907 research outputs found
Cirquent calculus deepened
Cirquent calculus is a new proof-theoretic and semantic framework, whose main
distinguishing feature is being based on circuits, as opposed to the more
traditional approaches that deal with tree-like objects such as formulas or
sequents. Among its advantages are greater efficiency, flexibility and
expressiveness. This paper presents a detailed elaboration of a deep-inference
cirquent logic, which is naturally and inherently resource conscious. It shows
that classical logic, both syntactically and semantically, is just a special,
conservative fragment of this more general and, in a sense, more basic logic --
the logic of resources in the form of cirquent calculus. The reader will find
various arguments in favor of switching to the new framework, such as arguments
showing the insufficiency of the expressive power of linear logic or other
formula-based approaches to developing resource logics, exponential
improvements over the traditional approaches in both representational and proof
complexities offered by cirquent calculus, and more. Among the main purposes of
this paper is to provide an introductory-style starting point for what, as the
author wishes to hope, might have a chance to become a new line of research in
proof theory -- a proof theory based on circuits instead of formulas.Comment: Significant improvements over the previous version
Shortening QBF Proofs with Dependency Schemes
We provide the first proof complexity results for QBF dependency calculi. By showing that the reflexive resolution path dependency scheme admits exponentially shorter Q-resolution proofs on a known family of instances, we answer a question first posed by Slivovsky and Szeider in 2014 [30]. Further, we conceive a method of QBF solving in which dependency recomputation is utilised as a form of inprocessing. Formalising this notion, we introduce a new calculus in which a dependency scheme is applied dynamically. We demonstrate the further potential of this approach beyond that of the existing static system with an exponential separation
A feasible algorithm for typing in Elementary Affine Logic
We give a new type inference algorithm for typing lambda-terms in Elementary
Affine Logic (EAL), which is motivated by applications to complexity and
optimal reduction. Following previous references on this topic, the variant of
EAL type system we consider (denoted EAL*) is a variant without sharing and
without polymorphism. Our algorithm improves over the ones already known in
that it offers a better complexity bound: if a simple type derivation for the
term t is given our algorithm performs EAL* type inference in polynomial time.Comment: 20 page
Some observations on the logical foundations of inductive theorem proving
In this paper we study the logical foundations of automated inductive theorem
proving. To that aim we first develop a theoretical model that is centered
around the difficulty of finding induction axioms which are sufficient for
proving a goal.
Based on this model, we then analyze the following aspects: the choice of a
proof shape, the choice of an induction rule and the language of the induction
formula. In particular, using model-theoretic techniques, we clarify the
relationship between notions of inductiveness that have been considered in the
literature on automated inductive theorem proving. This is a corrected version
of the paper arXiv:1704.01930v5 published originally on Nov.~16, 2017
Shaping Social Activity by Incentivizing Users
Events in an online social network can be categorized roughly into endogenous
events, where users just respond to the actions of their neighbors within the
network, or exogenous events, where users take actions due to drives external
to the network. How much external drive should be provided to each user, such
that the network activity can be steered towards a target state? In this paper,
we model social events using multivariate Hawkes processes, which can capture
both endogenous and exogenous event intensities, and derive a time dependent
linear relation between the intensity of exogenous events and the overall
network activity. Exploiting this connection, we develop a convex optimization
framework for determining the required level of external drive in order for the
network to reach a desired activity level. We experimented with event data
gathered from Twitter, and show that our method can steer the activity of the
network more accurately than alternatives
The symmetry rule in propositional logic
AbstractThe addition of the symmetry rule to the resolution system sometimes allows considerable shortening in the length of refutations. We prove exponential lower bounds on the size of resolution refutations using two forms of a global symmetry rule. The paper also discusses the relationship of symmetry rules to the extension rule that allows the use of abbreviative definitions in proofs
Investigations into Proof Structures
We introduce and elaborate a novel formalism for the manipulation and
analysis of proofs as objects in a global manner. In this first approach the
formalism is restricted to first-order problems characterized by condensed
detachment. It is applied in an exemplary manner to a coherent and
comprehensive formal reconstruction and analysis of historical proofs of a
widely-studied problem due to {\L}ukasiewicz. The underlying approach opens the
door towards new systematic ways of generating lemmas in the course of proof
search to the effects of reducing the search effort and finding shorter proofs.
Among the numerous reported experiments along this line, a proof of
{\L}ukasiewicz's problem was automatically discovered that is much shorter than
any proof found before by man or machine.Comment: This article is a continuation of arXiv:2104.1364
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