Proof-search in the λΠ-calculus

Pym, David J.; Wallen, Lincoln

doi:10.1017/cbo9780511569807.013

Cited by 13 publications

(6 citation statements)

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“…A natural deduction approach to operational semantics based on the ideas of LF has also been developed [5]. A general theory of search, including a unification algorithm for the LF type theory, has been developed [44,45]. Elliott has also developed a unification algorithm for LF [16] which has been used by Pfenning as the basis for a logic programming language based on the LF type theory [41].…”

Section: Related Workmentioning

confidence: 99%

A framework for defining logics

Harper¹,

Honsell²,

Plotkin³

1993

J. ACM

785

706

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The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed λ-calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Löf's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools such as proof editors and proof checkers can be constructed.

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Section: Related Workmentioning

confidence: 99%

A framework for defining logics

Harper¹,

Honsell²,

Plotkin³

1993

J. ACM

785

706

View full text Add to dashboard Cite

show abstract

“…Our move from G3 to LJT is also particularly convenient to capture the process of higherorder unification as a proof-search mechanism. Pym and Wallen address proof-search [PW91] in the particular case of λΠ, the type theory of the Edinburgh Logical Framework, using a black-box higher-order unification algorithm adapted from that of Huet. They discuss how well-typedness of meta-variable instantiations computed by unification can be exploited to control the search space.…”

Section: Introductionmentioning

confidence: 99%

A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Lengrand¹,

Dyckhoff²,

McKinna³

2011

Logical Methods in Computer Science

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Abstract. Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the issues of defining such sequent calculi for Pure Type Systems (PTS, which were originally presented in natural deduction style) and then organizing their rules for effective proof-search. We introduce the idea of Pure Type Sequent Calculus with meta-variables (PTSCα), by enriching the syntax of a permutation-free sequent calculus for propositional logic due to Herbelin, which is strongly related to natural deduction and already well adapted to proof-search. The operational semantics is adapted from Herbelin's and is defined by a system of local rewrite rules as in cut-elimination, using explicit substitutions. We prove confluence for this system. Restricting our attention to PTSC, a type system for the ground terms of this system, we obtain the Subject Reduction property and show that each PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising iff the latter is. We show how to make the logical rules of PTSC into a syntax-directed system PS for proof-search, by incorporating the conversion rules as in syntax-directed presentations of the PTS rules for type-checking. Finally, we consider how to use the explicitly scoped meta-variables of PTSCα to represent partial proof-terms, and use them to analyse interactive proof construction. This sets up a framework PE in which we are able to study proof-search strategies, type inhabitant enumeration and (higher-order) unification.1998 ACM Subject Classification: F.4.1.

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“…For example, systems based on type theory o er capabilities of representing proofs as terms in the logic which are built from constructive derivations 1] and the metalogic may be well suited for particular sorts of automated proof search, e.g. 14,16].…”

Section: Introductionmentioning

confidence: 99%

Adding metatheoretic facilities to first-order theories

Basin

Matthews

1996

Journal of Logic and Computation

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Generic proof systems like Isabelle provide some limited but useful metatheoretic facilities for declared logics; in particular, users can prove simple derived rules and also`solve' formulae that contain metavariables | a technique useful for, e.g., program synthesis. We show how an arbitrary rst order theory can be conservatively extended to provide similar facilities, without a supporting metatheory, and examine what the limitations of this approach are.

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Proof-search in the λΠ-calculus

Cited by 13 publications

References 0 publications

A framework for defining logics

A framework for defining logics

A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Adding metatheoretic facilities to first-order theories

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