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Probabilistic logic over equations and domain restrictions

Published online by Cambridge University Press:  08 March 2019

Andreia Mordido  and
Carlos Caleiro
Andreia Mordido*
Affiliation:
SQIG – Instituto de Telecomunicações, Lisbon, Portugal LASIGE, Faculdade de Ciências, Universidade de Lisboa, Portugal
Carlos Caleiro
Affiliation:
SQIG – Instituto de Telecomunicações, Lisbon, Portugal Department of Mathematics, IST – Universidade de Lisboa, Lisbon, Portugal
*
*Corresponding author. Email: afmordido@fc.ul.pt
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Abstract

We propose and study a probabilistic logic over an algebraic basis, including equations and domain restrictions. The logic combines aspects from classical logic and equational logic with an exogenous approach to quantitative probabilistic reasoning. We present a sound and weakly complete axiomatization for the logic, parameterized by an equational specification of the algebraic basis coupled with the intended domain restrictions.We show that the satisfiability problem for the logic is decidable, under the assumption that its algebraic basis is given by means of a convergent rewriting system, and, additionally, that the axiomatization of domain restrictions enjoys a suitable subterm property. For this purpose, we provide a polynomial reduction to Satisfiability Modulo Theories. As a consequence, we get that validity in the logic is also decidable. Furthermore, under the assumption that the rewriting system that defines the equational basis underlying the logic is also subterm convergent, we show that the resulting satisfiability problem is NP-complete, and thus the validity problem is coNP-complete.We test the logic with meaningful examples in information security, namely by verifying and estimating the probability of the existence of offline guessing attacks to cryptographic protocols.

Keywords

probabilistic logicequational logicsatisfiability problemofffline guessing attacks
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 6 , June 2019 , pp. 872 - 895
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Work done under the scope of R&D Unit 50008, financed by the applicable financial framework (FCT/MEC through national funds and when applicable co-funded by FEDER–PT2020). The first author was supported by FCT under the grant SFRH/BD/77648/2011 and by the Calouste Gulbenkian Foundation under Programa de Estímulo à Investigação 2011. The second author also acknowledges the support of EU FP7Marie Curie PIRSES-GA-2012-318986 project GeTFun: Generalizing Truth-Functionality. Work also supported by LASIGE Research Unit, ref. UID/CEC/00408/2019.

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