Raul Gutierrez

• Polytechnic University of Valencia

This article describes the Confluence Framework, a novel framework for proving and disproving confluence using a divide-and-conquer modular strategy, and its implementation in CONFident. Using this approach, we are able to automatically prove and disprove confluence of Generalized Term Rewriting Systems, where (i) only selected arguments of functio...

Presentation of the main results of the JLAMP 2022 paper "Proving and disproving confluence of context-sensitive rewriting"

This article describes the *Confluence Framework*, a novel framework for proving and disproving confluence using a divide-and-conquer modular strategy, and its implementation in CONFident. Using this approach, we are able to automatically prove and disprove confluence of *Generalized Term Rewriting Systems*, where (i) only selected arguments of fun...

This paper describes CONFident, a tool which is able to automatically prove and disprove confluence of variants of rewrite systems: term rewriting systems, conditional term rewriting systems (using join, oriented, or semi-equational semantics), and context-sensitive term rewriting systems. We introduce a new proof framework to generate proof trees...

Context-sensitive rewriting is a restriction of term rewriting where reductions are allowed on specific arguments of function symbols only, and then in particular positions of terms. Confluence is an abstract property of reduction relations guaranteeing that two diverging reduction sequences can always be joined into a common reduct. In this paper...

Slides of the presentation of the FSTTCS 2021 paper "Confluence of Conditional Rewriting in Logic Form"

We characterize conditional rewriting as satisfiability in a Herbrand-like model of terms where variables are also included as fresh constant symbols extending the original signature. Confluence of conditional rewriting and joinability of conditional critical pairs is characterized similarly. Joinability of critical pairs is then translated into co...

CONFident 1.0 is a tool for checking the confluence or non-confluence of systems based on rewriting by means of its logical representation. The tool is available here: http://zenon. dsic.upv.es/confident/. It is written in Haskell following a DP-framework structure, i.e., by defining problems and processors: • problems are tuples ⌧ = (T, G), where...

Proving termination of programs in 'real-life' rewriting-based languages like CafeOBJ, Haskell, Maude, etc., is an important subject of research. To advance this goal, faithfully capturing the impact in the termination behavior of the main language features (e.g., conditions in program rules) is essential. In Part I of this work, we have introduced...

Maude-NPA is an analysis tool for cryptographic security protocols that takes into account the algebraic properties of the cryptosystem. Maude-NPA can reason about a wide range of cryptographic properties. However, some algebraic properties, and protocols using them, have been beyond Maude-NPA capabilities, either because the cryptographic properti...

We report on the new version of mu-term, a tool for proving termination properties of variants of rewrite systems, including conditional, context-sensitive, equational, and order-sorted rewrite systems. We follow a unified, logic-based approach to describe rewriting computations. The automatic generation of logical models for suitable first-order t...

In the realm of term rewriting, given terms s and t, a reachability condition s ->* t is called *feasible* if there is a substitution \sigma such that sigma(s) rewrites into sigma(t) in zero or more steps; otherwise, it is called *infeasible*. Checking infeasibility of (sequences of) reachability conditions is important in the analysis of computati...

infChecker 1.0 is a tool for checking (in)feasibility of goals.The tool is available here: http://zenon.dsic.upv.es/infChecker/.

In order to automatically infer the resource consumption of programs, analyzers track how data sizes change along program’s execution. Typically, analyzers measure the sizes of data by applying norms which are mappings from data to natural numbers that represent the sizes of the corresponding data. When norms are defined by taking type information...

We describe a new tool, AGES, which can be used to automatically generate models for order-sorted first-order theories. The tool uses linear algebra techniques to associate finite or infinite domains to the different sorts. Function and predicate symbols are then interpreted by means of piecewise interpretations with matrix-based expressions and in...

In order to automatically infer the resource consumption of programs, analyzers track how data sizes change along program's execution. Typically, analyzers measure the sizes of data by applying norms which are mappings from data to natural numbers that represent the sizes of the corresponding data. When norms are defined by taking type information...

