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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
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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Cited by 11 (9 self)
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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.
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
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
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
A Systematic Approach to Canonicity in the Classical Sequent Calculus
"... The sequent calculus is often criticized for requiring proofs to be laden with large volumes of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, cutfree sequent proofs can separate closely related steps—such ..."
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The sequent calculus is often criticized for requiring proofs to be laden with large volumes of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, cutfree sequent proofs can separate closely related steps—such as instantiating a block of quantifiers—by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically noninterfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers revolt against the sequent calculus and replace it with proof structures that are more parallel or geometric. Proofnets, matings, and atomic flows are examples of such revolutionary formalisms. In this paper, we propose taking, instead, an evolutionary approach to recover canonicity within the sequent calculus, an approach we illustrate for classical firstorder logic. We use a multifocused sequent system as our means of abstracting away the details from classical sequent proofs. We then show that, among the focused sequent proofs, the maximally multifocused proofs, which make the foci as parallel as possible, are canonical. Moreover, such proofs are isomorphic to expansion tree proofs—a well known, simple, and parallel generalization of Herbrand disjunctions—for classical firstorder logic. We thus provide a systematic method of recovering the essence of any sequent proof without abandoning the sequent calculus. 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.
Using LJF as a Framework for Proof Systems
, 2009
"... In this work we show how to use the focused intuitionistic logic system LJF as a framework for encoding several different intuitionistic and classical proof systems. The proof systems are encoded in a strong level of adequacy, namely the level of (open) derivations. Furthermore we show how to prove ..."
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In this work we show how to use the focused intuitionistic logic system LJF as a framework for encoding several different intuitionistic and classical proof systems. The proof systems are encoded in a strong level of adequacy, namely the level of (open) derivations. Furthermore we show how to prove relative completeness between the different systems. By relative completeness we mean that the systems prove the same formulas. The proofs of relative completeness exploit the encodings to give, in most cases, fairly simple proofs. This work is heavily based on the recent work by Nigam and Miller, which uses the focused linear logic system LLF to encode the same proof systems as we do. Our work shows that the features of linear logic are not needed for the full adequacy result, and furthermore we show that even though encoding in LLF is more generic and streamlined, the encoding in LJF sometimes gives simpler, more natural encodings and easier proofs.