YES Problem: a(x1) -> b(x1) a(x1) -> c(x1) a(b(x1)) -> b(a(c(x1))) c(c(x1)) -> a(x1) Proof: String Reversal Processor: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) DP Processor: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(x1)) b#(a(x1)) -> c#(a(b(x1))) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) TDG Processor: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(x1)) b#(a(x1)) -> c#(a(b(x1))) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) graph: c#(c(x1)) -> a#(x1) -> a#(x1) -> c#(x1) c#(c(x1)) -> a#(x1) -> a#(x1) -> b#(x1) b#(a(x1)) -> c#(a(b(x1))) -> c#(c(x1)) -> a#(x1) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> c#(a(b(x1))) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> a#(b(x1)) b#(a(x1)) -> b#(x1) -> b#(a(x1)) -> b#(x1) b#(a(x1)) -> a#(b(x1)) -> a#(x1) -> c#(x1) b#(a(x1)) -> a#(b(x1)) -> a#(x1) -> b#(x1) a#(x1) -> c#(x1) -> c#(c(x1)) -> a#(x1) a#(x1) -> b#(x1) -> b#(a(x1)) -> c#(a(b(x1))) a#(x1) -> b#(x1) -> b#(a(x1)) -> a#(b(x1)) a#(x1) -> b#(x1) -> b#(a(x1)) -> b#(x1) Arctic Interpretation Processor: dimension: 4 usable rules: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) interpretation: [c#](x0) = [0 0 0 0]x0, [b#](x0) = [0 -& 0 -&]x0, [a#](x0) = [0 0 0 0]x0, [0 0 0 -&] [0 ] [0 0 0 0 ] [-&] [c](x0) = [0 1 0 0 ]x0 + [0 ] [0 0 0 0 ] [0 ], [0 -& 0 -&] [0 -& 0 -&] [b](x0) = [0 0 0 -&]x0 [0 0 0 -&] , [0 0 0 -&] [0 ] [0 0 0 0 ] [-&] [a](x0) = [1 1 1 0 ]x0 + [0 ] [0 0 0 0 ] [0 ] orientation: a#(x1) = [0 0 0 0]x1 >= [0 -& 0 -&]x1 = b#(x1) a#(x1) = [0 0 0 0]x1 >= [0 0 0 0]x1 = c#(x1) b#(a(x1)) = [1 1 1 0]x1 + [0] >= [0 -& 0 -&]x1 = b#(x1) b#(a(x1)) = [1 1 1 0]x1 + [0] >= [0 0 0 -&]x1 = a#(b(x1)) b#(a(x1)) = [1 1 1 0]x1 + [0] >= [1 1 1 -&]x1 + [0] = c#(a(b(x1))) c#(c(x1)) = [0 1 0 0]x1 + [0] >= [0 0 0 0]x1 = a#(x1) [0 0 0 -&] [0 ] [0 -& 0 -&] [0 0 0 0 ] [-&] [0 -& 0 -&] a(x1) = [1 1 1 0 ]x1 + [0 ] >= [0 0 0 -&]x1 = b(x1) [0 0 0 0 ] [0 ] [0 0 0 -&] [0 0 0 -&] [0 ] [0 0 0 -&] [0 ] [0 0 0 0 ] [-&] [0 0 0 0 ] [-&] a(x1) = [1 1 1 0 ]x1 + [0 ] >= [0 1 0 0 ]x1 + [0 ] = c(x1) [0 0 0 0 ] [0 ] [0 0 0 0 ] [0 ] [1 1 1 0] [0] [1 1 1 -&] [0] [1 1 1 0] [0] [1 1 1 -&] [0] b(a(x1)) = [1 1 1 0]x1 + [0] >= [1 1 1 -&]x1 + [0] = c(a(b(x1))) [1 1 1 0] [0] [1 1 1 -&] [0] [0 1 0 0] [0] [0 0 0 -&] [0 ] [0 1 0 0] [0] [0 0 0 0 ] [-&] c(c(x1)) = [1 1 1 1]x1 + [0] >= [1 1 1 0 ]x1 + [0 ] = a(x1) [0 1 0 0] [0] [0 0 0 0 ] [0 ] problem: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) b#(a(x1)) -> c#(a(b(x1))) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) Restore