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Axiomatizing Reflective Logics and Languages
 Proceedings of Reflection'96
, 1996
"... The very success and breadth of reflective techniques underscores the need for a general theory of reflection. At present what we have is a wideranging variety of reflective systems, each explained in its own idiosyncratic terms. Metalogical foundations can allow us to capture the essential aspects ..."
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Cited by 36 (20 self)
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The very success and breadth of reflective techniques underscores the need for a general theory of reflection. At present what we have is a wideranging variety of reflective systems, each explained in its own idiosyncratic terms. Metalogical foundations can allow us to capture the essential aspects of reflective systems in a formalismindependent way. This paper proposes metalogical axioms for reflective logics and declarative languages based on the theory of general logics [34]. In this way, several strands of work in reflection, including functional, equational, Horn logic, and rewriting logic reflective languages, as well as a variety of reflective theorem proving systems are placed within a common theoretical framework. General axioms for computational strategies, and for the internalization of those strategies in a reflective logic are also given. 1 Introduction Reflection is a fundamental idea. In logic it has been vigorously pursued by many researchers since the fundamental wor...
A proofproducing decision procedure for real arithmetic
 Automated deduction – CADE20. 20th international conference on automated deduction
, 2005
"... Abstract. We present a fully proofproducing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proofproducing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate conv ..."
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Cited by 34 (3 self)
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Abstract. We present a fully proofproducing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proofproducing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate convincing examples of its value in interactive theorem proving. 1 Overview and related work Arguably the first automated theorem prover ever written was for a theory of lineararithmetic [8]. Nowadays many theorem proving systems, even those normally classified as `interactive ' rather than `automatic', contain procedures to automate routinearithmetical reasoning over some of the supported number systems like N, Z, Q, R and C. Experience shows that such automated support is invaluable in relieving users ofwhat would otherwise be tedious lowlevel proofs. We can identify several very common limitations of such procedures: Often they are restricted to proving purely universal formulas rather than dealingwith arbitrary quantifier structure and performing general quantifier elimination. Often they are not complete even for the supported class of formulas; in particular procedures for the integers often fail on problems that depend inherently on divisibility properties (e.g. 8x y 2 Z. 2x + 1 6 = 2y) They seldom handle nontrivial nonlinear reasoning, even in such simple cases as 8x y 2 R. x> 0 ^ y> 0) xy> 0, and those that do [18] tend to use heuristicsrather than systematic complete methods. Many of the procedures are standalone decision algorithms that produce no certificate of correctness and do not produce a `proof ' in the usual sense. The earliest serious exception is described in [4]. Many of these restrictions are not so important in practice, since subproblems arising in interactive proof can still often be handled effectively. Indeed, sometimes the restrictions are unavoidable: Tarski's theorem on the undefinability of truth implies thatthere cannot even be a complete semidecision procedure for nonlinear reasoning over
A Reflective Functional Language for Hardware Design and Theorem Proving
"... This paper introduces reFLect, a functional programming language with reflection features intended for applications in hardware design and verification. The reFLect language is strongly typed and similar to ML, but has quotation and antiquotation constructs. These may be used to construct and decomp ..."
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Cited by 32 (7 self)
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This paper introduces reFLect, a functional programming language with reflection features intended for applications in hardware design and verification. The reFLect language is strongly typed and similar to ML, but has quotation and antiquotation constructs. These may be used to construct and decompose expressions in the reFLect language itself. The paper motivates and presents the syntax and type system of this language, which brings together a new combination of patternmatching and reflection features targeted specifically at our application domain. It also gives an operational semantics based on a new use of contexts as expression constructors, and it presents a scheme for compiling reFLect programs into the λcalculus using the same context mechanism.
Real number calculations and theorem proving
 Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. ..."
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Cited by 20 (5 self)
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Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs. 1
A mechanically verified, sound and complete theorem prover for first order logic
 In Theorem Proving in Higher Order Logics, 18th International Conference, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. We present a system of first order logic, together with soundness and completeness proofs wrt. standard first order semantics. Proofs are mechanised in Isabelle/HOL. Our definitions are computable, allowing us to derive an algorithm to test for first order validity. This algorithm may be e ..."
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Abstract. We present a system of first order logic, together with soundness and completeness proofs wrt. standard first order semantics. Proofs are mechanised in Isabelle/HOL. Our definitions are computable, allowing us to derive an algorithm to test for first order validity. This algorithm may be executed in Isabelle/HOL using the rewrite engine. Alternatively the algorithm has been ported to OCaML. 1
A verified runtime for a verified theorem prover
"... rely on the correctness of runtime systems for programming languages like ML, OCaml or Common Lisp. These runtime systems are complex and critical to the integrity of the theorem provers. In this paper, we present a new Lisp runtime which has been formally verified and can run the Milawa theorem pro ..."
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Cited by 14 (8 self)
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rely on the correctness of runtime systems for programming languages like ML, OCaml or Common Lisp. These runtime systems are complex and critical to the integrity of the theorem provers. In this paper, we present a new Lisp runtime which has been formally verified and can run the Milawa theorem prover. Our runtime consists of 7,500 lines of machine code and is able to complete a 4 gigabyte Milawa proof effort. When our runtime is used to carry out Milawa proofs, less unverified code must be trusted than with any other theorem prover. Our runtime includes a justintime compiler, a copying garbage collector, a parser and a printer, all of which are HOL4verified down to the concrete x86 code. We make heavy use of our previously developed tools for machinecode verification. This work demonstrates that our approach to machinecode verification scales to nontrivial applications. 1
Verifying and reflecting quantifier elimination for Presburger arithmetic
 LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING
, 2005
"... We present an implementation and verification in higherorder logic of Cooper’s quantifier elimination for Presburger arithmetic. Reflection, i.e. the direct execution in ML, yields a speedup of a factor of 200 over an LCFstyle implementation and performs as well as a decision procedure handcode ..."
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We present an implementation and verification in higherorder logic of Cooper’s quantifier elimination for Presburger arithmetic. Reflection, i.e. the direct execution in ML, yields a speedup of a factor of 200 over an LCFstyle implementation and performs as well as a decision procedure handcoded in ML.
Complex quantifier elimination in HOL
 TPHOLs 2001: Supplemental Proceedings
, 2001
"... Abstract. Building on a simple construction of the complex numbers and a proof of the Fundamental Theorem of Algebra, we implement, as a HOL derived inference rule, a decision method for the first order algebraic theory of C based on quantifier elimination. Although capable of solving some mildly in ..."
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Abstract. Building on a simple construction of the complex numbers and a proof of the Fundamental Theorem of Algebra, we implement, as a HOL derived inference rule, a decision method for the first order algebraic theory of C based on quantifier elimination. Although capable of solving some mildly interesting problems, we also implement a more efficient semidecision procedure for the universal fragment based on Gröbner bases. This is applied to examples including the automatic proof of some simple geometry theorems. The general and universal procedures present an interesting contrast in that the latter can exploit the findingchecking separation to achieve greater efficiency, though this feature is only partly exploited in the present implementation. 1
Formal Interoperability
, 1998
"... this paper I briefly sketch recent work on metalogical foundations that seems promising as a conceptual basis on which to achieve the goal of formal interoperability. Specificaly, I will briefly discuss: ..."
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Cited by 13 (3 self)
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this paper I briefly sketch recent work on metalogical foundations that seems promising as a conceptual basis on which to achieve the goal of formal interoperability. Specificaly, I will briefly discuss: