The computational complexity of the unconstrained limited domino problem (with implications for logical decision problems)
The unconstrained domino or tiling problem is the following. Given a finite set T (of tiles), sets H,V ⊑ T×T and a cardinal k ≤ ω, does there exist a function :k×k->T such that (t(i,j),t(i+1,j))εH and (t(i,j),t(i,j+1))εV for all i,j<k ?
The unlimited domino problem (k infinite) has played an important role in the process of finding the undecidability proof for the ∀ε∀ prefix class of predicate calculus. The limited domino problem is similarly connected to some decidable prefix classes.
The limited domino problem is NP-complete, if k is given in unary. The same was conjectured for k given in binary, but we show an Θ (c n ) nondeterministic time lower bound (upper bound O(d n )). As a consequence, the non-deterministic time complexity of the decision problem for the Вε ВВ class of predicate calculus lies between Θ (c n/log n ) and O(d n/log n ).
KeywordsFiling System Turing Machine Tiling System Binary Number Origin Constraint
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