In this paper, we propose a set theoretic approach for knowledge representation. While the syntax of an application domain is captured by set theoretic constructs including individuals, concepts and operators, knowledge is formalized by equality assertions. We first present a primitive form that uses minimal assumed knowledge and constructs. Then, assuming naive set theory, we extend it by defi...

Dependence logic is a new logic which incorporates the notion of “dependence”, as well as “independence” between variables into first-order logic. In this thesis, we study extensions and variants of dependence logic on the first-order, propositional and modal level. In particular, the role of intuitionistic connectives in this setting is emphasized. We obtain, among others, the following result...

The logic of equality with uninterpreted functions (EUF) and its extensions have been widely applied to processor verification, by means of a large variety of progressively more sophisticated (lazy or eager) translations into propositional SAT. Here we propose a new approach, namely a general DPLL(X) engine, whose parameter X can be instantiated with a specialized solver Solver T for a given th...

We investigate the system TRC of untyped illative combinatory logic that is equiconsistent with New Foundations. We prove that various unstratified combinators do not exist in TRC. Introduction. We prove some results in the axiomatic system TRC introduced in [3]. The system TRC (for ‘type-respecting combinators’) is an untyped system of combinatory logic, in the sense of [1], [2]. TRC is a firs...

We present a model of type theory with dependent product, sum, and identity, in cubical sets. We describe a universe and explain how to transform an equivalence between two types into an equality. We also explain how to model propositional truncation and the circle. While not expressed internally in type theory, the model is expressed in a constructive metalogic. Thus it is a step towards a com...

Abstract datatypes can be viewed as sorted ground term algebras. Unification can be used to solve conjunctions of equations. We give a new algorithm to extend this to the full quantifier free fragment, i.e. including formulas with disjunction and negation. The algorithm is based on unification (to deal with equality) and DPLL (to deal with propositional logic). In this paper we present our algo...

In this paper, we study logics of dependence on the propositional level. We prove that several interesting propositional logics of dependence, including propositional dependence logic, propositional intuitionistic dependence logic as well as propositional inquisitive logic, are expressively complete and have disjunctive or conjunctive normal forms. We provide deduction systems and prove the com...

We find a translation with particularly nice properties from intuitionistic propositional logic in countably many variables to intuitionistic propositional logic in two variables. In addition, the existence of a possibly-not-as-nice translation from any countable logic into intuitionistic propositional logic in two variables is shown. The nonexistence of a translation from classical logic into ...

The paper consists of two parts. The rst one is a concise introduction to epistemic (both propositional and predicate) logic with common knowl-ege operator. As the full common knowledge predicate logics are not even recursively enumerable, in the second part we introduce and investigate the monodic fragment of these logics which allows applications of the epistemic operators to formulas with at...