Path Ordering Research Articles

Light-like Wilson line in QCD without path ordering

Unlike the Wilson line in QED the Wilson line in QCD contains path ordering. In this paper we get rid of the path ordering in the light-like Wilson line in QCD by simplifying all the infinite number of noncommuting terms in the SU(3) pure gauge. We prove that the light-like Wilson line in QCD naturally emerges when path integral formulation of QCD is used to prove factorization of soft and collinear divergences at all order in coupling constant in QCD processes at high energy colliders.

On Order-Constrained Transitive Distance Clustering

We consider the problem of approximating order-constrained transitive distance (OCTD) and its clustering applications. Given any pairwise data, transitive distance (TD) is defined as the smallest possible "gap" on the set of paths connecting them. While such metric definition renders significant capability of addressing elongated clusters, it is sometimes also an over-simplified representation which loses necessary regularization on cluster structure and overfits to short links easily. As a result, conventional TD often suffers from degraded performance given clusters with "thick" structures. Our key intuition is that the maximum (path) order, which is the maximum number of nodes on a path, controls the level of flexibility. Reducing this order benefits the clustering performance by finding a trade-off between flexibility and regularization on cluster structure. Unlike TD, finding OCTD becomes an intractable problem even though the number of connecting paths is reduced. We therefore propose a fast approximation framework, using random samplings to generate multiple diversified TD matrices and a pooling to output the final approximated OCTD matrix. Comprehensive experiments on toy, image and speech datasets show the excellent performance of OCTD, surpassing TD with significant gains and giving state-of-the-art performance on several datasets.

The computability path ordering

This paper aims at carrying out termination proofs for simply typed higher-order calculi automatically by using ordering comparisons. To this end, we introduce the computability path ordering (CPO), a recursive relation on terms obtained by lifting a precedence on function symbols. A first version, core CPO, is essentially obtained from the higher-order recursive path ordering (HORPO) by eliminating type checks from some recursive calls and by incorporating the treatment of bound variables as in the com-putability closure. The well-foundedness proof shows that core CPO captures the essence of computability arguments \'a la Tait and Girard, therefore explaining its name. We further show that no further type check can be eliminated from its recursive calls without loosing well-foundedness, but for one for which we found no counterexample yet. Two extensions of core CPO are then introduced which allow one to consider: the first, higher-order inductive types; the second, a precedence in which some function symbols are smaller than application and abstraction.

Unified theory in the worldline approach

We explore unified field theories based on the gauge groups SU(5) and SO(10) using the worldline approach for chiral fermions with a Wilson loop coupling to a background gauge field. Representing path ordering and chiral projection operators with functional integrals has previously reproduced the sum over the chiralities and representations of standard model particles in a compact way. This paper shows that for SU(5) the 5¯ and 10 representations – into which the Georgi–Glashow model places the left-handed fermionic content of the standard model – appear naturally and with the familiar chirality. We carry out the same analysis for flipped SU(5) and uncover a link to SO(10) unified theory. We pursue this by exploring the SO(10) theory in the same framework, the less established unified theory based on SU(6) and briefly consider the Pati–Salam model using SU(4)×SU(2)×SU(2).

Formalizing Termination Proofs under Polynomial Quasi-interpretations

Usual termination proofs for a functional program require to check all the possible reduction paths. Due to an exponential gap between the height and size of such the reduction tree, no naive formalization of termination proofs yields a connection to the polynomial complexity of the given program. We solve this problem employing the notion of minimal function graph, a set of pairs of a term and its normal form, which is defined as the least fixed point of a monotone operator. We show that termination proofs for programs reducing under lexicographic path orders (LPOs for short) and polynomially quasi-interpretable can be optimally performed in a weak fragment of Peano arithmetic. This yields an alternative proof of the fact that every function computed by an LPO-terminating, polynomially quasi-interpretable program is computable in polynomial space. The formalization is indeed optimal since every polynomial-space computable function can be computed by such a program. The crucial observation is that inductive definitions of minimal function graphs under LPO-terminating programs can be approximated with transfinite induction along LPOs.

A new order-theoretic characterisation of the polytime computable functions

We propose a new order-theoretic characterisation of the class of polytime computable functions. To this avail we define the small polynomial path order (sPOP⁎ for short). This termination order entails a new syntactic method to analyse the innermost runtime complexity of term rewrite systems fully automatically: for any rewrite system compatible with sPOP⁎ that employs recursion up to depth d, the (innermost) runtime complexity is polynomially bounded of degree d. This bound is tight. Thus we obtain a direct correspondence between a syntactic (and easily verifiable) condition of a program and the asymptotic worst-case complexity of the program.

Parallelization of Termination Checkers for Algebraic Software

Algebraic software is modeled as a set of equations representing its specification, and when each equation is directed either from left to right or from right to left, the resultant set of directed equations (or rewrite rules) is called a term rewriting system, which can be interpreted as a functional program executed by the pattern matching and term rewriting. In the field of formal verification of information systems, most of the properties of such a system are formalized as inductive theorems, which are equations over terms which hold on recursively-defined data structure such as natural numbers, lists and trees. Well-known as a method for inductive theorem proving is the Rewriting Induction (RI) proposed by Reddy. Recently, this method was extended by Sato and Kurihara to the Multi-context Rewriting Induction with termination checker (MRIt), which is a variant of RI to try to find a suitable context for induction automatically. However, MRIt should perform a large amount of termination checking of term rewriting systems, causing a significant efficiency bottleneck. In this paper, we propose a method of parallelizing the termination checkers used in MRIt to improve its efficiency by focusing on the well-known typical termination checking method based on the lexicographic path orders. For implementation, we used the functional concurrent programming language Erlang. We discuss the efficiency of our implementation based on the experiments with the standard set of termination problems.

A unified ordering for termination proving

We introduce a reduction order called the weighted path order (WPO) that subsumes many existing reduction orders. WPO compares weights of terms as in the Knuth–Bendix order (KBO), while WPO allows weights to be computed by a wide class of interpretations. We investigate summations, polynomials and maximums for such interpretations. We show that KBO is a restricted case of WPO induced by summations, the polynomial order (POLO) is subsumed by WPO induced by polynomials, and the lexicographic path order (LPO) is a restricted case of WPO induced by maximums. By combining these interpretations, we obtain an instance of WPO that unifies KBO, LPO and POLO. In order to fit WPO in the modern dependency pair framework, we further provide a reduction pair based on WPO and partial statuses. As a reduction pair, WPO also subsumes matrix interpretations. We finally present SMT encodings of our techniques, and demonstrate the significance of our work through experiments.

The rigorous stochastic matrix multiplication scheme for the calculations of reduced equilibrium density matrices of open multilevel quantum systems

Understanding the roles of the temporary and spatial structures of quantum functional noise in open multilevel quantum molecular systems attracts a lot of theoretical interests. I want to establish a rigorous and general framework for functional quantum noises from the constructive and computational perspectives, i.e., how to generate the random trajectories to reproduce the kernel and path ordering of the influence functional with effective Monte Carlo methods for arbitrary spectral densities. This construction approach aims to unify the existing stochastic models to rigorously describe the temporary and spatial structure of Gaussian quantum noises. In this paper, I review the Euclidean imaginary time influence functional and propose the stochastic matrix multiplication scheme to calculate reduced equilibrium density matrices (REDM). In addition, I review and discuss the Feynman-Vernon influence functional according to the Gaussian quadratic integral, particularly its imaginary part which is critical to the rigorous description of the quantum detailed balance. As a result, I establish the conditions under which the influence functional can be interpreted as the average of exponential functional operator over real-valued Gaussian processes for open multilevel quantum systems. I also show the difference between the local and nonlocal phonons within this framework. With the stochastic matrix multiplication scheme, I compare the normalized REDM with the Boltzmann equilibrium distribution for open multilevel quantum systems.

Polynomial Path Orders

This paper is concerned with the complexity analysis of constructor term rewrite systems and its ramification in implicit computational complexity. We introduce a path order with multiset status, the polynomial path order POP*, that is applicable in two related, but distinct contexts. On the one hand POP* induces polynomial innermost runtime complexity and hence may serve as a syntactic, and fully automatable, method to analyse the innermost runtime complexity of term rewrite systems. On the other hand POP* provides an order-theoretic characterisation of the polytime computable functions: the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP*.

Jumping and escaping: Modular termination and the abstract path ordering

Combinatorial commutation properties for reordering a sequence consisting of two kinds of steps, and for separating the well-foundedness of their combination into well-foundedness of each, are investigated. A weak commutation property, called “jumping”, along with a weakened version of the lifting property, called “escaping” and requiring only an eventual lifting, are used for proving well-foundedness of a generic, abstract version of the recursive path orderings.

The recursive path and polynomial ordering for first-order and higher-order terms

In most termination tools two ingredients, namely recursive path orderings (RPOs) and polynomial interpretation orderings (POLOs), are used in a consecutive disjoint way to solve the final constraints generated from the termination problem. In this article we present a simple ordering that combines both RPO and POLO and defines a family of orderings that includes both, and extend them with the possibility of having, at the same time, an RPO-like treatment for some symbols and a POLO-like treatment for the others. The ordering is extended to higher-order terms, providing a new fully automatable use of polynomial interpretations in combination with beta-reduction.

