We show that descriptive complexity can serve as a foundation for formalizing computational complexity results in a proof assistant, by constructing a Lean library centered around the following concepts: decision problems are isomorphism-invariant predicates on finite structures; complexity classes are defined by their logical characterization; membership is shown by definability witnesses; hardness is shown by first-order reductions from a known hard problem. We also establish bridges to traditional machine models such as (non)deterministic Turing machines. The library proves 73 completeness results, on 68 problems or problem families, over 14 different classes; relations between the classes established inside the logic and not by machine simulation, among them NL = coNL and the Abiteboul-Vianu theorem; and unconditional lower bounds, among them $\mathrm{FO}(\leq) \subsetneq \mathrm{FO}(\leq, \mathrm{TC})$ and the failure of order-free FO(IFP) to capture PTIME.
The Ideal Membership Problem (IMP) asks whether a polynomial f belongs to an idealof Q[x_1, ..., x_n]. Polynomial Calculus (PC) certifies membership by deriving f from the generators, and a degree-d derivation needs at most n^O(d) steps. We write PC-IMPd for the problem of producing a degree-bounded PC certificate, and...
This work provides a categorical treatment of Canetti's Universal Composability (UC) framework for systems with a static number of parties and sessions, often termed UC for static systems, yielding four benefits, including graphically verified formulation of the composition theorem.
P. Farshim, M. Karvonen, Andre Knispel et al.· IACR Cryptology ePrint Archi...· 0 citations
It is proved that Medvedev logic ML, the intermediate logic of finite problems, is Pi-01-complete under computable many-one reductions, and is not recursively enumerable, a fortiori undecidable, and admits no recursively enumerable sound and complete proof calculus.
We present a domain-theoretical formalization of interaction trees in the Rocq prover. Unlike existing formalizations, ours does not rely on Rocq's built-in coinduction. Hence, we avoid complications occurring in earlier works, such as artificially including silent steps to comply with Rocq's productivity checker, trea...
David Nowak, Vlad Rusu· Electronic Proceedings in Th...· 0 citations
We study the reverse-mathematical strength of frame definability in modal logic. The central principle is the Valuation Extension Lemma (VEL), which asserts that every assignment of propositional variables on a frame extends to a full valuation. We show that, over $\mathrm{RCA}_0$, VEL is equivalent to $\mathrm{ACA}^{+...
For graphoid automata, these results give polynomial-time recognition without a graph-width restriction, effective boundary composition, and comparison of finite graph relations, and the quadratic boundary bounds are optimal in the worst case.
Antonios Kalampakas· 0 citations
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