Decision problems for propositional linear logic

Alternating Vector Addition Systems with States

“…We find further arguments in our own initial interest in such games, which comes from the study of simulation problems between Petri nets and finite-state systems [14,18] where they arise naturally-Abdulla et al [3] recently made a similar observation. Furthermore the model was already explicitly defined in the '90s in the study of substructural logics [20,15,25], and appears implicitly in recent proofs of complexity lower bounds in [10,4]. We show in this paper that determining the winner of an asymmetric VASS game with a state reachability objective is 2-ExpTime-complete.…”

“…VASS were originally called 'and-branching' counter machines by Lincoln, Mitchell, Scedrov, and Shankar [20], and were introduced to prove the undecidability of propositional linear logic. Kanovich [15] later identified a fragment of linear logic, called the (!, ⊕)-Horn fragment, that captures exactly alternation in VASS, and adopted a game viewpoint, see Section 6.…”

“…Reachability. The decision problem that originally motivated the definition of AVASS by Lincoln et al [20] is reachability: given an AVASS Q, d, T u , T f and two states q r and q in Q, does there exist a deduction tree with root labelled by (q r , 0) and every leaf labelled by (q , 0)? Equivalently, does Controller have a strategy that ensures that a play starting in (q r , 0) eventually visits (q , 0)?…”

“…Perhaps more importantly than those technical contributions, we single out in Section 2 a simple definition for alternation in VASS by way of 'fork' rules (following [20]), for which the complexity analyses of sections 3 and 4 are relatively easy, and establish it as a pivotal definition for VASS games. Indeed, we relate it to energy games in Section 5 (following [2]), to Horn fragments of affine linear logic in Section 6 (following [15]), and to regular simulation problems for VASS in Section 7.…”

The Ideal View on Rackoff’s Coverability Technique

“…The coverability problem, given a d-dimensional bottom-up ABVAS (A, B ∧ , B + ), and leaf and target vectors l, t ∈ N d , asks whether there exists a computation whose every leaf label is l and whose root label is t. We remark that the problem is equivalent to the reachability problem for lossy bottom-up ABVAS 4 , but that the variant of coverability in which the fork rules are applied with exact alternation is undecidable [20].…”