Symbolic Algebraic Manipulation Research Articles

Frugal LMs Trained to Invoke Symbolic Solvers Achieve Parameter-Efficient Arithmetic Reasoning

Large Language Models (LLM) exhibit zero-shot mathematical reasoning capacity as a behavior emergent with scale, commonly manifesting as chain-of-thoughts (CoT) reasoning. However, multiple empirical findings suggest that this prowess is exclusive to LLMs that have exorbitant sizes (beyond 50 billion parameters). Meanwhile, educational neuroscientists suggest that symbolic algebraic manipulation be introduced around the same time as arithmetic word problems so as to modularize language-to-formulation, symbolic manipulation of the formulation, and endgame arithmetic. In this paper, we start with the hypothesis that much smaller LMs, which are weak at multi-step reasoning, can achieve reasonable arithmetic reasoning if arithmetic word problems are posed as a formalize-then-solve task. In our architecture, which we call SyReLM, the LM serves the role of a translator to map natural language arithmetic questions into a formal language (FL) description. A symbolic solver then evaluates the FL expression to obtain the answer. A small frozen LM, equipped with an efficient low-rank adapter, is capable of generating FL expressions that incorporate natural language descriptions of the arithmetic problem (e.g., variable names and their purposes, formal expressions combining variables, etc.). We adopt policy-gradient reinforcement learning to train the adapted LM, informed by the non-differentiable symbolic solver. This marks a sharp departure from the recent development in tool-augmented LLMs, in which the external tools (e.g., calculator, Web search, etc.) are essentially detached from the learning phase of the LM. SyReLM shows massive improvements (e.g., +30.65 absolute point improvement in accuracy on the SVAMP dataset using GPT-J 6B model) over base LMs, while keeping our testbed easy to diagnose and interpret, and within the reach of most researchers.

Teaching with new technology: four ‘early majority’ teachers

This paper explores how four good teachers, who do not have a special interest in technology, meet the challenge of introducing the rapidly developing mathematics analysis software (e.g. spreadsheets, function graphers, symbolic algebra manipulation and dynamic geometry) into their classrooms. These teachers’ practice is viewed through the lens of Roger’s framework for the diffusion of innovation and Pierce and Stacey’s pedagogical opportunities map. Data on teachers, views and practices were collected over 2 years. ‘Pedagogical Maps’ give a picture of the teachers’ perception and uptake of pedagogical opportunities. New practices have been added slowly to each teacher’s repertoire and their increasing fluency in practical ability to teach with the technology resulted in some changes to the classroom didactic contract. Overall, new technology seemed to have been absorbed into current practice, more than changing practice. At this stage of their development, these teachers do not identify the distinctive new mathematical capabilities as contributing to the major relative advantage of the innovation. Instead, they see the relative advantage mostly in the incremental improvements to capabilities of earlier calculators, and meeting the need for students to be up to date. One of the current challenges is that significant changes in both software and hardware design have been happening so rapidly that these early majority teachers felt almost constantly hampered by the need to learn and teach new technical skills and so continue to make limited progress in taking advantage of opportunities to approach mathematics concepts in new ways.

On the generation and solution of the symbolic, open-loop characteristic equation for a double inverted pendulum

Symbolic algebra manipulation is applied computationally to the mechanical system described by the inverted pendulum of two links, to yield analytic roots of the open-loop characteristic equation obtained from the linearized, symbolic dynamic equations of motion of the system. Detailed analysis of the characteristic polynomial form is presented, together with conditions relating polynomial root properties to physical parameter inequalities. Finally, features of the theoretical form of the characteristic polynomial for the general n-link system are proposed, by simple explication of results provided by this low order system, and also those existing for the corresponding single-link case which may be obtained and checked directly.

Instability of flows in spatially developing media

This paper studies specific forms of partial differential equations which represent the amplitude evolution of disturbances to flows in media whose properties vary slowly in space (over a length scale ∊ -1 ). A method is described for finding series solutions in the small parameter ∊, and general results for the Green’s function and the time-periodic response are obtained. With the aid of a computer-based symbolic algebra manipulation package, the method permits results to be obtained to any desired accuracy for any specified spatial variation. The method is then applied to find the Green’s function for the linearized Ginzburg-Landau equation (which is a good simple model of instability growth in many marginally unstable fluid-dynamical systems) in the case where the instability parameter μ (normally a constant in the equation) varies with the spatial coordinate. Even though the method produces only power series solutions in ∊, it is shown that with its help exact analytic solutions, valid for all ∊ (however large), can be found in some cases. The exact solutions for the cases of linear and quadratic spatial variation of μ are obtained and discussed, and criteria are found for the onset of global instability. Comparisons are made between these criteria and the local absolute and convective instability criteria, and the results are found to be in qualitative agreement with other authors’ numerical and analytic results; in particular, it is shown that a small region of local absolute instability (up to a maximal size which is known analytically) may exist and yet that the flow may be globally stable. It is also shown that in the case of quadratic spatial variation of μ with an infinite region of local absolute instability, the Green’s function may (for some particular parameter ranges) become infinite at all spatial locations simultaneously at some finite time.

New approach to automatic symbolic analysis off electric circuits

A new approach to the problem of computer generation of symbolic network functions is presented. The method is based on the application of artificial intelligence techniques and, in particular, on symbolic algebraic manipulation. The computer programs developed by following this approach are presented. An application example is given to illustrate the features and potentials of such programs.

