Related papers: From proof theory to theories theory
According to some algorithmicists, algorithmics traditionally uses algorithm theory, which stems from mathematics. The growing need for innovative algorithms has caused increasing gaps between theory and practice. Originally, this motivated…
Mathematical theorems are human knowledge able to be accumulated in the form of symbolic representation, and proving theorems has been considered intelligent behavior. Based on the BHK interpretation and the Curry-Howard isomorphism, proof…
A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into…
The familiar theories of physics have the feature that the application of the theory to make predictions in specific circumstances can be done by means of an algorithm. We propose a more precise formulation of this feature --- one based on…
Recent years have seen the dramatic rise of the usage of AI algorithms in pure mathematics and fundamental sciences such as theoretical physics. This is perhaps counter-intuitive since mathematical sciences require the rigorous definitions,…
Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof…
A general theory of programs, programming and programming languages built up from a few concepts of elementary set theory. Derives, as theorems, properties treated as axioms by classic approaches to programming. Covers sequential and…
Proof theory began in the 1920's as a part of Hilbert's program, which aimed to secure the foundations of mathematics by modeling infinitary mathematics with formal axiomatic systems and proving those systems consistent using restricted,…
Mechanical reasoning is a key area of research that lies at the crossroads of mathematical logic and artificial intelligence. The main aim to develop mechanical reasoning systems (also known as theorem provers) was to enable mathematicians…
Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In…
In logic there is a clear concept of what constitutes a proof and what not. A proof is essentially defined as a finite sequence of formulae which are either axioms or derived by proof rules from formulae earlier in the sequence.…
We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are essentially the same program. The set of…
We lay the groundwork for a formal framework that studies scientific theories and can serve as a unified foundation for the different theories within physics. We define a scientific theory as a set of verifiable statements, assertions that…
We argue how AI can assist mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and theoretical physics in 2017, we summarize how this emerging field has…
A theory graph is a network of axiomatic theories connected with meaning-preserving mappings called theory morphisms. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not…
Although algorithm is one of the central subjects, there have been little common understandings of what an algorithm is. For example, Gurevich view algorithms as abstract state machines, while others view algorithms as recursors. We promote…
Engineering needs mathematics, but the converse is also increasingly evident. Indeed, mathematics is still recovering from the drawbacks of several "reforms". Encouraging is the revived interest in proofs indicated by various recent…
Decision theories offer principled methods for making choices under various types of uncertainty. Algorithms that implement these theories have been successfully applied to a wide range of real-world problems, including materials and drug…
Recent developments show that AI can prove research-level theorems in mathematics, both formally and informally. This essay urges mathematicians to stay up-to-date with the technology, to consider the ways it will disrupt mathematical…
The Curry-Howard correspondence is about a relationship between types and programs on the one hand and propositions and proofs on the other. The implications for programming language design and program verification is an active field of…