Related papers: Classical theorems in the Implicational Propositio…
The $P$ versus $NP$ problem is still unsolved. But there are several oracles with $P$ unequal $NP$ relative to them. Here we will prove, that $P\not=NP$ relative to a $P$-complete oracle. In this paper, we use padding arguments as the proof…
We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite.
We introduce a calculus for tuplices, which are expressions that generalize matrices and vectors. Tuplices have an underlying data type for quantities that are taken from a zero-totalized field. We start with the core tuplix calculus CTC…
Combinatory logic shows that bound variables can be eliminated without loss of expressiveness. It has applications both in the foundations of mathematics and in the implementation of functional programming languages. The original…
Computability theory is traditionally conceived as the theoretical basis of informatics. Nevertheless, numerous proposals transcend computability theory, in particular by emphasizing interaction of modules, or components, parts,…
Plausible reasoning concerns situations whose inherent lack of precision is not quantified; that is, there are no degrees or levels of precision, and hence no use of numbers like probabilities. A hopefully comprehensive set of principles…
System I is a proof language for a fragment of propositional logic where isomorphic propositions, such as $A\wedge B$ and $B\wedge A$, or $A\Rightarrow(B\wedge C)$ and $(A\Rightarrow B)\wedge(A\Rightarrow C)$ are made equal. System I enjoys…
Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally…
Conformal predictors are set predictors that are automatically valid in the sense of having coverage probability equal to or exceeding a given confidence level. Inductive conformal predictors are a computationally efficient version of…
Cyclic proof theory breaks tradition by allowing certain infinite proofs: those that can be represented by a finite graph, while satisfying a soundness condition. We reconcile cyclic proofs with traditional finite proofs: we extend abstract…
We introduce isotonic conditional laws (ICL) which extend the classical notion of conditional laws by the additional requirement that there exists an isotonic relationship between the random variable of interest and the conditioning random…
The Functional Machine Calculus (FMC), recently introduced by the authors, is a generalization of the lambda-calculus which may faithfully encode the effects of higher-order mutable store, I/O and probabilistic/non-deterministic input.…
Probabilistic Cell Decomposition (PCD) is a probabilistic path planning method combining the concepts of approximate cell decomposition with probabilistic sampling. It has been shown that the use of lazy evaluation techniques and supervised…
This is a short paper about the relationship between logic and computation. More specifically, it is about a relationship between the completeness proof for intuitionistic propositional logic within the form of proof-theoretic semantics…
In this paper we explore the application of methods for classical judgment aggregation in pooling probabilistic opinions on logically related issues. For this reason, we first modify the Boolean judgment aggregation framework in the way…
A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive…
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone's representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof…
An inductive logic can be formulated in which the elements are not propositions or probability distributions, but information systems. The logic is complete for information systems with binary hypotheses, i.e., it applies to all such…
In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar…
In this paper we present a formalization of Intuitionistic Propositional Logic in the Lean proof assistant. Our approach focuses on verifying two completeness proofs for the studied logical system, as well as exploring the relation between…