Automatically Exploring the Domain of Residue Classes Extended Abstract

We describe a module for exploring simple algebraic properties of operations on residue class sets over the integers. The framework is implemented within the mega theorem proving environment [1]. It employs computations of the computer algebra system Gap [3] to classify a given residue class set together with one or two operations in terms of its algebraic structure. During this classi cation process proof obligations for proving or refuting single properties are generated. These proof obligations are passed to mega's multi-strategy proof planner Multi [5] that constructs a proof with the help of Gap and Maple [7]. Since the presented exploration module has originated from work done in the context of tutor systems, the motivation is not to obtain new results in nite algebra. It shall rather enable a user to learn fundamental algebraic notions by ddling about with arbitrary residue class sets and combinations of operations. Moreover the module enables the automatic exploration of large testbeds. In our context a residue class set over the integers is either the set of all congruence classes modulo an integer n, i.e., ZZn, or an arbitrary subset of ZZn. Some examples are ZZ3;ZZ5;ZZ3nf 13g;ZZ5nf 05g, f 16; 36; 56g; : : : where 13 denotes the congruence class 1 modulo 3. An operation on a residue class set is an arbitrary combination of the constant operations, addition, multiplication, and subtraction on congruence classes.