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We describe a graph-theoretic syntax for self-referential formulas as well as a four-valued logic to include contradictory and independent formulas. We then explore the degree to which generalized truth tables can be realized in our theory,…
We introduce neural networks for end-to-end differentiable proving of queries to knowledge bases by operating on dense vector representations of symbols. These neural networks are constructed recursively by taking inspiration from the…
Applied process calculi include advanced programming constructs such as type systems, communication with pattern matching, encryption primitives, concurrent constraints, nondeterminism, process creation, and dynamic connection topologies.…
Logic rules and inference are fundamental in computer science and have been studied extensively. However, prior semantics of logic languages can have subtle implications and can disagree significantly, on even very simple programs,…
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…
In this paper, we establish the foundations of a novel logical framework for the {\pi}-calculus, based on the deduction-as-computation paradigm. Following the standard proof-theoretic interpretation of logic programming, we represent…
Focusing, introduced by Jean-Marc Andreoli in the context of classical linear logic, defines a normal form for sequent calculus derivations that cuts down on the number of possible derivations by eagerly applying invertible rules and…
This paper employs the linear nested sequent framework to design a new cut-free calculus LNIF for intuitionistic fuzzy logic--the first-order G\"odel logic characterized by linear relational frames with constant domains. Linear nested…
Double-negation translations are used to encode and decode classical proofs in intuitionistic logic. We show that, in the cut-free fragment, we can simplify the translations and introduce fewer negations. To achieve this, we consider 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…
Proofs (sequent calculus, natural deduction) and imperative algorithms (pseudocodes) are two well-known coexisting concepts. Then what is their relationship? Our answer is that \[ imperative\ algorithms\ =\ proofs\ with\ cuts \] This…
Answering logical queries over incomplete knowledge bases is challenging because: 1) it calls for implicit link prediction, and 2) brute force answering of existential first-order logic queries is exponential in the number of existential…
Introduced in 2006 by Japaridze, cirquent calculus is a refinement of sequent calculus. The advent of cirquent calculus arose from the need for a deductive system with a more explicit ability to reason about resources. Unlike the more…
Nested dichotomies are used as a method of transforming a multiclass classification problem into a series of binary problems. A tree structure is induced that recursively splits the set of classes into subsets, and a binary classification…
We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is…
Logic is the science of correct inferences and a logical system is a tool to prove assertions in a certain logic in a correct way. There are many logical systems, and many ways of formalizing them, e.g., using natural deduction or sequent…
We study nested conditions, a generalization of first-order logic to a categorical setting, and provide a tableau-based (semi-decision) procedure for checking (un)satisfiability and finite model generation. This generalizes earlier results…
We introduce labelled sequent calculi for quantified modal logics with definite descriptions. We prove that these calculi have the good structural properties of G3-style calculi. In particular, all rules are height-preserving invertible,…
We extend the simply-typed guarded $\lambda$-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming…
We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a…