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State Complexity Characterizations of Parameterized Degree-Bounded Graph Connectivity, Sub-Linear Space Computation, and the Linear Space Hypothesis

The linear space hypothesis is a practical working hypothesis, which originally states the insolvability of a restricted 2CNF Boolean formula satisfiability problem parameterized by the number of Boolean variables. From this hypothesis, it naturally follows that the degree-3 directed graph connectivity problem (3DSTCON) parameterized by the number of vertices in a given graph cannot belong to PsubLIN, composed of all parameterized decision problems computable by polynomial-time, sub-linear-space deterministic Turing machines. This hypothesis immediately implies L$\neq$NL and it was used as a solid foundation to obtain new lower bounds on the computational complexity of various NL search and NL optimization problems. The state complexity of transformation refers to the cost of converting one type of finite automata to another type, where the cost is measured in terms of the increase of the number of inner states of the converted automata from that of the original automata. We relate the linear space hypothesis to the state complexity of transforming restricted 2-way nondeterministic finite automata to computationally equivalent 2-way alternating finite automata having narrow computation graphs. For this purpose, we present state complexity characterizations of 3DSTCON and PsubLIN. We further characterize a nonuniform version of the linear space hypothesis in terms of the state complexity of transformation.