Narrowing-based Optimization of Rewrite Theories

Partial evaluation has been never investigated in the context of rewrite theories that allow concurrent systems to be specified by means of rules, with an underlying equational theory being used to model system states as terms of an algebraic data type. In this paper, we develop a symbolic, narrowing-driven partial evaluation framework for rewrite theories that supports sorts, subsort overloading, rules, equations, and algebraic axioms. Our partial evaluation scheme allows a rewrite theory to be optimized by specializing the plugged equational theory with respect to the rewrite rules that define the system dynamics. This can be particularly useful for automatically optimizing rewrite theories that contain overly general equational theories which perform unnecessary computations involving matching modulo axioms, because some of the axioms may be blown away after the transformation. The specialization is done by using appropriate unfolding and abstraction algorithms that achieve significant specialization while ensuring the correctness and termination of the specialization. Our preliminary results demonstrate that our transformation can speed up a number of benchmarks that are difficult to optimize otherwise.This work has been partially supported by the EU (FEDER) and the Spanish MCIU under grant RTI2018094403-B-C32,andbyGeneralitatValencianaundergrantPROMETEO/2019/098. JuliaSapiñahasbeensupported by the Generalitat Valenciana APOSTD/2019/127 grantAlpuente Frasnedo, M.; Ballis, D.; Escobar Román, S.; Sapiña Sanchis, J. (2020). Narrowing-based Optimization of Rewrite Theories. Universitat Politècnica de València. http://hdl.handle.net/10251/14557

Automated deduction with built-in theories: completeness results and constraint solving techniques

Postprint (published version

From Outermost Reduction Semantics to Abstract Machine

Reduction semantics is a popular format for small-step operational semantics of deterministic programming languages with computational effects.Each reduction semantics gives rise to a reduction-based normalization function where the reduction sequence is enumerated.Refocusing is a practical way to transform a reduction-based normalization function into a reduction-free one where the reduction sequence is not enumerated.This reduction-free normalization function takes the form of an abstract machine that navigates from one redex site to the next without systematically detouring via the root of the term to enumerate the reduction sequence, in contrast to the reduction-based normalization function.We have discovered that refocusing does not apply as readily for reduction semantics that use an outermost reduction strategy and have overlapping rules where a contractum can be a proper subpart of a redex.In this article, we consider such an outermost reduction semantics with backward-overlapping rules, and we investigate how to apply refocusing to still obtain a reduction-free normalization function in the form of an abstract machine

Adding homomorphisms to commutative/monoidal theories or : how algebra can help in equational unification

Two approaches to equational unification can be distinguished. The syntactic approach relies heavily on the syntactic structure of the identities that define the equational theory. The semantic approach exploits the structure of the algebras that satisfy the theory. With this paper we pursue the semantic approach to unification. We consider the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. This class has been introduced by the authors independently of each other as commutative theories (Baader) and monoidal theories (Nutt). The class encompasses important examples like the theories of abelian monoids, idempotent abelian monoids, and abelian groups. We identify a large subclass of commutative/monoidal theories that are of unification type zero by studying equations over the corresponding semiring. As a second result, we show with methods from linear algebra that unitary and finitary commutative/monoidal theories do not change their unification type when they are augmented by a finite monoid of homomorphisms, and how algorithms for the extended theory can be obtained from algorithms for the basic theory. The two results illustrate how using algebraic machinery can lead to general results and elegant proofs in unification theory

Basic Narrowing Revisited

In this paper we study basic narrowing as a method for solving equations in the initial algebra specified by a ground confluent and terminating term rewriting system. Since we are interested in equation solving, we don’t study basic narrowing as a reduction relation on terms but consider immediately its reformulation as an equation solving rule. This reformulation leads to a technically simpler presentation and reveals that the essence of basic narrowing can be captured without recourse to term unification. We present an equation solving calculus that features three classes of rules. Resolution rules, whose application is don’t care nondeterministic, are the basic rules and suffice for a complete solution procedure. Failure rules detect inconsistent parts of the search space. Simplification rules, whose application is don’t care nondeterministic, enhance the power of the failure rules and reduce the number of necessary don’t know steps. Three of the presented simplification rules are new. The rewriting rule allows for don’t care nondeterministic rewriting and thus yields a marriage of basic and normalizing narrowing. The safe blocking rule is specific to basic narrowing and is particulary useful in conjunction with the rewriting rule. Finally, the unfolding rule allows for a variety of search strategies that reduce the number of don’t know alternatives that need to be explored

Digital Ecosystems: Ecosystem-Oriented Architectures

We view Digital Ecosystems to be the digital counterparts of biological ecosystems. Here, we are concerned with the creation of these Digital Ecosystems, exploiting the self-organising properties of biological ecosystems to evolve high-level software applications. Therefore, we created the Digital Ecosystem, a novel optimisation technique inspired by biological ecosystems, where the optimisation works at two levels: a first optimisation, migration of agents which are distributed in a decentralised peer-to-peer network, operating continuously in time; this process feeds a second optimisation based on evolutionary computing that operates locally on single peers and is aimed at finding solutions to satisfy locally relevant constraints. The Digital Ecosystem was then measured experimentally through simulations, with measures originating from theoretical ecology, evaluating its likeness to biological ecosystems. This included its responsiveness to requests for applications from the user base, as a measure of the ecological succession (ecosystem maturity). Overall, we have advanced the understanding of Digital Ecosystems, creating Ecosystem-Oriented Architectures where the word ecosystem is more than just a metaphor.Comment: 39 pages, 26 figures, journa