Lexicographic Path Ordering Research Articles

Kruskal's Tree Theorem for Acyclic Term Graphs

In this paper we study termination of term graph rewriting, where we restrict our attention to acyclic term graphs. Motivated by earlier work by Plump we aim at a definition of the notion of simplification order for acyclic term graphs. For this we adapt the homeomorphic embedding relation to term graphs. In contrast to earlier extensions, our notion is inspired by morphisms. Based on this, we establish a variant of Kruskal's Tree Theorem formulated for acyclic term graphs. In proof, we rely on the new notion of embedding and follow Nash-Williams' minimal bad sequence argument. Finally, we propose a variant of the lexicographic path order for acyclic term graphs.

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.

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.

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.

A lexicographic path order with slow growing derivation bounds

AbstractThis paper is concerned with implicit computational complexity of the exptime computable functions. Modifying the lexicographic path order, we introduce a path order EPO. It is shown that a termination proof for a term rewriting system via EPO implies an exponential bound on the lengths of derivations. The path order EPO is designed so that every exptime function is representable as a term rewrite system compatible with EPO (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solving Partial Order Constraints for LPO Termination

This paper introduces a propositional encoding for lexicographic path orders (LPOs) and the corresponding LPO termination property of term rewrite systems. Given this encoding, termination analysis can be performed using a state-of-the-art Boolean satisfiability solver. Experimental results are unequivocal, indicating orders of magnitude speedups in comparison with previous implementations for LPO termination. The results of this paper have already had a direct impact on the design of several major termination analyzers for term rewrite systems. The contribution builds on a symbol-based approach towards reasoning about partial orders. The symbols in an unspecified partial order are viewed as variables that take integer values and are interpreted as indices in the order. For a partial order statement on n symbols, each index is represented in ⌈log2 n⌉ propositional variables and partial orderconstraints between symbols are modeled on the bit representations. The proposed encoding is general and relevant to other applications which involve propositional reasoning about partial orders.

Editor’s Introduction to the Special Volume on Application of Constraints to Formal Verification

Editor’s Introduction to the Special Volume on Application of Constraints to Formal Verification

THINGS TO KNOW WHEN IMPLEMENTING LPO

The Lexicographic Path Ordering (LPO) poses an interesting problem to the implementor: How to achieve a version that is both efficient and correct? The method of program transformation helps us to develop an efficient version step-by-step, making clear the essential ideas, while retaining correctness. By theoretical analysis we show that the worst-case behavior is thereby changed from exponential to polynomial. Detailed measurements show the practical improvements of the different variants. They allow us to assess experimentally various optimizations suggested for LPO.

Theories of orders on the set of words

It is shown that small fragments of the first-order theory of the subword order, the (partial) lexicographic path ordering on words, the homomorphism preorder, and the infix order are undecidable. This is in contrast to the decidability of the monadic second-order theory of the prefix order [M.O. Rabin, Trans. Amer. Math. Soc., 1969] and of the theory of the total lexicographic path ordering [P. Narendran and M. Rusinowitch, Lect. Notes Artificial Intelligence, 2000] and, in case of the subword and the lexicographic path order, improves upon a result by Comon & Treinen [H. Comon and R. Treinen, Lect. Notes Comp. Sci., 1994]. Our proofs rely on the undecidability of the positive Σ 1 -theory of (N, +,.) [Y. Matiyasevich, Hilbert's Tenth Problem, 1993] and on Treinen's technique [R. Treinen, J. Symbolic Comput., 1992] that allows to reduce Post's correspondence problem to logical theories.

Some results on cut-elimination, provable well-orderings, induction and reflection

We gather the following miscellaneous results in proof theory from the attic. 1. 1. A provably well-founded elementary ordering admits an elementary order preserving map. 2. 2. A simple proof of an elementary bound for cut elimination in propositional calculus and its applications to separation problem in relativized bounded arithmetic below S 2 1( X). 3. 3. Equivalents for Bar Induction, e.g., reflection schema for ω logic. 4. 4. Direct computations in an equational calculus PRE and a decidability problem for provable inequations in PRE. 5. 5. Intuitionistic fixed point theories which are conservative extensions of HA. 6. 6. Proof theoretic strengths of classical fixed points theories. 7. 7. An equivalence between transfinite induction rule and iterated reflection schema over IΣ n . 8. 8. Derivation lengths of finite rewrite rules reducing under lexicographic path orders and multiply recursive functions. Each section can be read separately in principle.

The first-order theory of lexicographic path orderings is undecidable

We show, under some assumption on the signature, that the ∃ ∗∀ ∗ fragment of the theory of a lexicographic path ordering is undecidable, both in the partial and in the total precedence cases. Our result implies in particular that the simplification rule of ordered completion is undecidable.

Termination proofs for term rewriting systems by lexicographic path orderings imply multiply recursive derivation lengths

It is shown that a termination proof for a term rewriting system using a lexicographic path ordering yields a multiply recursive bound on the length of derivations, measured in the depth of the starting term. This result is essentially optimal since for every multiply recursive function ƒ a rewrite system (which reduces under the lexicographic path ordering) can be found such that its derivation length cannot be bounded by ƒ.

A path ordering for proving termination of AC rewrite systems

We describe a method that extends the lexicographic recursive path ordering of Dershowitz and Kamin and Levy for proving termination of associative-commutative (AC) rewrite systems. Instead of comparing the arguments of an AC-operator using the multiset extension, wepartition them into disjoint subsets and use each subset only once for comparison. To preserve transitivity, we introduce two techniques —pseudocopying andelevating of arguments of an AC operator. This method imposesno restrictions at all on the underlying precedence relation on function symbols. It can therefore prove termination of a much more extensive class of AC rewrite systems than can previous methods, such as associative path ordering, that restrict AC operators to be minimal or subminimal in precedence. A number of examples illustrating the power of the approach are discussed. The method has been implemented inRRL, Rewrite Rule Laboratory, a theorem-proving environment based on rewrite techniques and completion.

A new method for undecidability proofs of first order theories

We claim that the reduction of Post's Correspondence Problem to the decision problem of a theory provides a useful tool for proving undecidability of first order theories given by some interpretation. The goal of this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering.

Associative-commutative reduction orderings

Rewrite systems are sets of directed equations used to compute by repeatedly replacing subterms in a given expression by equal terms until a simplest form possible (a normal form) is obtained. If a rewrite system is terminating (that is, allows no infinite sequence of rewrites), then every expression has a normal form. A variety of orderings, called reduction orderings, have been designed for proving termination, but most of them are not applicable to extended rewrite systems, where rewrites take into account such properties of functions as associativitty annd commutativity. In this paper we show how an ordering represented as a schematic rewrite system - the lexicographic path ordering - can be systematically modified into an ordering compatible with associativity and commutativity.

SOLVING SYMBOLIC ORDERING CONSTRAINTS

We show how to solve boolean combinations of inequations s>t in the Herbrand Universe, assuming that ≥ is interpreted as a lexicographic path ordering extending a total precedence. In other words, we prove that the existential fragment of the theory of a lexicographic path ordering which extends a total precedence is decidable.