719 research outputs found
Nominal Unification from a Higher-Order Perspective
Nominal Logic is a version of first-order logic with equality, name-binding,
renaming via name-swapping and freshness of names. Contrarily to higher-order
logic, bindable names, called atoms, and instantiable variables are considered
as distinct entities. Moreover, atoms are capturable by instantiations,
breaking a fundamental principle of lambda-calculus. Despite these differences,
nominal unification can be seen from a higher-order perspective. From this
view, we show that nominal unification can be reduced to a particular fragment
of higher-order unification problems: Higher-Order Pattern Unification. This
reduction proves that nominal unification can be decided in quadratic
deterministic time, using the linear algorithm for Higher-Order Pattern
Unification. We also prove that the translation preserves most generality of
unifiers
Unification in the Description Logic EL
The Description Logic EL has recently drawn considerable attention since, on
the one hand, important inference problems such as the subsumption problem are
polynomial. On the other hand, EL is used to define large biomedical
ontologies. Unification in Description Logics has been proposed as a novel
inference service that can, for example, be used to detect redundancies in
ontologies. The main result of this paper is that unification in EL is
decidable. More precisely, EL-unification is NP-complete, and thus has the same
complexity as EL-matching. We also show that, w.r.t. the unification type, EL
is less well-behaved: it is of type zero, which in particular implies that
there are unification problems that have no finite complete set of unifiers.Comment: 31page
Productive Corecursion in Logic Programming
Logic Programming is a Turing complete language. As a consequence, designing
algorithms that decide termination and non-termination of programs or decide
inductive/coinductive soundness of formulae is a challenging task. For example,
the existing state-of-the-art algorithms can only semi-decide coinductive
soundness of queries in logic programming for regular formulae. Another, less
famous, but equally fundamental and important undecidable property is
productivity. If a derivation is infinite and coinductively sound, we may ask
whether the computed answer it determines actually computes an infinite
formula. If it does, the infinite computation is productive. This intuition was
first expressed under the name of computations at infinity in the 80s. In
modern days of the Internet and stream processing, its importance lies in
connection to infinite data structure processing.
Recently, an algorithm was presented that semi-decides a weaker property --
of productivity of logic programs. A logic program is productive if it can give
rise to productive derivations. In this paper we strengthen these recent
results. We propose a method that semi-decides productivity of individual
derivations for regular formulae. Thus we at last give an algorithmic
counterpart to the notion of productivity of derivations in logic programming.
This is the first algorithmic solution to the problem since it was raised more
than 30 years ago. We also present an implementation of this algorithm.Comment: Paper presented at the 33nd International Conference on Logic
Programming (ICLP 2017), Melbourne, Australia, August 28 to September 1, 2017
16 pages, LaTeX, no figure
Revisiting Chase Termination for Existential Rules and their Extension to Nonmonotonic Negation
Existential rules have been proposed for representing ontological knowledge,
specifically in the context of Ontology- Based Data Access. Entailment with
existential rules is undecidable. We focus in this paper on conditions that
ensure the termination of a breadth-first forward chaining algorithm known as
the chase. Several variants of the chase have been proposed. In the first part
of this paper, we propose a new tool that allows to extend existing acyclicity
conditions ensuring chase termination, while keeping good complexity
properties. In the second part, we study the extension to existential rules
with nonmonotonic negation under stable model semantics, discuss the relevancy
of the chase variants for these rules and further extend acyclicity results
obtained in the positive case.Comment: This paper appears in the Proceedings of the 15th International
Workshop on Non-Monotonic Reasoning (NMR 2014
Variant-Based Satisfiability
Although different satisfiability decision procedures
can be combined by algorithms such as those of Nelson-Oppen or
Shostak, current tools typically can only support a finite number of
theories to use in such combinations. To make SMT solving more
widely applicable, generic satisfiability algorithms that can
allow a potentially infinite number of decidable theories to be
user-definable, instead of needing to be built in by the
implementers, are highly desirable. This work studies how
folding variant narrowing, a generic
unification algorithm that offers
good extensibility in unification theory, can be extended to
a generic variant-based satisfiability algorithm for the initial
algebras of its user-specified input theories when such theories
satisfy Comon-Delaune's finite variant property (FVP) and some
extra conditions. Several, increasingly larger infinite classes of
theories whose initial algebras enjoy decidable variant-based satisfiability
are identified, and a method based on descent maps to bring other theories
into these classes and to improve the generic
algorithm's efficiency is proposed and illustrated with examples.Partially supported by NSF Grant CNS 13-19109.Ope
Variant-based Equational Unification under Constructor Symbols
Equational unification of two terms consists of finding a substitution that,
when applied to both terms, makes them equal modulo some equational properties.
