Computing the Homology of the Lambda Algebra by Martin C. Tangora

By Martin C. Tangora

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Additional resources for Computing the Homology of the Lambda Algebra

Sample text

For p=3 we also use a two-digit string. With two kinds of generators (lambdas and mu's), the subscript alone is not sufficient. We decided to use the dimension r as the integer code for the generators; thus MI^I becomes 0403. Using the dimension as the code seems as natural as any other representation (it conforms to the case p=2), and it is convenient when we need to find the dimension of a monomial. We expected to change over to three-digit strings when we got up to t=100 (p=3), and indeed we did so when we were computing in the subalgebra of lambdas only, but in the full lambda algebra for p=3 the computations became so heavy as t approached 100 that we never crossed the line.

The pattern NAD (iNADmissible) detects inadmissible products of generators, while the pattern FRD (FRee D) looks for sufficiently small, visible, untagged cycles. The calls to CTAB and DTAB establish the tables of coefficients and differentials of generators. The function INS (iNSert) inserts a term into a polynomial in the proper place. It is recursive, which is convenient in coding, and has not proved to be a bottleneck. The function SUM merges two ordered polynomials recursively, treating one-term summands as special cases passed to INS.

We learned that eventually even 4 million bytes is not enough storage, and so we went with the LTO algorithm and did not keep full tags or complete cycles. This does certainly cost us dearly in execution time, although one should consider search time as a partially compensating effect. Differentials of polynomials would require vast storage and an early version of the program showed this impractical. However, differentials of monomials present an interesting decision. In proper form, the differential of a monomial can be distressingly long; for example, at p=2, the differential of ^21^40 ^as 51 terms.