146 research outputs found
A System of Interaction and Structure III: The Complexity of BV and Pomset Logic
Pomset logic and BV are both logics that extend multiplicative linear logic
(with Mix) with a third connective that is self-dual and non-commutative.
Whereas pomset logic originates from the study of coherence spaces and proof
nets, BV originates from the study of series-parallel orders, cographs, and
proof systems. Both logics enjoy a cut-admissibility result, but for neither
logic can this be done in the sequent calculus. Provability in pomset logic can
be checked via a proof net correctness criterion and in BV via a deep inference
proof system. It has long been conjectured that these two logics are the same.
In this paper we show that this conjecture is false. We also investigate the
complexity of the two logics, exhibiting a huge gap between the two. Whereas
provability in BV is NP-complete, provability in pomset logic is
-complete. We also make some observations with respect to possible
sequent systems for the two logics
BV and Pomset Logic Are Not the Same
BV and pomset logic are two logics that both conservatively extend unit-free multiplicative linear logic by a third binary connective, which (i) is non-commutative, (ii) is self-dual, and (iii) lies between the "par" and the "tensor". It was conjectured early on (more than 20 years ago), that these two logics, that share the same language, that both admit cut elimination, and whose connectives have essentially the same properties, are in fact the same. In this paper we show that this is not the case. We present a formula that is provable in pomset logic but not in BV
The Grail theorem prover: Type theory for syntax and semantics
As the name suggests, type-logical grammars are a grammar formalism based on
logic and type theory. From the prespective of grammar design, type-logical
grammars develop the syntactic and semantic aspects of linguistic phenomena
hand-in-hand, letting the desired semantics of an expression inform the
syntactic type and vice versa. Prototypical examples of the successful
application of type-logical grammars to the syntax-semantics interface include
coordination, quantifier scope and extraction.This chapter describes the Grail
theorem prover, a series of tools for designing and testing grammars in various
modern type-logical grammars which functions as a tool . All tools described in
this chapter are freely available
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
Pomset logic: a logical and grammatical alternative to the Lambek calculus
Thirty years ago, I introduced a non commutative variant of classical linear
logic, called POMSET LOGIC, issued from a particular denotational semantics or
categorical interpretation of linear logic known as coherence spaces. In
addition to the multiplicative connectives of linear logic, pomset logic
includes a non-commutative connective, "" called BEFORE, which is
associative and self-dual: (observe that there
is no swapping), and pomset logic handles Partially Ordered MultiSETs of
formulas. This classical calculus enjoys a proof net calculus, cut-elimination,
denotational semantics, but had no sequent calculus, despite my many attempts
and the study of closely related deductive systems like the calculus of
structures. At the same period, Alain Lecomte introduced me to Lambek calculus
and grammars. We defined a grammatical formalism based on pomset logic, with
partial proof nets as the deductive systems for parsing-as-deduction, with a
lexicon mapping words to partial proof nets. The study of pomset logic and of
its grammatical applications has been out of the limelight for several years,
in part because computational linguists were not too keen on proof nets.
However, recently Sergey Slavnov found a sequent calculus for pomset logic, and
reopened the study of pomset logic. In this paper we shall present pomset logic
including both published and unpublished material. Just as for Lambek calculus,
Pomset logic also is a non commutative variant of linear logic --- although
Lambek calculus appeared 30 years before linear logic ! --- and as in Lambek
calculus it may be used as a grammar. Apart from this the two calculi are quite
different, but perhaps the algebraic presentation we give here, with terms and
the semantic correctness criterion, is closer to Lambek's view
Normalisation Control in Deep Inference via Atomic Flows
We introduce `atomic flows': they are graphs obtained from derivations by
tracing atom occurrences and forgetting the logical structure. We study simple
manipulations of atomic flows that correspond to complex reductions on
derivations. This allows us to prove, for propositional logic, a new and very
general normalisation theorem, which contains cut elimination as a special
case. We operate in deep inference, which is more general than other syntactic
paradigms, and where normalisation is more difficult to control. We argue that
atomic flows are a significant technical advance for normalisation theory,
because 1) the technique they support is largely independent of syntax; 2)
indeed, it is largely independent of logical inference rules; 3) they
constitute a powerful geometric formalism, which is more intuitive than syntax
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