Although demonstrating remarkable performance on reasoning tasks, Large Language Models (LLMs) still tend to fabricate unreliable responses when confronted with problems that are unsolvable or beyond their capability, severely undermining the reliability. Prior studies of LLM reliability have primarily focused on knowledge tasks to identify unanswerable questions, while mathematical reasoning tasks have remained unexplored due to the dearth of unsolvable math problems. To systematically investigate LLM reliability in mathematical reasoning tasks, we formulate the reliability evaluation for both solvable and unsolvable problems. We then develop a ReliableMath dataset which incorporates open-source solvable problems and high-quality unsolvable problems synthesized by our proposed construction workflow with human evaluations. Experiments are conducted on various LLMs with several key findings uncovered. LLMs fail to directly identify unsolvable problems and always generate fabricated responses. When instructing LLMs to indicate unsolvability using a reliable prompt, the reliability of larger-sized LLMs remains on solvable problems, but notably improves on unsolvable problems yet still falls short of solvable problems. However, small LLMs rarely show any progress despite employing reliable prompts. Therefore, we further propose an alignment strategy to enhance small LLMs' reliability, which can significantly improve LLM reliability performances on both in-domain and out-of-domain tasks.

Machine learning evolved from left to right as shown in the above diagram. Initially, researchers started out with Supervised Learning. This is the case of housing price prediction discussed earlier. This was followed by unsupervised learning, where the machine is made to learn on its own without any supervision. Scientists discovered further that it may be a good idea to reward the machine when it does the job the expected way and there came the Reinforcement Learning.

Artificial intelligence (AI) is trending globally in commerce, science, health care, geopolitics, and more areas. Deep learning, a subset of machine learning, is the lever that launched the worldwide rush--an area of strategic interest for researchers, scientists, visionary CEOs, academics, geopolitical think tanks, pioneering entrepreneurs, astute venture capitalists, strategy consultants, and management executives from companies of all sizes. Yet in the midst of this AI renaissance, is a relatively fundamental unsolvable problem with machine learning that is not commonly known, nor frequently discussed outside of the small cadre of philosophers, and artificial intelligence experts. A global research team of researchers have recently demonstrated that machine learning has an unsolvable problem, and published their findings in Nature Machine Intelligence in January 2019. Researchers from Princeton University, the University of Waterloo, Technion-IIT, Tel Aviv University, and the Institute of Mathematics of the Academy of Sciences of the Czech Republic, proved that AI learnability cannot be proved nor refuted when using the standard axioms of mathematics.

Austrian mathematician Kurt Gödel is known for his'incompleteness' theorems.Credit: Alfred Eisenstaedt/ LIFE Picture Coll./Getty A team of researchers has stumbled on a question that is mathematically unanswerable because it is linked to logical paradoxes discovered by Austrian mathematician Kurt Gödel in the 1930s that can't be solved using standard mathematics. The mathematicians, who were working on a machine-learning problem, show that the question of'learnability' -- whether an algorithm can extract a pattern from limited data -- is linked to a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. The latest result appeared on 7 January in Nature Machine Intelligence1.

Have you ever wondered: What exactly is the device that you are reading this article on? Computational science dates back to a time long before these modern computing devices were even imagined. In an industry where the more frequently asked questions revolve around programming languages, frameworks, and libraries, we often taken for granted the fundamental concepts that make a computer tick. But these computers, which seem to possess endless potential--do they have any limitations? Are there problems that computers cannot be used to solve? In this article, we will address these questions by stepping away from the particulars of programming languages and computer architectures. By understanding the power and limitations of computers and algorithms, we can improve the way we think and better reason about different strategies. The abstract view of computing produces results that have stood the test of time, being as valuable to us today as they were when initially developed in the 1970s.

This paper investigates how readily an unsolvable constraint satisfaction problem can be reformulated so that it becomes solvable. We investigate small changes in the definitions of the problemís constraints, changes that alter neither the structure of its constraint graph nor the tightness of its constraints. Our results show that structured and unstructured problems respond differently to such changes, as do easy and difficult problems taken from the same problem class. Several plausible explanations for this behavior are discussed.

If you need to you can always contact us, otherwise we'll be back online soon.