I will show how to define, in any type system with dependent sums, products and Martin-Lof identity types, the notion of a homotopy equivalence between two types and how to formulate the Equivalence Axiom which provides a natural way to assert that ”two homotopy equivalent types are equal”. I will then sketch a construction of a model of one of the standard Martin-Lof type theories which satisf...

‎in this paper, a new class of fuzzy sets called fuzzy strongly ${{g}^{*}}$-closed sets is introduced and its properties are investigated. moreover, we study some more properties of this type of closed spaces.

In this paper all subdirectly irreducible pseudocomplemented distributive lattices are found. This result is used to establish a Stone-like representation theorem conjectured by G. Grätzer and to find all equational subclasses of the class of pseudocomplemented distributive lattices.

A new hierarchy of “exact” unification types is introduced, motivated by the study of admissible rules for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by instantiation, but rather by inclusion over the corresponding sets of unified identities. Minimal complete sets of unifiers under this new preordering always h...

The polynomial closure Pol(L) of a class of languages L of A is the set of languages that are finite unions of marked products of the form L0a1L1 · · · anLn, where the ai are letters and the Li are elements of L. The main result of this paper gives an equational description of Pol(L), given an equational description of L, when L is a lattice of regular languages closed under quotients, or a quo...

We introduce a class of counting problems that arise naturally in equational matching and study their computational complexity. If E is an equational theory, then #E-Matching is the problem of counting the number of complete minimal E-matchers of two given terms. #E-Matching is a well-deened algorithmic problem for every nitary equational theory. Moreover, it captures more accurately the comput...

Algebraic reasoning about programming language constructs has been a popular research topic for many years. At the propositional level, the theory of flowchart programs and linear recursion are well handled by such systems as Kleene algebra and iteration theories, systems that characterize the equational theory of the regular sets. To handle more general forms of recursion including procedures ...

We investigate the following classes of equational theories which are important in unification theory: permutative, finite, Noetherian, simple, almost collapse free, collapse free, regular, and O-free theories. We show some relationships between the particular classes and their connections to the unification hierarchy. Especially, we study conditions, under which minimal and complete sets of un...

Uniication in the presence of an equational theory is an important problem in theorem-proving and in the integration of functional and logic programming languages. This paper presents an improvement of the proposed lazy uniication methods by incorporating simpliication into the uniication process. Since simpliication is a deterministic computation process, more eecient uniication algorithms can...