Automating Inductive Proofs using Theory Exploration

HipSpec is a system for automatically deriving and proving properties about functional programs. It uses a novel approach, combining theory exploration, counterexample testing and inductive theorem proving. HipSpec automatically generates a set of equational theorems about the available recursive functions of a program. These equational properties make up an algebraic specification for the program and can in addition be used as a background theory for proving additional user-stated properties. Experimental results are encouraging: HipSpec compares favourably to other inductive theorem provers and theory exploration systems

Testing Division Rings and Fields Using a Computer Program

AbstractAbstract algebra is a branch of mathematics which is studying algebraic structures of sets with respect to some operations on them. Each structure has its own axioms and properties. In some cases, to test what kind of structure is a given set and operations is difficult to be done manually. To help this, an application program to test algebraic structures is developed. In this article, we focus on two structures: division ring and fields, and an application program to test them is created by using Java, an open-source based programming language. This application program provides testing of various input of finite set such as integers, matrices and alphabets. By this application, testing of algebraic structures can be done faster than the manual one and its results are accurate. In this article, we focus on testing division rings and fields, together with some examples

Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve

Deterministic Black-Box Identity Testing π\pi-Ordered Algebraic Branching Programs

In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. Given a permutation $\pi$ of $n$ variables, for a $\pi$-ordered ABP ($\pi$-OABP), for any directed path $p$ from source to sink, a variable can appear at most once on $p$, and the order in which variables appear on $p$ must respect $\pi$. An ABP $A$ is said to be of read $r$, if any variable appears at most $r$ times in $A$. Our main result pertains to the identity testing problem. Over any field $F$ and in the black-box model, i.e. given only query access to the polynomial, we have the following result: read $r$ $\pi$-OABP computable polynomials can be tested in \DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]. Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size $\Omega(2^n/n)$ and read $\Omega(2^n/n^2)$. We give a multilinear polynomial $p$ in $2n+1$ variables over some specifically selected field $G$, such that any OABP computing $p$ must read some variable at least $2^n$ times. We show that the elementary symmetric polynomial of degree $r$ in $n$ variables can be computed by a size $O(rn)$ read $r$ OABP, but not by a read $(r-1)$ OABP, for any $0 < 2r-1 \leq n$. Finally, we give an example of a polynomial $p$ and two variables orders $\pi \neq \pi'$, such that $p$ can be computed by a read-once $\pi$-OABP, but where any $\pi'$-OABP computing $p$ must read some variable at least $2^n