A type system for PSPACE derived from light linear logic

We present a polymorphic type system for lambda calculus ensuring that well-typed programs can be executed in polynomial space: dual light affine logic with booleans (DLALB). To build DLALB we start from DLAL (which has a simple type language with a linear and an intuitionistic type arrow, as well as one modality) which characterizes FPTIME functions. In order to extend its expressiveness we add two boolean constants and a conditional constructor in the same way as with the system STAB. We show that the value of a well-typed term can be computed by an alternating machine in polynomial time, thus such a term represents a program of PSPACE (given that PSPACE = APTIME). We also prove that all polynomial space decision functions can be represented in DLALB. Therefore DLALB characterizes PSPACE predicates.

💡 Research Summary

The paper introduces DLALB (Dual Light Affine Logic with Booleans), a polymorphic type system for the λ‑calculus that precisely captures the class PSPACE. DLALB builds on Dual Light Affine Logic (DLAL), a system originally designed to characterize FPTIME functions. DLAL’s type language distinguishes a linear arrow (⊸) from an intuitionistic arrow (→) and employs a single modality (!). These features enforce strict linear use of resources, guaranteeing that any well‑typed term can be evaluated within polynomial time.

To raise the expressive power to PSPACE, the authors extend DLAL with two boolean constants (true and false) and a conditional constructor (if‑then‑else), following the approach of the STAB system. The new typing rules for booleans and the conditional are crafted so that the condition does not consume linear resources and both branches must be typable under the same linear context. This preserves the linear discipline while allowing genuine control flow.

The core technical contribution is a two‑directional complexity characterization. First, the authors show that the value of any DLALB‑typed term can be computed by an alternating Turing machine (ATM) in polynomial time. They translate the operational semantics of a DLALB term into an ATM computation, where each function application, abstraction, and conditional corresponds to a bounded number of alternating steps. Because DLAL’s linear constraints bound the depth of the call stack and the number of active !‑boxes, the ATM’s space consumption remains polynomial. Since PSPACE = APTIME, this establishes that every DLALB program runs within PSPACE.

Second, they prove completeness: every decision problem in PSPACE can be represented by a DLALB term. The construction starts from a PSPACE‑complete problem such as Quantified Boolean Formula (QBF). Quantifiers are encoded as alternating choices in the term, and the evaluation of the quantified formula is expressed using the boolean constants and the conditional. The linear typing ensures that each quantifier level reuses resources without duplication, mirroring the alternating structure of an ATM. Consequently, any PSPACE predicate can be compiled into a well‑typed DLALB term.

The paper also discusses the implications of this result. DLALB demonstrates that a type system can serve as a static guarantee of polynomial‑space consumption, extending the line of research that links logical systems (especially variants of linear logic) with computational complexity classes. The addition of booleans and conditionals shows that modest extensions to a core linear logic can dramatically increase its expressive power while preserving tractable type checking. The authors note that further extensions (e.g., richer data structures, recursion schemes) would require additional modalities or type constructors, and they outline future work on type inference algorithms and practical language design.

In summary, DLALB provides a clean, polymorphic typing discipline that enforces PSPACE bounds on λ‑terms, proves both soundness (well‑typed terms are in PSPACE) and completeness (all PSPACE predicates are typable), and thereby offers a logical characterization of PSPACE analogous to DLAL’s characterization of FPTIME.