Abstract
AbstractWe define nominal equational problems of the form $$\exists \overline{W} \forall \overline{Y} : P$$
∃
W
¯
∀
Y
¯
:
P
, where $$P$$
P
consists of conjunctions and disjunctions of equations $$s\approx _\alpha t$$
s
≈
α
t
, freshness constraints $$a\#t$$
a
#
t
and their negations: $$s \not \approx _\alpha t$$
s
≉
α
t
and "Equation missing", where $$a$$
a
is an atom and $$s, t$$
s
,
t
nominal terms. We give a general definition of solution and a set of simplification rules to compute solutions in the nominal ground term algebra. For the latter, we define notions of solved form from which solutions can be easily extracted and show that the simplification rules are sound, preserving, and complete. With a particular strategy for rule application, the simplification process terminates and thus specifies an algorithm to solve nominal equational problems. These results generalise previous results obtained by Comon and Lescanne for first-order languages to languages with binding operators. In particular, we show that the problem of deciding the validity of a first-order equational formula in a language with binding operators (i.e., validity modulo $$\alpha $$
α
-equality) is decidable.
Publisher
Springer International Publishing
Reference23 articles.
1. Ayala-Rincón, M., Fernández, M., Nantes-Sobrinho, D., Vale, D.: On Solving Nominal Disunification Constraints. ENTCS 348, 3 – 22 (2020). https://doi.org/10.1016/j.entcs.2020.02.002, proc. 14th Int. Workshop on Logical and Semantic Frameworks, with Applications LSFA 2019
2. Baader, F., Schulz, K.U.: Combination techniques and decision problems for disunification. Theor. Comput. Sci. 142(2), 229–255 (1995). https://doi.org/10.1016/0304-3975(94)00277-0.
3. Buntine, W.L., Bürckert, H.J.: On solving equations and disequations. J. ACM 41(4), 591–629 (Jul 1994). https://doi.org/10.1145/179812.179813
4. Byrd, W.E., Friedman, D.P.: $$\alpha $$Kanren A Fresh Name in Nominal Logic Programming (2008), http://webyrd.net/alphamk/alphamk.pdf, earlier version available in the Proc. 2007 Workshop on Scheme and Functional Programming, Université Laval Technical Report DIUL-RT-0701
5. Calvès, C., Fernández, M.: Matching and alpha-equivalence check for nominal terms. Journal of Computer and System Sciences 76(5), 283 – 301 (2010). https://doi.org/10.1016/j.jcss.2009.10.003