A Path Ordering for Term Rewriting Systems with Polynomial Size Normal Forms

A Path Ordering for Term Rewriting Systems with Polynomial Size Normal Forms

Routing multi-class traffic flows in the plane

We study a class of multi-commodity flow problems in geometric domains: For a given planar domain P populated with obstacles (holes) of K ⩾ 2 types, compute a set of thick paths from a “source” edge of P to a “sink” edge of P for vehicles of K distinct classes. Each class k of vehicle has a given set, O k , of obstacles it must avoid and a certain width, w k , of path it requires. The problem is to determine if it is possible to route N k width- w k paths for class k vehicles from source to sink, with each path avoiding the requisite set O k of obstacles, and no two paths overlapping. This form of multi-commodity flow in two-dimensional domains arises in computing throughput capacity for multiple classes of aircraft in an airspace impacted by different types of constraints, such as those arising from weather hazards. We give both algorithmic theory results and experimental results. We show hardness of many versions of the problem by proving that two simple variants are NP-hard even in the case K = 2 . If w 1 = w 2 = 1 , then the problem is NP-hard even when O 1 = ∅ . If w 1 = 2 , w 2 = 3 , then the problem is NP-hard even when O 1 = O 2 . In contrast, the problem for a single width and a single type of obstacles is polynomially solvable. We present approximation algorithms for the multi-criteria optimization problems that arise when trying to maximize the number of routable paths. We also give a polynomial-time algorithm for the case in which the number of holes in the input domain is bounded. Finally, we give experimental results based on an implementation of our methods and experiment with enhanced heuristics for efficient solutions in practice. Our algorithms are being utilized in simulations with NASAʼs Future Air traffic management Concepts Evaluation Tool (FACET). We report on experimental results based on applying our algorithms to weather-impacted airspaces, comparing heuristic strategies for searching for feasible path orderings and for computing short multi-class routes. Our results show that multi-class routes can feasibly be computed on real weather data instances on the scale required in air traffic management applications.

Pyrolysis Modelling of PVC Cable Materials

One of the most commonly used plastics in electrical cables is flexible PVC, which includes 30-40 wt. % combustible plasticizers. In this work, the effects of the modelling decisions and parameter estimation methods on the pyrolysis modelling of two PVC cables were studied. The kinetic and thermal parameters were estimated from the TGA and cone calorimeter experiments. The role of the plasticizers was shown to be important for the early HRR. The effects of the reaction path and reaction order were only minor in the TGA results but significant effects were found in the cone calorimeter results, unless a specific set of thermal parameters was estimated. The results show that the thermal parameters estimated for one kinetic model should not be used for another, unless the difference between the kinetic models considers only fuel yields or different pairs of kinetic coefficients with same reaction order.

Pyrolysis Modelling of PVC Cable Materials

One of the most commonly used plastics in electrical cables is flexible PVC, which includes 30-40 wt. % combustible plasticizers. In this work, the effects of the modelling decisions and parameter estimation methods on the pyrolysis modelling of two PVC cables were studied. The kinetic and thermal parameters were estimated from the TGA and cone calorimeter experiments. The role of the plasticizers was shown to be important for the early HRR. The effects of the reaction path and reaction order were only minor in the TGA results but significant effects were found in the cone calorimeter results, unless a specific set of thermal parameters was estimated. The results show that the thermal parameters estimated for one kinetic model should not be used for another, unless the difference between the kinetic models considers only fuel yields or different pairs of kinetic coefficients with same reaction order.

SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs

This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPOS). This facilitates the application of SAT solvers for termination analysis of term rewrite systems (TRSs). We address four main inter-related issues and show how to encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (A) the lexicographic comparison w.r.t. a permutation of the arguments; (B) the multiset extension of a base order; (C) the combined search for a path order together with an argument filter to orient a set of inequalities; and (D) how the choice of the argument filter influences the set of inequalities that have to be oriented (so-called usable rules). We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power.

Observation of implicit complexity by non confluence

We propose to consider non confluence with respect to implicit complexity. We come back to some well known classes of first-order functional program, for which we have a characterization of their intentional properties, namely the class of cons-free programs, the class of programs with an interpretation, and the class of programs with a quasi-interpretation together with a termination proof by the product path ordering. They all correspond to PTIME. We prove that adding non confluence to the rules leads to respectively PTIME, NPTIME and PSPACE. Our thesis is that the separation of the classes is actually a witness of the intentional properties of the initial classes of programs.

Casimir forces from a loop integral formulation

We reformulate the Casimir force between bodies in non-trivial background media. The force may be written in terms of loop variables, the loop being a curve around the scattering sites. A natural path ordering of exponentials takes place when a particular representation of the scattering centres is given. The basic object to be evaluated is a reduced (or abbreviated) classical pseudo-action that can be operator valued, and can be obtained from a classical path integral description.

Coq formalization of the higher-order recursive path ordering

The recursive path ordering (RPO) is a well-known reduction ordering introduced by Dershowitz, that is useful for proving termination of term rewriting systems (TRSs). Jouannaud and Rubio generalized this ordering to the higher-order case thus creating the higher-order recursive path ordering (HORPO). They proved that this ordering can be used for proving termination of higher-order TRSs, which essentially comes down to proving well-foundedness of HORPO. This result entails well-foundedness of RPO and termination of simply typed lambda calculus (as the β-reduction relation is included in HORPO). This paper describes our undertaking of providing a complete, axiom-free, fully constructive formalization of those results in the proof assistant Coq. The formalization can be divided into three parts: finite multisets and two variants of multiset extensions of a relation, simply typed lambda calculus with termination of β-reduction as the main result, HORPO with a proof of its well-foundedness; also decidability of HORPO has been proved and, due to its constructive nature, a certified algorithm to verify whether two terms can be oriented with HORPO can be extracted from the proof.