5 research outputs found
Nominal Unification of Higher Order Expressions with Recursive Let
A sound and complete algorithm for nominal unification of higher-order
expressions with a recursive let is described, and shown to run in
non-deterministic polynomial time. We also explore specializations like nominal
letrec-matching for plain expressions and for DAGs and determine the complexity
of corresponding unification problems.Comment: Pre-proceedings paper presented at the 26th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh,
Scotland UK, 6-8 September 2016 (arXiv:1608.02534
Nominal C-Unification
Nominal unification is an extension of first-order unification that takes
into account the \alpha-equivalence relation generated by binding operators,
following the nominal approach. We propose a sound and complete procedure for
nominal unification with commutative operators, or nominal C-unification for
short, which has been formalised in Coq. The procedure transforms nominal
C-unification problems into simpler (finite families) of fixpoint problems,
whose solutions can be generated by algebraic techniques on combinatorics of
permutations.Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
On Nominal Syntax and Permutation Fixed Points
We propose a new axiomatisation of the alpha-equivalence relation for nominal
terms, based on a primitive notion of fixed-point constraint. We show that the
standard freshness relation between atoms and terms can be derived from the
more primitive notion of permutation fixed-point, and use this result to prove
the correctness of the new -equivalence axiomatisation. This gives rise
to a new notion of nominal unification, where solutions for unification
problems are pairs of a fixed-point context and a substitution. Although it may
seem less natural than the standard notion of nominal unifier based on
freshness constraints, the notion of unifier based on fixed-point constraints
behaves better when equational theories are considered: for example, nominal
unification remains finitary in the presence of commutativity, whereas it
becomes infinitary when unifiers are expressed using freshness contexts. We
provide a definition of -equivalence modulo equational theories that
take into account A, C and AC theories. Based on this notion of equivalence, we
show that C-unification is finitary and we provide a sound and complete
C-unification algorithm, as a first step towards the development of nominal
unification modulo AC and other equational theories with permutative
properties
Nominal unification of higher order expressions with recursive let
A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. As extension a nominal unification algorithm for higher-order expressions with recursive let and atom-variables is constructed, where we show that it also runs in non-deterministic polynomial time
Nominal Unification and Matching of Higher Order Expressions with Recursive Let
A sound and complete algorithm for nominal unification of higher-order
expressions with a recursive let is described, and shown to run in
nondeterministic polynomial time. We also explore specializations like nominal
letrec-matching for expressions, for DAGs, and for garbage-free expressions and
determine their complexity. We also provide a nominal unification algorithm for
higher-order expressions with recursive let and atom-variables, where we show
that it also runs in nondeterministic polynomial time. In addition we prove
that there is a guessing strategy for nominal unification with letrec and
atom-variable that is a trade-off between exponential growth and
non-determinism. Nominal matching with variables representing partial
letrec-environments is also shown to be in NP.Comment: 37 pages, 9 figures, This paper is an extended version of the
conference publication: Manfred Schmidt-Schau{\ss} and Temur Kutsia and Jordi
Levy and Mateu Villaret and Yunus Kutz, Nominal Unification of Higher Order
Expressions with Recursive Let, LOPSTR-16, Lecture Notes in Computer Science
10184, Springer, p 328 -344, 2016. arXiv admin note: text overlap with
arXiv:1608.0377