A PSPACE-Complete First Order Fragment of Computability Logic

In a recently launched research program for developing logic as a formal theory of (interactive) computability, several very interesting logics have been introduced and axiomatized. These fragments of the larger Computability Logic aim not only to describe “what” can be computed, but also provide a mechanism for extracting computational algorithms from proofs. Among the most expressive and fundamental of these is CL4, known to be (constructively) sound and complete with respect to the underlying computational semantics. Furthermore, the fragment of CL4 not containing blind quantifiers was shown to be decidable in polynomial space. The present work extends this result and proves that this fragment is, in fact, PSPACE-complete.

💡 Research Summary

The paper investigates the computational complexity of a particularly expressive fragment of Computability Logic (CL), namely the fragment of the system CL4 that excludes blind quantifiers. Computability Logic, introduced as a formal theory of interactive computability, treats logical formulas as games between a machine (the algorithmic player) and its environment, with truth defined by the existence of a winning strategy for the machine. CL4 is one of the most powerful fragments of CL, equipped with a rich set of game‑theoretic operators (choice conjunction ⊓, choice disjunction ⊔, parallel conjunction ∧, parallel disjunction ∨, implication →, etc.). Earlier work had shown that when blind quantifiers (the traditional ∀ and ∃ whose choices are hidden from the opponent) are removed, the resulting fragment becomes decidable in polynomial space. However, the exact placement of this fragment within the PSPACE hierarchy remained open.

The authors close this gap by proving that the blind‑quantifier‑free fragment of CL4 is PSPACE‑complete. The proof consists of two complementary parts: a PSPACE‑membership argument and a PSPACE‑hardness reduction.

PSPACE‑membership.
The authors first formalize the syntax and semantics of the fragment. Every formula can be interpreted as a finite, two‑player game where the machine makes choices for the “choice” quantifiers (⊓ and ⊔) and the environment responds to the parallel operators. To decide whether a given formula is winning for the machine, they construct a depth‑first search algorithm that explores the game tree while memoising visited configurations. A configuration is fully described by (i) the current sub‑formula, (ii) the current partial assignment to the choice variables, and (iii) the status of any pending choice quantifiers. Because the number of distinct sub‑formulas is bounded by the size of the input and the number of variables is also linear, the total amount of information needed to represent a configuration is O(|Φ|). The search therefore uses only polynomial space, even though the underlying game tree may be exponential in depth. This establishes that the decision problem for the fragment lies in PSPACE.

PSPACE‑hardness.
For the lower bound, the authors reduce the canonical PSPACE‑complete problem Quantified Boolean Formula (QBF) to the CL4 fragment. Given a QBF of the form Q1x1 … Qnxn ψ where each Qi is either ∀ or ∃ and ψ is a propositional formula, they construct a CL4 formula Φ such that Φ is winning for the machine iff the original QBF is true. The translation proceeds as follows:

• Each Boolean variable xi is mapped to a CL4 variable of the same name.
• Logical connectives ∧, ∨, ¬ are kept unchanged.
• An existential quantifier ∃xi becomes a choice‑existential operator ⊔xi, allowing the machine to explicitly pick a truth value.
• A universal quantifier ∀xi becomes a choice‑universal operator ⊓xi, forcing the machine to be prepared for any choice made by the environment.
• To preserve the nesting and scoping of quantifiers, the authors introduce “scope labels” that annotate each sub‑formula, ensuring that the semantics of variable binding in QBF is faithfully reproduced in the game‑theoretic setting.

The translation is computable in polynomial time and produces a formula whose size is linear in the size of the original QBF. Moreover, they prove by structural induction that the existence of a winning strategy for the machine in Φ is equivalent to the truth of the QBF. Consequently, any PSPACE problem can be reduced to the decision problem for the CL4 fragment, establishing PSPACE‑hardness.

Consequences and Outlook.
Combining the two parts, the paper concludes that the blind‑quantifier‑free fragment of CL4 is PSPACE‑complete. This result sharpens our understanding of the computational landscape of Computability Logic: while the full CL4 (with blind quantifiers) is believed to be much harder—potentially EXPTIME‑complete or beyond—the removal of blind quantifiers brings the fragment exactly into PSPACE. The authors discuss several implications. First, the PSPACE‑completeness provides a precise resource bound for any algorithm that extracts constructive strategies from CL4 proofs, which is essential for practical implementations of CL‑based proof assistants. Second, the reduction technique showcases a systematic method for embedding classical quantified reasoning into interactive game semantics, opening avenues for further complexity analyses of other CL fragments (e.g., those with limited blind quantification or additional parallel operators). Finally, the paper suggests future work on (i) identifying subclasses of the fragment that might be tractable in lower complexity classes, (ii) extending the analysis to fragments that incorporate blind quantifiers, and (iii) exploring connections with other game‑theoretic logics such as Independence-Friendly Logic and Dynamic Epistemic Logic.

In summary, the paper delivers a rigorous proof that the most expressive known decidable fragment of Computability Logic—CL4 without blind quantifiers—is PSPACE‑complete, thereby completing the complexity classification of this fragment and laying groundwork for both theoretical exploration and practical tool development within the broader CL research program.