A theorem is presented which has applications in the numerical computation of fixed points of recursive functions. If a sequence of functions {fn} is convergent on a metric space I subset of R, then it is possible to ...
详细信息
A theorem is presented which has applications in the numerical computation of fixed points of recursive functions. If a sequence of functions {fn} is convergent on a metric space I subset of R, then it is possible to observe this behaviour on the set D subset of Q of all numbers represented in a computer. However, as D is not complete, the representation of fn on D is subject to an error. Then f(n) and f(m) are considered equal when its differences computed on D are equal or lower than the sum of error of each f(n) and f(m). An example is given to illustrate the use of the theorem.
Basing on an original Coq implementation of unbounded linear search for partially decidable predicates, we study the computational contents of µ-recursive functions via their syntactic representation, and a corre...
详细信息
The term substitution theorem is a vital theorem in mathematical logic that concerns the replacement operation of variables within terms. In this study, we present a comprehensive formalization and verification of the...
详细信息
In this paper, we present a linear and reversible programming language with inductives types and recursion. The semantics of the languages is based on pattern-matching;we show how ensuring syntactical exhaustivity and...
详细信息
In this paper we present a novel termination order the predicative lexicographic path order (PLPO for short), a syntactic restriction of the lexicographic path order. As well as lexicographic path orders, several non-...
详细信息
ISBN:
(纸本)9783319124667;9783319124650
In this paper we present a novel termination order the predicative lexicographic path order (PLPO for short), a syntactic restriction of the lexicographic path order. As well as lexicographic path orders, several non-trivial primitive recursive equations, e.g., primitive recursion with parameter substitution, unnested multiple recursion, or simple nested recursion, can be oriented with PLPOs. It can be shown that the PLPO however only induces primitive recursive upper bounds on derivation lengths of compatible rewrite systems. This yields an alternative proof of a classical fact that the class of primitive recursive functions is closed under those non-trivial primitive recursive equations.
We describe a learning-based approach for verifying recursive functions. The Boolean formula learning algorithm CDNF is used to automatically infer function summaries for recursive functions. In contrast to traditiona...
详细信息
Reversible Primitive Permutations (RPP) are recursively defined functions designed to model Reversible Computation. We illustrate a proof, fully developed with the proof-assistant Lean, certifying that: "RPP can ...
详细信息
ISBN:
(纸本)9783031090059;9783031090042
Reversible Primitive Permutations (RPP) are recursively defined functions designed to model Reversible Computation. We illustrate a proof, fully developed with the proof-assistant Lean, certifying that: "RPP can encode every Primitive recursive Function". Our reworking of the original proof of that statement is conceptually simpler, fixes some bugs, suggests a new more primitive reversible iteration scheme for RPP, and, in order to keep formalization and semi-automatic proofs simple, led us to identify a single pattern that can generate some useful reversible algorithms in RPP: Cantor Pairing, Quotient/Reminder of integer division, truncated Square Root. Our Lean source code is available for experiments on Reversible Computation whose properties can be certified.
In classic program synthesis algorithms, such as counter-example-guided inductive synthesis (CEGIS), the algorithms alternate between a synthesis phase and an oracle (verification) phase. Many synthesis algorithms use...
详细信息
ISBN:
(数字)9783030945831
ISBN:
(纸本)9783030945824;9783030945831
In classic program synthesis algorithms, such as counter-example-guided inductive synthesis (CEGIS), the algorithms alternate between a synthesis phase and an oracle (verification) phase. Many synthesis algorithms use a white-box oracle based on satisfiability modulo theory (SMT) solvers to provide counter examples. But what if a white box oracle is either not available or not easy to work with? We present a framework for solving a general class of oracle-guided synthesis problems which we term synthesis modulo oracles (SyMo). In this setting, oracles are black boxes with a query-response interface defined by the synthesis problem. As a necessary component of this framework, we also formalize the problem of satisfiability modulo theories and oracles (SMTO), and present an algorithm for solving this problem. We implement a prototype solver for satisfiability and synthesis modulo oracles and demonstrate that, by using oracles that execute functions not easily modeled in SMT-constraints, such as recursive functions or oracles that incorporate compilation and execution of code, SMTO and SyMO can solve problems beyond the abilities of standard SMT and synthesis solvers.
Hofstadter’s G function is recursively defined via G(0) = 0 and then G(n) = n − G(G(n − 1)). Following Hofstadter, a family (Fk) of similar functions is obtained by varying the number k of nested recursive calls in t...
详细信息
BSS RAMs over first-order structures help to characterize algorithms for processing objects by means of useful operations and relations. They are the result of a generalization of several types of abstract machines. W...
详细信息
暂无评论