There has been an increased interest in the decision problems for linear logic and its fragments. Here, we give a fully self-contained, easy-to-follow, but fully detailed, direct and constructive proof of the undecidability of a very simple Horn-like fragment of linear logic, which is accessible to a wide range of people. Namely, we show that there is a direct correspondence between terminated computations of a Minsky machine M and cut-free linear logic derivations for a Horn-like sequent of the form
Neither negation, nor &, nor constants, nor embedded implications/bangs are used here.
Furthermore, our particular correspondence constructed above provides decidability for each of the Horn-like fragments whenever we confine ourselves to any two forms of the above Horn-like implications, along with the complexity bounds that come from the proof.
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