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A Turing machine with a fixed finite number of tapes and a head for each tape. One transition reads the scanned symbols, updates them and moves the heads. This model accepts the same computably enumerable languages as a single-tape Turing machine, but step counts can differ.

${\mathcal{L}}_{\ell 3}$ is undecidable. The condition depends only on the accepted formal language. It holds for a Turing machine accepting the empty language, and fails for a Turing machine accepting the singleton $\{aaaa\}$, where $a$ is an alphabet symbol. It is therefore a nontrivial extensional property of programs, and Rice's theorem applies. In particular, the condition does not assert that the machine halts on every input; nonacceptance of long words over an alphabet may result from rejection or from an infinite computation.

${\mathcal{L}}_{1\mathrm{T}}$ is a decidable language: it consists of every valid Turing machine encoding. Every multitape Turing machine has an equivalent single-tape Turing machine. To see this, represent the finitely used portions of its tapes on one tape, separated by delimiters, and mark each simulated head position. To simulate one step, scan these portions to read the marked symbols, retain this fixed number of symbols and the control state in finite control, and scan again to update the symbols and head marks. Extend a represented portion with a blank cell when necessary. Each simulated step takes finitely many real steps.

The single-tape Turing machine accepts exactly when the simulated multitape Turing machine accepts; infinite computations remain infinite. The simulation may increase time complexity, but the question only asks about accepted formal languages. Hence a decider simply checks the syntax of the Turing machine encoding and accepts every valid description. Rice's theorem does not make this property undecidable, because the property is true of every computably enumerable language and is therefore trivial in the theorem's sense.

A deterministic single-tape Turing machine that leaves every scanned symbol unchanged and moves one cell right after every nonhalting transition. Such machines decide exactly the regular languages. A deterministic finite automaton simulates the finite control, with accepting states determined by the blank-only continuation after the input; conversely, a machine simulating a deterministic finite automaton halts on the first blank.

A Turing machine with one read-write tape. Every multitape Turing machine can be simulated by a single-tape Turing machine, using delimited tape portions and marked head positions. The simulation preserves the accepted formal language and whether a computation halts, while possibly increasing time complexity.