420 research outputs found
Representing scope in intuitionistic deductions
AbstractIntuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semantics-based translation methods for modal deduction. This calculus is simple and is justified by direct proof-theoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoever — rules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how λ-terms are to be extracted from its deductions
Representing Scope in Intuitionistic Deductions
Intuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semantics-based translation methods for modal deduction. This calculus is simple and is justified by direct proof-theoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoever—rules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how λ-terms are to be extracted from its deductions
Generating collection transformations from proofs
Nested relations, built up from atomic types via product and set types, form a rich data model. Over the last decades the nested relational calculus, NRC, has emerged as a standard language for defining transformations on nested collections. NRC is a strongly-typed functional language which allows building up transformations using tupling and projections, a singleton-former, and a map operation that lifts transformations on tuples to transformations on sets.In this work we describe an alternative declarative method of describing transformations in logic. A formula with distinguished inputs and outputs gives an implicit definition if one can prove that for each input there is only one output that satisfies it. Our main result shows that one can synthesize transformations from proofs that a formula provides an implicit definition, where the proof is in an intuitionistic calculus that captures a natural style of reasoning about nested collections. Our polynomial time synthesis procedure is based on an analog of Craig’s interpolation lemma, starting with a provable containment between terms representing nested collections and generating an NRC expression that interpolates between them.We further show that NRC expressions that implement an implicit definition can be found when there is a classical proof of functionality, not just when there is an intuitionistic one. That is, whenever a formula implicitly defines a transformation, there is an NRC expression that implements it
A bridge between constructive logic and computer programming
AbstractSome logic notions have their analogies among programming concepts and vice versa. But people often try to understand these analogies in too straightforward a manner. A collection of analogies arising between constructive logics and programming is summarized and illustrated here. Some examples of complexities usually not taken into account are shown
From coinductive proofs to exact real arithmetic: theory and applications
Based on a new coinductive characterization of continuous functions we
extract certified programs for exact real number computation from constructive
proofs. The extracted programs construct and combine exact real number
algorithms with respect to the binary signed digit representation of real
numbers. The data type corresponding to the coinductive definition of
continuous functions consists of finitely branching non-wellfounded trees
describing when the algorithm writes and reads digits. We discuss several
examples including the extraction of programs for polynomials up to degree two
and the definite integral of continuous maps
Optimization techniques for propositional intuitionistic logic and their implementation
AbstractThis paper presents some techniques which bound the proof search space in propositional intuitionistic logic. These techniques are justified by Kripke semantics and are the backbone of a tableau based theorem prover (PITP) implemented in C++. PITP and some known theorem provers are compared using the formulas of ILTP benchmark library. It turns out that PITP is, at the moment, the propositional prover that solves most formulas of the library
Deductive Pluralism
This paper proposes an approach to the philosophy of
mathematics, deductive pluralism, that is designed to
satisfy the criteria of inclusiveness of and consistency
with mathematical practice.
Deductive pluralism views mathematical statements as
assertions that a result follows from logical and
mathematical foundations and that there are a variety of
incompatible foundations such as standard foundations,
constructive foundations, or univalent foundations.
The advantages of this philosophy include the elimination
of ontological problems, epistemological clarity, and
objectivity.
Possible objections and relations with some other
philosophies of mathematics are also considered
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