### 26` `Glossary

The bandwidth between two network nodes is the quantity of data that can be transferred in a unit of time between the nodes.

A cache is an instance of a ☛ space-time tradeoff: it trades space for time by using the space to avoid recomputing an answer. The act of using a cache is called caching. The word “cache” is often used loosely; I use it only for information that can be perfectly reconstructed even if it were lost: this enables a program that needs to reverse the trade—

i.e., use less space in return for more time— to do so safely, knowing it will lose no information and thus not sacrifice correctness.

Coinduction is a proof principle for mathematical structures that are equipped with methods of observation rather than of construction. Conversely, functions over inductive data take them apart; functions over coinductive data construct them. The classic tutorial on the topic will be useful to mathematically sophisticated readers.

An idempotent operator is one whose repeated application to any value in its domain yields the same result as a single application (note that this implies the range is a subset of the domain). Thus, a function \(f\) is idempotent if, for all \(x\) in its domain, \(f(f(x)) = f(x)\) (and by induction this holds for additional applications of \(f\)).

Invariants are assertions about programs that are intended to always be true (“in-vary-ant”—

never varying). For instance, a sorting routine may have as an invariant that the list it returns is sorted.

The latency between two network nodes is the time it takes for packets to get between the nodes.

A metasyntactic variable is one that lives outside the language, and ranges over a fragment of syntax. For instance, if I write “for expressions e1 and e2, the sum e1 + e2”, I do not mean the programmer literally wrote “e1” in the program; rather I am using e1 to refer to whatever the programmer might write on the left of the addition sign. Therefore, e1 is metasyntax.

At the machine level, a packed representation is one that ignores traditional alignment boundaries (in older or smaller machines, bytes; on most contemporary machines, words) to let multiple values fit inside or even spill over the boundary.

For instance, say we wish to store a vector of four values, each of which represents one of four options. A traditional representation would store one value per alignment boundary, thereby consuming four units of memory. A packed representation would recognize that each value requires two bits, and four of them can fit into eight bits, so a single byte can hold all four values. Suppose instead we wished to store four values representing five options each, therefore requiring three bits for each value. A byte- or word-aligned representation would not fundamentally change, but the packed representation would use two bytes to store the twelve bits, even permitting the third value’s three bytes to be split across a byte boundary.

Of course, packed representations have a cost. Extracting the values requires more careful and complex operations. Thus, they represent a classic ☛ space-time tradeoff: using more time to shrink space consumption. More subtly, packed representations can confound certain run-time systems that may have expected data to be aligned.

Reduction is a relationship between a pair of situations—

problems, functions, data structures, etc.— where one is defined in terms of the other. A reduction R is a function from situations of the form P to ones of the form Q if, for every instance of P, R can construct an instance of Q such that it preserves the meaning of P. Note that the converse strictly does not need to hold.

Suppose you have an expensive computation that always produces the same answer for a given set of inputs. Once you have computed the answer once, you now have a choice: store the answer so that you can simply look it up when you need it again, or throw it away and re-compute it the next time. The former uses more space, but saves time; the latter uses less space, but consumes more time. This, at its heart, is the space-time tradeoff. Memoization [REF], using a ☛ cache, environments (From Substitution to Environments), etc. are all instances of it.

Type variables are identifiers in the type language that (usually) range over actual types.