Results 1  10
of
40
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 ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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
Structural focalization
, 2011
"... Focusing, introduced by JeanMarc Andreoli in the context of classical linear logic, defines a normal form for sequent calculus derivations that cuts down on the number of possible derivations by eagerly applying invertible rules and grouping sequences of noninvertible rules. A focused sequent calc ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Focusing, introduced by JeanMarc Andreoli in the context of classical linear logic, defines a normal form for sequent calculus derivations that cuts down on the number of possible derivations by eagerly applying invertible rules and grouping sequences of noninvertible rules. A focused sequent calculus is defined relative to some nonfocused sequent calculus; focalization is the property that every nonfocused derivation can be transformed into a focused derivation. In this paper, we present a focused sequent calculus for polarized propositional intuitionistic logic and prove the focalization property relative to a standard presentation of propositional intuitionistic logic. Compared to existing approaches, the proof is quite concise, depending only on the internal soundness and completeness of the focused logic. In turn, both of these properties can be established (and mechanically verified) by structural induction in the style of Pfenning’s structural cut elimination without the need for any tedious and repetitious invertibility lemmas. The proof of cut admissibility for the focused system, which establishes internal soundness, is not particularly novel. The proof of identity expansion, which establishes internal completeness, is the principal contribution of this work. 1
Compact Proof Certificates For Linear Logic
"... Abstract. Linear logic is increasingly being used as a tool for communicating reasoning agents in domains such as authorization, access control, electronic voting, etc., where proof certificates represent evidence that must be verified by proof consumers as part of higher protocols. Controlling the ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Linear logic is increasingly being used as a tool for communicating reasoning agents in domains such as authorization, access control, electronic voting, etc., where proof certificates represent evidence that must be verified by proof consumers as part of higher protocols. Controlling the size of these certificates is critical. We assume that the proof consumer is allowed to do some search to reconstruct details of the full proof that are omitted from the certificates. Because the decision problem for linear logic is unsolvable, the certificate must contain at least enough information to bound the search: we show how to use the sequence of contractions in the sequent proof for this bound. The remaining content of the proof, in particular the information about resource divisions, can then be omitted from the certificate. We also describe a technique for giving a variable amount of additional search hints to the proof consumer to limit its nondeterminism. 1