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Formal proof—theory and practice
 Notices AMS
, 2008
"... Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are nume ..."
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Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are numerous computer programs known as proof assistants that can check, or even partially construct, formal proofs written in their preferred proof language. These can be considered as practical, computerbased realizations of the traditional systems of formal symbolic logic and set theory proposed as foundations for mathematics. Why should we wish to create formal proofs?
A computational approach to pocklington certificates in type theory
 In Proc. of the 8th Int. Symp. on Functional and Logic Programming, volume 3945 of LNCS
, 2006
"... Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on ..."
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Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on computations inside and outside of the system (twolevel approach). 1 Formal Computational Proofs 1.1 Machines and the Quest for Correctness It is generally considered that modern mathematical logic was born towards the end of 19 th century, with the work of logicians like Frege, Peano, Russell or Zermelo, which lead to the precise definition of the notion of logical deduction and to formalisms like arithmetic, set theory or early type theory. From then on, a mathematical proof could be understood as a mathematical object itself, whose correction obeys some welldefined syntactical rules. In most formalisms, a formal proof is viewed as some treestructure; in natural deduction for instance, given to formal proofs σA and σB respectively of propositions A and B, these can be combined in order to build a proof of A ∧ B: σA σB ⊢ A ⊢ B ⊢ A ∧ B To sum things up, the logical point of view is that a mathematical statement holds in a given formalism if there exists a formal proof of this statement which follows the syntactical rules of the formalism. A traditional mathematical text can then be understood as an informal description of the formal proof. Things changed in the 1960ties, when N.G. de Bruijn’s team started to use computers to actually build formal proofs and verify their correctness. Using the fact that datastructures like formal proofs are very naturally represented in a computer’s memory, they delegated the proofverification work to the machine; their software Automath is considered as the first proofsystem and is the common
Integration of Deduction and Computation
 Applications of Computer Algebra
, 2000
"... We outline some of our approaches to the integration of Computer Algebra Systems and Automated Theorem Provers. Experimental couplings led to the development of the OMSCS framework, an architecture to specify the coupling of computational and reasoning systems. A model defining the context of a ..."
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We outline some of our approaches to the integration of Computer Algebra Systems and Automated Theorem Provers. Experimental couplings led to the development of the OMSCS framework, an architecture to specify the coupling of computational and reasoning systems. A model defining the context of a computation is proposed next. Finally, a multiagent approach, built upon our KOMET project, is then outlined through the integration of Mathematica.
System Description: Interface between Theorema And External Automated Deduction Systems
 In Linton and Sebastiani [175
, 2001
"... The interface between the Theorema system and external automated deduction systems is described. It provides a tool to access external provers within a Theorema session in the same way as \internal" Theorema provers. Currently 11 external systems are supported. The design of the interface a ..."
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The interface between the Theorema system and external automated deduction systems is described. It provides a tool to access external provers within a Theorema session in the same way as \internal" Theorema provers. Currently 11 external systems are supported. The design of the interface allows combining external systems with each other as well as with \internal" Theorema provers.
AUTOMATIC PROOF OF GRAPH NONISOMORPHISM
"... Abstract. We describe automated methods for constructing nonisomorphism proofs for pairs of graphs. The proofs can be humanreadable or machinereadable. We have developed a proof generator for graph nonisomorphism, which allows users to input graphs and construct a proof of (non)isomorphism. 1. ..."
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Abstract. We describe automated methods for constructing nonisomorphism proofs for pairs of graphs. The proofs can be humanreadable or machinereadable. We have developed a proof generator for graph nonisomorphism, which allows users to input graphs and construct a proof of (non)isomorphism. 1.
Edinburgh, Scotland
"... Abstract. This paper presents an ongoing effort to integrate the AXIOM family of computer algebra systems with Poly/MLbased proof assistants in the same framework. A longterm goal is to make a large set of efficient implementations of algebraic algorithms available to popular proof assistants, and ..."
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Abstract. This paper presents an ongoing effort to integrate the AXIOM family of computer algebra systems with Poly/MLbased proof assistants in the same framework. A longterm goal is to make a large set of efficient implementations of algebraic algorithms available to popular proof assistants, and also to bring the power of mechanized formal verification to a family of strongly typed computer algebra systems at a modest cost. Our approach is based on retargeting the code generator of the OpenAxiom compiler to the Poly/ML abstract machine.
