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...

In this work, we wish to determine the complexity of deciding whether a given multiset of symbols, chosen from some finite set of symbols, can be derived according to the rules of a given context-free grammar. Providing an upper bound for this problem is of interest to computational linguistics, one goal of which is to classify the complexity of parsing and understanding natural language. We co...

This thesis focuses on automatic theorem proving. In this area there is usually a tradeoff between the "generality" of a prover and the amount of user-guidance it requires to find a proof. This thesis demonstrates that the amount of guidance needed can be reduced by employing reasoning procedures for specialized theories. It presents an enhancement to the Larch Prover (LP) with specialized theo...

Propositional Product Logic is known to be standard finite-strong complete but not a strong complete, that is, it is complete for deductions only from finite sets of premises with respect to evaluations on the standard product chain over the real unit interval. On the other hand, Montagna has defined a logical system, an axiomatic extension of the Hájek’s Basic Fuzzy Logic BL with an storage op...

Ordered semantic hyper-linking (OSHL) is a theorem prover for first-order logic that generates models and uses them to instantiate first-order clauses to ground clauses, and applies a propositional prover to these ground clauses. OSHL-U extends OSHL with rules for unit clauses and with heuristics. The unit rules in OHSL-U are designed to minimize the use of blind instantiation to generate groun...

We consider proof systems for effectively propositional logic. First, we show that propositional resolution for effectively propositional logic may have exponentially longer refutations than resolution for this logic. This shows that methods based on ground instantiation may be weaker than non-ground methods. Second, we introduce a generalisation rule for effectively propositional logic and sho...