Comment by LegionMammal978

> Binary Lambda Calculus seems to spend its bits much more wisely than a Turing Machine encoding, working at a much higher level of abstraction. Defining and applying functions performs more useful work than moving a tape head around and jumping from state to state.

That's what I mean by β-reduction being a more powerful operation: it can copy a term into arbitrary points in the program. (In the BB context, I like to think of BLC in terms of substitution rather than functions.) So I wonder if the comparison is somewhat biased, since applying a TM transition is logically simpler than applying a BTC β-reduction, which involves recursively parsing the current state, substituting, and reindexing.

> The Turing Machine is also severely hampered by the pure sequential nature of the tape, where lambda calculus has access to all kinds of easily defined data structures.

I'd say TMs have plenty of data structures, but most of the useful ones are weird and alien since the the control flow has to be encoded on the tape alongside the data, vs. BLC which can precisely direct where a term should be substituted. The real hamper IMO is the locality of the tape: a TM can't move a block of data across another block of data on the tape without wasting valuable states.

> The universal variant https://oeis.org/A361211 of the lambda busy beaver can easily simulate any binary encoded TM with fixed overhead.

Of course; the trick is to go from n + c bits to n bits.