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Large Language Models (LLMs) have demonstrated significant promise in formal theorem proving. In this study, we investigate the ability of LLMs to discover novel theorems and produce verified proofs. We propose a pipeline called…
Since Austin introduced the term ``infelicity'', the linguistic literature has been flooded with its use, but no formal or computational explanation has been given for it. This thesis provides one for those infelicities that occur when a…
"[M]athematicians care no more for logic than logicians for mathematics." Augustus de Morgan, 1868. Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional…
A major reason behind the success of probability calculus is that it possesses a number of valuable tools, which are based on the notion of probabilistic independence. In this paper, I identify a notion of logical independence that makes…
In traditional justification logic, evidence terms have the syntactic form of polynomials, but they are not equipped with the corresponding algebraic structure. We present a novel semantic approach to justification logic that models…
We show how to extract a monotonic learning algorithm from a classical proof of a geometric statement by interpreting the proof by means of interactive realizability, a realizability sematics for classical logic. The statement is about the…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
We present an implementation of a probabilistic first-order logic called TensorLog, in which classes of logical queries are compiled into differentiable functions in a neural-network infrastructure such as Tensorflow or Theano. This leads…
As mathematical induction is applied to prove statements on natural numbers, {\it continuous induction} (or, {\it real induction}) is a tool to prove some statements in real analysis.(Although, this comparison is somehow an overstatement.)…
This paper is a brief and informal presentation of cirquent calculus, a novel proof system for resource-conscious logics. As such, it is a refinement of sequent calculus with mechanisms that allow to explicitly account for the possibility…
A proof procedure, in the spirit of the sequent calculus, is proposed to check the validity of entailments between Separation Logic formulas combining inductively defined predicates denoted structures of bounded tree width and theory…
A prototype system is described whose core functionality is, based on propositional logic, the elimination of second-order operators, such as Boolean quantifiers and operators for projection, forgetting and circumscription. This approach…
The computational properties of modal and propositional dependence logics have been extensively studied over the past few years, starting from a result by Sevenster showing NEXPTIME-completeness of the satisfiability problem for modal…
A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. In this paper, we give a brief overview of a theory exploration system called QuickSpec, which is able to…
Logics of limited belief aim at enabling computationally feasible reasoning in highly expressive representation languages. These languages are often dialects of first-order logic with a weaker form of logical entailment that keeps reasoning…
Probability theory, epistemically interpreted, provides an excellent, if not the best available account of inductive reasoning. This is so because there are general and definite rules for the change of subjective probabilities through…
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…
Human mathematicians are often good at recognizing modular and reusable theorems that make complex mathematical results within reach. In this paper, we propose a novel method called theoREm-from-prooF extrACTOR (REFACTOR) for training…
Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…
Proof-theoretic methods are developed for subsystems of Johansson's logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems.…