Related papers: Beyond Nonexpansive Operations in Quantitative Alg…
We explore the possibility of extending Mardare et al. quantitative algebras to the structures which naturally emerge from Combinatory Logic and the lambda-calculus. First of all, we show that the framework is indeed applicable to those…
In this paper we show several similarities among logic systems that deal simultaneously with deductive and quantitative inference. We claim it is appropriate to call the tasks those systems perform as Quantitative Logic Reasoning. Analogous…
The use of exponentials in linear logic greatly enhances its expressive power. In this paper we focus on nonassociative noncommutative multiplicative linear logic, and systematically explore modal axioms K, T, and 4 as well as the…
Fuzzy logic extends the classical truth values "true" and "false" with additional truth degrees in between. More specifically, fuzzy modal logics in this sense are given by a choice of fuzzy modalities and a fuzzy propositional base. It has…
A non-commutative, non-associative weakening of Girard's linear logic is developed for multiplicative and additive connectives. Additional assumptions capture the logic of quantic measurements.
This paper explores the space of (propositional) probabilistic logical languages, ranging from a purely `qualitative' comparative language to a highly `quantitative' language involving arbitrary polynomials over probability terms. While…
Quantitative logic reasons about the degree to which formulas are satisfied. This paper studies the fundamental reasoning principles of higher-order quantitative logic and their application to reasoning about probabilistic programs and…
Some critical open problems of epistemic logics can be investigated in the framework of a quantum computational approach. The basic idea is to interpret sentences - like Alice knows that Bob does not understand that Pi is irrational - as…
We propose a two-parametric non-distributive algebraic structure that follows from $(q,q')$-logarithm and $(q,q')$-exponential functions. Properties of generalized $(q,q')$-operators are analyzed. We also generalize the proposal into a…
The analysis and control of stochastic dynamical systems rely on probabilistic models such as (continuous-space) Markov decision processes, but large or continuous state spaces make exact analysis intractable and call for principled…
We present quantitative separation logic ($\mathsf{QSL}$). In contrast to classical separation logic, $\mathsf{QSL}$ employs quantities which evaluate to real numbers instead of predicates which evaluate to Boolean values. The connectives…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
The commutative algebra of functions on a manifold is extended to a noncommutative algebra by considering its tensor product with the algebra of nxn complex matrices. Noncommutative geometry is used to formulate an extension of the…
The rational quantum algebraically integrable systems are non-trivial generalizations of Laplacian operators to the case of elliptic operators with variable coefficients. We study corresponding extensions of Laplacian growth connected with…
This paper aims at connecting the various classes that provide an algebraic semantics for three different conservative expansions of Lukasiewicz logic, using algebraic and category-theoretical techniques. We connect such classes of algebras…
We study which standard operators of probabilistic process calculi allow for compositional reasoning with respect to bisimulation metric semantics. We argue that uniform continuity (generalizing the earlier proposed property of…
In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties…
We use computational linear algebra and commutative algebra to study spaces of relations satisfied by quadrilinear operations. The relations are analogues of associativity in the sense that they are quadratic (every term involves two…
Calder\'on-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov…
We propose modal Markov logic as an extension of propositional Markov logic to reason under the principle of maximum entropy for modal logics K45, KD45, and S5. Analogous to propositional Markov logic, the knowledge base consists of…