Results 1 
9 of
9
Heterogeneous fibring of deductive systems via abstract proof systems
, 2005
"... Fibring is a metalogical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Fibring is a metalogical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have different presentations (e.g. one presented by a Hilbert calculus and the other by a sequent calculus), has been an open problem. Herein, consequence systems are shown to be a good solution for heterogeneous fibring when one of the logics is presented in a semantic way and the other by a calculus and also a solution for the heterogeneous fibring of calculi. The new notion of abstract proof system is shown to provide a better solution to heterogeneous fibring of calculi namely because derivations in the fibring keep the constructive nature of derivations in the original logics. Preservation of compactness and semidecidability is investigated.
Fibring logics: Past, present and future
 We Will Show Them: Essays in Honour of Dov Gabbay, Volume One
, 2005
"... abstract. This paper is a guided tour through the theory of fibring as a general mechanism for combining logics. We present the main ideas, constructions and difficulties of fibring, from both a model and a prooftheoretic perspective, and give an outline of soundness, completeness and interpolation ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
abstract. This paper is a guided tour through the theory of fibring as a general mechanism for combining logics. We present the main ideas, constructions and difficulties of fibring, from both a model and a prooftheoretic perspective, and give an outline of soundness, completeness and interpolation preservation results. Along the way, we show how the current algebraic semantics of fibring relates with the original ideas of Dov Gabbay. We also analyze the collapsing problem, the challenges it raises, and discuss a number of future research directions. 1
Graphtheoretic fibring of logics
 Part II  Completeness preservation. Preprint, SQIG  IT and IST  TU Lisbon
, 2008
"... A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as an mgraph where the nodes and the medges include the sorts and the constructors of the signatu ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as an mgraph where the nodes and the medges include the sorts and the constructors of the signatures at hand. Fibring of two models is an mgraph where the nodes and the medges are the values and the operations in the models, respectively. Fibring of two deductive systems is an mgraph whose nodes are language expressions and the medges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graphtheoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with nondeterministic semantics, logics with an algebraic semantics, logics with partial semantics, and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided. 1
Parallel composition of logic calculi with proofs as generalized 2cells
 Preprint, SQIG  IT and IST  TU Lisbon
, 2010
"... Recent graphtheoretic developments [26, 27] in the semantic theory of combination of logics look at a signature as a multigraph (of sorts and connectives) and obtain the induced language as the category of multipaths where formulas appear as morphisms. This idea is carried over to Hilbertstyle de ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Recent graphtheoretic developments [26, 27] in the semantic theory of combination of logics look at a signature as a multigraph (of sorts and connectives) and obtain the induced language as the category of multipaths where formulas appear as morphisms. This idea is carried over to Hilbertstyle deductive systems by looking at an inference rule as a metaedge from its premises to its conclusion. From such a deductive system, a generalized 2category is induced where proofs appear as 2cells. Vertical composition is used for concatenating proofs and horizontal composition for instantiation. The workabilityofthe approachis illustrated by defining freeandsynchronizedvariantsofparallelcompositionofdeductivesystems and proving the conservative nature of the free variant.
Graphtheoretic Fibring of Logics Part I Completeness
, 2008
"... It is well known that interleaving presentations is at the heart of fibring, as shown by the mechanism of fibring languages and deduction systems. This idea is abstractly introduced herein at the level of the general notion of mgraph (that is, a graph where each edge can have a finite sequence of n ..."
Abstract
 Add to MetaCart
It is well known that interleaving presentations is at the heart of fibring, as shown by the mechanism of fibring languages and deduction systems. This idea is abstractly introduced herein at the level of the general notion of mgraph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as mgraphs. After defining a category freely generated by a mgraph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness results are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics and subsume all logics endowed with an algebraic semantics. 1
Graphtheoretic Fibring of Logics Part II Completeness Preservation
, 2008
"... A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring. Signatures, interpretation structures and deductive systems are defined as enriched graphs. This graphtheoretic view is general enough to accommodate very different propositiona ..."
Abstract
 Add to MetaCart
A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring. Signatures, interpretation structures and deductive systems are defined as enriched graphs. This graphtheoretic view is general enough to accommodate very different propositional based logics encompassing logics with nondeterministic semantics, logics with an algebraic semantics, logics with partial semantics, substructural logics, among others. Fibring is seen as a universal construction in the category of logic systems. Graphtheoretic fibring allows the explicit construction of the interpretation structure resulting from the fibring of a pair of interpretation structures. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems is avoided. 1
Interpolation via translations
"... A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties betwe ..."
Abstract
 Add to MetaCart
A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a KolmogorovGentzenGödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a KiriyamaOno style translation with features of a KolmogorovGentzenGödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics. AMS Classification: 03C40, 03F03, 03B22