Some conditions implying if P=NP then P=PSPACE

We identify a few conditions $X$ such that $(P=NP \wedge X) ;\Rightarrow; P=PSPACE$.

💡 Research Summary

The paper investigates a family of sufficient conditions under which the equality P = NP would automatically imply P = PSPACE. The author begins by recalling the well‑known inclusions P ⊆ NP ⊆ PSPACE and observes that while neither inclusion is known to be strict, the hypothesis P = NP is generally considered weaker than P = PSPACE. The goal, therefore, is to identify concrete properties X such that (P = NP ∧ X) ⇒ P = PSPACE.

In Section 2 the author formalises the computational model. A deterministic Turing machine M = (S, Σ, Γ, s₁, s_A, Δ) is used, with configurations encoded as binary strings via a coding function σ. For any PSPACE‑complete decision problem Q there exists a deterministic machine M′ that solves Q using space T(n) for some polynomial T. By adding a trivial “erase‑and‑stop” wrapper, the author obtains a machine M that has a unique accepting configuration c_A.

The decision procedure for Q is split into two steps:

1. From M and an input x construct a finite Boolean function f_{M,x} that tells, for any configuration y, whether M can reach y from the initial configuration σ(x). The domain of f_{M,x} consists of all configurations whose size is bounded by the space bound T(n), so its size is at most 2^{T(n)} = O(2^{n^k}) for some constant k.

2. Evaluate f_{M,x} on the accepting configuration c_A; output “yes” iff f_{M,x}(c_A)=1.

If both steps can be performed in polynomial time, then Q (and therefore every PSPACE problem) would be solvable in P. The crux is therefore whether step 1 can be carried out efficiently.

To capture this, three conditions are introduced:

(a) There exists a polynomial R such that step 1 can be executed in time R(n).

(b) There exists a polynomial P such that the constructed computation device runs in time P(n) on every configuration whose size is bounded by T(n).

(c) There exist polynomials D and P such that for every input x there is a computation device of size ≤ D(n) that satisfies (b).

Condition (c) is the strongest; the author shows that if (c) holds and P = NP, then both (a) and (b) follow automatically. The argument proceeds by defining an auxiliary predicate “chained_P(M, x, f, c)”. This predicate checks whether a candidate Boolean function f is locally consistent with the transition relation of M at configuration c: if f(c)=1 then f must also be 1 on at least one predecessor of c (unless c is the initial configuration) and on the unique successor of c (unless c is the accepting configuration). The predicate can be evaluated in polynomial time because the number of predecessors is bounded by a constant (the transition function is fixed) and the size of each configuration is bounded by T(n).

The author proves that any function f that satisfies chained_P for all configurations of size ≤ T(n) can be used in step 2: even if f contains “spurious cycles” (sets of configurations that are mutually consistent but never actually visited by M), the value of f on c_A correctly reflects whether M accepts x. Consequently, the original condition (c) can be weakened to (c′): there exist polynomials D and P and a function f of size ≤ D(n) such that (i) f satisfies chained_P for all relevant configurations, and (ii) f runs in time P(n) on those configurations.

Two global conditions are then defined:

A : For every deterministic machine M that runs in polynomial space T(n), condition (c) holds.

A′ : For every such M, condition (c′) holds.

Since (c) ⇒ (c′), A ⇒ A′. The paper shows that (P = NP ∧ A′) ⇒ P = PSPACE. The proof introduces three languages:

• G_{P,T} = { (M, x, f) | ∃c (|c| ≤ T(n) ∧ chained_P(M, x, f, c)=0) }.
  This language is in NP because a nondeterministic machine can guess c and verify the predicate in polynomial time. Assuming P = NP, G_{P,T} ∈ P.

• H_{P,T} = { (M, x, f, s) | (M, x, f) ∈ G_{P,T} ∧ representation(f) <lex s }.
  Since G{P,T} ∈ P and the lexicographic comparison is trivial, H_{P,T} ∈ P.

• W_{P,D,T} = { (M, x, s) | ∃f (|f| ≤ D(n) ∧ (M, x, f, s) ∈ H_{P,T}) }.
  This language is in NP (guess f, verify H_{P,T}); again P = NP yields W_{P,D,T} ∈ P.

Using W_{P,D,T}, the author describes a binary‑search procedure that, given M and x, finds a suitable f in polynomial time: repeatedly query whether there exists an f whose representation is ≤ s, adjusting s according to the answer. Because the number of queries is polynomial and each query is answered in polynomial time, the whole search runs in polynomial time. Once such an f is found, evaluating f(c_A) decides membership in Q within polynomial time. Hence any PSPACE problem can be solved in P under (P = NP ∧ A′).

Section 3 discusses the plausibility of A′. The author admits that the arguments are “vague” and that A′ would be trivially true if either of the two sub‑conditions in (c′) were dropped. No concrete construction of the required f is provided; the feasibility remains an open question.

Section 4 sketches three possible approaches to prove A′:

(I) Augment configurations with a step counter to eliminate spurious cycles, ensuring that chained_P enforces a strict linear order.

(II) Modify M so that after reaching the accepting configuration it rewrites the tape with the initial configuration and returns to the start state, turning the accepting path into a cycle; this forces any valid f to represent a set of cycles that includes the accepting configuration.

(III) Use the languages G, H, and W defined earlier to perform a binary search for f, as already described.

Sections 5 and 6 propose additional conditions and note a curious consequence of the first condition, but these are presented only at a high level without formal statements or proofs.

The conclusion reiterates that the paper has identified several sufficient conditions (A, A′, and variants) under which P = NP would imply P = PSPACE, and calls for future work to either prove or refute these conditions.

Overall, the contribution is conceptual: it frames the implication “P = NP ⇒ P = PSPACE” as contingent on the existence of compact, efficiently‑verifiable representations of the configuration‑reachability relation of polynomial‑space machines. The technical machinery (chained_P, the three languages, binary search) is sound under the assumption P = NP, but the central open problem—whether such compact representations actually exist for all polynomial‑space machines—remains unresolved. Consequently, the paper’s results are conditional and speculative, offering a new perspective but no definitive progress on the long‑standing P vs. NP vs. PSPACE questions.