MATH.11:11 - SoTA-Echoing

For deriving a condition valid after arbitrarily many steps, adopt the induction-based Invariant Principle in Lehman, Leighton and Meyer’s Mathematics for Computer Science, §5.4.3. Compared with inspecting more executions, :4.4 uses one preservation proof to cover every finite sequence. This pattern constructs an invariant function; the book’s principle also covers broader preserved predicates. Stop when the resulting relation answers the question, and reopen if an allowed transition changes. MIT text.

For constructing polynomial expressions from update rules, adapt Bayarmagnai, Mohammadi and Prébet’s Algebraic and Algorithmic Methods for Computing Polynomial Loop Invariants, §5, Corollary 5.4 and Algorithm 6. The adopted move substitutes an expression with unknown coefficients and solves the resulting linear equations. It improves on unstructured guessing in :4.2-:4.3 and the sum construction in :5.2. The source’s stronger algorithms distinguish invariant form, initial-value conditions and search space. Here the polynomial-identity construction has its stated scope; a restricted reachable-set question can require a different method. Reopen when a richer family or specialized algorithm obtains a more useful relation at acceptable cost. Extended paper.

For count transformations, adapt the conserved linear expression described by Gopalkrishnan in Autocatalysis in Reaction Networks. The weighted-exchange derivation in :5.1 makes such a quantity available instead of assuming an unweighted total. Equal values still require a reachability construction. The source’s chemical-network results need their own premises; they are not used to identify the counted kinds in this mathematical example. A changed exchange rule reopens the weight equations. Author’s account.

The three worked constructions and their changed conditions are derived here. The source algorithms do not establish that these mathematical rules describe a particular physical system or implementation.