اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدCut Elimination in Multifocused Linear Logic
81
0
0.0
(
0
)
تأليف
Taus Brock-Nannestad
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study cut elimination for a multifocused variant of full linear logic in the sequent calculus. The multifocused normal form of proofs yields problems that do not appear in a standard focused system, related to the constraints in grouping rule instances in focusing phases. We show that cut elimination can be performed in a sensible way even though the proof requires some specific lemmas to deal with multifocusing phases, and discuss the difficulties arising with cut elimination when considering normal forms of proofs in linear logic.
قيم البحث
اقرأ أيضاً
Linear Logic was introduced by Girard as a resource-sensitive refinement of classical logic. It turned out that full propositional Linear Logic is undecidable (Lincoln, Mitchell, Scedrov, and Shankar) and, hence, it is more expressive than (modalized
) classical or intuitionistic logic. In this paper we focus on the study of the simplest fragments of Linear Logic, such as the one-literal and constant-only fragments (the latter contains no literals at all). Here we demonstrate that all these extremely simple fragments of Linear Logic (one-literal, $bot$-only, and even unit-only) are exactly of the same expressive power as the corresponding fu
First-order logic is typically presented as the study of deduction in a setting with elementary quantification. In this paper, we take another vantage point and conceptualize first-order logic as a linear space that encodes plausibility. Whereas a de
ductive perspective emphasizes how (i.e., process), a space perspective emphasizes where (i.e., location). We explore several consequences that a shift in perspective to signals in space has for first-order logic, including (1) a notion of proof based on orthogonal decomposition, (2) a method for assigning probabilities to sentences that reflects logical uncertainty, and (3) a models as boundary principle that relates the models of a theory to its size.
The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the pro
blems fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our main result is the following meta-theorem: for every graph property P expressible by a first order-logic formula phiin Sigma_3, that is, of the form phi=exists x_1exists x_2cdots exists x_r forall y_1forall y_2cdots forall y_s exists z_1exists z_2cdots exists z_t psi, where psi is a quantifier-free first-order formula, checking whether the elimination distance of a graph to P does not exceed k, is fixed-parameter tractable parameterized by k. Properties of graphs expressible by formulas from Sigma_3 include being of bounded degree, excluding a forbidden subgraph, or containing a bounded dominating set. We complement this theorem by showing that such a general statement does not hold for formulas with even slightly more expressive prefix structure: there are formulas phiin Pi_3, for which computing elimination distance is W[2]-hard.
Process calculi based on logic, such as $pi$DILL and CP, provide a foundation for deadlock-free concurrent programming. However, in previous work, there is a mismatch between the rules for constructing proofs and the term constructors of the $pi$-cal
culus: the fundamental operator for parallel composition does not correspond to any rule of linear logic. Kokke et al. (2019) introduced Hypersequent Classical Processes (HCP), which addresses this mismatch using hypersequents (collections of sequents) to register parallelism in the typing judgements. However, the step from CP to HCP is a big one. As of yet, HCP does not have reduction semantics, and the addition of delayed actions means that CP processes interpreted as HCP processes do not behave as they would in CP. We introduce HCP-, a variant of HCP with reduction semantics and without delayed actions. We prove progress, preservation, and termination, and show that HCP- supports the same communication protocols as CP.
The problem we want to solve is how to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. We start by filtering for linearity the proof terms associated by our Prolog-based theorem prover
for Implicational Intuitionistic Logic. This works, but using for each formula a PSPACE-complete algorithm limits it to very small formulas. We take a few walks back and forth over the bridge between proof terms and theorems, provided by the Curry-Howard isomorphism, and derive step-by-step an efficient algorithm requiring a low polynomial effort per generated theorem. The resulting Prolog program runs in O(N) space for terms of size N and generates in a few hours 7,566,084,686 theorems in the implicational fragment of Linear Intuitionistic Logic together with their proof terms in normal form. As applications, we generate datasets for correctness and scalability testing of linear logic theorem provers and training data for neural networks working on theorem proving challenges. The results in the paper, organized as a literate Prolog program, are fully replicable. Keywords: combinatorial generation of provable formulas of a given size, intuitionistic and linear logic theorem provers, theorems of the implicational fragment of propositional linear intuitionistic logic, Curry-Howard isomorphism, efficient generation of linear lambda terms in normal form, Prolog programs for lambda term generation and theorem proving.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
