356 research outputs found
Testing for the ground (co-)reducibility property in term-rewriting systems
AbstractGiven a term-rewriting system R, a term t is ground-reducible by R if every ground instance tÏ of it is R-reducible. A pair (t, s) of terms is ground-co-reducible by R if every ground instance (tÏ, sÏ] of it for which tÏ and sÏ are distinct is R-reducible. Ground (co-)reducibility has been proved to be the fundamental tool for mechanizing inductive proofs, together with the Knuth-Bendix completion procedure presented by Jouannaud and Kounalis (1986, 1989).Jouannaud and Kounalis (1986, 1989) also presented an algorithm for testing ground reducibility which is tractable in practical cases but restricted to left-linear term-rewriting systems. The solution of the ground (co-)reducibility problem, for the general case, turned out to be surprisingly complicated. Decidability of ground reducibility for arbitrary term-rewriting systems has been first proved by Plaisted (1985) and independently by Kapur (1987). However, the algorithms of Plaisted and Kapur amount to intractable computation, even in very simple cases.We present here a new algorithm for the general case which outperforms the algorithms of Plaisted and Kapur and even our previous algorithm in case of left-linear term-rewriting systems. We then show how to adapt it to check for ground co-reducibility
Inductive Theorem Proving Using Refined Unfailing Completion Techniques
We present a brief overview on completion based inductive theorem proving techniques, point out the key concepts for the underlying "proof by consistency" - paradigm and isolate an abstract description of what is necessary for an algorithmic realization of such methods.
In particular, we give several versions of proof orderings, which - under certain conditions - are well-suited for that purpose. Together with corresponding notions of (positive and negative) covering sets we get abstract "positive" and "negative" characterizations of inductive validity. As a consequence we can generalize known criteria for inductive validity, even for the cases where some of the conjectures may not be orientable or where the base system is terminating but not necessarily ground conïŹuent.
Furthermore we consider several reïŹnements and optimizations of completion based inductive theorem proving techniques. In particular, sufïŹcient criteria for being a covering set including restrictions of critical pairs (and the usage of non-equational inductive knowledge) are discussed.
Moreover a couple of lemma generation methods are brieïŹy summarized and classiïŹed. A new techniques of save generalization is particularly interesting, since it provides means for syntactic generalizations, i.e. simpliïŹcations, of conjectures without loosing semantic equivalence.
Finally we present the main features and characteristics of UNICOM, an inductive theorem prover with reïŹned unfailing completion techniques and built on top of TRSPEC, a term rewriting based system for investigating algebraic speciïŹcations
Computing Constructor Forms with Non Terminating Rewrite Programs
In the context of the study of rule-based programming, we focus in this paper on the property of C-reducibility, expressing that every term reduces to a constructor term on at least one of its rewriting derivations. This property implies completeness of function definitions, and enables to stop evaluations of a program on a constructor form, even if the program is not terminating. We propose an inductive procedure proving C-reducibility of rewriting. The rewriting relation on ground terms is simulated through an abstraction mechanism and narrowing. The induction hypothesis allows assuming that terms smaller than the starting terms rewrite into a constructor term. The existence of the induction ordering is checked during the proof process, by ensuring satisfiability of ordering constraints. The proof is constructive, in the sense that the branch leading to a constructor term can be computed from the proof trees establishing C-reducibility for every term
Computing Constructor Forms with Non Terminating Rewrite Programs - Extended version -
In the context of the study of rule-based programming, we focus in this paper on the property of C-reducibility, expressing that every term reduces to a constructor term on at least one of its rewriting derivations. This property implies completeness of function definitions, and enables to stop evaluations of a program on a constructor form, even if the program is not terminating. We propose an inductive procedure proving C-reducibility of rewriting. The rewriting relation on ground terms is simulated through an abstraction mechanism and narrowing. The induction hypothesis allows assuming that terms smaller than the starting terms rewrite into a constructor term. The existence of the induction ordering is checked during the proof process, by ensuring satisfiability of ordering constraints. The proof is constructive, in the sense that the branch leading to a constructor term can be computed from the proof trees establishing C-reducibility for every term
Strategic Issues, Problems and Challenges in Inductive Theorem Proving
Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP
Ground Reducibility is EXPTIME-complete
