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Inductionless Induction
, 1994
"... Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 A few words explaining the title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Some examples of the problem we are considering . . . . . . . . . . . . . . . . . . . ..."
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Cited by 23 (0 self)
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 A few words explaining the title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Some examples of the problem we are considering . . . . . . . . . . . . . . . . . . . 3 1.3 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Formal background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Terms and clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Equational deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Inductive theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Constructors and sufficient completeness . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Term Rewriting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Standar
Hidden Algebra for Software Engineering
 PROCEEDINGS COMBINATORICS, COMPUTATION AND LOGIC
, 1999
"... This paper is an introduction to recent research on hidden algebra and its application to software engineering; it is intended to be informal and friendly, but still precise. We first review classical algebraic specification for traditional "Platonic" abstract data types like integers, ve ..."
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Cited by 14 (0 self)
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This paper is an introduction to recent research on hidden algebra and its application to software engineering; it is intended to be informal and friendly, but still precise. We first review classical algebraic specification for traditional "Platonic" abstract data types like integers, vectors, matrices, and lists. Software engineering also needs changeable "abstract machines," recently called "objects," that can communicate concurrently with other objects through visible "attributes" and statechanging "methods." Hidden algebra is a new development in algebraic semantics designed to handle such systems. Equational theories are used in both cases, but the notion of satisfaction for hidden algebra is behavioral, in the sense that equations need only appear to be true under all possible experiments; this extra flexibility is needed to accommodate the clever implementations that software engineers often use to conserve space and/or time. The most important results in hidden algebra are ...
Alternating TwoWay ACTree Automata
 IN PREPARATION
, 2002
"... We explore the notion of alternating twoway tree automata modulo the theory of finitely many associativecommutative (AC) symbols, some of them with a unit (AC1). This was prompted by questions arising in cryptographic protocol verification, where the emptiness question for intersections of such au ..."
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Cited by 11 (5 self)
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We explore the notion of alternating twoway tree automata modulo the theory of finitely many associativecommutative (AC) symbols, some of them with a unit (AC1). This was prompted by questions arising in cryptographic protocol verification, where the emptiness question for intersections of such automata is fundamental. We show that the use of conditional push clauses, or of alternation, leads to undecidability, already in the case of one AC or AC1 symbol, with only functions of arity zero. On the other hand, emptiness is decidable in the general case of many function symbols, including many AC or AC1 symbols, provided push clauses are unconditional and intersection clauses are final. To this end, extensive use of refinements of resolution is made.
Automated Induction with Constrained Tree Automata
, 2008
"... We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values), and may express syntactic equality, disequality, ordering ..."
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Cited by 7 (2 self)
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We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values), and may express syntactic equality, disequality, ordering and also membership in a fixed tree language. Constrained equational axioms between constructor terms are supported and can be used in order to specify complex data structures like sets, sorted lists, trees, powerlists... Our procedure is based on tree grammars with constraints, a formalism which can describe exactly the initial model of the given specification (when it is sufficiently complete and terminating). They are used in the inductive proofs first as an induction scheme for the generation of subgoals at induction steps, second for checking validity and redundancy criteria by reduction to an emptiness problem, and third for defining and solving membership constraints. We show that the procedure is sound and refutationally complete. It generalizes former test set induction techniques and yields natural proofs for several nontrivial examples presented in the paper, these examples are difficult (if not impossible) to specify and carry on automatically with other induction procedures.
Superposition for fixed domains
, 2009
"... Superposition is an established decision procedure for a variety of firstorder logic theories represented by sets of clauses. A satisfiable theory, saturated by superposition, implicitly defines a minimal termgenerated model for the theory. Proving universal properties with respect to a saturated ..."
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Superposition is an established decision procedure for a variety of firstorder logic theories represented by sets of clauses. A satisfiable theory, saturated by superposition, implicitly defines a minimal termgenerated model for the theory. Proving universal properties with respect to a saturated theory directly leads to a modification of the minimal model’s termgenerated domain, as new Skolem functions are introduced. For many applications, this is not desired. Therefore, we propose the first superposition calculus that can explicitly represent existentially quantified variables and can thus compute with respect to a given domain. This calculus is sound and refutationally complete for a firstorder fixed domain semantics. For some classes of formulas and theories, we can even employ the calculus to prove properties of the minimal model itself, going beyond the scope of known superpositionbased approaches.
Automated induction for complex data structures. Research Report LSV0511, Laboratoire Spécification et Vérification, 2005. personal communication
 of Joe Hendrix Adel Bouhoula and Florent Jacquemard inria00579017, version 1  22 Mar 2011
"... Abstract. We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values), and may express syntactic equality, disequality, o ..."
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Cited by 4 (3 self)
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Abstract. We propose a procedure for automated implicit inductive theorem proving for equational specifications made of rewrite rules with conditions and constraints. The constraints are interpreted over constructor terms (representing data values), and may express syntactic equality, disequality, ordering and also membership in a fixed tree language. Constrained equational axioms between constructor terms are supported and can be used in order to specify complex data structures like sets, sorted lists, trees, powerlists... Our procedure is based on tree grammars with constraints, a formalism which can describe exactly the initial model of the given specification (when it is sufficiently complete and terminating). They are used in the inductive proofs first as an induction scheme for the generation of subgoals at induction steps, second for checking validity and redundancy criteria by reduction to an emptiness problem, and third for defining and solving membership constraints. We show that the procedure is sound and refutationally complete. It generalizes former test set induction techniques and yields natural proofs for several nontrivial examples presented in the paper, these examples are difficult to specify and carry on automatically with related induction procedures.
