778 research outputs found
Automatic Generation of Proof Tactics for Finite-Valued Logics
A number of flexible tactic-based logical frameworks are nowadays available
that can implement a wide range of mathematical theories using a common
higher-order metalanguage. Used as proof assistants, one of the advantages of
such powerful systems resides in their responsiveness to extensibility of their
reasoning capabilities, being designed over rule-based programming languages
that allow the user to build her own `programs to construct proofs' - the
so-called proof tactics.
The present contribution discusses the implementation of an algorithm that
generates sound and complete tableau systems for a very inclusive class of
sufficiently expressive finite-valued propositional logics, and then
illustrates some of the challenges and difficulties related to the algorithmic
formation of automated theorem proving tactics for such logics. The procedure
on whose implementation we will report is based on a generalized notion of
analyticity of proof systems that is intended to guarantee termination of the
corresponding automated tactics on what concerns theoremhood in our targeted
logics
Heuristic Ranking in Tightly Coupled Probabilistic Description Logics
The Semantic Web effort has steadily been gaining traction in the recent
years. In particular,Web search companies are recently realizing that their
products need to evolve towards having richer semantic search capabilities.
Description logics (DLs) have been adopted as the formal underpinnings for
Semantic Web languages used in describing ontologies. Reasoning under
uncertainty has recently taken a leading role in this arena, given the nature
of data found on theWeb. In this paper, we present a probabilistic extension of
the DL EL++ (which underlies the OWL2 EL profile) using Markov logic networks
(MLNs) as probabilistic semantics. This extension is tightly coupled, meaning
that probabilistic annotations in formulas can refer to objects in the
ontology. We show that, even though the tightly coupled nature of our language
means that many basic operations are data-intractable, we can leverage a
sublanguage of MLNs that allows to rank the atomic consequences of an ontology
relative to their probability values (called ranking queries) even when these
values are not fully computed. We present an anytime algorithm to answer
ranking queries, and provide an upper bound on the error that it incurs, as
well as a criterion to decide when results are guaranteed to be correct.Comment: Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty
in Artificial Intelligence (UAI2012
Ruitenburg's Theorem mechanized and contextualized
In 1984, Wim Ruitenburg published a surprising result about periodic
sequences in intuitionistic propositional calculus (IPC). The property
established by Ruitenburg naturally generalizes local finiteness
(intuitionistic logic is not locally finite, even in a single variable).
However, one of the two main goals of this note is to illustrate that most
"natural" non-classical logics failing local finiteness also do not enjoy the
periodic sequence property; IPC is quite unique in separating these properties.
The other goal of this note is to present a Coq formalization of Ruitenburg's
heavily syntactic proof. Apart from ensuring its correctness, the formalization
allows extraction of a program providing a certified implementation of
Ruitenburg's algorithm.Comment: This note has been prepared for the informal (pre-)proceedings of
FICS 2024. The version to be submitted to the post-proceedings volume is
going to be significantly different, focusing on the Coq formalization, as
requested by referees and the P
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Effective Interactive Proofs for Higher-Order Imperative Programs
We present a new approach for constructing and verifying higher-order, imperative programs using the Coq proof assistant. We build on the past work on the Ynot system, which is based on Hoare Type Theory. That original system was a proof of concept, where every program verification was accomplished via laborious manual proofs, with much code devoted to uninteresting low-level details. In this paper, we present a re-implementation of Ynot which makes it possible to implement fully-verified, higher-order imperative programs with reasonable proof burden. At the same time, our new system is implemented entirely in Coq source files, showcasing the versatility of that proof assistant as a platform for research on language design and verification. Both versions of the system have been evaluated with case studies in the verification of imperative data structures, such as hash tables with higher-order iterators. The verification burden in our new system is reduced by at least an order of magnitude compared to the old system, by replacing manual proof with automation. The core of the automation is a simplification procedure for implications in higher-order separation logic, with hooks that allow programmers to add domain-specific simplification rules.
We argue for the effectiveness of our infrastructure by verifying a number of data structures and a packrat parser, and we compare to similar efforts within other projects. Compared to competing approaches to data structure verification, our system includes much less code that must be trusted; namely, about a hundred lines of Coq code defining a program logic. All of our theorems and decision procedures have or build machine-checkable correctness proofs from first principles, removing opportunities for tool bugs to create faulty verifications.Engineering and Applied Science
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Zero-one laws with respect to models of provability logic and two Grzegorczyk logics
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems
Learning-Assisted Automated Reasoning with Flyspeck
The considerable mathematical knowledge encoded by the Flyspeck project is
combined with external automated theorem provers (ATPs) and machine-learning
premise selection methods trained on the proofs, producing an AI system capable
of answering a wide range of mathematical queries automatically. The
performance of this architecture is evaluated in a bootstrapping scenario
emulating the development of Flyspeck from axioms to the last theorem, each
time using only the previous theorems and proofs. It is shown that 39% of the
14185 theorems could be proved in a push-button mode (without any high-level
advice and user interaction) in 30 seconds of real time on a fourteen-CPU
workstation. The necessary work involves: (i) an implementation of sound
translations of the HOL Light logic to ATP formalisms: untyped first-order,
polymorphic typed first-order, and typed higher-order, (ii) export of the
dependency information from HOL Light and ATP proofs for the machine learners,
and (iii) choice of suitable representations and methods for learning from
previous proofs, and their integration as advisors with HOL Light. This work is
described and discussed here, and an initial analysis of the body of proofs
that were found fully automatically is provided
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