4,381 research outputs found
Rewriting modulo in Deduction modulo
International audienceWe study the termination of rewriting modulo a set of equations in the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In a previous work, we defined general syntactic conditions based on the notion of computable closure for ensuring the termination of the combination of rewriting and beta-reduction. Here, we show that this result is preserved when considering rewriting modulo a set of equations if the equivalence classes generated by these equations are finite, the equations are linear and satisfy general syntactic conditions also based on the notion of computable closure. This includes equations like associativity and commutativity, and provides an original treatment of termination modulo equations
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation
AC-completion efficiently handles equality modulo associative and commutative
function symbols. When the input is ground, the procedure terminates and
provides a decision algorithm for the word problem. In this paper, we present a
modular extension of ground AC-completion for deciding formulas in the
combination of the theory of equality with user-defined AC symbols,
uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our
algorithm, called AC(X), is obtained by augmenting in a modular way ground
AC-completion with the canonizer and solver present for the theory X. This
integration rests on canonized rewriting, a new relation reminiscent to
normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is
proved sound, complete and terminating, and is implemented to extend the core
of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized
Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for
publication by LMCS (Logical Methods in Computer Science
Unification modulo a 2-sorted Equational theory for Cipher-Decipher Block Chaining
We investigate unification problems related to the Cipher Block Chaining
(CBC) mode of encryption. We first model chaining in terms of a simple,
convergent, rewrite system over a signature with two disjoint sorts: list and
element. By interpreting a particular symbol of this signature suitably, the
rewrite system can model several practical situations of interest. An inference
procedure is presented for deciding the unification problem modulo this rewrite
system. The procedure is modular in the following sense: any given problem is
handled by a system of `list-inferences', and the set of equations thus derived
between the element-terms of the problem is then handed over to any
(`black-box') procedure which is complete for solving these element-equations.
An example of application of this unification procedure is given, as attack
detection on a Needham-Schroeder like protocol, employing the CBC encryption
mode based on the associative-commutative (AC) operator XOR. The 2-sorted
convergent rewrite system is then extended into one that fully captures a block
chaining encryption-decryption mode at an abstract level, using no AC-symbols;
and unification modulo this extended system is also shown to be decidable.Comment: 26 page
Key Substitution in the Symbolic Analysis of Cryptographic Protocols (extended version)
Key substitution vulnerable signature schemes are signature schemes that
permit an intruder, given a public verification key and a signed message, to
compute a pair of signature and verification keys such that the message appears
to be signed with the new signature key. A digital signature scheme is said to
be vulnerable to destructive exclusive ownership property (DEO) If it is
computationaly feasible for an intruder, given a public verification key and a
pair of message and its valid signature relatively to the given public key, to
compute a pair of signature and verification keys and a new message such that
the given signature appears to be valid for the new message relatively to the
new verification key. In this paper, we prove decidability of the insecurity
problem of cryptographic protocols where the signature schemes employed in the
concrete realisation have this two properties
A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems
We give a method to prove confluence of term rewriting systems that contain
non-terminating rewrite rules such as commutativity and associativity. Usually,
confluence of term rewriting systems containing such rules is proved by
treating them as equational term rewriting systems and considering E-critical
pairs and/or termination modulo E. In contrast, our method is based solely on
usual critical pairs and it also (partially) works even if the system is not
terminating modulo E. We first present confluence criteria for term rewriting
systems whose rewrite rules can be partitioned into a terminating part and a
possibly non-terminating part. We then give a reduction-preserving completion
procedure so that the applicability of the criteria is enhanced. In contrast to
the well-known Knuth-Bendix completion procedure which preserves the
equivalence relation of the system, our completion procedure preserves the
reduction relation of the system, by which confluence of the original system is
inferred from that of the completed system
Towards Static Analysis of Functional Programs using Tree Automata Completion
This paper presents the first step of a wider research effort to apply tree
automata completion to the static analysis of functional programs. Tree
Automata Completion is a family of techniques for computing or approximating
the set of terms reachable by a rewriting relation. The completion algorithm we
focus on is parameterized by a set E of equations controlling the precision of
the approximation and influencing its termination. For completion to be used as
a static analysis, the first step is to guarantee its termination. In this
work, we thus give a sufficient condition on E and T(F) for completion
algorithm to always terminate. In the particular setting of functional
programs, this condition can be relaxed into a condition on E and T(C) (terms
built on the set of constructors) that is closer to what is done in the field
of static analysis, where abstractions are performed on data.Comment: Proceedings of WRLA'14. 201
Sequential Composition in the Presence of Intermediate Termination (Extended Abstract)
The standard operational semantics of the sequential composition operator
gives rise to unbounded branching and forgetfulness when transparent process
expressions are put in sequence. Due to transparency, the correspondence
between context-free and pushdown processes fails modulo bisimilarity, and it
is not clear how to specify an always terminating half counter. We propose a
revised operational semantics for the sequential composition operator in the
context of intermediate termination. With the revised operational semantics, we
eliminate transparency, allowing us to establish a close correspondence between
context-free processes and pushdown processes. Moreover, we prove the reactive
Turing powerfulness of TCP with iteration and nesting with the revised
operational semantics for sequential composition.Comment: In Proceedings EXPRESS/SOS 2017, arXiv:1709.00049. arXiv admin note:
substantial text overlap with arXiv:1706.0840
Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories
Cyclic data structures, such as cyclic lists, in functional programming are
tricky to handle because of their cyclicity. This paper presents an
investigation of categorical, algebraic, and computational foundations of
cyclic datatypes. Our framework of cyclic datatypes is based on second-order
algebraic theories of Fiore et al., which give a uniform setting for syntax,
types, and computation rules for describing and reasoning about cyclic
datatypes. We extract the "fold" computation rules from the categorical
semantics based on iteration categories of Bloom and Esik. Thereby, the rules
are correct by construction. We prove strong normalisation using the General
Schema criterion for second-order computation rules. Rather than the fixed
point law, we particularly choose Bekic law for computation, which is a key to
obtaining strong normalisation. We also prove the property of "Church-Rosser
modulo bisimulation" for the computation rules. Combining these results, we
have a remarkable decidability result of the equational theory of cyclic data
and fold.Comment: 38 page
- …