54 research outputs found
A Characterisation of Medial as Rewriting Rule
International audienceMedial is an inference rule scheme that appears in various deductive systems based on deep inference. In this paper we investigate the properties of medial as rewriting rule independently from logic. We present a graph theoretical criterion for checking whether there exists a medial rewriting path between two formulas. Finally, we return to logic and apply our criterion for giving a combinatorial proof for a decomposition theorem, i.e., proof theoretical statement about syntax
From Proof Nets to the Free *-Autonomous Category
In the first part of this paper we present a theory of proof nets for full
multiplicative linear logic, including the two units. It naturally extends the
well-known theory of unit-free multiplicative proof nets. A linking is no
longer a set of axiom links but a tree in which the axiom links are subtrees.
These trees will be identified according to an equivalence relation based on a
simple form of graph rewriting. We show the standard results of
sequentialization and strong normalization of cut elimination. In the second
part of the paper we show that the identifications enforced on proofs are such
that the class of two-conclusion proof nets defines the free *-autonomous
category.Comment: LaTeX, 44 pages, final version for LMCS; v2: updated bibliograph
On Nested Sequents for Constructive Modal Logics
We present deductive systems for various modal logics that can be obtained
from the constructive variant of the normal modal logic CK by adding
combinations of the axioms d, t, b, 4, and 5. This includes the constructive
variants of the standard modal logics K4, S4, and S5. We use for our
presentation the formalism of nested sequents and give a syntactic proof of cut
elimination.Comment: 33 page
Non-crossing tree realizations of ordered degree sequences
We investigate the enumeration of non-crossing tree realizations of integer sequences, and we consider a special case in four parameters, that can be seen as a four-dimensional tetrahedron that generalizes Pascal's triangle and the Catalan numbers
A System of Interaction and Structure IV: The Exponentials and Decomposition
We study a system, called NEL, which is the mixed commutative/non-commutative
linear logic BV augmented with linear logic's exponentials. Equivalently, NEL
is MELL augmented with the non-commutative self-dual connective seq. In this
paper, we show a basic compositionality property of NEL, which we call
decomposition. This result leads to a cut-elimination theorem, which is proved
in the next paper of this series. To control the induction measure for the
theorem, we rely on a novel technique that extracts from NEL proofs the
structure of exponentials, into what we call !-?-Flow-Graphs
No complete linear term rewriting system for propositional logic
International audienceRecently it has been observed that the set of all sound linear inference rules in propositional logic is already coNP-complete, i.e. that every Boolean tautology can be written as a (left-and right-) linear rewrite rule. This raises the question of whether there is a rewriting system on linear terms of propositional logic that is sound and complete for the set of all such rewrite rules. We show in this paper that, as long as reduction steps are polynomial-time decidable, such a rewriting system does not exist unless coNP = NP. We draw tools and concepts from term rewriting, Boolean function theory and graph theory in order to access the required intermediate results. At the same time we make several connections between these areas that, to our knowledge, have not yet been presented and constitute a rich theoretical framework for reasoning about linear TRSs for propositional logic. 1998 ACM Subject Classification F.4 Mathematical Logic and Formal Language
On linear rewriting systems for Boolean logic and some applications to proof theory
Linear rules have played an increasing role in structural proof theory in
recent years. It has been observed that the set of all sound linear inference
rules in Boolean logic is already coNP-complete, i.e. that every Boolean
tautology can be written as a (left- and right-)linear rewrite rule. In this
paper we study properties of systems consisting only of linear inferences. Our
main result is that the length of any 'nontrivial' derivation in such a system
is bound by a polynomial. As a consequence there is no polynomial-time
decidable sound and complete system of linear inferences, unless coNP=NP. We
draw tools and concepts from term rewriting, Boolean function theory and graph
theory in order to access some required intermediate results. At the same time
we make several connections between these areas that, to our knowledge, have
not yet been presented and constitute a rich theoretical framework for
reasoning about linear TRSs for Boolean logic.Comment: 27 pages, 3 figures, special issue of RTA 201
On the Axiomatisation of Boolean Categories with and without Medial
In its most general meaning, a Boolean category is to categories what a Boolean algebra is to posets. In a more specific meaning a Boolean category should provide the abstract algebraic structure underlying the proofs in Boolean Logic, in the same sense as a Cartesian closed category captures the proofs in intuitionistic logic and a *-autonomous category captures the proofs in linear logic. However, recent work has shown that there is no canonical axiomatisation of a Boolean category. In this work, we will see a series (with increasing strength) of possible such axiomatisations, all based on the notion of *-autonomous category. We will particularly focus on the medial map, which has its origin in an inference rule in KS, a cut-free deductive system for Boolean logic in the calculus of structures. Finally, we will present a category proof nets as a particularly well-behaved example of a Boolean category
A Characterisation of Medial as Rewriting Rule
International audienceMedial is an inference rule scheme that appears in various deductive systems based on deep inference. In this paper we investigate the properties of medial as rewriting rule independently from logic. We present a graph theoretical criterion for checking whether there exists a medial rewriting path between two formulas. Finally, we return to logic and apply our criterion for giving a combinatorial proof for a decomposition theorem, i.e., proof theoretical statement about syntax
What is a logic, and what is a proof ?
International audienceI will discuss the two problems of how to define identity between logics and how to define identity between proofs. For the identity of logics, I propose to simply use the notion of preorder equivalence. This might be considered to be folklore, but is exactly what is needed from the viewpoint of the problem of the identity of proofs: If the proofs are considered to be part of the logic, then preorder equivalence becomes equivalence of categories, whose arrows are the proofs. For identifying these, the concept of proof nets is discussed
- …