Mandayam K. Srivas

Conditional specifications with inequational assumptions

Equational-Inequational Conditional Term Rewriting Systems (EI-CTRS) are a natural formalism for expressing data type and function specifications. EI-CTRS are sets of oriented conditional equations (rules) (eg.a=b Λ c ≠ d ::lhs → rhs), whose antecedents are conjunctions of equations and inequations. The EI-CTRS formalism extends existing equational languages such as OBJ2, by allowing within the rules the direct use of the = (or ≠) relation being defined. Using = and ≠ in rules clearly enhances the expressive power of the formalism, but it poses termination and semantic problems in executing a specification. We define a conditional rewriting mechanism, called EI-rewriting, which assumes two terms to be equal if they can be rewritten to a common term and inequal otherwise. We show that ground EI-rewriting is sound and complete, and can hence be used to correctly execute specifications expressed as EI-CTRS which satisfy a sufficient completeness-like property We give syntactic conditions and procedures to check that a specification in the formalism denotes a well-defined total function. We illustrate that useful and meaningful specifications can be constructed using EI-CTRS.

Automatic inductive theorem proving using PROLOG

Abstract Although P rolog is a programming language based on techniques from theorem proving, its use as a base for a theorem prover has not been explored until recently (Stickel, 1984). In this paper, we introduce a P rolog -based deductive theorem proving method for proving theorems in a first-order inductive theory representable in Horn clauses. The method has the following characteristics: 1. It automatically partitions the domains over which the variables range into subdomains according to the manner in which the predicate symbols in the theorem are defined. 2. For each of the subdomains the prover returns a lemma. If the lemma is true , then the target theorem is true for this subdomain. The lemma could also be an induction hypothesis for the theorem. 3. The method does not explicitly use any inductive inference rule. The induction hypothesis, if needed for a certain subdomain, will sometimes be generated from a (limited) forward chaining mechanism in the prover and not from employing any particular inference rule. In addition to the backward chaining and backtracking facilities of P rolog , our method introduces three new mechanism— skolemization by need, suspended evaluation , and limited forward chaining . These new mechanisms are simple enough to be easily implemented or even incorporated into P rolog . We describe how the theorem prover can be used to prove properties of P rolog programs by showing two simple examples.

Computability and implementability issues in abstract data types

In a strongly typed system supporting user-defined data abstractions, the designer of a data abstraction ought to be careful in choosing the operations for the abstraction. If the operation set chosen is not expressive enough, it might be impossible or inconvenient to implement certain useful functions on the values of the data abstraction. In this paper, two properties of the operation set of a data abstraction, expressive completeness and expressive richness, are defined to formally characterize the expressive power of the operation set.For an expressively complete data abstraction, the operation set is powerful enough to implement in principle all computable properties of the values, whereas for an expressively rich data abstraction, the operation set can be used to implement the properties in a `simple and natural? fashion. It is shown that if the equality predicate on the values of a data abstraction can be implemented in terms of its operations, then the data abstraction is expressively complete.For expressive richness, we identify a finite set of functions that represent certain basic kinds of manipulations of the values, and require them to be implemented in terms of the operation set as `straight line? programs. The relation between these formal properties and the intuitive notions are considered. We argue that it is important to consider both expressive completeness and expressive richness while designing the operation set of a data abstraction. Practical applications of the properties of expressiveness introduced are also discussed.

A PROLOG environment for developing and reasoning about data types

PROLOG is a programming language based on first order logic. The feature that distinguishes PROLOG from most other programming languages is that the execution of PROLOG programs is based on subgoal reduction and unification. Unfortunately, the reliance on unification for execution has also inhibited PROLOG from utilizing some recently developed concepts in programming languages such as abstract data types. In this paper we introduce a discipline for incorporating abstract data types into PROLOG, and study the use of PROLOG as a uniform programming environment for the specification, implementation, and verification of PROLOG programs. We illustrate the application of the environment to the development of abstract data types in PROLOG.

