The statement "w | w is an even-length binary string and contains an odd number of 1's" describes a language or a set of strings. In this case, the language is regular. Regular languages can be defined and recognized by regular expressions, finite automata, or regular grammar.
A regular language can be represented by a regular expression such as:
(0+1)*(1(0+1)1(0+1)*(1(0+1)1)*(0+1)*)
This regular expression matches strings that have an even length and contain an odd number of 1's. The expression (0+1)* matches any number of 0's and 1's, and the rest of the expression ensures that there is at least one group of 1's with an odd count.
Regular languages are a subset of context-free languages, which means that any regular language is also a context-free language. Context-free languages can be defined and recognized by context-free grammars or pushdown automata.
The statement does not involve recursion, as there is no self-reference or nesting of language rules.
As for decidability, determining whether a given string belongs to the described language (i.e., whether it is an even-length binary string with an odd number of 1's) is a decidable problem. It can be checked in a finite number of steps by counting the number of 1's and checking the length of the string.
In summary, the language described by "w | w is an even-length binary string and contains an odd number of 1's" is a regular language, which is a subset of the context-free languages. The problem of determining membership in this language is decidable.