Related papers: Evidence, Proofs, and Derivations
This paper proposes new derivations of three well-known sorting algorithms, in their functional formulation. The approach we use is based on three main ingredients: first, the algorithms are derived from a simpler algorithm, i.e. the…
Viewing formal mathematical proofs as logical terms provides a powerful and elegant basis for analyzing how human experts tend to structure proofs and how proofs can be structured by automated methods. We pursue this approach by (1)…
In this essay, I argue that mathematics is a natural science---just like physics, chemistry, or biology---and that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for…
The notion of argumentation and the one of belief stand in a problematic relation to one another. On the one hand, argumentation is crucial for belief formation: as the outcome of a process of arguing, an agent might come to (justifiably)…
It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an…
It has been shown that a functional interpretation of proofs in mathematical analysis can be given by the product of selection functions, a mode of recursion that has an intuitive reading in terms of the computation of optimal strategies in…
Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial…
Indirect evidence is crucial for successful statistical practice. Sometimes, however, it is better used informally. Future efforts should be directed toward understanding better the connection between statistical methods and scientific…
Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis…
Proofs (sequent calculus, natural deduction) and imperative algorithms (pseudocodes) are two well-known coexisting concepts. Then what is their relationship? Our answer is that \[ imperative\ algorithms\ =\ proofs\ with\ cuts \] This…
Theorem proving is a fundamental aspect of mathematics, spanning from informal reasoning in natural language to rigorous derivations in formal systems. In recent years, the advancement of deep learning, especially the emergence of large…
In fact-checking applications, a common reason to reject a claim is to detect the presence of erroneous cause-effect relationships between the events at play. However, current automated fact-checking methods lack dedicated causal-based…
The use of argumentation in education has been shown to improve critical thinking skills for end-users such as students, and computational models for argumentation have been developed to assist in this process. Although these models are…
General acceptance of a mathematical proposition $P$ as a theorem requires convincing evidence that a proof of $P$ exists. But what constitutes "convincing evidence?" I will argue that, given the types of evidence that are currently…
The development of logic has largely been through the 'deductive' paradigm: conclusions are inferred from established premisses. However, the use of logic in the context of both human and machine reasoning is typically through the dual…
Dynamic evidence logics are logics for reasoning about the evidence and evidence-based beliefs of agents in a dynamic environment. In this paper, we introduce a family of logics for reasoning about relational evidence: evidence that…
Understanding and creating mathematics using natural mathematical language - the mixture of symbolic and natural language used by humans - is a challenging and important problem for driving progress in machine learning. As a step in this…
The traditional foundation of science lies on the cornerstones of theory and experiment. Theory is used to explain experiment, which in turn guides the development of theory. Since the advent of computers and the development of…
Even with impressive advances in automated formal methods, certain problems in system verification and synthesis remain challenging. Examples include the verification of quantitative properties of software involving constraints on timing…
Mathematical proofs are a cornerstone of control theory, and it is important to get them right. Deduction systems can help with this by mechanically checking the proofs. However, the structure and level of detail at which a proof is…