A decision procedure for term algebras with queues

The algebra of stream processing functions

Journal Article HOMOLOGICAL SYSTEMS IN MODULE CATEGORIES OVER PRE-ORDERED SETS Get access Octavio Mendoza, Octavio Mendoza Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D.F. 04510, México Search for other works by this author on: Oxford Academic Google Scholar Corina Sáenz, Corina Sáenz Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México D.F. 04510, México Search for other works by this author on: Oxford Academic Google Scholar Changchang Xi Changchang Xi § School of Mathematical Sciences; Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, 100875 Beijing, P.R. China §Corresponding author. E-mail: xicc@bnu.edu.cn Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 60, Issue 1, March 2009, Pages 75–103, https://doi.org/10.1093/qmath/han003 Published: 10 April 2008 Article history Received: 21 August 2006 Revision received: 07 January 2008 Published: 10 April 2008

In the theories of groups and Lie algebras, investigations of the properties of the non-abelian tensor product and their relations to the second homology groups are worthwhile. It is the purpose of the present paper to exhibit such investigations about the non-abelian tensor product of Leibniz algebras. The isomorphism between the non-abelian tensor square and non-abelian exterior square of a Lie algebra L, will enable us to set a simple connection between and . Furthermore, we shall relate the concepts of capability and solvability of a Leibniz algebra to its tensor square. Finally, we give an upper bound for the dimension of the non-abelian tensor square and the second homology of a nilpotent Leibniz algebra in terms of the dimension of its center and derived subalgebra.

Term algebras have been widely used in the theory of abstract data types. Here, the concept of canonical term algebra is generalized to the notion of canonical term functor, which is useful for various aspects of abstract data type specifications. In particular, we show how canonical term functors provide a constraint mechanism in loose specifications and how they constitute a junction between axiomatic and constructive approaches. These concepts are the semantic base for the specification development language ASPIK which has been implemented as a core component of an integrated software development and verification system.

We present LIAMC, a novel decision procedure for (quantifier-free) linear arithmetic over both integers modulo 2N (LIA n ) and integers (LIA). There is no need to explain our motivation to design a new efficient decision procedure for the widely used LIA logic. A LIA n decision procedure can be extremely useful in the context of software (SW) verification. SW verification usually requires to reason about arithmetic constraints over finite integers. To that end, modern SW verification tools commonly use fixed-width bit-vector (BV) solvers. However, BV solvers' efficiency drops dramatically as the width increases. To solve the performance problem, LIA solvers are applied, but they are imprecise as they cannot handle integer overflow. An efficient LIA N solver would be the ideal solution in this context. Our decision procedure LIAMC is based on a transformation of linear arithmetic into safety verification. We treat integers as unbounded streams of bits over time. More precisely, for each input integer, the least significant bit (LSB) corresponds to time 0 in the corresponding stream, and the k-th bit corresponds to the bit received at time k. LIAMC then uses SAT-based model checking (SATMC) to solve the resulting problem. In order to achieve efficiency, LIAMC uses two forms of generalization. First, if it finds a formula to be unsatisfiable for width N, it tries to generalize this result for all the widths. Second, if LIAMC finds a formula to be satisfiable for width N, it tries to “extend” and thus generalize the assignment to a wider target width. To evaluate LIAMC we used the QF_LIA subset of SMT-COMP'16, and ran two sets of experiments. First, we reinterpreted the QF_LIA over fixed-width bit-vectors of varying widths and compared LIAMC in LIAn mode to both Boolector and Z3. LIAMC solved the most satisfiable instances out of the three even for the shortest width 32. Second, we compared LIAMC to CVC4 and Z3 on the original QF_LIA benchmarks. LIAMC was able to solve many instances that had not been solved by the other solvers.

The study of computational agents capable of rational behaviour has received increasing attention in recent years. A number of theoretical formalizations for such multi-agent systems have been proposed. However, most of these formalizations do not have a strong semantic basis nor a sound and complete axiomatization. Hence, it has not been clear as to how these formalizations could assist in building agents in practice. This paper explores a particular type of multi-agent system, in which each agent is viewed as having the three mental attitudes of belief (B), desire (D), and intention (I). It provides a family of multi-modal branching-time BDI logics with a possible-worlds semantics, categorizes them, provides sound and complete axiomatizations, and gives constructive tableau-based decision procedures for testing the satisfiability and validity of formulas. The computational complexity of these decision procedures is no greater than the complexity of their underlying temporal logic component.

