On models of a nondeterministic computation

In this paper we consider a nondeterministic computation by deterministic multi-head 2-way automata having a read-only access to an auxiliary memory. The memory contains additional data (a guess) and computation is successful iff it is successful for some memory content. Also we consider the case of restricted guesses in which a guess should satisfy some constraint. We show that the standard complexity classes such as L, NL, P, NP, PSPACE can be characterized in terms of these models of nondeterministic computation. These characterizations differ from the well-known ones by absence of alternation.

💡 Research Summary

The paper introduces a novel way to model nondeterministic computation using deterministic multi‑head two‑way automata equipped with a read‑only auxiliary memory. Instead of the traditional definition where nondeterminism is expressed by a transition relation that may branch, the authors shift the source of nondeterminism to a “guess” stored in an auxiliary memory. The automaton operates deterministically on the input, but it may succeed for some memory content; acceptance is defined as the existence of a memory content that leads to an accepting state.

The auxiliary memory is formalized as a directed graph (M, E) with a distinguished initial cell m₀ and a marking function g that assigns a unique label from a finite set G to each outgoing edge of a vertex. A memory content is a labeling μ: M → Δ where Δ is a finite alphabet. The automaton has h heads that read the input, a finite set of states Q, and a transition function δ that, given the current state, the symbols under the heads, and the symbol in the current memory cell, produces a new state, head movements (−1, 0, +1), and a command to move to a neighboring memory cell according to the label g.

Three concrete memory topologies are examined:

1. 1‑way tape (W₁) – a linear chain that can be read only forward. The nondeterministic automaton can use the single symbol read from the guess tape to make a nondeterministic choice, which exactly captures NL. Deterministic automata ignore the guess tape, yielding L. Hence W₁‑NF A = NL and W₁‑DF A = L.

2. 2‑way tape (W₂) – a bidirectional infinite tape where edges are labeled ‘+’ (right) and ‘−’ (left). This model is essentially equivalent to nondeterministic stack automata (NENSA) and recognizes PSPACE. Both nondeterministic and deterministic versions have the same power: W₂‑NF A = W₂‑DF A = PSPACE.

3. 1.5‑way tape (W₁.₅) – a 1‑way tape with an additional “return‑to‑origin” move. The authors prove two central theorems:

   • Deterministic W₁.₅‑automata recognize exactly P. The proof relies on a bound on the number of returns (≤ |Q|) and on the fact that between two returns the automaton can visit at most n·k·|Q| new cells (n = input length, k = number of heads). Consequently the total number of steps is polynomial, and a polynomial‑time Turing machine can simulate the automaton.
   • Nondeterministic W₁.₅‑automata recognize PSPACE. The inclusion W₁.₅‑NF A ⊆ W₂‑NF A = PSPACE is immediate. For the reverse direction, the automaton stores a polynomial‑size computation history of a space‑bounded Turing machine on its guess tape, and verifies its correctness using modular arithmetic and the Chinese Remainder Theorem, all doable in logarithmic space.

Beyond unrestricted guesses, the paper studies restricted guesses, focusing on sparse guesses: the guess tape contains at most one (or finitely many) non‑blank symbols. Under this restriction, nondeterministic automata capture exactly NP, i.e., M‑NF A(sparse) = NP. This mirrors the classic certificate view of NP but is expressed purely in terms of the auxiliary memory.

The authors also discuss a deterministic variant called WORM‑memory (Write‑Once‑Read‑Many). A deterministic automaton with WORM‑memory can be simulated by a nondeterministic automaton (Lemma 1), showing that the deterministic class is a subset of the nondeterministic one.

Overall, the paper achieves a characterization of the classic complexity classes L, NL, P, NP, and PSPACE without invoking alternation. The key parameters are the structure of the auxiliary memory (fan‑out, directionality, and return capability) and the restriction on the guess content. By demonstrating that modest changes to the memory model (e.g., adding a single “return” edge) elevate deterministic power from L to P, and that allowing nondeterministic access to a 1.5‑way tape yields PSPACE, the work provides a fresh perspective on the relationship between memory access patterns and computational complexity. It suggests that many complexity‑theoretic separations can be understood in terms of simple automata‑theoretic resources rather than the more abstract notion of alternation.