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28
Completion of Rewrite Systems with Membership Constraints Part II: Constraint Solving
 J. Symbolic Computation
, 1992
"... this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an ordersorted logic with context variables and can be read independently of the first part. Thi ..."
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Cited by 66 (2 self)
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this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an ordersorted logic with context variables and can be read independently of the first part. This can also be compared with unification of term schemes of various kind (Chen & Hsiang, 1991; Salzer, 1992; Comon, 1995; R. Galbav'y and M. Hermann, 1992). Indeed,
Towards an Automatic Analysis of Security Protocols in FirstOrder Logic
, 1999
"... . The NeumanStubblebine key exchange protocol is formalized in rstorder logic and analyzed by the automated theorem prover Spass. In addition to the analysis, we develop the necessary theoretical background providing new (un)decidability results for monadic rstorder fragments involved in the a ..."
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Cited by 62 (4 self)
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. The NeumanStubblebine key exchange protocol is formalized in rstorder logic and analyzed by the automated theorem prover Spass. In addition to the analysis, we develop the necessary theoretical background providing new (un)decidability results for monadic rstorder fragments involved in the analysis. The approach is applicable to a variety of security protocols and we identify possible extensions leading to future directions of research. 1 Introduction The growing importance of the internet causes a growing need for security protocols that protect transactions and communication. It turns out that the design of such protocols is highly errorprone. Therefore, a variety of dierent methods have been described that analyze security protocols to discover aws. The topic of this paper is to add a further, new method that is based on automated theorem proving in rstorder logic. In the context of rstorder automated theorem proving, Schumann (1997) implemented the wellknown ...
Structural Subtyping of NonRecursive Types is Decidable
, 2003
"... We show that the firstorder theory of structural subtyping of nonrecursive types is decidable, as a consequence of a more general result on the decidability of term powers of decidable theories. ..."
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Cited by 26 (6 self)
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We show that the firstorder theory of structural subtyping of nonrecursive types is decidable, as a consequence of a more general result on the decidability of term powers of decidable theories.
Unification of Infinite Sets of Terms Schematized by Primal Grammars
 Theoretical Computer Science
, 1996
"... Infinite sets of terms appear frequently at different places in computer science. On the other hand, several practically oriented parts of logic and computer science require the manipulated objects to be finite or finitely representable. Schematizations present a suitable formalism to manipulate fin ..."
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Cited by 23 (3 self)
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Infinite sets of terms appear frequently at different places in computer science. On the other hand, several practically oriented parts of logic and computer science require the manipulated objects to be finite or finitely representable. Schematizations present a suitable formalism to manipulate finitely infinite sets of terms. Since schematizations provide a different approach to solve the same kind of problems as constraints do, they can be viewed as a new type of constraints. The paper presents a new recurrent schematization called primal grammars. The main idea behind the primal grammars is to use primitive recursion as the generating engine of infinite sets. The evaluation of primal grammars is based on substitution and rewriting, hence no particular semantics for them is necessary. This fact allows also a natural integration of primal grammars into Prolog, into functional languages or into other rewritebased applications. Primal grammars have a decidable unification problem and ...
Decision procedures for algebraic data types with abstractions
 IN 37TH ACM SIGACTSIGPLAN SYMPOSIUM ON PRINCIPLES OF PROGRAMMING LANGUAGES (POPL), 2010. DECISION PROCEDURES FOR ORDERED COLLECTIONS 15 SHE75. SAHARON SHELAH. THE MONADIC THEORY OF ORDER. THA ANNALS OF MATHEMATICS OF MATHEMATICS
, 2010
"... We describe a family of decision procedures that extend the decision procedure for quantifierfree constraints on recursive algebraic data types (term algebras) to support recursive abstraction functions. Our abstraction functions are catamorphisms (term algebra homomorphisms) mapping algebraic data ..."
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Cited by 23 (11 self)
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We describe a family of decision procedures that extend the decision procedure for quantifierfree constraints on recursive algebraic data types (term algebras) to support recursive abstraction functions. Our abstraction functions are catamorphisms (term algebra homomorphisms) mapping algebraic data type values into values in other decidable theories (e.g. sets, multisets, lists, integers, booleans). Each instance of our decision procedure family is sound; we identify a widely applicable manytoone condition on abstraction functions that implies the completeness. Complete instances of our decision procedure include the following correctness statements: 1) a functional data structure implementation satisfies a recursively specified invariant, 2) such data structure conforms to a contract given in terms of sets, multisets, lists, sizes, or heights, 3) a transformation of a formula (or lambda term) abstract syntax tree changes the set of free variables in the specified way.