Continuous KAOS, ASM, and Formal Control System Design Across the Continuous/Discrete Modeling Interface: A Simple Train Stopping Application
 UNDER CONSIDERATION FOR PUBLICATION IN FORMAL ASPECTS OF COMPUTING
"... A very simple model for train stopping is used as a vehicle for investigating how the development of a control system, initially designed in the continuous domain and subsequently discretized, can be captured within a formal development process compatible with standard model based refinement method ..."
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A very simple model for train stopping is used as a vehicle for investigating how the development of a control system, initially designed in the continuous domain and subsequently discretized, can be captured within a formal development process compatible with standard model based refinement methodologies. Starting with a formalized requirements analysis using KAOS, an abstract model of the continuous system is created in the ASM formalism. This requires extensions of the KAOS and ASM formalisms, capable of dealing with quantities evolving continuously over real time, which are developed. After considering how the continuous system, described as a continuous control system in the state space framework, can be discretized, a discrete control system is created in the state space framework. This is reexpressed in the ASM formalism. The rigorous results on the relationship between continuous and discrete control system models that are needed to establish provable properties of the discretization, then become the ingredients of a retrenchment between continuous and discrete ASM models, and are thus fully integrated into the
DOI 10.1007/s117860140175z Mathematics in Computer Science Formal Analysis of Optical Systems
"... Abstract Optical systems are becoming increasingly important by resolving many bottlenecks in today’s communication, electronics, and biomedical systems. However, given the continuous nature of optics, the inability to efficiently analyze optical system models using traditional paperandpencil and ..."
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Abstract Optical systems are becoming increasingly important by resolving many bottlenecks in today’s communication, electronics, and biomedical systems. However, given the continuous nature of optics, the inability to efficiently analyze optical system models using traditional paperandpencil and computer simulation approaches sets limits especially in safetycritical applications. In order to overcome these limitations, we propose to employ higherorderlogic theorem proving as a complement to computational and numerical approaches to improve optical model analysis in a comprehensive framework. The proposed framework allows formal analysis of optical systems at four abstraction levels, i.e., ray, wave, electromagnetic, and quantum.
www.elsevier.com/locate/jal Error analysis of digital filters using HOL theorem proving ✩
, 2005
"... When a digital filter is realized with floatingpoint or fixedpoint arithmetics, errors and constraints due to finite word length are unavoidable. In this paper, we show how these errors can be mechanically analysed using the HOL theorem prover. We first model the ideal real filter specification an ..."
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When a digital filter is realized with floatingpoint or fixedpoint arithmetics, errors and constraints due to finite word length are unavoidable. In this paper, we show how these errors can be mechanically analysed using the HOL theorem prover. We first model the ideal real filter specification and the corresponding floatingpoint and fixedpoint implementations as predicates in higherorder logic. We use valuation functions to find the real values of the floatingpoint and fixedpoint filter outputs and define the error as the difference between these values and the corresponding output of the ideal real specification. Fundamental analysis lemmas have been established to derive expressions for the accumulation of roundoff error in parametric Lthorder digital filters, for each of the three canonical forms of realization: direct, parallel, and cascade. The HOL formalization and proofs are found to be in a good agreement with existing theoretical paperandpencil counterparts.
An Approach for the Formal Verification of DSP Designs Using Theorem Proving
"... Abstract—This paper proposes a framework for the incorporation of formal methods in the design flow of digital signal processing (DSP) systems in a rigorous way. In the proposed approach, DSP descriptions were modeled and verified at different abstraction levels using higher order logic based on the ..."
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Abstract—This paper proposes a framework for the incorporation of formal methods in the design flow of digital signal processing (DSP) systems in a rigorous way. In the proposed approach, DSP descriptions were modeled and verified at different abstraction levels using higher order logic based on the higher order logic (HOL) theorem prover. This framework enables the formal verification of DSP designs that in the past could only be done partially using conventional simulation techniques. To this end, a shallow embedding of DSP descriptions in HOL at the floatingpoint (FP), fixedpoint (FXP), behavioral, register transfer level (RTL), and netlist gate levels is provided. The paper made use of existing formalization of FP theory in HOL and a parallel one developed for FXP arithmetic. The high ability of abstraction in HOL allows a seamless hierarchical verification encompassing the whole DSP design path, starting from toplevel FP and FXP algorithmic descriptions down to RTL, and gate level implementations. The paper illustrates the new verification framework on the fast Fourier transform (FFT) algorithm as a case study. Index Terms—Design automation, digital signal processors, error analysis, fast Fourier transforms, finite wordlength effects,