Intractability of decision problems for finite-memory automata

Bisimilarity in Fresh-Register Automata

“…Sakamoto [21] conjectured that the language inequivalence problem is not in NP. Below we refute the conjecture, showing that, for RA(S# 0 ), the complexity of deterministic language inequivalence actually matches that of nonemptiness [22].…”

Reachability in Pushdown Register Automata

“…Since the latter allows for storage of identical values in different registers, their hardness can also be established more directly by encoding relative to two fixed data values for 0 and 1. These different policies for register management are known to lead to different complexity bounds for emptiness testing in the absence of pushdown store: NP-completeness [21] 8 (injective assignment) vs PSPACE-completeness (non-injective assignment) [10]. Perhaps surprisingly, we have shown the presence of pushdown store cushions these differences and there is no analogous complexity gap.…”

“…A full assignment discipline, on the other hand, requires that at all times every register must contain some letter from the infinite alphabet, whereas an initially empty discipline allows registers to be empty at the start of a run. The complexity of emptiness for register automata according to the assumed register discipline is given in the first row of Table 1: in the S F case the problem is NL-complete, as it coincides with emptiness of the underlying finite-state automaton; in the S# 0 case it becomes NP-complete [21], as one is able to use the registers to encode boolean assignments; while in the M F case one is able to encode a linear-size tape with the registers, and therefore the problem is PSPACE-complete [10]. In contrast, in the pushdown case, we show that such distinctions do not affect the complexity: even if identical elements can be kept in different registers, the problem can still be solved in EXPTIME and it is EXPTIME-hard already in the case where only distinct elements are allowed.…”

LTL with the Freeze Quantifier and Register Automata

“…Interestingly, the same general mechanism is central to the notion of register automata [14,26,6,21], which recognise words over infinite alphabets. Indeed, a letter can be stored in a register and tested later against the current letter.…”

“…In a data word, at each index, there is a letter from a finite alphabet Σ, and an element of an infinite domain D. As in [14,26,21,7,4,8,17], elements of D can only be compared for equality, so it is equivalent and simpler to define a data word as a word over Σ equipped with an equivalence relation on its indices: i ∼ j iff the elements of D at indices i and j are equal. In common with [7,4], we take this latter approach.…”