Related papers: Tableaux for First Order Logic of Proofs
For any first order theory T we construct a Boolean valued model M, in which precisely the T--provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a first order…
Automated theorem proving in first-order logic is an active research area which is successfully supported by machine learning. While there have been various proposals for encoding logical formulas into numerical vectors -- from simple…
The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for first order linear logic specifications. The fragment of linear logic we consider in this paper is based on the linear…
We present a sequent-style proof system for provability logic GL that admits so-called circular proofs. For these proofs, the graph underlying a proof is not a finite tree but is allowed to contain cycles. As an application, we establish…
Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called ``topological semantics''. The first is classical higher-order logic, with…
By Solovay's celebrated completeness result on formal provability we know that the provability logic $\mathrm GL$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable…
We develop a semantics for logics of imperfect information with respect to general models. Then we build a proof system and prove its soundness and completeness with respect to this semantics.
Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires…
This note discusses proofs for convergence of first-order methods based on simple potential-function arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants.
Large Language Models (LLMs) have demonstrated impressive reasoning capabilities, leading to their adoption in high-stakes domains such as healthcare, law, and scientific research. However, their reasoning often contains subtle logical…
In this paper we give an ordinal analysis of the theory of second order arithmetic. We do this by working with proof trees -- that is, "deductions" which may not be well-founded. Working in a suitable theory, we are able to represent…
This paper presents a combination of several automated reasoning and proof presentation tools with the Mizar system for formalization of mathematics. The combination forms an online service called MizAR, similar to the SystemOnTPTP service…
Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary…
The introduction of first-class type classes in the Coq system calls for re-examination of the basic interfaces used for mathematical formalization in type theory. We present a new set of type classes for mathematics and take full advantage…
A cyclic proof system is a proof system whose proof figure is a tree with cycles. The cut-elimination in a proof system is fundamental. It is conjectured that the cut-elimination in the cyclic proof system for first-order logic with…
This paper tackles the problem of formulating and proving the completeness of focused-like proof systems in an automated fashion. Focusing is a discipline on proofs which structures them into phases in order to reduce proof search…
We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.
Description Logics (DLs) are a family of languages used for the representation and reasoning on the knowledge of an application domain, in a structured and formal manner. In order to achieve this objective, several provers, such as RACER…
In this paper we present a theorem proving methodology for a restricted but significant fragment of the conditional language made up of (boolean combinations of) conditional statements with unnested antecedents. The method is based on the…
Recursively defined linked data structures embedded in a pointer-based heap and their properties are naturally expressed in pure first-order logic with least fixpoint definitions (FO+lfp) with background theories. Such logics, unlike pure…