A narrowing-based equational unification algorithm relying on the concept of
the variants of a term is available in the most recent version of Maude,
version 3.0, which provides quite sophisticated unification features. A variant
of a term t is a pair consisting of a substitution sigma and the canonical form
of tsigma. Variant-based unification is decidable when the equational theory
satisfies the finite variant property. However, this unification procedure does
not take into account constructor symbols and, thus, may compute many more
unifiers than the necessary or may not be able to stop immediately. In this
paper, we integrate the notion of constructor symbol into the variant-based
unification algorithm. Our experiments on positive and negative unification
problems show an impressive speedup.Comment: In Proceedings ICLP 2020, arXiv:2009.09158. arXiv admin note:
substantial text overlap with arXiv:1909.0824
Asymmetric Unification: A New Unification Paradigm for Cryptographic Protocol Analysis
The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-38574-2_16We present a new paradigm for unification arising out of a technique commonly used in cryptographic protocol analysis tools that employ unification modulo equational theories. This paradigm relies on: (i) a decomposition of an equational theory into (R,E) where R is confluent, terminating, and coherent modulo E, and (ii) on reducing unification problems to a set of problems s=?ts=?t under the constraint that t remains R/E-irreducible. We call this method asymmetric unification. We first present a general-purpose generic asymmetric unification algorithm. and then outline an approach for converting special-purpose conventional unification algorithms to asymmetric ones, demonstrating it for exclusive-or with uninterpreted function symbols. We demonstrate how asymmetric unification can improve performanceby running the algorithm on a set of benchmark problems. We also give results on the complexity and decidability of asymmetric unification.S. Escobar and S. Santiago were partially supported by EU (FEDER) and the Spanish MEC/MICINN under grant TIN 2010-21062-C02-02, and by Generalitat Valenciana PROMETEO2011/052. The following authors were partially supported by
NSF: S. Escobar, J. Meseguer, and R. Sasse under CNS 09-04749 and CCF 09-
05584; D. Kapur under CNS 09-05222; C. Lynch, Z. Liu, and C. Meadows under
CNS 09-05378, and P. Narendran and S. Erbatur under CNS 09-05286. Part of the
S. Erbatur’s work was supported while with the Department of Computer Science,
University at Albany, and part of R. Sasse’s work was supported while with the Department of Computer Science, University of Illinois at Urbana-Champaign.Erbatur, S.; Escobar Román, S.; Kapur, D.; Liu, Z.; Lynch, CA.; Meadows, C.; Meseguer, J.... (2013). Asymmetric Unification: A New Unification Paradigm for Cryptographic Protocol Analysis. En Automated Deduction – CADE-24. Springer. 231-248. https://doi.org/10.1007/978-3-642-38574-2_16S231248IEEE 802.11 Local and Metropolitan Area Networks: Wireless LAN Medium Access Control (MAC) and Physical (PHY) Specifications (1999)Basin, D., Mödersheim, S., Viganò, L.: An on-the-fly model-checker for security protocol analysis. In: Snekkenes, E., Gollmann, D. (eds.) ESORICS 2003. LNCS, vol. 2808, pp. 253–270. Springer, Heidelberg (2003)Blanchet, B.: An efficient cryptographic protocol verifier based on Prolog rules. In: CSFW, pp. 82–96. IEEE Computer Society (2001)Bürckert, H.-J., Herold, A., Schmidt-Schauß, M.: On equational theories, unification, and (un)decidability. Journal of Symbolic Computation 8(1/2), 3–49 (1989)Comon-Lundh, H., Delaune, S.: The finite variant property: How to get rid of some algebraic properties. