, ?y m . ? such that for all i ? [1, m] and all formulae ? 1 ? * ? 2 occurring at polarity 1 in ?, we have y i ? var(? 1 ) ? var(? 2 ), 2. BSR fin (SL k ) as the set of sentences ?y 1, BSR inf (SL k ) as the set of sentences ?y 1

, Note that BSR fin (SL k ) BSR inf (SL k ) BSR(SL k ), for any k ? 1

, Definition 7 imposes no constraint on the occurrences of separating implications at the left of a ? * 8 . The decidability result of this paper is stated below: Theorem 2. For any integer k ? 1 not depending on the input, the infinite satisfiability problem for BSR inf (SL k ) and the finite satisfiability problem for BSR fin

, We provide a brief sketch of the proof (all details are available in [8]). In both cases, PSPACE-hardness is an immediate consequence of the fact that the quantifier-free fragment of SL k , without the separating implication, but with the separating conjunction and negation

, For PSPACE-membership, consider a formula ? in BSR inf (SL k ), and its equivalent disjunction of minterms ? (of exponential size)

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