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A Focused Approach to Combining Logics
, 2010
"... We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicativeadditive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structura ..."
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We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicativeadditive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut elimination holds in such fragments. From cut elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classicallinear hybrid logics.
A proposal for broad spectrum proof certificates
"... Abstract. Recent developments in the theory of focused proof systems provide flexible means for structuring proofs within the sequent calculus. This structuring is organized around the construction of “macro” level inference rules based on the “micro ” inference rules which introduce single logical ..."
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Abstract. Recent developments in the theory of focused proof systems provide flexible means for structuring proofs within the sequent calculus. This structuring is organized around the construction of “macro” level inference rules based on the “micro ” inference rules which introduce single logical connectives. After presenting focused proof systems for firstorder classical logics (one with and one without fixed points and equality) we illustrate several examples of proof certificates formats that are derived naturally from the structure of such focused proof systems. In principle, a proof certificate contains two parts: the first part describes how macro rules are defined in terms of micro rules and the second part describes a particular proof object using the macro rules. The first part, which is based on the vocabulary of focused proof systems, describes a collection of macro rules that can be used to directly present the structure of proof evidence captured by a particular class of computational logic systems. While such proof certificates can capture a wide variety of proof structures, a proof checker can remain simple since it must only understand the microrules and the discipline of focusing. Since proofs and proof certificates are often likely to be large, there must be some flexibility in allowing proof certificates to elide subproofs: as a result, proof checkers will necessarily be required to perform (bounded) proof search in order to reconstruct missing subproofs. Thus, proof checkers will need to do unification and restricted backtracking search. 1
Finding Unity in Computational Logic
"... While logic was once developed to serve philosophers and mathematicians, it is increasingly serving the varied needs of computer scientists. In fact, recent decades have witnessed the creation of the new discipline of Computational Logic. While Computation Logic can claim involvement in many, divers ..."
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While logic was once developed to serve philosophers and mathematicians, it is increasingly serving the varied needs of computer scientists. In fact, recent decades have witnessed the creation of the new discipline of Computational Logic. While Computation Logic can claim involvement in many, diverse areas of computing, little has been done to systematize the foundations of this new discipline. Here, we envision a unity for Computational Logic organized around recent developments in the theory of sequent calculus proofs. We outline how new tools and methodologies can be developed around a boarder approach to computational logic. Computational logic, unity of logic, proof theory 1. SOFTWARE AND HARDWARE CORRECTNESS IS CRITICALLY IMPORTANT Computer systems are everywhere in our societies and their integration with all parts of our lives is constantly increasing. There are a host of computer systems—such as those in cars, airplanes, missiles, hospital equipment—where correctness of software is
Substructural Logical Specifications
, 2012
"... Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author and A logical framework and its implementation should serve as a flexible tool for specifying, simulating, and reasoning about formal systems. When the formal systems we are interested in exh ..."
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Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author and A logical framework and its implementation should serve as a flexible tool for specifying, simulating, and reasoning about formal systems. When the formal systems we are interested in exhibit state and concurrency, however, existing logical frameworks fall short of this goal. Logical frameworks based on a rewriting interpretation of substructural logics, ordered and linear logic in particular, can help. To this end, this dissertation introduces and demonstrates four methodologies for developing and using substructural logical frameworks for specifying and reasoning about stateful and concurrent systems. Structural focalization is a synthesis of ideas from Andreoli’s focused sequent calculi and Watkins’s hereditary substitution. We can use structural focalization to take a logic and define a restricted form of derivations, the focused derivations, that form the basis of a logical framework. We apply this methodology to define SLS, a logical framework for substructural logical specifications, as a fragment of ordered
Kripke semantics and proof systems for combining intuitionistic logic and classical logic. Submitted
, 2011
"... We combine intuitionistic logic and classical logic into a new, firstorder logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities ..."