Modifier: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) b#(a(x1)) -> c#(a(b(x1))) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) EDG Processor: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) b#(a(x1)) -> c#(a(b(x1))) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) graph: c#(c(x1)) -> a#(x1) -> a#(x1) -> b#(x1) c#(c(x1)) -> a#(x1) -> a#(x1) -> c#(x1) b#(a(x1)) -> c#(a(b(x1))) -> c#(c(x1)) -> a#(x1) a#(x1) -> c#(x1) -> c#(c(x1)) -> a#(x1) a#(x1) -> b#(x1) -> b#(a(x1)) -> c#(a(b(x1))) Arctic Interpretation Processor: dimension: 4 usable rules: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) interpretation: [c#](x0) = [0 0 0 -&]x0 + [0], [b#](x0) = [0 0 0 0]x0 + [0], [a#](x0) = [0 0 0 0]x0 + [0], [0 0 0 0 ] [0 ] [0 0 0 0 ] [0 ] [c](x0) = [-& 0 0 0 ]x0 + [0 ] [0 1 0 0 ] [-&], [0 0 0 0 ] [0] [0 0 0 0 ] [0] [b](x0) = [-& 0 0 0 ]x0 + [0] [-& 0 0 0 ] [0], [0 0 0 0 ] [0] [0 0 0 0 ] [0] [a](x0) = [-& 0 0 0 ]x0 + [0] [1 1 1 1 ] [1] orientation: a#(x1) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = b#(x1) a#(x1) = [0 0 0 0]x1 + [0] >= [0 0 0 -&]x1 + [0] = c#(x1) b#(a(x1)) = [1 1 1 1]x1 + [1] >= [0 0 0 0]x1 + [0] = c#(a(b(x1))) c#(c(x1)) = [0 0 0 0]x1 + [0] >= [0 0 0 0]x1 + [0] = a#(x1) [0 0 0 0 ] [0] [0 0 0 0 ] [0] [0 0 0 0 ] [0] [0 0 0 0 ] [0] a(x1) = [-& 0 0 0 ]x1 + [0] >= [-& 0 0 0 ]x1 + [0] = b(x1) [1 1 1 1 ] [1] [-& 0 0 0 ] [0] [0 0 0 0 ] [0] [0 0 0 0 ] [0 ] [0 0 0 0 ] [0] [0 0 0 0 ] [0 ] a(x1) = [-& 0 0 0 ]x1 + [0] >= [-& 0 0 0 ]x1 + [0 ] = c(x1) [1 1 1 1 ] [1] [0 1 0 0 ] [-&] [1 1 1 1] [1] [1 1 1 1] [1] [1 1 1 1] [1] [1 1 1 1] [1] b(a(x1)) = [1 1 1 1]x1 + [1] >= [1 1 1 1]x1 + [1] = c(a(b(x1))) [1 1 1 1] [1] [1 1 1 1] [1] [0 1 0 0] [0] [0 0 0 0 ] [0] [0 1 0 0] [0] [0 0 0 0 ] [0] c(c(x1)) = [0 1 0 0]x1 + [0] >= [-& 0 0 0 ]x1 + [0] = a(x1) [1 1 1 1] [1] [1 1 1 1 ] [1] problem: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) Restore Modifier: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) EDG Processor: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) graph: c#(c(x1)) -> a#(x1) -> a#(x1) -> c#(x1) c#(c(x1)) -> a#(x1) -> a#(x1) -> b#(x1) a#(x1) -> c#(x1) -> c#(c(x1)) -> a#(x1) CDG Processor: DPs: a#(x1) -> b#(x1) a#(x1) -> c#(x1) c#(c(x1)) -> a#(x1) TRS: a(x1) -> b(x1) a(x1) -> c(x1) b(a(x1)) -> c(a(b(x1))) c(c(x1)) -> a(x1) graph: c#(c(x1)) -> a#(x1) -> a#(x1) -> b#(x1) c#(c(x1)) -> a#(x1) -> a#(x1) -> c#(x1) SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/9