Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · cs/0612085

The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems

Roberto Bagnara·Patricia M. Hill·Enea Zaffanella
View original source ↗Read source document ↗Export metadata ↓

Abstract

Since its inception as a student project in 2001, initially just for the handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library has been continuously improved and extended by joining scrupulous research on the theoretical foundations of (possibly non-convex) numerical abstractions to a total adherence to the best available practices in software development. Even though it is still not fully mature and functionally complete, the Parma Polyhedra Library already offers a combination of functionality, reliability, usability and performance that is not matched by similar, freely available libraries. In this paper, we present the main features of the current version of the library, emphasizing those that distinguish it from other similar libraries and those that are important for applications in the field of analysis and verification of hardware and software systems.

Explore related subjects

cs.MScs.PL

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Roberto Bagnara, Patricia M. Hill, Enea Zaffanella. 2006-12-18. The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems. https://arxiv.org/abs/cs/0612085

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

AWE: Adaptive Weight Encoding for Exact Integer Matrix Products with Fewer GEMMs on FP4 Tensor Cores

Emulation of high-accuracy floating-point matrix multiplication, as in the Ozaki scheme, splits the inputs into low-precision components and multiplies them pairwise. These products must be error-free, and each is an integer matrix product times a scale factor. FP4 Tensor Cores are the fastest on the NVIDIA B200 and B300 but cannot hold INT8 operands. The FP4 values scaled by 2 form the set $S = \{0, \pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12\}$, which contains every residue modulo 13, so with carries any integer splits into base-13 digits that FP4 can store. Prior work splits each INT8 operand into 3 such digits (limbs) with weights $(1, 13, 169)$ and multiplies them pairwise, 9 FP4 matrix multiplications (GEMMs) for INT8$\times$INT8. The classical ways to reduce products, such as the Karatsuba and Toom--Cook methods, do not apply as they stand: sums of limbs reach $\pm 24$ and leave $S$. This paper asks how many FP4 GEMMs are needed for one integer matrix product. We propose Adaptive Weight Encoding (AWE): the limbs take freely chosen integer weights, the stored planes are linear combinations of limbs, and the exact product is the sum of the FP4 GEMMs scaled by reconstruction coefficients. For each input range, we searched these choices for encodings with fewer products and found INT8$\times$INT8 in 6 products and INT4$\times$INT8 in 4. The formulation also holds modulo $m$, which covers the residue number systems of Ozaki scheme II: for the FP64 significand, the 75 products of prior work are reduced to 59. The boundary in product count between encodings with and without residues lies near input width 15. We release the encodings found.

cs.MS↗
arXiv2026

The Pauli Lightcone: Information-Theoretic Error Mitigation Beyond the Autocorrelation

We introduce the wavemap: a spatial portrait of noise effects that assigns each site a per-noise-level arrival delay l_γ(v) and cross-entropy loss L_γ(v). These observables are exact at the lightcone frontier, where bond dimension χis small and the simulation is most faithful. Eigenvalue analysis of the composed gate-plus-noise Pauli transfer matrices confirms that the studied noise is pure amplitude damping: the spatial propagation pattern is entirely determined by the gate, making the wavemap a model-free noise diagnostic. We apply the multi-product formula (MPF) to recover the noiseless Pauli weight field from the noisy samples, subject to the Lieb-Robinson causal constraint nMPF <= nnl . Fitting time-adaptive coefficients α(t) over the frontier recovers up to 55% of the information loss relative to the best noisy sample, exploiting the fact that the frontier is where truncation error is smallest. On an IBM heavy-hex lattice with heterogeneous hardware noise the method identifies an information-starved regime, pointing to calibrated synthetic noise as the next required experiment.

cs.MS↗
arXiv2026

Ozaki Scheme II: A GEMM-oriented emulation of floating-point matrix multiplication using an integer modular technique

This paper addresses emulation algorithms for matrix multiplication. General Matrix-Matrix Multiplication (GEMM), a fundamental operation in the Basic Linear Algebra Subprograms (BLAS), is typically optimized for specific hardware architectures. The Ozaki scheme is a well-established GEMM-based emulation method for matrix multiplication, wherein input matrices are decomposed into several low-precision components to ensure that the resulting matrix product is computed exactly through numerical operations. This study proposes a novel GEMM-based emulation method for matrix multiplication that leverages the Chinese Remainder Theorem. The proposed method inherits the computational efficiency of highly optimized GEMM routines and further enables control over the number of matrix multiplications, which can enhance computational accuracy. We present numerical experiments featuring INT8 Tensor Core operations on GPUs and FP64 arithmetic on CPUs as case studies. The results demonstrate that FP64 emulation using the proposed method achieves performance levels of up to 7.4 to 9.8 TFLOPS on the NVIDIA RTX 4090 and 56.6 to 80.2 TFLOPS on the NVIDIA GH200, exceeding the measured performance of native FP64 arithmetic. Furthermore, for FP64 computations on CPUs, the proposed method achieved up to a 2.3x speedup in emulating quadruple-precision arithmetic compared to the conventional Ozaki scheme.

cs.MS↗