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We combine intuitionistic logic and classical logic into a new, firstorder logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined modeltheoretically by a Kripkestyle semantics for the logic. Two proof systems are also formulated. The first system extends Gentzen’s intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girard’s LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalin’s multipleconclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable. 1
Communicating and trusting proofs: The case for broad spectrum proof certificates. Available from author’s website
, 2011
"... Abstract. Proofs, both formal and informal, are documents that are intended to circulate within societies of humans and machines distributed across time and space in order to provide trust. Such trust might lead one mathematician to accept a certain statement as true or it might help convince a cons ..."
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Abstract. Proofs, both formal and informal, are documents that are intended to circulate within societies of humans and machines distributed across time and space in order to provide trust. Such trust might lead one mathematician to accept a certain statement as true or it might help convince a consumer that a certain software system is secure. Using this general characterization of proofs, we examine a range of perspectives about proofs and their roles within mathematics and computer science that often appear contradictory. We then consider the possibility of defining a broad spectrum proof certificate format that is intended as a universal language for communicating formal proofs among computational logic systems. We identify four desiderata for such proof certificates: they must be (i) checkable by simple proof checkers, (ii) flexible enough that existing provers can conveniently produce such certificates from their internal evidence of proof, (iii) directly related to proof formalisms used within the structural proof theory literature, and (iv) permit certificates to elide some proof information with the expectation that a proof checker can reconstruct the missing information using bounded and structured proof search. We consider various consequences of these desiderata, including how they can mix computation and deduction and what they mean for the establishment of marketplaces and libraries of proofs. In a companion paper we proposal a specific framework for achieving all four of these desiderata. 1
A formal framework for specifying sequent calculus proof systems
, 2012
"... Intuitionistic logic and intuitionistic type systems are commonly used as frameworks for the specification of natural deduction proof systems. In this paper we show how to use classical linear logic as a logical framework to specify sequent calculus proof systems and to establish some simple consequ ..."
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Intuitionistic logic and intuitionistic type systems are commonly used as frameworks for the specification of natural deduction proof systems. In this paper we show how to use classical linear logic as a logical framework to specify sequent calculus proof systems and to establish some simple consequences of the specified sequent calculus proof systems. In particular, derivability of an inference rule from a set of inference rules can be decided by bounded (linear) logic programming search on the specified rules. We also present two simple and decidable conditions that guarantee that the cut rule and nonatomic initial rules can be eliminated.
Checking foundational proof certificates for firstorder logic
"... We present the design philosophy of a proof checker based on a notion of foundational proof certificates. This checker provides a semantics of proof evidence using recent advances in the theory of proofs for classical and intuitionistic logic. That semantics is then performed by a (higherorder) log ..."
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We present the design philosophy of a proof checker based on a notion of foundational proof certificates. This checker provides a semantics of proof evidence using recent advances in the theory of proofs for classical and intuitionistic logic. That semantics is then performed by a (higherorder) logic program: successful performance means that a formal proof of a theorem has been found. We describe how the λProlog programming language provides several features that help guarantee such a soundness claim. Some of these features (such as strong typing, abstract datatypes, and higherorder programming) were features of the ML programming language when it was first proposed as a proof checker for LCF. Other features of λProlog (such as support for bindings, substitution, and backtracking search) turn out to be equally important for describing and checking the proof evidence encoded in proof certificates. Since trusting our proof checker requires trusting a programming language implementation, we discuss various avenues for enhancing one’s trust of such a checker. 1
Classical and Intuitionistic Subexponential Logics are Equally Expressive
"... Abstract. It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by doublenegation, while the other direction has no truthpreserving propositional ..."
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Abstract. It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by doublenegation, while the other direction has no truthpreserving propositional encodings. We show here that subexponential logic, which is a family of substructural refinements of classical logic, each parametric over a preorder over the subexponential connectives, does not suffer from this asymmetry if the preorder is systematically modified as part of the encoding. Precisely, we show a bijection between synthetic (i.e., focused) partial sequent derivations modulo a given encoding. Particular instances of our encoding for particular subexponential preorders give rise to both known and novel adequacy theorems for substructural logics. 1