable Computing; a substantially revised and enlarged version of quant-ph/990402

Idempotent/tropical analysis, the Hamilton-Jacobi and Bellman equations

Tropical and idempotent analysis with their relations to the Hamilton-Jacobi and matrix Bellman equations are discussed. Some dequantization procedures are important in tropical and idempotent mathematics. In particular, the Hamilton-Jacobi-Bellman equation is treated as a result of the Maslov dequantization applied to the Schr\"{o}dinger equation. This leads to a linearity of the Hamilton-Jacobi-Bellman equation over tropical algebras. The correspondence principle and the superposition principle of idempotent mathematics are formulated and examined. The matrix Bellman equation and its applications to optimization problems on graphs are discussed. Universal algorithms for numerical algorithms in idempotent mathematics are investigated. In particular, an idempotent version of interval analysis is briefly discussed.Comment: 70 pages, 5 figures, CIME lectures (2011), to be published in Lecture Notes in Mathematics (Springer

Finitely Additive, Modular and Probability Functions on Pre-semirings

In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. We also generalize some classical results in probability theory such as the Law of Total Probability, Bayes' Theorem, the Equality of Parallel Systems, and Poincar\'{e}'s Inclusion-Exclusion Theorem. While we prove that modular functions over a couple of known semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring $(Id(D), + ,\cdot)$, where $D$ is a Dedekind domain.Comment: Minor update, 24 pages, Published online on 15 Dec 2017 in the Journal of Communications in Algebr

Partial Conway and iteration semirings

A Conway semiring is a semiring $S$ equipped with a unary operation $^*:S \to S$, always called 'star', satisfying the sum star and product star identities. It is known that these identities imply a Kleene type theorem. Some computationally important semirings, such as $N$ or N^{\rat}\llangle \Sigma^* \rrangle of rational power series of words on $\Sigma$ with coefficients in $N$, cannot have a total star operation satisfying the Conway identities. We introduce here partial Conway semirings, which are semirings $S$ which have a star operation defined only on an ideal of $S$; when the arguments are appropriate, the operation satisfies the above identities. We develop the general theory of partial Conway semirings and prove a Kleene theorem for this generalization