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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…
Consequential decision-making incentivizes individuals to strategically adapt their behavior to the specifics of the decision rule. While a long line of work has viewed strategic adaptation as gaming and attempted to mitigate its effects,…
We present a method for generating possible proofs of a query with respect to a given Answer Set Programming (ASP) rule set using an abductive process where the space of abducibles is automatically constructed just from the input rules…
Transformers have been shown to emulate logical deduction over natural language theories (logical rules expressed in natural language), reliably assigning true/false labels to candidate implications. However, their ability to generate…
We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof…
The definition is a common form of human expert knowledge, a building block of formal science and mathematics, a foundation for database theory and is supported in various forms in many knowledge representation and formal specification…
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…
When teaching an elementary logic course to students who have a general scientific background but have never been exposed to logic, we have to face the problem that the notions of deduction rule and of derivation are completely new to them,…
Uniform proofs are sequent calculus proofs with the following characteristic: the last step in the derivation of a complex formula at any stage in the proof is always the introduction of the top-level logical symbol of that formula. We…
The development of logic has largely been through the 'deductive' paradigm: conclusions are inferred from established premisses. However, the use of logic in the context of both human and machine reasoning is typically through the dual…
The calculus of constructions (CC) is a core theory for dependently typed programming and higher-order constructive logic. Originally introduced in Coquand's 1985 thesis, CC has inspired 25 years of research in programming languages and…
The computational complexity of reasoning within the Dempster-Shafer theory of evidence is one of the main points of criticism this formalism has to face. To overcome this difficulty various approximation algorithms have been suggested that…
We study the termination of rewriting modulo a set of equations in the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In a previous…
We propose an abductive diagnosis theory that integrates probabilistic, causal and taxonomic knowledge. Probabilistic knowledge allows us to select the most likely explanation; causal knowledge allows us to make reasonable independence…
Reasoning with defeasible and conflicting knowledge in an argumentative form is a key research field in computational argumentation. Reasoning under various forms of uncertainty is both a key feature and a challenging barrier for automated…
Transformers have recently been shown to be capable of reliably performing logical reasoning over facts and rules expressed in natural language, but abductive reasoning - inference to the best explanation of an unexpected observation - has…
When engaging in argumentative discourse, skilled human debaters tailor claims to the beliefs of the audience, to construct effective arguments. Recently, the field of computational argumentation witnessed extensive effort to address the…
The purpose of this paper is to give an easy to understand with step-by-step explanation to allow interested people to fully appreciate the power of natural deduction for first-order logic. Natural deduction as a proof system can be used to…
In this paper we consider propositional calculi, which are finitely axiomatizable extensions of intuitionistic implicational propositional calculus together with the rules of modus ponens and substitution. We give a proof of undecidability…
Deep inference is a proof theoretic methodology that generalizes the standard notion of inference of the sequent calculus, whereby inference rules become applicable at any depth inside logical expressions. Deep inference provides more…