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Functional Distributional Semantics is a framework that aims to learn, from text, semantic representations which can be interpreted in terms of truth. Here we make two contributions to this framework. The first is to show how a type of…
Probabilistic sentential decision diagrams are a class of structured-decomposable probabilistic circuits especially designed to embed logical constraints. To adapt the classical LearnSPN scheme to learn the structure of these models, we…
Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this paper, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using…
Natural logic offers a powerful relational conception of meaning that is a natural counterpart to distributed semantic representations, which have proven valuable in a wide range of sophisticated language tasks. However, it remains an open…
Both algebraic and computational approaches for dealing with similarity spaces are well known in generalized rough set theory. However, these studies may be said to have been confined to particular perspectives of distinguishability in the…
Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called ``topological semantics''. The first is classical higher-order logic, with…
Standard methods of using categorical variables as predictors either endow them with an ordinal structure or assume they have no structure at all. However, categorical variables often possess structure that is more complicated than a linear…
We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into…
These are lecture notes on the algebraic approach to regular languages. The classical algebraic approach is for finite words; it uses semigroups instead of automata. However, the algebraic approach can be extended to structures beyond…
This paper explores relational syllogistic logics, a family of logical systems related to reasoning about relations in extensions of the classical syllogistic. These are all decidable logical systems. We prove completeness theorems and…
We propose a presentation of classical propositional tableaux elaborated by application of methods that are noteworthy in program design, namely program derivation with separation of concerns. We start by deriving from a straightforward…
Knowledge representation is a popular research field in IT. As mathematical knowledge is most formalized, its representation is important and interesting. Mathematical knowledge consists of various mathematical theories. In this paper we…
When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL…
Within classical propositional logic, assigning probabilities to formulas is shown to be equivalent to assigning probabilities to valuations. A novel notion of probabilistic entailment enjoying desirable properties of logical consequence is…
We obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial non-deterministic…
The process of cognition is analysed to adjust the set theory to physical description. Postulates and basic definitions are revised. The specific sets of predicates, called presets, corresponding to the physical objects identified by an…
This invited paper is a passionate pitch for the significance of logic in scientific education. Logic helps focus on the essential core to identify the foundations of ideas and provides corresponding longevity with the resulting approach to…
Linear logic (LL) is a resource-aware, abstract logic programming language that refines both classical and intuitionistic logic. Linear logic semantics is typically presented in one of two ways: by associating each formula with the set of…
The use of logical systems for problem-solving may be as diverse as in proving theorems in mathematics or in figuring out how to meet up with a friend. In either case, the problem solving activity is captured by the search for an…
As the key to sentiment analysis, sentiment composition considers the classification of a constituent via classifications of its contained sub-constituents and rules operated on them. Such compositionality has been widely studied previously…