In this paper, we study McKay’s E7 observation on the Baby Monster. By investigating the so-called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and certain subalgebras of the Baby Monster vertex operator algebra which are constructed by the nodes of the affine E7 diagram. This allows us to reinterpret McKay’s E7 observation via the theory of vertex operator algebras. For a class of vertex operator algebras including the Moonshine module, we will show that the product of two Miyamoto involutions associated to derived c=7/10 Virasoro vectors in certain commutant vertex operator algebras is an element of order at most 4. For the case of the Moonshine module, we obtain the Baby monster vertex operator algebra as the commutant and we can identify the group generated by these Miyamoto involutions with the Baby Monster and recover the {3,4}-transposition property of the Baby Monster in terms of vertex operator algebras.

Graded-division algebras and Galois extensions

This paper presents an algebraic theory of linear signal processing. At the core of algebraic signal processing is the concept of a linear signal model defined as a triple (A, M, phi), where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively, and phi generalizes the concept of the z-transform to bijective linear mappings from a vector space of, e.g., signal samples, into the module M. A signal model provides the structure for a particular linear signal processing application, such as infinite and finite discrete time, or infinite or finite discrete space, or the various forms of multidimensional linear signal processing. As soon as a signal model is chosen, basic ingredients follow, including the associated notions of filtering, spectrum, and Fourier transform. The shift operator is a key concept in the algebraic theory: it is the generator of the algebra of filters A. Once the shift is chosen, a well-defined methodology leads to the associated signal model. Different shifts correspond to infinite and finite time models with associated infinite and finite z-transforms, and to infinite and finite space models with associated infinite and finite C-transforms (that we introduce). In particular, we show that the 16 discrete cosine and sine transforms are Fourier transforms for the finite space models. Other definitions of the shift naturally lead to new signal models and to new transforms as associated Fourier transforms in one and higher dimensions, separable and non-separable. We explain in algebraic terms shift-invariance (the algebra of filters A is commutative), the role of boundary conditions and signal extensions, the connections between linear transforms and linear finite Gauss-Markov fields, and several other concepts and connections.

The notion of Gorenstein rings in the commutative ring theory is generalized to that of Noetherian algebras which are not necessarily commutative. We faithfully follow in the steps of the commutative case: Gorenstein algebras will be defined using the not

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by B\^ocher contractions of the conformal Lie algebra $\mathfrak{so}(4,\mathbb {C})$ to itself. In 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between B\^ocher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all B\^ocher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric B\^ocher contractions.

A specification, as we shall use it here, consists of a signature together with a collection of (non-conditional) equations; these equations involve terms in the ‘distributive type theory’ which is built on top of the signature. This type theory has finite product (x, 1) and coproduct (+, 0) types. Particular simple examples of such specifications are Hagino specifications, which are used to describe inductively defined types. Models of specifications are described in arbitrary distributive categories. In a more categorical approach, one describes models as structure preserving functors. It enables us to define in general what are (a) models of parametrized spefications (in terms of Kan extensions) and (b) models with parameters (in terms of so-called ‘simple slice’ categories). It is shown that in the special case of Hagino specifications, these general definitions specialize to ones in terms of algebras or coalgebras for associated ‘strong’ polynomial functors. Models with parameters of Hagino specifications were described earlier by Cockett and Spencer.

I present a formal connection between algebraic effects and game semantics, two important lines of work in programming languages semantics with applications in compositional software verification. Specifically, the algebraic signature enumerating the possible side-effects of a computation can be read as a game, and strategies for this game constitute the free algebra for the signature in a category of complete partial orders (cpos). Hence, strategies provide a convenient model of computations with uninterpreted side-effects. In particular, the operational flavor of game semantics carries over to the algebraic context, in the form of the coincidence between the initial algebras and the terminal coalgebras of cpo endofunctors. Conversely, the algebraic point of view sheds new light on the strategy constructions underlying game semantics. Strategy models can be reformulated as ideal completions of partial strategy trees (free dcpos on the term algebra). Extending the framework to multi-sorted signatures would make this construction available for a large class of games.

Recursive algebraic data types (term algebras, ADTs) are one of the most well-studied theories in logic, and find application in contexts including functional programming, modelling languages, proof assistants, and verification. At this point, several state-of-the-art theorem provers and SMT solvers include tailor-made decision procedures for ADTs, and version~2.6 of the SMT-LIB standard includes support for ADTs. We study an extremely simple approach to decide satisfiability of ADT constraints, the reduction of ADT constraints to equisatisfiable constraints over uninterpreted functions (EUF) and linear integer arithmetic (LIA). We show that the reduction approach gives rise to both decision and Craig interpolation procedures in (extensions of) ADTs.

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.