Given a Conditional Term Rewriting System (CTRS) R and terms s and t, we say that the reachability condition s → * t is feasible if there is a substitution σ instantiating the variables in s and t such that the reachability test σ (s) → * R σ (t) succeeds; otherwise, we call it infeasible. Checking infeasibility of such (sequences of) reachability...

Given a Conditional Term Rewriting System (CTRS) R and terms s and t, we say that the reachability condition s → * t is feasible if there is a substitution σ instantiating the variables in s and t such that the reachability test σ(s) → * R σ(t) succeeds; otherwise, we call it infeasible. Checking infeasibility of such (sequences of) reachability co...

1 Overview mu-term 5.18 is a tool which can be used to verify termination and operational termination of variants of Term Rewriting Systems (TRSs) using dierent variants of the Dependency Pair (DP) Framework: Termination and innermost termination of TRSs using the DP Framework for TRSs [4]. This framework is also used to prove termination of String...

Decision procedures can be either theory-specific, e.g., Presburger arithmetic, or theory-generic, applying to an infinite number of user-definable theories. Variant satisfiability is a theory-generic procedure for quantifier-free satisfiability in the initial algebra of an order-sorted equational theory \((\varSigma ,E \cup B)\) under two conditio...

Different termination properties of conditional term rewriting systems have been recently described emphasizing the bidimensional nature of the termination behavior of conditional rewriting. The absence of infinite sequences of rewriting steps (termination in the usual sense), provides the horizontal dimension. The absence of infinitely many attemp...

Given a (Conditional) Rewrite System R and terms s and t, we consider the following problem: is there a substitution σ instantiating the variables in s and t such that the reachability test σ(s) → * R σ(t) succeeds? If such a substitution does not exist, we say that the problem is infeasible; otherwise, we call it feasible. Similarly, we can consid...

In program analysis, the *synthesis of models* of logical theories representing the program semantics is often useful to prove program properties. We use *Order-Sorted First-Order Logic* as an appropriate framework to describe the semantics and properties of programs as given theories. Then we investigate the \emph{automatic synthesis} of models fo...

Given a Conditional Term Rewriting System R, a sequence s1 ->* t1 ,..., sn ->* tn is R-feasible if there is a substitution \sigma such that for all i, 1 ≤ i ≤ n, \sigma(si) ->* \sigma(ti). Otherwise, it is called R-infeasible. In this paper we use a semantic approach to the analysis of R-infeasibility. An R-infeasibility problem is translated into...

Semantics-based program analysis guarantees that the obtained knowledge about focused program features matches the real behaviour of the program. Automation of the analyses requires abstraction mechanisms to approximate the (usually undecidable) program semantics and targeted properties. In this setting, the logical notions of interpretation of a l...

Context-sensitive rewriting (CSR) is a variant of rewriting where only selected arguments of function symbols can be rewritten. Consequently, the subterm positions of a term are classified as either active, i.e., positions of subterms that can be rewritten; or frozen, i.e., positions that cannot. Frozen positions can be used to denote subexpres-sio...

Context-sensitive rewriting (CSR) is a variant of rewriting where only selected arguments of function symbols can be rewritten. Consequently, the subterm positions of a term are classified as either active, i.e., positions of subterms that can be rewritten; or frozen, i.e., positions that cannot. Frozen positions can be used to denote subexpres-sio...

Recently, a new dependency pair framework for proving operational termination of Conditional Term Rewriting Systems (CTRSs) has been introduced. We call it 2D Dependency Pair (DP) Framework for CTRSs because it makes explicit and exploits the bidimensional nature of the termination behavior of conditional rewriting, where rewriting steps s → t and...

Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction...

Modularity is essential in software development, where a piece of software is often designed and implemented as a composition of simpler modules. So, if we want to prove that a program satisfies a given property, a modular approach becomes natural. With the development and successful use of the Dependency Pair Framework, which rather focuses on the...

