Results 1  10
of
18
Noncommutative logic I : the multiplicative fragment
, 1998
"... INTRODUCTION Unrestricted exchange rules of Girard's linear logic [8] force the commutativity of the multiplicative connectives\Omega (times, conjunction) and & (par, disjunction) , and henceforth the commutativity of all logic. This a priori commutativity is not always desirable  it is ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
INTRODUCTION Unrestricted exchange rules of Girard's linear logic [8] force the commutativity of the multiplicative connectives\Omega (times, conjunction) and & (par, disjunction) , and henceforth the commutativity of all logic. This a priori commutativity is not always desirable  it is quite problematic in applications like linguistics or computer science , and actually the desire of a noncommutative logic goes back to the very beginning of LL [9]. Previous works on noncommutativity deal essentially with noncommutative fragments of LL, obtained by removing the exchange rule at all. At that point, a simple remark on the status of exchange in the sequent calculus is necessary to be clear: there are two presentations of exchange in commutative LL, either sequents are finite sets of occurrences of formulas and exchange is obviously implicit, or sequents are fini
Lambek calculus is npcomplete
 Theoretical Computer Science
, 2003
"... We prove that for both the Lambek calculus L and the Lambek calculus allowing empty premises L ∗ the derivability problem is NPcomplete. It follows that also for the multiplicative fragments of cyclic linear logic and noncommutative linear logic the derivability problem is NPcomplete. ..."
Abstract

Cited by 23 (0 self)
 Add to MetaCart
We prove that for both the Lambek calculus L and the Lambek calculus allowing empty premises L ∗ the derivability problem is NPcomplete. It follows that also for the multiplicative fragments of cyclic linear logic and noncommutative linear logic the derivability problem is NPcomplete.
Types as graphs: Continuations in type logical grammar
, 2005
"... Using the programminglanguage concept of CONTINUATIONS, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of insitu quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as c ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
Using the programminglanguage concept of CONTINUATIONS, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of insitu quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, typelogical way to model evaluation order and side effects in natural language. We illustrate with an improved account of quantificational binding, weak crossover, whquestions, superiority, and polarity licensing.
Model Theoretic Syntax
 The Glot International State of the Article Book 1, Studies in Generative Grammar 48, Mouton de Gruyter
, 1998
"... this article appeared in Glot, the main issue agitating researchers in model theoretic syntax was the problem of the contextfree barrier. We have seen that the hierarchy of logics collapses, when applied to trees, at the border of the tree languages strongly generated by context free (string) gramm ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
this article appeared in Glot, the main issue agitating researchers in model theoretic syntax was the problem of the contextfree barrier. We have seen that the hierarchy of logics collapses, when applied to trees, at the border of the tree languages strongly generated by context free (string) grammars, in the sense that distinctions between the different tree logics reduce to apparently superficial distinctions in how much memory allocation is hidden in the logic. The problem which researchers set themselves was not just breaking the context free barrier but remaining decidable in the process. This is a very difficult problem, and it must be admitted right off that it is somewhat artificial in that there is no a priori reason to suppose that natural languages can be described in a decidable logic. The arguments on either side are something like the following. First, the rather slight increases in computational complexity required to get the "mildly context sensitive" languages do suggest that this might be possible. The hunch here would be that the qualities that characterize the mildly context sensitive languages (polynomial parsability, constant growth property) as being like the contextfree languages are going to turn out to be reflections of decidability. The problems must not be underestimated, however! It is well known that the monadic second order logic of trees is one of the most powerful decidable logics known. It seems unlikely that any primitive relations can be added to the repertoire of tree description primitives that we have already seen, without making the logic undecidable. Many attempts have been made within logic and all have failed. So it is equally tempting to conjecture that the contextfree boundary coincides in some deep sense with the bounda...
Geometry of LexicoSyntactic Interaction
, 1999
"... Interaction of lexical and derivational semanticsfor example substitution and lambda conversion is typically a part of the online interpretation process. Proofnets are to categorial grammar what phrase markers are to phrase structure grammar: unique graphical structures underlying equivalenc ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Interaction of lexical and derivational semanticsfor example substitution and lambda conversion is typically a part of the online interpretation process. Proofnets are to categorial grammar what phrase markers are to phrase structure grammar: unique graphical structures underlying equivalence classes of sequential syntactic derivations; but the role of proofnets is deeper since they integrate also semantics. In this paper we show how interaction of lexical and derivational semantics at the lexicosyntactic interface can be precomputed as a process of offline lexical compilation comprising Cut elimination in partial proofnets.
A Survey of Linear Logic Programming
 Computational Logic
, 1995
"... Abstract. Logic programming can be given a foundation in sequent calculus, viewing computation as the process of building a cutfree sequent proof from the bottomup. Earliest accounts of logic programming were based in classical logic and then later in intuitionistic logic. The use of linear logic ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. Logic programming can be given a foundation in sequent calculus, viewing computation as the process of building a cutfree sequent proof from the bottomup. Earliest accounts of logic programming were based in classical logic and then later in intuitionistic logic. The use of linear logic to design new logic programming languages was inevitable given that it allows for more dynamics in the way sequents change during the search for a proof and since it can account for logic programming in these other logics. We overview how linear logic has been used to design new logic programming languages and describe some applications and implementation issues for such languages. §1. Introduction. It is now common place to recognize the important role of logic in the foundations of computer science in general and programming languages more specifically. For this reason, when a major new advance is made in our understanding of logic, we can expect to see that advance ripple into other areas of computer science. Such rippling has been observed during the years
A Substructural Type System for Delimited Continuations
"... We propose type systems that abstractly interpret smallstep rather than bigstep operational semantics. We treat an expression or evaluation context as a structure in a linear logic with hypothetical reasoning. Evaluation order is not only regulated by familiar focusing rules in the operational sem ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We propose type systems that abstractly interpret smallstep rather than bigstep operational semantics. We treat an expression or evaluation context as a structure in a linear logic with hypothetical reasoning. Evaluation order is not only regulated by familiar focusing rules in the operational semantics, but also expressed by structural rules in the type system, so the types track control flow more closely. Binding and evaluation contexts are related, but the latter are linear. We use these ideas to build a type system for delimited continuations. It lets control operators change the answer type or act beyond the nearest dynamicallyenclosing delimiter, yet needs no extra fields in judgments and arrow types to record answer types. The typing derivation of a directstyle program desugars it into continuationpassing style.
On generalized Ajdukiewicz and Lambek calculi and grammars
 Fundamenta Informaticae 30
, 1997
"... Investigations concerning the description of languages in terms of categorial grammar started with examining concatenation as the only syntactic operation. Type reduction systems such as associative Ajdukiewicz or Lambek calculi, which admit concatenation of types... ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Investigations concerning the description of languages in terms of categorial grammar started with examining concatenation as the only syntactic operation. Type reduction systems such as associative Ajdukiewicz or Lambek calculi, which admit concatenation of types...
3.2.1. Formal Grammars 2 3.2.2. HighLevel Syntactic Formalisms 3
"... Linguistic signs, grammar and meaning: computational logic for natural language ..."
Abstract
 Add to MetaCart
Linguistic signs, grammar and meaning: computational logic for natural language