98 research outputs found
Nested Term Graphs (Work In Progress)
We report on work in progress on 'nested term graphs' for formalizing
higher-order terms (e.g. finite or infinite lambda-terms), including those
expressing recursion (e.g. terms in the lambda-calculus with letrec). The idea
is to represent the nested scope structure of a higher-order term by a nested
structure of term graphs.
Based on a signature that is partitioned into atomic and nested function
symbols, we define nested term graphs both in a functional representation, as
tree-like recursive graph specifications that associate nested symbols with
usual term graphs, and in a structural representation, as enriched term graph
structures. These definitions induce corresponding notions of bisimulation
between nested term graphs. Our main result states that nested term graphs can
be implemented faithfully by first-order term graphs.
keywords: higher-order term graphs, context-free grammars, cyclic
lambda-terms, higher-order rewrite systemsComment: In Proceedings TERMGRAPH 2014, arXiv:1505.0681
On Causal Equivalence by Tracing in String Rewriting
We introduce proof terms for string rewrite systems and, using these, show
that various notions of equivalence on reductions known from the literature can
be viewed as different perspectives on the notion of causal equivalence. In
particular, we show that permutation equivalence classes (as known from the
lambda-calculus and term rewriting) are uniquely represented both by trace
graphs (known from physics as causal graphs) and by so-called greedy multistep
reductions (as known from algebra). We present effective maps from the former
to the latter, topological multi-sorting TM, and vice versa, the proof term
algebra [[ ]].Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421
Proof Orders for Decreasing Diagrams
We present and compare some well-founded proof orders for decreasing diagrams. These proof orders order a conversion above another conversion if the latter is obtained by filling any peak in the former by a (locally) decreasing diagram. Therefore each such proof order entails the decreasing diagrams technique for proving confluence. The proof orders differ with respect to monotonicity and complexity. Our results are developed in the setting of involutive monoids. We extend these results to obtain a decreasing diagrams technique for confluence modulo
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