Algebraic Algorithm Design and Local Search

Formal, mathematically-based techniques promise to play an expanding role in the development and maintenance of the software on which our technological society depends. Algebraic techniques have been applied successfully to algorithm synthesis by the use of algorithm theories and design tactics, an approach pioneered in the Kestrel Interactive Development System (KIDS). An algorithm theory formally characterizes the essential components of a family of algorithms. A design tactic is a specialized procedure for recognizing in a problem specification the structures identified in an algorithm theory and then synthesizing a program. Design tactics are hard to write, however, and much of the knowledge they use is encoded procedurally in idiosyncratic ways. Algebraic methods promise a way to represent algorithm design knowledge declaratively and uniformly. We describe a general method for performing algorithm design that is more purely algebraic than that of KIDS. This method is then applied to local search. Local search is a large and diverse class of algorithms applicable to a wide range of problems; it is both intrinsically important and representative of algorithm design as a whole. A general theory of local search is formalized to describe the basic properties common to all local search algorithms, and applied to several variants of hill climbing and simulated annealing. The general theory is then specialized to describe some more advanced local search techniques, namely tabu search and the Kernighan-Lin heuristic

Automatic Generation of Proof Tactics for Finite-Valued Logics

A number of flexible tactic-based logical frameworks are nowadays available that can implement a wide range of mathematical theories using a common higher-order metalanguage. Used as proof assistants, one of the advantages of such powerful systems resides in their responsiveness to extensibility of their reasoning capabilities, being designed over rule-based programming languages that allow the user to build her own `programs to construct proofs' - the so-called proof tactics. The present contribution discusses the implementation of an algorithm that generates sound and complete tableau systems for a very inclusive class of sufficiently expressive finite-valued propositional logics, and then illustrates some of the challenges and difficulties related to the algorithmic formation of automated theorem proving tactics for such logics. The procedure on whose implementation we will report is based on a generalized notion of analyticity of proof systems that is intended to guarantee termination of the corresponding automated tactics on what concerns theoremhood in our targeted logics

The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques

Verifying Safety Properties With the TLA+ Proof System

TLAPS, the TLA+ proof system, is a platform for the development and mechanical verification of TLA+ proofs written in a declarative style requiring little background beyond elementary mathematics. The language supports hierarchical and non-linear proof construction and verification, and it is independent of any verification tool or strategy. A Proof Manager uses backend verifiers such as theorem provers, proof assistants, SMT solvers, and decision procedures to check TLA+ proofs. This paper documents the first public release of TLAPS, distributed with a BSD-like license. It handles almost all the non-temporal part of TLA+ as well as the temporal reasoning needed to prove standard safety properties, in particular invariance and step simulation, but not liveness properties

A Proof Strategy Language and Proof Script Generation for Isabelle/HOL

We introduce a language, PSL, designed to capture high level proof strategies in Isabelle/HOL. Given a strategy and a proof obligation, PSL's runtime system generates and combines various tactics to explore a large search space with low memory usage. Upon success, PSL generates an efficient proof script, which bypasses a large part of the proof search. We also present PSL's monadic interpreter to show that the underlying idea of PSL is transferable to other ITPs.Comment: This paper has been submitted to CADE2

Proof-Pattern Recognition and Lemma Discovery in ACL2

We present a novel technique for combining statistical machine learning for proof-pattern recognition with symbolic methods for lemma discovery. The resulting tool, ACL2(ml), gathers proof statistics and uses statistical pattern-recognition to pre-processes data from libraries, and then suggests auxiliary lemmas in new proofs by analogy with already seen examples. This paper presents the implementation of ACL2(ml) alongside theoretical descriptions of the proof-pattern recognition and lemma discovery methods involved in it

UTP2: Higher-Order Equational Reasoning by Pointing

We describe a prototype theorem prover, UTP2, developed to match the style of hand-written proof work in the Unifying Theories of Programming semantical framework. This is based on alphabetised predicates in a 2nd-order logic, with a strong emphasis on equational reasoning. We present here an overview of the user-interface of this prover, which was developed from the outset using a point-and-click approach. We contrast this with the command-line paradigm that continues to dominate the mainstream theorem provers, and raises the question: can we have the best of both worlds?Comment: In Proceedings UITP 2014, arXiv:1410.785