A Rewrite Rule Based Approach for Synthesizing Abstract Data Types

An approach for synthesizing data type implementations based on the theory of term rewriting systems is presented. A specification is assumed to be given as a system of equations; an implementation is derived from the specification as another system of equations. The proof based approach used for the synthesis consists of reversing the process of proving theorems (i.e. searching for appropriate theorems rather than proving the given ones). New tools and concepts to embody this reverse process are developed. In particular, the concept of expansion, which is a reverse of rewriting (or reduction), is defined and analyzed. The proposed system consists of a collection of inference rules — instantiation, simplification, expansion and hypothesis tesing, and two strategies for searching for theorems depending upon whether the theorem being looked for is in the equational theory or in the inductive theory of the specification.

Function definitions in term rewriting and applicative programming

The frameworks of unconditional and conditional Term Rewriting and Applicative systems are explored with the objective of using them for defining functions. In particular, a new operational semantics, Tue-Reduction , is elaborated for conditional term rewriting systems. For each framework, the concept of evaluation of terms invoking defined functions is formalized. We then discuss how it may be ensured that a function definition in each of these frameworks is meaningful, by defining restrictions that may be imposed to guarantee termination, unambiguity , and completeness of definition. The three frameworks are then compared, studying when a definition may be translated from one formalism to another.

Negation with logical variables in conditional rewriting

We give a general formalism for conditional rewriting, with systems containing conditional rules whose antecedents contain literals to be shown satisfiable and/or unsatisfiable. We explore semantic issues, addressing when the associated operational rewriting mechanism is sound and complete. We then give restrictions on the formalism which enable us to construct useful and meaningful specifications using the proposed operational mechanism.

Inference rules and proof procedures for inequations

Abstract The negation of equality is an important relation that arises naturally in the study of equational programming languages and logic programming with equality. Proving and solving equations and inequations may also constitute subtasks in constraint logic programming. In this paper, we give forward (i.e., nonrefutational) techniques for proving the negation of equality in a theory. We develop a complete inference system to check whether an inequation is a logical consequence of a given system of equations and inequations. The inference system is used to develop a goal-directed semidecision procedure which uses a narrowing technique for proving inequations. A decision procedure is obtained when certain additional conditions are satisfied. The semidecision procedure for proving inequations is also modified to obtain a semidecision procedure for solving inequations in a theory, i.e., finding a substitution such that the corresponding instance of the given inequation is a logical consequence of the given system.

Reasoning in Systems of Equations and Inequations

Reasoning in purely equational systems has been studied quite extensively in recent years. Equational reasoning has been applied to several interesting problems, such as, development of equational programming languages, automating induction proofs, and theorem proving. However, reasoning in the presence of explicit inequations is still not as well understood. The expressive power of equational languages will be greatly enhanced if one is allowed to state inequations of terms explicitly. In this paper, we study reasoning in systems which consist of equations as well as inequations, emphasizing the development of forward, i.e., non-refutational, techniques for deducing valid inequations, similar to those for equational inference. Such techniques can be used as a basis for developing execution strategies for equational and declarative languages. We develop an inference system and show that it is complete for deducing all valid inequations. The inference system is used to develop a goal-directed semi-decision procedure which uses a narrowing technique for proving inequations. This semi-decision procedure can be converted into a decision procedure when certain additional conditions are satisfied.

PROLOG-Based Inductive Theorem Proving

Although PROLOG is a programming language based on techniques from theorem proving its use as a base for a theorem prover has not been explored until recently ([Sti84]). In this paper, we introduce a PROLOG-based deductive theorem proving method for proving first order inductive theory representable in Horn clauses. The method has the following characteristics: (1) It automatically partitions the domains over which the variables range into subdomains according to the manner in which the predicate symbols in the theorem are defined. (2) For each subdomain of the domain the prover returns a lemma. If the lemma is true, then the target theorem is true for this subdomain. The lemma could also be an induction hypothesis for the theorem. (3) The method does not explicitly use any inductive inference rule. The induction hypothesis, if needed for a certain subdomain, will sometimes be generated from a (limited) forward chaining mechanism in the prover and not from employing any particular inference rule.