Related papers: Reflection principles, propositional proof systems…
In this paper, we consider the problem of learning a first-order theorem prover that uses a representation of beliefs in mathematical claims to construct proofs. The inspiration for doing so comes from the practices of human mathematicians…
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.…
Proving proof-size lower bounds for $\mathbf{LK}$, the sequent calculus for classical propositional logic, remains a major open problem in proof complexity. We shed new light on this challenge by isolating the power of structural rules,…
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a…
Resolution and superposition are common techniques which have seen widespread use with propositional and first-order logic in modern theorem provers. In these cases, resolution proof production is a key feature of such tools; however, the…
Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs…
The objective of this paper is to present general, mechanically verified, refinement rules for reasoning about recursive programs and while loops in the context of concurrency. Unlike many approaches to concurrency, we do not assume that…
Liquid Haskell's refinement-reflection feature augments the Haskell language with theorem proving capabilities, allowing programmers to retrofit their existing code with proofs. But many of these proofs require routine, boilerplate code…
This work was intended to be an attempt to introduce the meta-language for working with multiple-conclusion inference rules that admit asserted propositions along with the rejected propositions. The presence of rejected propositions, and…
A typical kind of question in mathematical logic is that for the necessity of a certain axiom: Given a proof of some statement $\phi$ in some axiomatic system $T$, one looks for minimal subsystems of $T$ that allow deriving $\phi$. In…
In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2 and that…
Supervised fine-tuning enhances the problem-solving abilities of language models across various mathematical reasoning tasks. To maximize such benefits, existing research focuses on broadening the training set with various data augmentation…
Recently, there has been considerable progress on designing algorithms with provable guarantees -- typically using linear algebraic methods -- for parameter learning in latent variable models. But designing provable algorithms for inference…
We bring forward a logical system of transition algebras that enhances many-sorted first-order logic using features from dynamic logics. The sentences we consider include compositions, unions, and transitive closures of transition…
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…
A central problem in proof-theory is that of finding criteria for identity of proofs, that is, for when two distinct formal derivations can be taken as denoting the same logical argument. In the literature one finds criteria which are…
We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…
Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable.…
Argumentation is a promising model for reasoning with uncertain knowledge. The key concept of acceptability enables to differentiate arguments and counterarguments: The certainty of a proposition can then be evaluated through the most…
For Hilbert, the consistency of a formal theory T is an infinite series of statements "D is free of contradictions" for each derivation D and a consistency proof is i) an operation that, given D, yields a proof that D is free of…