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Substructural Operational Semantics as Ordered Logic Programming
"... We describe a substructural logic with ordered, linear, and persistent propositions and then endow a fragment with a committed choice forwardchaining operational interpretation. Exploiting higherorder terms in this metalanguage, we specify the operational semantics of a number of object language f ..."
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We describe a substructural logic with ordered, linear, and persistent propositions and then endow a fragment with a committed choice forwardchaining operational interpretation. Exploiting higherorder terms in this metalanguage, we specify the operational semantics of a number of object language features, such as callbyvalue, callbyname, callbyneed, mutable store, parallelism, communication, exceptions and continuations. The specifications exhibit a high degree of uniformity and modularity that allows us to analyze the structural properties required for each feature in isolation. Our substructural framework thereby provides a new methodology for language specification that synthesizes structural operational semantics, abstract machines, and logical approaches. 1
Reasoning with HigherOrder Abstract Syntax and Contexts: A Comparison
"... Abstract. A variety of logical frameworks support the use of higherorder abstract syntax (HOAS) in representing formal systems given via axioms and inference rules and reasoning about them. In such frameworks, objectlevel binding is encoded directly using metalevel binding. Although these systems ..."
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Abstract. A variety of logical frameworks support the use of higherorder abstract syntax (HOAS) in representing formal systems given via axioms and inference rules and reasoning about them. In such frameworks, objectlevel binding is encoded directly using metalevel binding. Although these systems seem superficially the same, they differ in a variety of ways; for example, in how they handle a context of assumptions and in what theorems about a given formal system can be expressed and proven. In this paper, we present several case studies which highlight a variety of different aspects of reasoning using HOAS, with the intention of providing a basis for comparison of different systems. We then carry out such a comparison among three systems: Twelf, Beluga, and Hybrid. We also develop a general set of criteria for comparing such systems. We hope that others will implement these challenge problems, apply these criteria, and further our understanding of the tradeoffs involved in choosing one system over another for this kind of reasoning. 1
Cut elimination for a logic with induction and coinduction
 JOURNAL OF APPLIED LOGIC
, 2012
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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
Contributions to the Theory of Syntax with Bindings and to Process Algebra
, 2010
"... We develop a theory of syntax with bindings, focusing on: methodological issues concerning the convenient representation of syntax; techniques for recursive definitions and inductive reasoning. Our approach consists of a combination of FOAS (FirstOrder Abstract Syntax) and HOAS (HigherOrder Abst ..."
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We develop a theory of syntax with bindings, focusing on: methodological issues concerning the convenient representation of syntax; techniques for recursive definitions and inductive reasoning. Our approach consists of a combination of FOAS (FirstOrder Abstract Syntax) and HOAS (HigherOrder Abstract Syntax) and tries to take advantage of the best of both worlds. The connection between FOAS and HOAS follows some general patterns and is presented as a (formally certified) statement of adequacy. We also develop a general technique for proving bisimilarity in process algebra Our technique, presented as a formal proof system, is applicable to a wide range of process algebras. The proof system is incremental, in that it allows building incrementally an a priori unknown bisimulation, and patternbased, in that it works on equalities of process patterns (i.e., universally quantified equations of process terms containing process variables), thus taking advantage of equational reasoning in a “circular ” manner, inside coinductive proof loops. All the work presented here has been formalized in the Isabelle theorem prover. The formalization is performed in a general setting: arbitrary manysorted syntax with bindings and arbitrary SOSspecified process algebra in de Simone format. The usefulness of our techniques is illustrated by several formalized case studies: a development of callbyname and callbyvalue λcalculus with constants, including ChurchRosser theorems, connection with de Bruijn representation, connection with other Isabelle formalizations, HOAS representation, and contituationpassingstyle (CPS) transformation; a proof in HOAS of strong normalization for the polymorphic secondorder λcalculus (a.k.a. System F). We also indicate the outline and some details of the formal development. ii to Leili R. Marleene iii
Recursion principles for syntax with bindings and substitution
 In ICFP
, 2011
"... We characterize the data type of terms with bindings, freshness and substitution, as an initial model in a suitable Horn theory. This characterization yields a convenient recursive definition principle, which we have formalized in Isabelle/HOL and employed in a series of case studies taken from the ..."
