21,798 research outputs found
A Characterisation of Medial as Rewriting Rule
International audienceMedial is an inference rule scheme that appears in various deductive systems based on deep inference. In this paper we investigate the properties of medial as rewriting rule independently from logic. We present a graph theoretical criterion for checking whether there exists a medial rewriting path between two formulas. Finally, we return to logic and apply our criterion for giving a combinatorial proof for a decomposition theorem, i.e., proof theoretical statement about syntax
Proof Certificates for Equality Reasoning
International audienceThe kinds of inference rules and decision procedures that one writes for proofs involving equality and rewriting are rather different from proofs that one might write in first-order logic using, say, sequent calculus or natural deduction. For example, equational logic proofs are often chains of replacements or applications of oriented rewriting and normal forms. In contrast, proofs involving logical connectives are trees of introduction and elimination rules. We shall illustrate here how it is possible to check various equality-based proof systems with a programmable proof checker (the kernel checker) for first-order logic. Our proof checker's design is based on the implementation of focused proof search and on making calls to (user-supplied) clerks and experts predicates that are tied to the two phases found in focused proofs. It is the specification of these clerks and experts that provide a formal definition of the structure of proof evidence. As we shall show, such formal definitions work just as well in the equational setting as in the logic setting where this scheme for proof checking was originally developed. Additionally, executing such a formal definition on top of a kernel provides an actual proof checker that can also do a degree of proof reconstruction. We shall illustrate the flexibility of this approach by showing how to formally define (and check) rewriting proofs of a variety of designs
A Characterisation of Medial as Rewriting Rule
International audienceMedial is an inference rule scheme that appears in various deductive systems based on deep inference. In this paper we investigate the properties of medial as rewriting rule independently from logic. We present a graph theoretical criterion for checking whether there exists a medial rewriting path between two formulas. Finally, we return to logic and apply our criterion for giving a combinatorial proof for a decomposition theorem, i.e., proof theoretical statement about syntax
Methods for Proving Termination of Rewriting-based Programming Languages by Transformation
AbstractDespite the remarkable development of the theory of termination of rewriting, its application to high-level (rewriting-based) programming languages is far from being optimal. This is due to the need for features such as conditional equations and rules, types and subtypes, (possibly programmable) strategies for controlling the execution, matching modulo axioms, and so on, that are used in many programs and tend to place such programs outside the scope of current termination tools. The operational meaning of such features is often formalized in a proof theoretic manner by means of an inference system rather than just by a rewriting relation. The corresponding termination notions can also differ from the standard ones. During the last years we have introduced and implemented different notions and transformation techniques which have been proved useful for proving and disproving termination of such programs by using existing tools for proving termination of (variants of) rewriting. In this paper we provide an overview of our main contributions
Termination of rewriting strategies: a generic approach
We propose a generic termination proof method for rewriting under strategies,
based on an explicit induction on the termination property. Rewriting trees on
ground terms are modeled by proof trees, generated by alternatively applying
narrowing and abstracting steps. The induction principle is applied through the
abstraction mechanism, where terms are replaced by variables representing any
of their normal forms. The induction ordering is not given a priori, but
defined with ordering constraints, incrementally set during the proof.
Abstraction constraints can be used to control the narrowing mechanism, well
known to easily diverge. The generic method is then instantiated for the
innermost, outermost and local strategies.Comment: 49 page
Proof Diagrams for Multiplicative Linear Logic
The original idea of proof nets can be formulated by means of interaction
nets syntax. Additional machinery as switching, jumps and graph connectivity is
needed in order to ensure correspondence between a proof structure and a
correct proof in sequent calculus.
In this paper we give an interpretation of proof nets in the syntax of string
diagrams. Even though we lose standard proof equivalence, our construction
allows to define a framework where soundness and well-typeness of a diagram can
be verified in linear time.Comment: In Proceedings LINEARITY 2016, arXiv:1701.0452
On the relative proof complexity of deep inference via atomic flows
We consider the proof complexity of the minimal complete fragment, KS, of
standard deep inference systems for propositional logic. To examine the size of
proofs we employ atomic flows, diagrams that trace structural changes through a
proof but ignore logical information. As results we obtain a polynomial
simulation of versions of Resolution, along with some extensions. We also show
that these systems, as well as bounded-depth Frege systems, cannot polynomially
simulate KS, by giving polynomial-size proofs of certain variants of the
propositional pigeonhole principle in KS.Comment: 27 pages, 2 figures, full version of conference pape
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