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 294–307. Springer, Heidelberg (2005)Durán, F., Meseguer, J.: A Maude coherence checker tool for conditional order-sorted rewrite theories. In: Ölveczky, P.C. (ed.) WRLA 2010. LNCS, vol. 6381, pp. 86–103. Springer, Heidelberg (2010)Erbatur, S., Escobar, S., Kapur, D., Liu, Z., Lynch, C., Meadows, C., Meseguer, J., Narendran, P., Santiago, S., Sasse, R.: Effective symbolic protocol analysis via equational irreducibility conditions. In: Foresti, S., Yung, M., Martinelli, F. (eds.) ESORICS 2012. LNCS, vol. 7459, pp. 73–90. Springer, Heidelberg (2012)Erbatur, S., Escobar, S., Kapur, D., Liu, Z., Lynch, C., Meadows, C., Meseguer, J., Narendran, P., Sasse, R.: Asymmetric unification: A new unification paradigm for cryptographic protocol analysis. In: UNIF 2011 (2011), https://sites.google.com/a/cs.uni.wroc.pl/unif-2011/programEscobar, S., Sasse, R., Meseguer, J.: Folding variant narrowing and optimal variant termination. J. Log. Algebr. Program. 81(7-8), 898–928 (2012)Harju, T., Karhumäki, J., Krob, D.: Remarks on generalized post correspondence problem. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 39–48. Springer, Heidelberg (1996)Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to automata theory, languages, and computation - international edition, 2nd edn. Addison-Wesley (2003)Jouannaud, J.-P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM J. Comput. 15(4), 1155–1194 (1986)Liu, Z., Lynch, C.: Efficient general unification for XOR with homomorphism. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 407–421. Springer, Heidelberg (2011)Liu, Z.: Dealing Efficiently with Exclusive OR, Abelian Groups and Homomorphism in Cryptographic Protocol Analysis. PhD thesis, Clarkson University (2012), http://people.clarkson.edu/~clynch/papers/Dissertation_of_Zhiqiang_Liu.pdfLowe, G., Roscoe, A.W.R.: Using CSP to detect errors in the TMN protocol. IEEE Transactions on Software Engineering 23, 659–669 (1997)Meseguer, J.: Conditional rewriting logic as a united model of concurrency. Theor. Comput. Sci. 96(1), 73–155 (1992)Schmidt, B., Meier, S., Cremers, C.J.F., Basin, D.A.: Automated analysis of Diffie-Hellman protocols and advanced security properties. In: Proc. CSF 2012, pp. 78–94. IEEE (2012)Tatebayashi, M., Matsuzaki, N., Newman Jr., D.B.: Key distribution protocol for digital mobile communication systems. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 324–334. Springer, Heidelberg (1990)TeReSe, editor. Term Rewriting Systems. Cambridge University Press (2003)Viry, P.: Equational rules for rewriting logic. Theor. Comp. Sci. 285(2), 487–517 (2002
Finitary Deduction Systems
Cryptographic protocols are the cornerstone of security in distributed
systems. The formal analysis of their properties is accordingly one of the
focus points of the security community, and is usually split among two groups.
In the first group, one focuses on trace-based security properties such as
confidentiality and authentication, and provides decision procedures for the
existence of attacks for an on-line attackers. In the second group, one focuses
on equivalence properties such as privacy and guessing attacks, and provides
decision procedures for the existence of attacks for an offline attacker. In
all cases the attacker is modeled by a deduction system in which his possible
actions are expressed. We present in this paper a notion of finitary deduction
systems that aims at relating both approaches. We prove that for such deduction
systems, deciding equivalence properties for on-line attackers can be reduced
to deciding reachability properties in the same setting.Comment: 30 pages. Work begun while in the CASSIS Project, INRIA Nancy Grand
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