International audienceWe prove that ground reducibility is EXPTIME-complete in the general case. EXPTIME-hardness is proved by encoding the emptiness problem for the intersection of recognizable tree languages. It is more difficult to show that ground reducibility belongs to DEXPTIME. We associate first an automaton with disequality constraints A(R,t) to a rewrite system R and a term t. This automaton is deterministic and accepts at least one term iff t is not ground reducible by R. The number of states of A(R,t) is O(2^|R|x|t|) and the size of its constraints is polynomial in the size of R, t. Then we prove some new pumping lemmas, using a total ordering on the computations of the automaton. Thanks to these lemmas, we can show that emptiness for an automaton with disequality constraints can be decided in a time which is polynomial in the number of states and exponential in the size of the constraints. Altogether, we get a simply exponential time deterministic algorithm for ground reducibility decision
Computability in constructive type theory
We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Riceâs theorem, the Myhill isomorphism theorem, and the existence of Postâs simple and hypersimple predicates relying on no other axioms such as Markovâs principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type â â â is L-computable.Wir behandeln eine formalisierte und maschinengeprĂŒfte Betrachtung von Berechenbarkeitstheorie im Calculus of Inductive Constructions (CIC), der konstruktiven Typtheorie die dem Beweisassistenten Coq zugrunde liegt. Wir entwickeln erst synthetische Berechenbarkeitstheorie, vorbereitet durch die Arbeit von Richman, Bridges und Bauer, wobei alle Funktionen als berechenbar behandelt werden, ohne Notwendigkeit eines Berechnungsmodells. Wir nehmen ein neues, parametrisches Axiom fĂŒr synthetische Berechenbarkeit an und beweisen Resultate wie das Theorem von Rice, das Isomorphismus Theorem von Myhill und die Existenz von Postâs simplen und hypersimplen PrĂ€dikaten ohne Annahme von anderen Axiomen wie Markovâs Prinzip oder Auswahlaxiomen. Als zweiten Schritt fĂŒhren wir Berechnungsmodelle ein. Wir geben einen kompakten Ăberblick ĂŒber die Definition von verschiedenen Berechnungsmodellen und erklĂ€ren maschinengeprĂŒfte Simulationsbeweise zwischen diesen Modellen, welche einen hohen Konstruktionsaufwand beinhalten. Wir identifizieren einen Begriff von synthetischer Unentscheidbarkeit relativ zu einem fixierten Halteproblem welcher axiomenfreie maschinengeprĂŒfte Unentscheidbarkeitsbeweise erlaubt. Wir erklĂ€ren solche Beweise fĂŒr die historisch grundlegenden Probleme der Berechenbarkeitstheorie, die das Identifizieren von Invarianten die normalerweise in der Literatur ausgelassen werden benötigen und nun die Basis der Coq Library of Undecidability Proofs bilden. Wir identifizieren dann den call-by-value λ-KalkĂŒl L als sweet spot fĂŒr die Programmierung in einem Berechnungsmodell. Wir fĂŒhren ein zertifizierendes Extraktionsframework ein und analysieren ein Axiom welches postuliert dass jede Funktion vom Typ NâN L-berechenbar ist
Construction of topological field theories using BV
We discuss in detail the construction of topological field theories using the
Batalin--Vilkovisky (BV) quantisation scheme. By carefully examining the
dependence of the antibracket on an external metric, we show that
differentiating with respect to the metric and the BRST charge do not commute
in general. We introduce the energy momentum tensor in this scheme and show
that it is BRST invariant, both for the classical and quantum BRST operators.
It is antifield dependent, guaranteeing gauge independence. For topological
field theories, this energy momentum has to be quantum BRST exact. This leads
to conditions at each order in . As an example of this procedure, we
consider topological Yang--Mills theory. We show how the reducible set of
symmetries used in topological Yang--Mills can be recovered by means of trivial
systems and canonical transformations. Self duality of the antighosts is
properly treated by introducing an infinite tower of auxiliary fields. Finally,
it is shown that the full energy momentum tensor is classically BRST exact in
the antibracket sense.Comment: 15
POPLMark reloaded: Mechanizing proofs by logical relations
We propose a new collection of benchmark problems in mechanizing the metatheory of programming languages, in order to compare and push the state of the art of proof assistants. In particular, we focus on proofs using logical relations (LRs) and propose establishing strong normalization of a simply typed calculus with a proof by Kripke-style LRs as a benchmark. We give a modern view of this well-understood problem by formulating our LR on well-typed terms. Using this case study, we share some of the lessons learned tackling this problem in different dependently typed proof environments. In particular, we consider the mechanization in Beluga, a proof environment that supports higher-order abstract syntax encodings and contrast it to the development and strategies used in general-purpose proof assistants such as Coq and Agda. The goal of this paper is to engage the community in discussions on what support in proof environments is needed to truly bring mechanized metatheory to the masses and engage said community in the crafting of future benchmarks
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