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We characterize the data type of terms with bindings, freshness and substitution, as an initial model in a suitable Horn theory. This characterization yields a convenient recursive definition principle, which we have formalized in Isabelle/HOL and employed in a series of case studies taken from the λcalculus literature.
Reasoning with Hypothetical Judgments and Open Terms in Hybrid
"... Hybrid is a system developed to specify and reason about logics, programming languages, and other formal systems expressed in higherorder abstract syntax (HOAS). An important goal of Hybrid is to exploit the advantages of HOAS within the wellunderstood setting of higherorder logic as implemented ..."
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Hybrid is a system developed to specify and reason about logics, programming languages, and other formal systems expressed in higherorder abstract syntax (HOAS). An important goal of Hybrid is to exploit the advantages of HOAS within the wellunderstood setting of higherorder logic as implemented by systems such as Isabelle and Coq. In this paper, we add new capabilities for reasoning by induction on encodings of objectlevel inference rules. Elegant and succinct specifications of such inference rules can often be given using hypothetical and parametric judgments, which are represented by embedded implication and universal quantification. Induction over such judgments is wellknown to be problematic. In previous work, we showed how to express this kind of judgment using a twolevel approach, but reasoning by induction on such judgments was restricted to closed terms. The new capabilities we add include techniques for adding arbitrary “new ” variables to contexts and inductively reasoning about open terms. Very little overhead is required, namely a small library of definitions and lemmas, yet the reasoning power of the system and the class of properties that can be proved is significantly increased. We illustrate the approach using PCF, a simple programming language that serves as the core of a variety of functional languages. We encode the typing judgment, and prove by induction on this judgment that welltyped PCF terms have unique types.
Abella: A system for reasoning about relational specifications
 Journal of Formalized Reasoning
"... The Abella interactive theorem prover is based on an intuitionistic logic that allows for inductive and coinductive reasoning over relations. Abella supports the λtree approach to treating syntax containing binders: it allows simply typed λterms to be used to represent such syntax and it provides ..."
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The Abella interactive theorem prover is based on an intuitionistic logic that allows for inductive and coinductive reasoning over relations. Abella supports the λtree approach to treating syntax containing binders: it allows simply typed λterms to be used to represent such syntax and it provides higherorder (pattern) unification, the ∇ quantifier, and nominal constants for reasoning about these representations. As such, it is a suitable vehicle for formalizing the metatheory of formal systems such as logics and programming languages. This tutorial exposes Abella incrementally, starting with its capabilities at a firstorder logic level and gradually presenting more sophisticated features, ending with the support it offers to the twolevel logic approach to metatheoretic reasoning. Along the way, we show how Abella can be used prove theorems involving natural numbers, lists, and automata, as well as involving typed and untyped λcalculi and the picalculus. Contents
The Representational Adequacy of HYBRID
"... The Hybrid system (Ambler et al., 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general and principles of (co)induction in particular. The form of HOAS provided by Hybrid ..."
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The Hybrid system (Ambler et al., 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general and principles of (co)induction in particular. The form of HOAS provided by Hybrid is essentially a lambda calculus with constants. Of fundamental interest is the form of the lambda abstractions provided by Hybrid. The user has the convenience of writing lambda abstractions using names for the binding variables. However each abstraction is actually a definition of a de Bruijn expression, and Hybrid can unwind the user’s abstractions (written with names) to machine friendly de Bruijn expressions (without names). In this sense the formal system contains a hybrid of named and nameless bound variable notation. In this paper, we present a formal theory in a logical framework which can be viewed as a model of core Hybrid, and state and prove that the model is representationally adequate for HOAS. In particular, it is the canonical translation function from λexpressions to Hybrid that witnesses adequacy. We also prove two results that characterise how Hybrid represents certain classes of λexpressions. The Hybrid system contains a number of different syntactic classes of expression, and associated abstraction mechanisms. Hence this paper also aims to provide a selfcontained theoretical introduction to both the syntax and key ideas of the system; background in automated theorem proving is not essential, although this paper will be of considerable interest to those who wish to work with Hybrid in Isabelle/HOL.