In all the years I regularly glanced at our peg solitaire on the board game shelf, seldomly even playing it and attempting a solve — moving 31 out of the total 32 pegs such that the final peg ends up at the board’s center —, yet never succeeded. The closest I got always left two or three pegs standing far between each other, calling doom for my solve.

One lobe cleared.

Not being able to solve this game, the other day I had the idea of pseudo-random game space exploration, my optimism driven by both the moderate number of pegs as well as the strict monotonicity of the game leading to all solutions being of the same length and thus in a sense equal — bearing in mind consecutive hops in the same direction are not considered as a singular move.

Sparked by a vague idea to query the space of all brainfuck programs — which indeed came to fruition and is detailed in my recent paper Going for a miniKanren implementation. — I thought to capture a miniscule glimpse of said space in a two-dimensional visualization:

All generalized 𝟨-long brainfuck programs’ initial three bytes of output.

Working on a minimal XML parser which does both build a data structure and remember its nodes’ source origin, I discarded the second part of a two-tuple by not requiring anything to be present. Yet instead of informing me that a mapping cannot be deconstructed, GHC 8.10.4 gave me an obscure error message:

Thankfully I sometimes adhere to a mindfully incremental approach and was not as bewilderingly disoriented as I could have been — after all, a there is an error message possesses little more value than solely stating an executable cannot be built.

Inspecting the probable location I had intelligence on, I managed to extract a seven long byte sequence which triggers a <no location info> compilation error:

j(0,)=0

Yet compilers are squeamish and shapeshifting entities — especially with regards to diagnostic capabilities. Whilst j(0,)=0 cannot be traced, j(,0)=0 induces the expected and appropriate error. As such, I tested the phenomenon using varying GHC versions:

To my surprise, I seemed to have found a GHC regression — triggered by only seven bytes.

Alas, this regression is a known one: Another <no location info> error (Tuple section in pattern context) has been opened on the 8th of March 2021, with its comments identifying it as a regression. Yet the fix outlined and performed is as much resolving an aspect of the issue as is it making a stylistic decision — myself facing the same problem of source location corseness, I am leaning further and further towards the approach of less pedantically accurate reporting allowing for a more minimal implementation.

Following digital excavation efforts at my disk’s deep directories, I stumbled upon a collection of colorful pseudo-random walk graphics. Since they are dated September of 2015 and were generated using unidiomatic slow snake script, their only property of note is a visually jolly aura; the following three possess a particularly vibrant one:

Inspired by grid worlds, non-linear notation and two-dimensional esolangs, I have attempted to design a few ASCII-art languages myself, none satisfactory enough for publication. Without the toolchain to interact in a non-typewriter manner — both on the software as well as on the hardware side — paired with the need for an apt encoding to facilitate higher-order capabilities, I could not manage to create something which stands on its own feet as a proper language, as opposed to nothing more than a convoluted yet primitive processor emulator.

In the fall of 2020, when I was tasked to teach elementary binary semantics courtesy of a brand new mandatory lecture at my university — constructing half-adders from basic gates and combining them to build full adders —, I thought that exactly this bare-bones grid world might be a fruitful endeavor for constructing and combining gates with a visualization of the entropy’s movement across the circuit (one might foolishly think of a bit meandering across a wire, although this interpretation has no physical merit to it). Within a few hours, I had managed to settle on a grid world definition together with an under 200 lines long interpreter for it. As I opted for an ASCII-CLI-look — significantly boosting development time —, I added image output facilities for this blog post (for which I swiftly designed a few pixel glyphs; only those used by the grid world), avoiding the need to take screenshots of my terminal emulator.

As in most grid worlds, non-inert characters are kept to a minimum: there are two entropy sources 0 and 1, the unary negation gate ! and three binary gates &, | and ^. All other characters except the space allow entropic bits to replicate, the special jumpers < and > allow to cross a gap of three characters, rendering interleaving wires possible.

Designing a 𝟥-bit adder

Calculating 0b110 + 0b011 == 0b1001 using a 𝟥-bit adder (input bits are interleaved, less significant bits reside on the left).

A matrix of discretely representable entries may be linearly layed out in memory using row-major order, concatenating successive rows into a contiguous -bytes long array. Such a representation, however, is disruptive to the matrix’ two-dimensional nature: whilst horizontally neighboring elements remain neighbors, vertically neighboring entries are torn apart by insertion of non-neighboring elements. As such, on matrices naturally defined operations get distorted by this encoding. One such inherently two-dimensional operation is matrix transposition. In the realm of matrices, are for nonsquare dimensions semantically different, being mapped to one another by transposition. Projecting onto their encoding, this semantic is lost and one is left with a permutation on memory . To visualize this permutation, its cycle decomposition is computed of which each cycle is given a color of the rainbow dyeing this cycle’s corresponding two-dimensional pixels when interpreting its path on the underlying array in the semantics of the original matrix.

Initial transposition cycles

Above listed are all visualizations for , shuffled. Whilst some behave extremely regularly — for example square matrices’ transposition permutations decompose into transpositions —, others are wildly intricate. Each of them adheres to a rotational symmetry; the top left and bottom right are fixed points.

Proof.Iota. Define via and observe any sequence’s reconstructibility by dyadic expansion.Pi. Define via and observe any real’s constructibility by dyadic expansion.