Simultaneous checking of completeness and ground confluence
 In the Fifteenth IEEE International Conference on Automated Software Engineering. IEEE Computer
"... Algebraic specifications provide a powerful method for the specification of abstract data types in programming languages and software systems. Completeness and ground confluence are fundamental notions for building algebraic specifications in a correct and modular way. Related works for checking gro ..."
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Algebraic specifications provide a powerful method for the specification of abstract data types in programming languages and software systems. Completeness and ground confluence are fundamental notions for building algebraic specifications in a correct and modular way. Related works for checking ground confluence are based on the completion techniques or on the test that all critical pairs between axioms are valid w.r.t. a sufficient criterion for ground confluence. It is generally accepted that such techniques may be very inefficient even for very small specifications. Indeed, the completion procedure often diverges and there often exist many critical pairs of the axioms. In this paper, we present a procedure for simultaneously checking completeness and ground confluence for specifications with free/nonfree constructors and parameterized specifications. If the specification is not complete or not ground confluent, then our procedure will output the set of patterns on whose ground instances a function is not defined and it can easily identify the rules that break ground confluence. In contrast to previous work, our method does not rely on completion techniques and does not require the computation of critical pairs of the axioms. The method is entirely implemented and allowed us to prove the completeness and the ground confluence of many specifications in a completely automatic way where related techniques diverge
Hidden Algebraic Engineering
 Conference on Semigroups and Algebraic Engineering
, 1997
"... : This paper outlines a research programme in algebraic engineering. It starts with a review of classical algebraic specification for abstract data types, such as integers, vectors, booleans, and lists. Software engineering also needs abstract machines, recently called "objects," that can ..."
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: This paper outlines a research programme in algebraic engineering. It starts with a review of classical algebraic specification for abstract data types, such as integers, vectors, booleans, and lists. Software engineering also needs abstract machines, recently called "objects," that can communicate concurrently with other objects, and that have local states with visible "attributes" that are changed by inputs. Hidden algebra is a new development in algebraic semantics for such systems; its most important results are powerful hidden coinduction principles for proving behavioral properties, especially behavioral refinement. 1 Introduction In view of the title of this conference, I should confess to being an algebraic engineer in (perhaps) the following four different senses: 1. I use algebra to build real software systems. 2. I build huge algebras to help build software systems. 3. I build software tools to help deal with these huge algebras. 4. I build new kinds of algebra, to get be...
Sufficient Completeness Verification for Conditional and Constrained TRS
 JOURNAL OF APPLIED LOGIC
, 2011
"... We present a procedure for checking sufficient completeness of conditional and constrained term rewriting systems containing axioms for constructors which may be constrained (by e.g. equalities, disequalities, ordering, membership...). Such axioms allow to specify complex data structures like e.g. s ..."
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We present a procedure for checking sufficient completeness of conditional and constrained term rewriting systems containing axioms for constructors which may be constrained (by e.g. equalities, disequalities, ordering, membership...). Such axioms allow to specify complex data structures like e.g. sets, sorted lists or powerlists. Our approach is integrated into a framework for inductive theorem proving based on tree grammars with constraints, a formalism which permits an exact representation of languages of ground constructor terms in normal form. The procedure is presented by an inference system which is shown sound and complete. A precondition of one inference of this system refers to a (undecidable) property called strong ground reducibility which is discharged to the above inductive theorem proving system. We have successfully applied our method to several examples, yielding readable proofs and, in case of negative answer, a counterexample suggesting how to complete the specification. Moreover, we show that it is a decision procedure when the TRS is unconditional but constrained, for an expressive class of constrained constructor axioms.
Designing a rewriting induction prover with an increased capability of nonorientable equations
 In Proc. of Symbolic Computation in Software Science AustrianJapanese Workshop, volume 0808 of RISC Technical Report
, 2008
"... Abstract. Rewriting induction (Reddy, 1990) is an automated proof method for inductive theorems of term rewriting systems. Reasoning by the rewriting induction is based on the noetherian induction on some reduction order and the original rewriting induction is not capable of proving theorems which a ..."
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Abstract. Rewriting induction (Reddy, 1990) is an automated proof method for inductive theorems of term rewriting systems. Reasoning by the rewriting induction is based on the noetherian induction on some reduction order and the original rewriting induction is not capable of proving theorems which are not orientable by that reduction order. To deal with such theorems, Bouhoula (1995) as well as Dershowitz & Reddy (1993) used the ordered rewriting. However, even using ordered rewriting, the weak capability of nonorientable theorems is considered one of the weakness of rewriting induction approach compared to other automated methods for proving inductive theorems. We present a rened system of rewriting induction with an increased capability of nonorientable theorems and a capability of disproving incorrect conjectures. Soundness for proving/disproving are shown and eectiveness of our system is demonstrated through some examples. 1