13,327 research outputs found
Cut formulae and logic programming
In this paper we present a mechanism to define names for proof-witnesses of formulae and thus to use Gentzen's cut-rule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proof-search by indicating when an instance of the cut-rule should be attempted. By using the cut-rule there are proofs that can be made dramatically shorter. We explain how this idea of using the cut-rule can be applied to the logic of hereditary Harrop formulae.Junta Nacional de Investigação Científica e Tecnológica (JNICT).União Europeia (UE) - Programa ESPRIT - grant BRA 7232 GENTZEN
On Structuring Proof Search for First Order Linear Logic
Full first order linear logic can be presented as an abstract logic
programming language in Miller's system Forum, which yields a sensible
operational interpretation in the 'proof search as computation' paradigm.
However, Forum still has to deal with syntactic details that would normally be
ignored by a reasonable operational semantics. In this respect, Forum improves
on Gentzen systems for linear logic by restricting the language and the form of
inference rules. We further improve on Forum by restricting the class of
formulae allowed, in a system we call G-Forum, which is still equivalent to
full first order linear logic. The only formulae allowed in G-Forum have the
same shape as Forum sequents: the restriction does not diminish expressiveness
and makes G-Forum amenable to proof theoretic analysis. G-Forum consists of two
(big) inference rules, for which we show a cut elimination procedure. This does
not need to appeal to finer detail in formulae and sequents than is provided by
G-Forum, thus successfully testing the internal symmetries of our system.Comment: Author website at http://alessio.guglielmi.name/res
The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them
In sequent calculi, cut elimination is a property that guarantees that any
provable formula can be proven analytically. For example, Gentzen's classical
and intuitionistic calculi LK and LJ enjoy cut elimination. The property is
less studied in coinductive extensions of sequent calculi. In this paper, we
use coinductive Horn clause theories to show that cut is not eliminable in a
coinductive extension of LJ, a system we call CLJ. We derive two further
practical results from this study. We show that CoLP by Gupta et al. gives rise
to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a
novel method of coinductive theory exploration that provides several heuristics
for discovery of cut formulae in CLJ.Comment: Paper presented at the 36th International Conference on Logic
Programming (ICLP 2019), University Of Calabria, Rende (CS), Italy, September
2020, 16 page
Session Types in Abelian Logic
There was a PhD student who says "I found a pair of wooden shoes. I put a
coin in the left and a key in the right. Next morning, I found those objects in
the opposite shoes." We do not claim existence of such shoes, but propose a
similar programming abstraction in the context of typed lambda calculi. The
result, which we call the Amida calculus, extends Abramsky's linear lambda
calculus LF and characterizes Abelian logic.Comment: In Proceedings PLACES 2013, arXiv:1312.221
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