Mauricio Ayala-Rincón

TL;DR: A theory for Abstract Reduction Systems (ARS) in the proof assistant PVS (Prototype Verification System) called ars is described and adequate specifications of basic definitions and notions of the theory of ARSs such as reduction, confluence and normal form are given and well-known results formalized.

TL;DR: This work presents a formalization in PVS of the computational theory for a computational model given as a class of partial recursive functions called PVS0, which results in a proven (formalized) Turing complete model.

TL;DR: This work presents a formalization of the theorem of existence of most general unifiers in first-order signatures in the higher-order proof assistant PVS that provides a complete formalized of the Knuth-Bendix Critical Pair theorem, among other relevant theorems of the theory of rewriting.

TL;DR: An automated program synthesis system that is based on the paradigm of programming by proofs is investigated, which would enable the incorporation of termination techniques used in other areas while still extracting programs.

Abstract: Orthogonality is a discipline of programming that in a syntactic manner guarantees determinism of functional specifications. Essentially, orthogonality avoids, on the one side, the inherent ambiguity of non determinism, prohibiting the existence of different rules that specify the same function and that may apply simultaneously (non-ambiguity), and, on the other side, it eliminates the possibility of occurrence of repetitions of variables in the left-hand side of these rules (left linearity). In the theory of term rewriting systems (TRSs) determinism is captured by the well-known property of confluence, that basically states that whenever different computations or simplifications from a term are possible, the computed answers or the obtained reduced terms should coincide. Although the proof is technically elaborated, confluence is well-known to be a consequence of orthogonality. Thus, orthogonality is an important mathematical discipline intrinsic to the specification of recursive functions that is naturally applied in functional programming and specification. Starting from a formalization of the theory of TRSs in the proof assistant PVS, this work describes how confluence of orthogonal TRSs is being formalized in this proof assistant. Substantial progress has been done in this research, obtaining until now complete formalizations for some similar, but restricted properties, such as a complete formalization for the property of confluence of non-ambiguous and (left and right) linear TRSs.