New insight into superfast testing
In a program with potential bugs, self-correctors are correcting the function for any given input and self-testers are checking whether the function is correct. Scientists within the CORTST-PROGRAMS (Testing and correcting programs with applications to codes) project found new relations between testing and correcting of functions/programs to seemingly unrelated areas. These include symmetric groups and symmetric codes, expander graphs, and high-dimension expansion and topology. Combining these, scientists can now construct better locally testable/correctable codes that are beyond the state of the art. The team solved an important open problem regarding the possible existence of symmetric codes that are good. These codes have constant rate and constant relative distance and they can be tested with a constant number of queries. Providing that locally testable codes (LTCs) and proofs – which allow for superfast testing – are symmetric, the project constructed the first symmetric low-density parity-check good code. These codes can potentially be testable. Work also showed that every LTC can be decomposed into a constant number of basic codes whose associated graph is an expander. This was a surprising finding since expander codes cannot yield LTCs. In addition, scientists constructed non-linear codes from higher-dimension expanders that are locally testable due to augmenting the original information with a certain amount of redundancy. Considering that high-dimension expansion is a form of testability, and that expander graphs are connected bounded-degree graphs, the team found bounded-degree high-dimension expanders of every dimension. Bounded-degree high-dimension expanders were also found to be LTCs, and their point configuration has overlapping topology. All these newly discovered relations are expected to enhance testing and correcting of functions and programs.
Keywords
Superfast testing, testable codes, CORTST-PROGRAMS, symmetric codes, expander graphs
