Related papers: Essential Incompleteness of Arithmetic Verified by…
An invaluable feature of computer algebra systems is their ability to plot the graph of functions. Unfortunately, when one is trying to design a library of mathematical functions, this feature often falls short, producing incorrect and…
Matching logic is a formalism for specifying, and reasoning about, mathematical structures, using patterns and pattern matching. Growing in popularity, it has been used to define many logical systems such as separation logic with recursive…
In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of…
Sharing of notations and theories across an inheritance hierarchy of mathematical structures, e.g., groups and rings, is important for productivity when formalizing mathematics in proof assistants. The packed classes methodology is a…
CoqQ is a framework for reasoning about quantum programs in the Coq proof assistant. Its main components are: a deeply embedded quantum programming language, in which classic quantum algorithms are easily expressed, and an expressive…
We prove, and mechanize in Rocq, an abstract obstruction theorem for primitive closure predicates, defined as $C : \mathsf{Form} \to \mathsf{Prop}$ over the closed implication-falsity fragment $A,B ::= \bot \mid A \to B$. Two structurally…
By affine arithmetic is meant the set of affine consequences of Peano arithmetic. This is a continuous theory which is studied in the framework of affine logic, a sublogic of continuous logic. Affine arithmetic is undecidable. Also, its…
We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting…
We exploit (co)inductive specifications and proofs to approach the evaluation of low-level programs for the Unlimited Register Machine (URM) within the Coq system, a proof assistant based on the Calculus of (Co)Inductive Constructions type…
Theory of choreographic languages typically includes a number of complex results that are proved by structural induction. The high number of cases and the subtle details in some of them lead to long reviewing processes, and occasionally to…
Recursive coalgebras provide an elegant categorical tool for modelling recursive algorithms and analysing their termination and correctness. By considering coalgebras over categories of suitably indexed families, the correctness of the…
In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…
Goedel Incompleteness Theorem leaves open a way around it, vaguely perceived for a long time but not clearly identified. (Thus, Goedel believed informal arguments can answer any math question.) Closing this loophole does not seem obvious…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
A new characterization of provably recursive functions of first-order arithmetic is described. Its main feature is using only terms consisting of 0, the successor S and variables in the quantifier rules, namely, universal elimination and…
We extend the work of A. Ciaffaglione and P. Di Gianantonio on mechanical verification of algorithms for exact computation on real numbers, using infinite streams of digits implemented as co-inductive types. Four aspects are studied: the…
One of the elegant achievements in the history of proof theory is the characterization of the provably total recursive functions of an arithmetical theory by its proof-theoretic ordinal as a way to measure the time complexity of the…
Given a first-order theory $T$ formulated in the usual language of first-order arithmetic, we say that $T$ is of *restricted complexity* if there is some natural number $n$ and some set $\mathcal A$ of $\Sigma_n$-sentences such that $T$ can…
Equational Unification is a critical problem in many areas such as automated theorem proving and security protocol analysis. In this paper, we focus on XOR-Unification, that is, unification modulo the theory of exclusive-or. This theory…
Consider a decision problem whose instance is a function. Its degree of undecidability, measured by the corresponding class of the arithmetic (or Kleene-Mostowski) hierarchy hierarchy, may depend on whether the instance is a partial…