arXiv subjects

Jiatu Li

Publications and source records attributed to Jiatu Li.

6 recordsLinked to original sources

arXiv2026

A Theory for Probabilistic Polynomial-Time Reasoning

In this work, we propose a new bounded arithmetic theory, denoted $APX_1$, designed to formalize a broad class of probabilistic arguments commonly used in theoretical computer science. Under plausible assumptions, $APX_1$ is strictly weaker than previously proposed frameworks, such as the theory $APC_1$ introduced in the seminal work of Jerabek (2007). From a computational standpoint, $APX_1$ is closely tied to approximate counting and to the central question in derandomization, the prBPP versus prP problem, whereas $APC_1$ is linked to the dual weak pigeonhole principle and to the existence of Boolean functions with exponential circuit complexity. A key motivation for introducing $APX_1$ is that its weaker axioms expose finer proof-theoretic structure, making it a natural setting for several lines of research, including unprovability of complexity conjectures and reverse mathematics of randomized lower bounds. In particular, the framework we develop for $APX_1$ enables the formulation of precise questions concerning the provability of prBPP=prP in deterministic feasible mathematics. Since the (un)provability of P versus NP in bounded arithmetic has long served as a central theme in the field, we expect this line of investigation to be of particular interest. Our technical contributions include developing a comprehensive foundation for probabilistic reasoning from weaker axioms, formalizing non-trivial results from theoretical computer science in $APX_1$, and establishing a tailored witnessing theorem for its provably total TFNP problems. As a byproduct of our analysis of the minimal proof-theoretic strength required to formalize statements arising in theoretical computer science, we resolve an open problem regarding the provability of $AC^0$ lower bounds in $PV_1$, which was considered in earlier works by Razborov (1995), Krajicek (1995), and Muller and Pich (2020).

cs.CC↗

arXiv2025

Diffusion Generative Models Meet Compressed Sensing, with Applications to Imaging and Finance

In this study we develop dimension-reduction techniques to accelerate diffusion model inference in the context of synthetic data generation. The idea is to integrate compressed sensing into diffusion models (hence, CSDM): First, compress the dataset into a latent space (from an ambient space), and train a diffusion model in the latent space; next, apply a compressed sensing algorithm to the samples generated in the latent space for decoding back to the original space; and the goal is to facilitate the efficiency of both model training and inference. Under certain sparsity assumptions on data, our proposed approach achieves provably faster convergence, via combining diffusion model inference with sparse recovery. It also sheds light on the best choice of the latent space dimension. To illustrate the effectiveness of this approach, we run numerical experiments on a range of datasets, including handwritten digits, medical and climate images, and financial time series for stress testing.

stat.ML↗

arXiv2025

Identity Testing for Circuits with Exponentiation Gates

Motivated by practical applications in the design of optimization compilers for neural networks, we initiated the study of identity testing problems for arithmetic circuits augmented with \emph{exponentiation gates} that compute the real function $x\mapsto e^x$. These circuits compute real functions of form $P(\vec x)/P'(\vec x)$, where both $P(\vec x)$ and $P'(\vec x)$ are exponential polynomials \[ \sum_{i=1}^k f_i(\vec x)\cdot \exp\left(\frac{g_i(\vec x)}{h_i(\vec x)}\right), \] for polynomials $f_i(\vec x),g_i(\vec x)$, and $h_i(\vec x)$. We formalize a black-box query model over finite fields for this class of circuits, which is mathematical simple and reflects constraints faced by real-world neural network compilers. We proved that a simple and efficient randomized identity testing algorithm achieves perfect completeness and non-trivial soundness. Concurrent with our work, the algorithm has been implemented in the optimization compiler Mirage by Wu et al.~(OSDI 2025), demonstrating promising empirical performance in both efficiency and soundness error. Finally, we propose a number-theoretic conjecture under which our algorithm is sound with high probability.

cs.CC↗

arXiv2024

On the Unprovability of Circuit Size Bounds in Intuitionistic $\mathsf{S}^1_2$

We show that there is a constant $k$ such that Buss's intuitionistic theory $\mathsf{IS}^1_2$ does not prove that SAT requires co-nondeterministic circuits of size at least $n^k$. To our knowledge, this is the first unconditional unprovability result in bounded arithmetic in the context of worst-case fixed-polynomial size circuit lower bounds. We complement this result by showing that the upper bound $\mathsf{NP} \subseteq \mathsf{coNSIZE}[n^k]$ is unprovable in $\mathsf{IS}^1_2$. In order to establish our main result, we obtain new unconditional lower bounds against refuters that might be of independent interest. In particular, we show that there is no efficient refuter for the lower bound $\mathsf{NP} \nsubseteq \mathsf{i.o.}\text{-}\mathsf{coNP}/\mathsf{poly}$, addressing in part a question raised by Atserias (2006).

cs.LO↗

arXiv2023

Unprovability of Strong Complexity Lower Bounds in Bounded Arithmetic

While there has been progress in establishing the unprovability of complexity statements in lower fragments of bounded arithmetic, understanding the limits of Je\v{r}\'abek's theory $APC_1$ (2007) and of higher levels of Buss's hierarchy $S^i_2$ (1986) has been a more elusive task. Even in the more restricted setting of Cook's theory PV (1975), known results often rely on a less natural formalization that encodes a complexity statement using a collection of sentences instead of a single sentence. This is done to reduce the quantifier complexity of the resulting sentences so that standard witnessing results can be invoked. In this work, we establish unprovability results for stronger theories and for sentences of higher quantifier complexity. In particular, we unconditionally show that $APC_1$ cannot prove strong complexity lower bounds separating the third level of the polynomial hierarchy. In more detail, we consider non-uniform average-case separations, and establish that $APC_1$ cannot prove a sentence stating that $\forall n \ge n_0\;\exists\,f_n \in \Pi_{3}$-$SIZE[n^d]$ that is $(1/n)$-far from every $\Sigma_{3}$-$SIZE[2^{n^{\delta}}]$ circuit. This is a consequence of a much more general result showing that, for every $i \geq 1$, strong separations for $\Pi_{i}$-$SIZE[poly(n)]$ versus $\Sigma_{i}$-$SIZE[2^{n^{\Omega(1)}}]$ cannot be proved in the theory $T_{PV}^i$ consisting of all true $\forall \Sigma^b_{i-1}$-sentences in the language of Cook's theory PV. Our argument employs a convenient game-theoretic witnessing result that can be applied to sentences of arbitrary quantifier complexity. We combine it with extensions of a technique introduced by Kraj\'i\v{c}ek (2011) that was recently employed by Pich and Santhanam (2021) to establish the unprovability of lower bounds in PV (i.e., the case $i=1$ above, but under a weaker formalization) and in a fragment of $APC_1$.

cs.CC↗

arXiv2020

Formalization of PAL$\cdot$S5 in Proof Assistant

As an experiment to the application of proof assistant for logic research, we formalize the model and proof system for multi-agent modal logic S5 with PAL-style dynamic modality in Lean theorem prover. We provide a formal proof for the reduction axiom of public announcement, and the soundness and completeness of modal logic S5, which can be typechecked with Lean 3.19.0. The complete proof is now available at Github.

cs.LO↗