AutomataDriven Automated Induction
 Information and Computation
, 1996
"... . This work investigates inductive theorem proving techniques for firstorder functions whose meaning and domains can be specified by Horn Clauses built up from the equality and finitely many unary membership predicates. In contrast with other works in the area, constructors are not assumed to be fr ..."
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Cited by 22 (9 self)
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. This work investigates inductive theorem proving techniques for firstorder functions whose meaning and domains can be specified by Horn Clauses built up from the equality and finitely many unary membership predicates. In contrast with other works in the area, constructors are not assumed to be free. Techniques originating from tree automata are used to describe ground constructor terms in normal form, on which the induction proofs are built up. Validity of (free) constructor clauses is checked by an original technique relying on the recent discovery of a complete axiomatisation of finite trees and their rational subsets. Validity of clauses with defined symbols or nonfree constructor terms is reduced to the latter case by appropriate inference rules using a notion of ground reducibility for these symbols. We show how to check this property by generating proof obligations which can be passed over to the inductive prover. 1 Introduction The need for large formal proofs has lead to t...
On the theory of structural subtyping
, 2003
"... We show that the firstorder theory of structural subtyping of nonrecursive types is decidable. Let Σ be a language consisting of function symbols (representing type constructors) and C a decidable structure in the relational language L containing a binary relation ≤. C represents primitive types; ..."
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Cited by 18 (8 self)
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We show that the firstorder theory of structural subtyping of nonrecursive types is decidable. Let Σ be a language consisting of function symbols (representing type constructors) and C a decidable structure in the relational language L containing a binary relation ≤. C represents primitive types; ≤ represents a subtype ordering. We introduce the notion of Σtermpower of C, which generalizes the structure arising in structural subtyping. The domain of the Σtermpower of C is the set of Σterms over the set of elements of C. We show that the decidability of the firstorder theory of C implies the decidability of the firstorder theory of the Σtermpower of C. This result implies the decidability of the firstorder theory of structural subtyping of nonrecursive types.
Feature Automata and Recognizable Sets of Feature Trees
 In TAPSOFT'93, pages 356375. SpringerVerlag LNCS 668
, 1993
"... Feature trees generalize firstorder trees whereby argument positions become keywords ("features") from an infinite symbol set F . Constructor symbols can occur with any argument positions, in any finite number. Feature trees are used to model flexible records; the assumption on the infiniteness ..."
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Cited by 15 (3 self)
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Feature trees generalize firstorder trees whereby argument positions become keywords ("features") from an infinite symbol set F . Constructor symbols can occur with any argument positions, in any finite number. Feature trees are used to model flexible records; the assumption on the infiniteness of F accounts for dynamic record field updates.
Implementing Computational Systems with Constraints
 BROWN UNIVERSITY
, 1993
"... The paper presents a framework to describe, experiment and study the combination of different computational systems including the constraint solving paradigm. Computational systems are interpreted in a firstorder setting thanks to an evaluator that rewrites formulas. ..."
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Cited by 14 (2 self)
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The paper presents a framework to describe, experiment and study the combination of different computational systems including the constraint solving paradigm. Computational systems are interpreted in a firstorder setting thanks to an evaluator that rewrites formulas.
Sorted Unification Using Set Constraints
, 1992
"... . This paper describes a new representation for sortal constraints and a unification algorithm for the corresponding constrained terms. Variables range over sets of terms described by systems of set constraints that can express limited intervariable dependencies. These sets of terms are more genera ..."
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Cited by 10 (0 self)
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. This paper describes a new representation for sortal constraints and a unification algorithm for the corresponding constrained terms. Variables range over sets of terms described by systems of set constraints that can express limited intervariable dependencies. These sets of terms are more general than regular tree languages, but are still closed under intersection. The new unification algorithm shows sorted unification to be decidable for a broad class of sorted signatures, which we call semilinear , and, more generally, for sort theories with a least Herbrand model that can be represented using the new constraints. A finite representation of a complete set of wellsorted unifiers can always be found, even in those cases where this set is infinite. 1 Introduction Sorts are widely used to add more "structure" to first order logic, improving the efficiency of deductive systems [ Cohn, 1989 ] . Increasingly, sorted logic is being viewed as an instance of constraint logic [ Comon and ...