1,426 research outputs found
On the confluence of lambda-calculus with conditional rewriting
The confluence of untyped \lambda-calculus with unconditional rewriting is
now well un- derstood. In this paper, we investigate the confluence of
\lambda-calculus with conditional rewriting and provide general results in two
directions. First, when conditional rules are algebraic. This extends results
of M\"uller and Dougherty for unconditional rewriting. Two cases are
considered, whether \beta-reduction is allowed or not in the evaluation of
conditions. Moreover, Dougherty's result is improved from the assumption of
strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. We
also provide examples showing that outside these conditions, modularity of
confluence is difficult to achieve. Second, we go beyond the algebraic
framework and get new confluence results using a restricted notion of
orthogonality that takes advantage of the conditional part of rewrite rules
Rewriting Modulo \beta in the \lambda\Pi-Calculus Modulo
The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with
dependent types where beta-conversion is extended with user-defined rewrite
rules. It is an expressive logical framework and has been used to encode logics
and type systems in a shallow way. Basic properties such as subject reduction
or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo.
However, they hold if the rewrite system generated by the rewrite rules
together with beta-reduction is confluent. But this is too restrictive. To
handle the case where non confluence comes from the interference between the
beta-reduction and rewrite rules with lambda-abstraction on their left-hand
side, we introduce a notion of rewriting modulo beta for the lambda-Pi-calculus
Modulo. We prove that confluence of rewriting modulo beta is enough to ensure
subject reduction and uniqueness of types. We achieve our goal by encoding the
lambda-Pi-calculus Modulo into Higher-Order Rewrite System (HRS). As a
consequence, we also make the confluence results for HRSs available for the
lambda-Pi-calculus Modulo.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
Deciding Confluence and Normal Form Properties of Ground Term Rewrite Systems Efficiently
It is known that the first-order theory of rewriting is decidable for ground
term rewrite systems, but the general technique uses tree automata and often
takes exponential time. For many properties, including confluence (CR),
uniqueness of normal forms with respect to reductions (UNR) and with respect to
conversions (UNC), polynomial time decision procedures are known for ground
term rewrite systems. However, this is not the case for the normal form
property (NFP). In this work, we present a cubic time algorithm for NFP, an
almost cubic time algorithm for UNR, and an almost linear time algorithm for
UNC, improving previous bounds. We also present a cubic time algorithm for CR
Uniqueness of Normal Forms is Decidable for Shallow Term Rewrite Systems
Uniqueness of normal forms (UN=) is an important property of term rewrite systems. UN= is decidable for ground (i.e., variable-free) systems and undecidable in general. Recently it was shown to be decidable for linear, shallow systems. We generalize this previous result and show that this property is decidable for shallow rewrite systems, in contrast to confluence, reachability and other properties, which are all undecidable for flat systems. Our result is also optimal in some sense, since we prove that the UN= property is undecidable for two superclasses of flat systems: left-flat, left-linear systems in which right-hand sides are of depth at most two and right-flat, right-linear systems in which left-hand sides are of depth at most two
On the Expressivity and Applicability of Model Representation Formalisms
A number of first-order calculi employ an explicit model representation
formalism for automated reasoning and for detecting satisfiability. Many of
these formalisms can represent infinite Herbrand models. The first-order
fragment of monadic, shallow, linear, Horn (MSLH) clauses, is such a formalism
used in the approximation refinement calculus. Our first result is a finite
model property for MSLH clause sets. Therefore, MSLH clause sets cannot
represent models of clause sets with inherently infinite models. Through a
translation to tree automata, we further show that this limitation also applies
to the linear fragments of implicit generalizations, which is the formalism
used in the model-evolution calculus, to atoms with disequality constraints,
the formalisms used in the non-redundant clause learning calculus (NRCL), and
to atoms with membership constraints, a formalism used for example in decision
procedures for algebraic data types. Although these formalisms cannot represent
models of clause sets with inherently infinite models, through an additional
approximation step they can. This is our second main result. For clause sets
including the definition of an equivalence relation with the help of an
additional, novel approximation, called reflexive relation splitting, the
approximation refinement calculus can automatically show satisfiability through
the MSLH clause set formalism.Comment: 15 page
Automatically Proving and Disproving Feasibility Conditions
[EN] In the realm of term rewriting, given terms s and t, a reachability condition s>>t is called feasible if there is a substitution O such that O(s) rewrites into O(t) in zero or more steps; otherwise, it is called infeasible. Checking infeasibility of (sequences of) reachability conditions is important in the analysis of computational properties of rewrite systems like confluence or (operational) termination. In this paper, we generalize this notion of feasibility to arbitrary n-ary relations on terms defined by first-order theories. In this way, properties of computational systems whose operational semantics can be given as a first-order theory can be investigated. We introduce a framework for proving feasibility/infeasibility, and a new tool, infChecker, which implements it.Supported by EU (FEDER), and projects RTI2018-094403-B-C32, PROMETEO/2019/098, and SP20180225. Also by INCIBE program "Ayudas para la excelencia de los equipos de investigación avanzada en ciberseguridad" (Raul Gutiérrez).Gutiérrez Gil, R.; Lucas Alba, S. (2020). Automatically Proving and Disproving Feasibility Conditions. Springer Nature. 416-435. https://doi.org/10.1007/978-3-030-51054-1_27S416435Andrianarivelo, N., Réty, P.: Over-approximating terms reachable by context-sensitive rewriting. In: Bojańczyk, M., Lasota, S., Potapov, I. (eds.) RP 2015. LNCS, vol. 9328, pp. 128–139. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24537-9_12Dershowitz, N.: Termination of rewriting. J. Symb. Comput. 3(1/2), 69–116 (1987). https://doi.org/10.1016/S0747-7171(87)80022-6Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reasoning 37(3), 155–203 (2006). https://doi.org/10.1007/s10817-006-9057-7Goguen, J.A., Meseguer, J.: Models and equality for logical programming. In: Ehrig, H., Kowalski, R., Levi, G., Montanari, U. 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Sci. 96, 74–106 (2018). https://doi.org/10.1016/j.jcss.2018.04.002Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems—Part II: advanced processors and implementation techniques. J. Autom. Reasoning (2020, in press)McCune, W.: Prover9 and Mace4. https://www.cs.unm.edu/~mccune/mace4/Meßner, F., Sternagel, C.: nonreach – a tool for nonreachability analysis. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11427, pp. 337–343. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17462-0_19Middeldorp, A., Nagele, J., Shintani, K.: Confluence competition 2019. In: Beyer, D., Huisman, M., Kordon, F., Steffen, B. (eds.) TACAS 2019. LNCS, vol. 11429, pp. 25–40. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17502-3_2Nishida, N., Maeda, Y.: Narrowing trees for syntactically deterministic conditional term rewriting systems. In: Kirchner, H. (ed.) 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Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17462-0_15Winkler, S., Moser, G.: MædMax: a maximal ordered completion tool. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 472–480. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94205-6_3
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