On the equivalence of schemes

Journal of Computer and System Sciences, 1973

This paper presents general methods for studying the problems of translatability between classes of schemes and equivalence of schemes in a given class. There are four methods: applying the theory of formal languages, programming, measuring the complexity of a computation, and "cutting and pasting." These methods are used to answer several questions of translatability and equivalence for classes of program schemes, program schemes augmented with counters, and recursively defined schemes. In particular, it is shown that (i) the quasirational recursion schemes are translatable into strongly equivalent program schemes, (ii) monadic recursion schemes are translatable into strongly equivalent program schemes with two counters, (iii) there is a monadic recursion scheme not strongly equivalent to any program scheme with one counter.

1972

1977

I t is proved that in the general case of arbitrary context-free schemes a program is (partially) correct with respect to given initial and final assertions if and only if a suitable finite system of intermediate assertions can be found. Assertions are allowed from the extended state space V x V. This result contrasts with the results of [2], where it is proved that if assertions are taken from the original state space V. then in the general case an infinite system of intermediate assertions is needed. The extension of the state space allows a unification in the relational framework of [2], of the (essence of the) results of [2], and of [4], 151 and [6], and provides a semantic counterpart of the use of auxiliary variables.

Lecture Notes in Computer Science, 2000

In this paper, we explore the testing-verification relationship with the objective of mechanizing the generation of test data. We consider program classes defined as recursive program schemes and we show that complete and finite test data sets can be associated with such classes, that is to say that these test data sets allow us to distinguish every two different functions in these schemes. This technique is applied to the verification of simple properties of programs.

Theoretical Computer Science, 1979

We study transformations and equivalences of recursive program schemes. We give an optimization algorithm which recognizes and removes all the parts of a program scheme which do not affect its final output. This result leads to a syntactic way of suppressing some erroneous loops in programs and can be used to prove that equivalence of recursive program schemes is solvable under particular conditions.

SIAM Journal on Computing, 1980

I t is proved that in the general case of arbitrary context-free schemes a program is (partially) correct with respect to given initial and final assertions if and only if a suitable finite system of intermediate assertions can be found. Assertions are allowed from the extended state space V x V. This result contrasts with the results of [2], where it is proved that if assertions are taken from the original state space V. then in the general case an infinite system of intermediate assertions is needed. The extension of the state space allows a unification in the relational framework of [2], of the (essence of the) results of [2], and of [4], 151 and [6], and provides a semantic counterpart of the use of auxiliary variables.

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16, 2016

A non-deterministic recursion scheme recognizes a language of finite trees. This very expressive model can simulate, among others, higher-order pushdown automata with collapse. We show decidability of the diagonal problem for schemes. This result has several interesting consequences. In particular, it gives an algorithm that computes the downward closure of languages of words recognized by schemes. In turn, this has immediate application to separability problems and reachability analysis of concurrent systems.

A program schema defines a class of programs, all of which have identical statement structure, but whose expressions may differ. We define a class of syntactic similarity binary relations between linear structured schemas and show that these relations characterise schema equivalence for structured schemas which are linear, free and liberal. In this paper we prove that similarity implies equivalence for linear schemas; the proof of a near-converse for schemas that are linear, free and liberal (LFL), which is much longer, is given in a Technical Report, which also contains the results of this paper. Our main result considerably extends the class of program schemas for which equivalence is known to be decidable, and suggests that linearity is a constraint worthy of further investigation.

Theoretical Computer Science, 2006

This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the category-theoretic version of the classical area of algebraic semantics. The overall assumptions needed are small indeed: working only in categories with "enough final coalgebras" we show how to formulate, solve, and study recursive program schemes. Our general theory is algebraic and so avoids using ordered or metric structures. Our work generalizes the previous approaches which do use this extra structure by isolating the key concepts needed to study substitution in infinite trees, including second-order substitution. As special cases of our interpreted solutions we obtain the usual denotational semantics using complete partial orders, and the one using complete metric spaces. Our theory also encompasses implicitly defined objects which are not usually taken to be related to recursive program schemes. For example, the classical Cantor two-thirds set falls out as an interpreted solution (in our sense) of a recursive program scheme.

2003

A program schema defines a class of programs, all of which have identical statement structures, but whose expressions may differ. We prove that given any two structured schemas which are conservative, linear and free, it is decidable whether they are equivalent.