64 research outputs found
Automated Confluence Proof by Decreasing Diagrams based on Rule-Labelling
Decreasing diagrams technique (van Oostrom, 1994) is a technique that
can be widely applied to prove confluence of rewrite systems. To
directly apply the decreasing diagrams technique to prove confluence
of rewrite systems, rule-labelling heuristic has been proposed by van
Oostrom (2008). We show how constraints for ensuring confluence of
term rewriting systems constructed based on the rule-labelling
heuristic are encoded as linear arithmetic constraints suitable for
solving the satisfiability of them by external SMT solvers. We point
out an additional constraint omitted in (van Oostrom, 2008) that is
needed to guarantee the soundness of confluence proofs based on the
rule-labelling heuristic extended to deal with non-right-linear rules.
We also present several extensions of the rule-labelling heuristic by
which the applicability of the technique is enlarged
Sound Lemma Generation for Proving Inductive Validity of Equations
In many automated methods for proving inductive theorems, finding a
suitable generalization of a conjecture is a key for the success of
proof attempts. On the other hand, an obtained generalized conjecture
may not be a theorem, and in this case hopeless proof attempts for the
incorrect conjecture are made, which is against the success and
efficiency of theorem proving. Urso and Kounalis (2004) proposed a
generalization method for proving inductive validity of equations,
called sound generalization, that avoids such an over-generalization.
Their method guarantees that if the original conjecture is an
inductive theorem then so is the obtained generalization. In this
paper, we revise and extend their method. We restore a condition on
one of the characteristic argument positions imposed in their previous
paper and show that otherwise there exists a counterexample to their
main theorem. We also relax a condition imposed in their framework
and add some flexibilities to some of other characteristic argument
positions so as to enlarge the scope of the technique
Ground Confluence Prover based on Rewriting Induction
Ground confluence of term rewriting systems guarantees that all ground
terms are confluent. Recently, interests in proving confluence of
term rewriting systems automatically has grown, and confluence provers
have been developed. But they mainly focus on confluence and not
ground confluence. In fact, little interest has been paid to
developing tools for proving ground confluence automatically. We
report an implementation of a ground confluence prover based on
rewriting induction, which is a method originally developed for
proving inductive theorems
Simple Derivation Systems for Proving Sufficient Completeness of Non-Terminating Term Rewriting Systems
A term rewriting system (TRS) is said to be sufficiently complete when each function yields some value for any input. Proof methods for sufficient completeness of terminating TRSs have been well studied. In this paper, we introduce a simple derivation system for proving sufficient completeness of possibly non-terminating TRSs. The derivation system consists of rules to manipulate a set of guarded terms, and sufficient completeness of a TRS holds if there exists a successful derivation for each function symbol. We also show that variations of the derivation system are useful for proving special cases of local sufficient completeness of TRSs, which is a generalised notion of sufficient completeness
A Fast Decision Procedure For Uniqueness of Normal Forms w.r.t. Conversion of Shallow Term Rewriting Systems
Uniqueness of normal forms w.r.t. conversion (UNC) of term rewriting systems (TRSs) guarantees that there are no distinct convertible normal forms. It was recently shown that the UNC property of TRSs is decidable for shallow TRSs (Radcliffe et al., 2010). The existing procedure mainly consists of testing whether there exists a counterexample in a finite set of candidates; however, the procedure suffers a bottleneck of having a sheer number of such candidates. In this paper, we propose a new procedure which consists of checking a smaller number of such candidates and enumerating such candidates more efficiently. Correctness of the proposed procedure is proved and its complexity is analyzed. Furthermore, these two procedures have been implemented and it is experimentally confirmed that the proposed procedure runs much faster than the existing procedure
Improving Rewriting Induction Approach for Proving Ground Confluence
In (Aoto&Toyama, FSCD 2016), a method to prove ground confluence of many-sorted term rewriting systems based on rewriting induction is given. In this paper, we give several methods that add wider flexibility to the rewriting induction approach for proving ground confluence. Firstly, we give a method to deal with the case in which suitable rules are not presented in the input system. Our idea is to construct additional rewrite rules that supplement or replace existing rules in order to obtain a set of rules that is adequate for applying rewriting induction. Secondly, we give a method to deal with non-orientable constructor rules. This is accomplished by extending the inference system of rewriting induction and giving a sufficient criterion for the correctness of the system. Thirdly, we give a method to deal with disproving ground confluence. The presented methods are implemented in our ground confluence prover AGCP and experiments are reported. Our experiments reveal the presented methods are effective to deal with problems for which state-of-the-art ground confluence provers can not handle
Confluence of Orthogonal Nominal Rewriting Systems Revisited
Nominal rewriting systems (Fernandez, Gabbay, Mackie, 2004;
Fernandez, Gabbay, 2007) have been introduced as a new framework
of higher-order rewriting systems based on the nominal approach
(Gabbay, Pitts, 2002; Pitts, 2003), which deals with variable
binding via permutations and freshness conditions on atoms.
Confluence of orthogonal nominal rewriting systems has been shown in
(Fernandez, Gabbay, 2007). However, their definition of
(non-trivial) critical pairs has a serious weakness so that the
orthogonality does not actually hold for most of standard nominal
rewriting systems in the presence of binders. To overcome this
weakness, we divide the notion of overlaps into the self-rooted and
proper ones, and introduce a notion of alpha-stability which
guarantees alpha-equivalence of peaks from the self-rooted
overlaps. Moreover, we give a sufficient criterion for uniformity and alpha-stability. The new definition of orthogonality and the
criterion offer a novel confluence condition effectively applicable to many standard nominal rewriting systems. We also report on an
implementation of a confluence prover for orthogonal nominal rewriting systems based on our framework
Rewriting Transfinite Terms
We define rewriting over terms with positions of transfinite length
Ultra-Shallow DoF Imaging Using Faced Paraboloidal Mirrors
Computer Vision - ACCV 2016: 13th Asian Conference on Computer Vision, Nov 20-24, 2016, Taipei, TaiwanWe propose a new imaging method that achieves an ultra-shallow depth of field (DoF) to clearly visualize a particular depth in a 3-D scene. The key optical device consists of a pair of faced paraboloidal mirrors with holes around their vertexes. In the device, a lens-less image sensor is set at one side of their holes and an object is set at the opposite side. The characteristic of the device is that the shape of the point spread function varies depending on both the positions of the target 3-D point and the image sensor. By leveraging this characteristic, we reconstruct a clear image for a particular depth by solving a linear system involving position-dependent point spread functions. In experiments, we demonstrate the effectiveness of the proposed method using both simulation and an actually developed prototype imaging system
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