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In a digital quantum simulator, basic two-qubit interactions are manipulated by means of fast local control operations to establish a desired target Hamiltonian. Here we consider a quantum simulator based on logical systems, i.e. where several physical qubits are used to represent a single logical two-level system to obtain enhanced and simple control over effective interactions between logical systems. Fixed, distance-dependent pairwise interactions between the physical qubits lead to effective interactions between the logical systems, which can be fully controlled solely by the choice of their internal state. This allows one to directly manipulate the topology and strength of effective interactions between logical systems. We show how to choose and generate the required states of logical systems for any desired interaction pattern and topology, how to perform arbitrary logical measurements, and how to obtain full control over single logical systems using only the intrinsic two-body interactions and control of individual physical qubits. This leads to a universal quantum simulator based on logical systems. We discuss the advantages of such a logical quantum simulator over standard ones, including the possibility to reach target topologies that are only accessible with large overheads otherwise. We provide several examples of how to obtain different target interaction patterns and topologies from initial long-ranged or short-ranged qubit-qubit interactions with a specific distance dependence.