Here's a Turing machine that uses finite space but infinite time: https://samwho.dev/christopher/?code=H4sIAAAAAAAAE8tNzMxTqFHQUqhRCNAw1FQIUtC1UyhKLSktyuOCUHBZA

Here's a Turing machine that uses finite space but infinite time: https://samwho.dev/christopher/?code=H4sIAAAAAAAAE8tNzMxTqFHQUqhRCNAw1FQIUtC1UyhKLSktyuOCUHBZA00FH5BsbmJmHgBlULd%2BNwAAAA%3D%3D https://samwho.dev/christopher/?code=H4sIAAAAAAAAE8tNzMxTqFH... (from Sam Rose's wonderful article on Turing machines [1]) Your last point is correct, though: if you can detect a "cycle" (like the one the Turing machine I linked to goes through) then you can conclude that the machine won't halt. [1] https://samwho.dev/turing-machines/ https://samwho.dev/turing-machines/