Related papers: Inductive Reasoning with Equality Predicates, Cont…
Causal inference is central to scientific discovery, yet choosing appropriate methods remains challenging because of the complexity of both statistical methodology and real-world data. Inspired by the success of artificial intelligence in…
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.)…
The purpose of the present work is to provide short and supple teaching notes for a $30$ hours introductory course on elementary \textit{Enumerative Algebraic Combinatorics}. We fully adopt the \textit{Rota way}. The themes are organized…
This paper proposes an alternative to standard first-order logic that seeks greater naturalness, generality, and semantic self-containment. The system removes the first-order restriction, avoids type hierarchies, and dispenses with external…
We present a first-order theorem proving framework for establishing the correctness of functional programs implementing sorting algorithms with recursive data structures. We formalize the semantics of recursive programs in many-sorted…
We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…
In this paper we relate different formulations of the DPLL(T) procedure. The first formulation is based on a system of rewrite rules, which we denote DPLL(T). The second formulation is an inference system of, which we denote LKDPLL(T). The…
Reasoning under uncertainty is a fundamental challenge in Artificial Intelligence. As with most of these challenges, there is a harsh dilemma between the expressive power of the language used, and the tractability of the computational…
The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation…
We develop a second-order extension of intuitionistic modal logic, allowing quantification over propositions, both syntactically and semantically. A key feature of second-order logic is its capacity to define positive connectives from the…
We propose a "formula slicing" method for finding inductive invariants. It is based on the observation that many loops in the program affect only a small part of the memory, and many invariants which were valid before a loop are still valid…
While model checking has often been considered as a practical alternative to building formal proofs, we argue here that the theory of sequent calculus proofs can be used to provide an appealing foundation for model checking. Since the…
This book can be seen either as a text on theorem proving that uses techniques from general algebra, or else as a text on general algebra illustrated and made concrete by practical exercises in theorem proving. The book considers several…
We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to…
In current Large Language Models we can trust the production of smoothly flowing prose on the basis of the principles of machine learning. However, there is no comparably principled basis to justify trust in the content of the text…
In an earlier paper, a new theory of measurefree "conditional" objects was presented. In this paper, emphasis is placed upon the motivation of the theory. The central part of this motivation is established through an example involving a…
Standard chain-of-thought reasoning generates a solution in a single forward pass, committing irrevocably to each token and lacking a mechanism to recover from early errors. We introduce Inference-Time Rethinking, a generative framework…
We establish various complexity results for the entailment problem between formulas in Separation Logic with user-defined predicates denoting recursive data structures. The considered fragments are characterized by syntactic conditions on…
Largely adopted by proof assistants, the conventional induction methods based on explicit induction schemas are non-reductive and local, at schema level. On the other hand, the implicit induction methods used by automated theorem provers…
It is well-known that extending the Hilbert axiomatic system for first-order intuitionistic logic with an exclusion operator, that is dual to implication, collapses the domains of models into a constant domain. This makes it an interesting…