Recently, a new dependency pair framework for proving operational termination of Conditional Term Rewriting Systems (CTRSs) has been introduced. We call it 2D DP Framework for CTRSs because it makes explicit and exploits the bidimensional nature of the termination behavior of conditional rewriting: a horizontal component concerning infinite sequenc...

Syntactic annotations are a powerful mechanism to avoid un-desired reductions. In term rewriting, context-sensitive rewriting (CSR) defines the reduction relation which only rewrites the so-called active subterms, i.e., those occurring in replacing arguments of function sym-bols, as indicated by a replacement map µ, which specifies those argu-ments...

In order to automatically infer the resource consumption of programs, analyzers track how data sizes change along a program’s execution. Typically, analyzers measure the sizes of data by applying norms which are mappings from data to natural numbers that represent the sizes of the corresponding data. When norms are defined by taking type informatio...

Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction...

Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction...

Termination of context-sensitive rewriting (CSR) is an interesting problem with several applications in the fields of term rewriting and in the analysis of programming languages like CafeOBJ, Maude, OBJ, etc. The dependency pair approach, one of the most powerful techniques for proving termination of rewriting, has been adapted to be used for provi...

mu-term is a tool which can be used to verify a number of termination properties of (variants of) Term Rewriting Systems (TRSs): termination of rewriting, termination of innermost rewriting, termination of order-sorted rewriting, termination of context-sensitive rewriting, termination of innermost context-sensitive rewriting and termination of rewr...

The dependency pairs approach, one of the most powerful techniques for proving termination of rewrit-ing, has been recently adapted to be used for proving termination of context-sensitive rewriting (CSR). The notion of context-sensitive dependency pair (CS-DP) is different from the standard one in that collapsing de-pendency pairs (i.e., rules whos...

Context-sensitive dependency pairs (CS-DPs) are currently the most powerful method for automated termination analysis of context- sensitive rewriting. However, compared to DPs for ordinary rewriting, CS-DPs suffer from two main drawbacks: (a) CS-DPs can be collapsing. This complicates the handling of CS-DPs and makes them less powerful in practice....

Recently, the dependency pairs (DP) approach has been generalized to context-sensitive rewriting (CSR). Although the context-sensitive dependency pairs (CS-DP) approach provides a very good basis for proving termination of CSR, the current developments basically correspond to a ten-years-old DP approach. Thus, the task of adapting all recently intr...

The dependency pairs method is one of the most powerful technique for proving termination of rewriting and it is currently central in most automatic termination provers. Recently, it has been adapted to be used in proofs of termination of context-sensitive rewriting. The use of collapsing dependency pairs i.e., having a single variable in the right...

Context-sensitive rewriting (CSR) is a restriction of rewriting which forbids reductions on selected arguments of functions. Proving termination of CSR is an interesting problem with several applications in the fields of term rewriting and programming languages. Several methods have been developed for proving termination of CSR. The new version of...

During the last decade, the impressive ad-vances in techniques for proving termination of rewriting have succeeded in solving termi-nation problems that were out of reach for a long time. Since incrementing the size of the problems directly affects the possibility of get-ting an answer, treating problems in a modu-lar way appears like a key issue....

Termination is one of the most interesting problems when dealing with context-sensitive rewrite systems. Although there is a good number of techniques for proving termination of context-sensitive rewriting (CSR), the dependency pair approach, one of the most powerful techniques for proving termination of rewriting, has not been investigated in conn...

Termination is one of the most interesting problems when dealing with context-sensitive rewrite systems. Although a good number of techniques for proving termination of context-sensitive rewriting (CSR) have been proposed so far, the adaptation to CSR of the dependency pair approach, one of the most powerful techniques for proving termination of re...

In this work a detailed study about the implementation of genetic algorithms (GAs) using parallelism and field programmable gate arrays (FPGAs) is presented. Concretely, we use the traveling salesman problem (TSP) as case study. First at all, the TSP is described as well as the GA used for solving it. Afterwards, we present the hardware implementat...