Results 1  10
of
17
Coherent bicartesian and sesquicartesian categories, R. Kahle et al
 eds, Proof Theory in Computer Science, Lecture
"... Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover the first and the second projection from the product of the initial object with itself are the same. (Every ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
Coherence is here demonstrated for sesquicartesian categories, which are categories with nonempty finite products and arbitrary finite sums, including the empty sum, where moreover the first and the second projection from the product of the initial object with itself are the same. (Every
Weak Typed Böhm Theorem on IMLL
 Annals of Pure and Applied Logic
, 2007
"... In the Böhm theorem workshop on Crete island, Zoran Petric called Statman’s “Typical Ambiguity theorem ” typed Böhm theorem. Moreover, he gave a new proof of the theorem based on settheoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Böhm ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
In the Böhm theorem workshop on Crete island, Zoran Petric called Statman’s “Typical Ambiguity theorem ” typed Böhm theorem. Moreover, he gave a new proof of the theorem based on settheoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Böhm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment of intuitionistic linear logic without the multiplicative unit 1 (for short IMLL) weak typed Böhm theorem holds. The system IMLL exactly corresponds to the linear lambda calculus without exponentials, additives and logical constants. The system IMLL also exactly corresponds to the free symmetric monoidal closed category without the unit object. As far as we know, our separation result is the first one with regard to these systems in a purely syntactical manner. 1
Z.: Bicartesian coherence revisited
 Logic in Computer Science, Zbornik Radova. Volume
"... A survey is given of results about coherence for categories with finite products and coproducts. For these results, which were published previously by the authors in several places, some formulations and proofs are here corrected, and matters are updated. The categories investigated in this paper fo ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
A survey is given of results about coherence for categories with finite products and coproducts. For these results, which were published previously by the authors in several places, some formulations and proofs are here corrected, and matters are updated. The categories investigated in this paper formalize equality of proofs in classical and intuitionistic conjunctivedisjunctive logic without distribution of conjunction over disjunction.
On Varieties of Closed Categories and Dependency of Diagrams of Canonical Maps ∗.
"... We present a series of diagrams Dn in Symmetric Monoidal Closed Categories such that there is infinitely many different varieties of SMCC (in the sense of universal algebra) defined by diagrams of this series as equations. Similar result will hold for weaker closed categories. We discuss the notion ..."
Abstract
 Add to MetaCart
We present a series of diagrams Dn in Symmetric Monoidal Closed Categories such that there is infinitely many different varieties of SMCC (in the sense of universal algebra) defined by diagrams of this series as equations. Similar result will hold for weaker closed categories. We discuss the notion of dependency of diagrams in connection with this result. 1 Introduction. Canonical maps in closed categories may be seen as instances of morphisms of the free closed category generated by an infinite set of atoms. There exist many types of closed categories, for example Cartesian Closed Categories (CCC), Symmetric Monoidal Closed Categories (SMCC) etc. Closed
The Maximality of Cartesian Categories
, 1999
"... It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, w ..."
Abstract
 Add to MetaCart
It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categories, which is related to modeltheoretical methods of normalization. The equations between arrows assumed for cartesian categories are maximal in the sense that extending them with new equations collapses the categories into preorders (i.e. categories in which between any two objects there is at most one arrow). The equations envisaged for the extension are in the language of free cartesian categories generated by sets of objects, and variables for arrows don’t occur in them. If such an equation doesn’t hold in the free cartesian category generated by a set of objects, then any cartesian category in which this equation holds is a preorder. An analogous result is provable for categories with binary products, which differ from cartesian categories in not necessarily having a terminal object. The proof of these results, which we are going to present below, is based on a coherence property of cartesian categories. This coherence, which is ultimately inspired by the geometric modelling of categories of [3], is related to modeltheoretic methods of normalization. It permits to establish uniqueness of normal form for arrow terms without proceeding via the ChurchRosser property for reductions. It also yields an easy decision procedure for the commuting of diagrams in free cartesian categories.
Identity of Proofs Based on Normalization and Generality
, 2002
"... Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, bas ..."
Abstract
 Add to MetaCart
Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, based on reduction to cutfree form in sequent systems, but he also suggested to understand identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it didn’t fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to lowdimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.
Identity of Proofs Based on Normalization and Generality
, 2002
"... Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, bas ..."
Abstract
 Add to MetaCart
Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, based on reduction to cutfree form in sequent systems, but he also suggested to understand identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it didn’t fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to lowdimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.
Identity of Proofs Based on Normalization and Generality
, 2002
"... Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, bas ..."
Abstract
 Add to MetaCart
Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, based on reduction to cutfree form in sequent systems, but he also suggested to understand identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it didn’t fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to lowdimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.
Identity of Proofs Based on Normalization and Generality
, 2002
"... Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, bas ..."
Abstract
 Add to MetaCart
Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, based on reduction to cutfree form in sequent systems, but he also suggested to understand identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it didn’t fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to lowdimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.
Identity of Proofs Based on Normalization and Generality
, 2003
"... Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, bas ..."
Abstract
 Add to MetaCart
Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz’s, based on reduction to cutfree form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to lowdimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic.