Showing papers by "Lawrence C. Paulson published in 2015"

The Higher-Order Prover Leo-II

Abstract: Leo-II is an automated theorem prover for classical higher-order logic. The prover has pioneered cooperative higher-order---first-order proof automation, it has influenced the development of the TPTP THF infrastructure for higher-order logic, and it has been applied in a wide array of problems. Leo-II may also be called in proof assistants as an external aid tool to save user effort. For this it is crucial that Leo-II returns proof information in a standardised syntax, so that these proofs can eventually be transformed and verified within proof assistants. Recent progress in this direction is reported for the Isabelle/HOL system.

A Mechanised Proof of Gödel's Incompleteness Theorems Using Nominal Isabelle

Abstract: An Isabelle/HOL formalisation of Godel's two incompleteness theorems is presented. The work follows ?wierczkowski's detailed proof of the theorems using hereditarily finite (HF) set theory (Dissertationes Mathematicae 422, 1---58, 2003). Avoiding the usual arithmetical encodings of syntax eliminates the necessity to formalise elementary number theory within an embedded logical calculus. The Isabelle formalisation uses two separate treatments of variable binding: the nominal package (Logical Methods in Computer Science 8(2:14), 1---35, 2012) is shown to scale to a development of this complexity, while de Bruijn indices (Indagationes Mathematicae 34, 381---392, 1972) turn out to be ideal for coding syntax. Critical details of the Isabelle proof are described, in particular gaps and errors found in the literature.

Deciding Univariate Polynomial Problems Using Untrusted Certificates in Isabelle/HOL

Abstract: We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition. We follow the approach of untrusted certificates, separating solving from verifying: efficient external tools perform expensive real algebraic computations, producing evidence that is formally checked within Isabelle's logic. This allows us to exploit highly-tuned computer algebra systems like Mathematica to guide our procedure without impacting the correctness of its results. We present experiments demonstrating the efficacy of this approach, in many cases yielding orders of magnitude improvements over previous methods.

A Formalisation of Finite Automata using Hereditarily Finite Sets

Abstract: Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode theorem and Brzozowski's minimisation algorithm. The states of an automaton are HF sets, possibly constructed by product, sum, powerset and similar operations.

Verifying multicast-based security protocols using the inductive method

Abstract: Multicast, originally designed as an efficient way of broadcasting content, is being used in security protocols. Multicast security protocols are difficult to verify using model checking because they typically involve a large number of participants. Likewise, the exponential growth of knowledge being distributed during protocol run is a challenge. From a specification point of view, multicast is also a general way of representing message casting in protocol verification, with unicast, anycast and broadcast as special cases. Using the inductive method of protocol verification and Isabelle/HOL, we have devised techniques for specifying multicast protocols and proving many of their essential properties. We show backwards compatibility revisiting a well-known protocol and secrecy proofs for a mixed environment protocol as a case study. Our contributions are twofold: a usable multicast specification using the inductive method and the assertion that protocols should be verified by default using a multicast specification.

A Formalisation of Finite Automata Using Hereditarily Finite Sets

Abstract: Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode theorem and Brzozowski’s minimisation algorithm. The states of an automaton are HF sets, possibly constructed by product, sum, powerset and similar operations.

Proofs and Reconstructions

Abstract: Implementing proof reconstruction is difficult because it involves symbolic manipulations of formal objects whose representation varies between different systems. It requires significant knowledge of the source and target systems. One cannot simply re-target to another logic. We present a modular proof reconstruction system with separate components, specifying their behaviour and describing how they interact. This system is demonstrated and evaluated through an implementation to reconstruct proofs generated by Leo-II and Satallax in Isabelle HOL, and is shown to work better than the current method of rediscovering proofs using a select set of provers.

A Complete Decision Procedure for Univariate Polynomial Problems in Isabelle/HOL

Abstract: We present a complete, certificate-based decision procedure for first-order univariate polynomial problems in Isabelle. It is built around an executable function to decide the sign of a univariate polynomial at a real algebraic point. The procedure relies on no trusted code except for Isabelle's kernel and code generation. This work is the first step towards integrating the MetiTarski theorem prover into Isabelle.

A Comparison of Three Heuristics to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

Abstract: Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples.