Related papers: The Sources of Certainty in Computation and Formal…
Precise rules are developed in order to formalize the reasoning processes involved in standard non-relativistic quantum mechanics, with the help of analogies from classical physics. A classical or quantum description of a mechanical system…
In La G\'eom\'etrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation.…
The syntactic nature of logic and computation separates them from other fields of mathematics. Nevertheless, syntax has been the only way to adequately capture the dynamics of proofs and programs such as cut-elimination, and the finiteness…
There is a lack of formalism for some key foundational concepts in systems engineering. One of the most recently acknowledged deficits is the inadequacy of systems engineering practices for engineering intelligent systems. In our previous…
Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring…
Currently it is widely accepted that the language of science is mathematics. This book explores an alternative idea where the future of science is based on the language of algorithms and programs. How such a language can actually be…
What is reasoning? This question has driven centuries of philosophical inquiry, from Aristotle's syllogisms to modern computational complexity theory. In the age of large language models achieving superhuman performance on benchmarks like…
Algorithmic robustness refers to the sustained performance of a computational system in the face of change in the nature of the environment in which that system operates or in the task that the system is meant to perform. Below, we motivate…
In 1927 Heisenberg discovered that the ``more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa''. Four years later G\"odel showed that a finitely specified, consistent formal…
Computational mechanisms for uncertainty management must support interactive and incremental problem formulation, inference, hypothesis testing, and decision making. However, most current uncertainty inference systems concentrate primarily…
Despite significant advancements in the conception of (formal) integrated development environments, applying formal methods in software industry is still perceived as a difficult task. To make the task easier, providing tools that help…
We advocate a declarative approach to proving properties of logic programs. Total correctness can be separated into correctness, completeness and clean termination; the latter includes non-floundering. Only clean termination depends on the…
Formal Methods are mathematically-based techniques for software design and engineering, which enable the unambiguous description of and reasoning about a system's behaviour. Autonomous systems use software to make decisions without human…
Implicit computational complexity, which aims at characterizing complexity classes by machine-independent means, has traditionally been based, on the one hand, on programs and deductive formalisms for free algebras, and on the other hand on…
Dialectical systems are a mathematical formalism for modeling an agent updating a knowledge base seeking consistency. Introduced in the 1970s by Roberto Magari, they were originally conceived to capture how a working mathematician or a…
The question "What is real?" can be traced back to the shadows in Plato's cave. Two thousand years later, Rene Descartes lacked knowledge about arguing against an evil deceiver feeding us the illusion of sensation. Descartes'…
Though many safety-critical software systems use floating point to represent real-world input and output, programmers usually have idealized versions in mind that compute with real numbers. Significant deviations from the ideal can cause…
Context is a rich concept and is an elusive concept to define. The concept of context has been studied by philosophers, linguists, psychologists, and recently by computer scientists. Within each research community the term context was…
We propose axioms governing the interaction of constructive assertibility and meaningfulness predicates with a self-applicative truth predicate characterized by the T-scheme, and we prove the consistency of the resulting formal system.
Conceptual Integrity is the most important consideration for software system design, as stated by Frederick Brooks. Brooks also suggested that Conceptual Integrity can be attained by means of design principles, such as Propriety, and…