A symbolic algorithm for finding exactly soluble statistical mechanical models

In general it is a very difficult task to find statistical mechanical models which satisfy the Yang-Baxter equations and thus are completely integrable. We propose a new approach leading to a (overdetermined) set oflinear equations. The formalism is applied to the Ising and the Ashkin-Teller models, which are both self-duals in two dimensions. Preliminary results for a symbolic algebra manipulation program is given, which would derive the relevant set of equations for an arbitrary internal spin symmetry group.

Solving the problems of kinetics of complex reactions by symbolic algebraic manipulation methods

Application of graph-theoretical methods to solving some problems of kinetics of complex reactions is considered. The possibility of reducing the graph-theoretical problems to the multivariate polynomial operations and use of a symbolic manipulation system is shown. The implemented symbolic manipulation system for solving the problems of kinetics of complex reactions is described and example of its usage given.

The rational canonical form of a matrix

The purpose of this paper is to provide an efficient algorithmic means of determining the rational canonical form of a matrix using computational symbolic algebraic manipulation packages, and is in fact the practical implementation of a classical mathematical method.

Abstracts of computer algebra papers from 1984 national meeting of American Chemical Society

The 188th National Meeting of American Chemical Society held in Philadelphia, Pennsylvania, during August 26-31, 1984, had a special symposium on symbolic algebraic manipulation. Following are the abstracts of computer algebra papers in that symposium (courtesy Dr. Richard Pavelle of Symbolics, Inc).

Analytic Estimates for the Dynamic Aperture of Non-Linear Lattices

For an accelerator lattice containing non-linear elements, particle amplitudes outside a "dynamic aperture" are unstable. In this paper an analytic procedure for estimating this aperture not depending on resonance assumptions is described. The amplitude x(N) after N iterations is expanded in powers of the initial amplitude x0. Low order coefficients are obtained using the symbolic algebraic manipulation program REDUCE. For intermediate values of N (e.g., 100) these terms give an asymptotic description which, by particle tracking, is shown to be valid. This leads to an estimate of the dynamic aperture and to its possible enlargement.

Some aspects of Euler-Newton equations of motion

Euler-Newton's equations for a system of connected rigid bodies, written in a special state space form, provide a systematic method of arriving at the differential equations of the system. This method is amenable to programming and symbolic algebraic manipulation. The elimination of some or all forces of constraint is by projection, implementing the principle of virtual work, and is done by inner products. The computation of these forces requires symbolic inversion of a matrix for which an iterative scheme is proposed here. A method for construction of Lyapunov functions for stability of such systems in the vicinity of an arbitrary operating point is proposed. This construction may be achieved by symbolic manipulations and supplements applications of the Euler-Newton method.

Quadratic and cubic transition elements

AbstractQuadratic and cubic quadrilateral isoparametric elements are investigated for use as transition elements for use a crack tip elements. Computations for a single element are explicitly carried out using a symbolic algebraic manipulation program and in addition to the desired square root singularity, additional singularities are revealed. Cubic transitional elements were used for a double‐edge crack problem and only marginal improvement is found for a well‐laid out grid.

A note about Laplace transform tables for computer use

The purpose of this note is to give another illustration of the fact that the best way for a human being to represent or process information is not necessarily the best way for a computer. The example concerns the use of a table of inverse Laplace transforms within a program, written in the REDUCE language [1] for symbolic algebraic manipulation, which solves linear ordinary differential equations with constant coefficients using Laplace transform methods. (See [2] for discussion of an earlier program which solved such equations.)

Applications of a symbolic algebra manipulation language for composite structures analysis

The computer is frequently used in the solution of structural problems involving composite materials. The algebraic manipulations required to cast the problem into a form suitable for direct computer solutions have become the most tedious and error-prone parts of the solution process. This step may now be markedly simplified by using a symbolic aglebra manipulation language. The use of PL/I FORMAC for this purpose is described.

Artificial intelligence

There are numerous indications that the field of Artificial intelligence (AI) is now well established: a specialized journal is entering its third year of publication; the third major biennial conference organized by an ad hoc international council will be held in August; several textbooks and paper collections have been published, and several Computer Science Departments have AI specialties; and some important disciplines whose roots are in AI, such as Pattern Recognition and Symbolic Algebraic Manipulation, have already spun off as independent areas. Still, when I identify my field as Artificial intelligence, I frequently am met by blank stares, snickers, or questions concerning whether that means building simulated rats or planting false military rumors. I hope this issue of COMPUTER will clear up some of the confusion.

An overlay facility for LISP programs

An overlay facility is described which enables large LISP programs such as are frequently encountered in symbolic algebraic manipulation to be used in a memory environment considerably smaller than required for the entire program. This facility has been implemented in the University of Texas LISP for CDC 6000 series machines, and has been so far applied to Joel Moses' integration package with good results.

GRAD Assistant

The General Recursive Algebra and Differentiation Assistant (GRAD Assistant) now under development, is a LISP-based system which can perform symbolic algebraic manipulation and differentiation. It is designed to produce a simplified expression of the result of any, sequence of symbolic manipulations involving addition, multiplication, exponentiation, and their inverses, and differentiation. In particular, it will never over- look that an expression is in fact independent of a particular variable. It does these things without requiring any externally supplied hints about difficult expressions. It can read and write a Fortran-like infix notation. It has been used in connection with a heuristic ordinary-differential-equation-solving program and a program that performs calculations arising from Ricmannian geometry.