2,196 research outputs found
Non-linear Real Arithmetic Benchmarks derived from Automated Reasoning in Economics
We consider problems originating in economics that may be solved
automatically using mathematical software. We present and make freely available
a new benchmark set of such problems. The problems have been shown to fall
within the framework of non-linear real arithmetic, and so are in theory
soluble via Quantifier Elimination (QE) technology as usually implemented in
computer algebra systems. Further, they all can be phrased in prenex normal
form with only existential quantifiers and so are also admissible to those
Satisfiability Module Theory (SMT) solvers that support the QF_NRA. There is a
great body of work considering QE and SMT application in science and
engineering, but we demonstrate here that there is potential for this
technology also in the social sciences.Comment: To appear in Proc. SC-Square 2018. Dataset described is hosted by
Zenodo at: https://doi.org/10.5281/zenodo.1226892 . arXiv admin note:
substantial text overlap with arXiv:1804.1003
Local Search For SMT On Linear and Multilinear Real Arithmetic
Satisfiability Modulo Theories (SMT) has significant application in various
domains. In this paper, we focus on quantifier-free Satisfiablity Modulo Real
Arithmetic, referred to as SMT(RA), including both linear and non-linear real
arithmetic theories. As for non-linear real arithmetic theory, we focus on one
of its important fragments where the atomic constraints are multi-linear. We
propose the first local search algorithm for SMT(RA), called LocalSMT(RA),
based on two novel ideas. First, an interval-based operator is proposed to
cooperate with the traditional local search operator by considering the
interval information. Moreover, we propose a tie-breaking mechanism to further
evaluate the operations when the operations are indistinguishable according to
the score function. Experiments are conducted to evaluate LocalSMT(RA) on
benchmarks from SMT-LIB. The results show that LocalSMT(RA) is competitive with
the state-of-the-art SMT solvers, and performs particularly well on
multi-linear instances
Algorithmically generating new algebraic features of polynomial systems for machine learning
There are a variety of choices to be made in both computer algebra systems
(CASs) and satisfiability modulo theory (SMT) solvers which can impact
performance without affecting mathematical correctness. Such choices are
candidates for machine learning (ML) approaches, however, there are
difficulties in applying standard ML techniques, such as the efficient
identification of ML features from input data which is typically a polynomial
system. Our focus is selecting the variable ordering for cylindrical algebraic
decomposition (CAD), an important algorithm implemented in several CASs, and
now also SMT-solvers. We created a framework to describe all the previously
identified ML features for the problem and then enumerated all options in this
framework to automatically generation many more features. We validate the
usefulness of these with an experiment which shows that an ML choice for CAD
variable ordering is superior to those made by human created heuristics, and
further improved with these additional features. We expect that this technique
of feature generation could be useful for other choices related to CAD, or even
choices for other algorithms with polynomial systems for input.Comment: To appear in Proc SC-Square Workshop 2019. arXiv admin note:
substantial text overlap with arXiv:1904.1106
TheoryGuru: A Mathematica Package to Apply Quantifier Elimination Technology to Economics
We consider the use of Quantifier Elimination (QE) technology for automated
reasoning in economics. There is a great body of work considering QE
applications in science and engineering but we demonstrate here that it also
has use in the social sciences. We explain how many suggested theorems in
economics could either be proven, or even have their hypotheses shown to be
inconsistent, automatically via QE.
However, economists who this technology could benefit are usually unfamiliar
with QE, and the use of mathematical software generally. This motivated the
development of a Mathematica Package TheoryGuru, whose purpose is to lower the
costs of applying QE to economics. We describe the package's functionality and
give examples of its use.Comment: To appear in Proc ICMS 201
Tools and Algorithms for the Construction and Analysis of Systems
This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers
Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition
There has been recent interest in the use of machine learning (ML) approaches
within mathematical software to make choices that impact on the computing
performance without affecting the mathematical correctness of the result. We
address the problem of selecting the variable ordering for cylindrical
algebraic decomposition (CAD), an important algorithm in Symbolic Computation.
Prior work to apply ML on this problem implemented a Support Vector Machine
(SVM) to select between three existing human-made heuristics, which did better
than anyone heuristic alone. The present work extends to have ML select the
variable ordering directly, and to try a wider variety of ML techniques.
We experimented with the NLSAT dataset and the Regular Chains Library CAD
function for Maple 2018. For each problem, the variable ordering leading to the
shortest computing time was selected as the target class for ML. Features were
generated from the polynomial input and used to train the following ML models:
k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision
tree (DT) and SVM, as implemented in the Python scikit-learn package. We also
compared these with the two leading human constructed heuristics for the
problem: Brown's heuristic and sotd. On this dataset all of the ML approaches
outperformed the human made heuristics, some by a large margin.Comment: Accepted into CICM 201
ARB: Advanced Reasoning Benchmark for Large Language Models
Large Language Models (LLMs) have demonstrated remarkable performance on
various quantitative reasoning and knowledge benchmarks. However, many of these
benchmarks are losing utility as LLMs get increasingly high scores, despite not
yet reaching expert performance in these domains. We introduce ARB, a novel
benchmark composed of advanced reasoning problems in multiple fields. ARB
presents a more challenging test than prior benchmarks, featuring problems in
mathematics, physics, biology, chemistry, and law. As a subset of ARB, we
introduce a challenging set of math and physics problems which require advanced
symbolic reasoning and domain knowledge. We evaluate recent models such as
GPT-4 and Claude on ARB and demonstrate that current models score well below
50% on more demanding tasks. In order to improve both automatic and assisted
evaluation capabilities, we introduce a rubric-based evaluation approach,
allowing GPT-4 to score its own intermediate reasoning steps. Further, we
conduct a human evaluation of the symbolic subset of ARB, finding promising
agreement between annotators and GPT-4 rubric evaluation scores.Comment: Submitted to NeurIPS Datasets and Benchmarks Trac
Proceedings of the 21st Conference on Formal Methods in Computer-Aided Design – FMCAD 2021
The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing
Tools and Algorithms for the Construction and Analysis of Systems
This open access book constitutes the proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2022, which was held during April 2-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 46 full papers and 4 short papers presented in this volume were carefully reviewed and selected from 159 submissions. The proceedings also contain 16 tool papers of the affiliated competition SV-Comp and 1 paper consisting of the competition report. TACAS is a forum for researchers, developers, and users interested in rigorously based tools and algorithms for the construction and analysis of systems. The conference aims to bridge the gaps between different communities with this common interest and to support them in their quest to improve the utility, reliability, exibility, and efficiency of tools and algorithms for building computer-controlled systems
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