MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { minus(x, 0()) -> x , minus(minus(x, y), z) -> minus(x, plus(y, z)) , minus(0(), y) -> 0() , minus(s(x), s(y)) -> minus(x, y) , plus(0(), y) -> y , plus(s(x), y) -> plus(x, s(y)) , plus(s(x), y) -> s(plus(y, x)) , zero(0()) -> true() , zero(s(x)) -> false() , p(0()) -> 0() , p(s(x)) -> x , div(x, y) -> quot(x, y, 0()) , quot(0(), s(y), z) -> z , quot(s(x), s(y), z) -> quot(minus(p(ack(0(), x)), y), s(y), s(z)) , ack(0(), x) -> plus(x, s(0())) , ack(0(), x) -> s(x) , ack(s(x), 0()) -> ack(x, s(0())) , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { minus^#(x, 0()) -> c_1(x) , minus^#(minus(x, y), z) -> c_2(minus^#(x, plus(y, z))) , minus^#(0(), y) -> c_3() , minus^#(s(x), s(y)) -> c_4(minus^#(x, y)) , plus^#(0(), y) -> c_5(y) , plus^#(s(x), y) -> c_6(plus^#(x, s(y))) , plus^#(s(x), y) -> c_7(plus^#(y, x)) , zero^#(0()) -> c_8() , zero^#(s(x)) -> c_9() , p^#(0()) -> c_10() , p^#(s(x)) -> c_11(x) , div^#(x, y) -> c_12(quot^#(x, y, 0())) , quot^#(0(), s(y), z) -> c_13(z) , quot^#(s(x), s(y), z) -> c_14(quot^#(minus(p(ack(0(), x)), y), s(y), s(z))) , ack^#(0(), x) -> c_15(plus^#(x, s(0()))) , ack^#(0(), x) -> c_16(x) , ack^#(s(x), 0()) -> c_17(ack^#(x, s(0()))) , ack^#(s(x), s(y)) -> c_18(ack^#(x, ack(s(x), y))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(x, 0()) -> c_1(x) , minus^#(minus(x, y), z) -> c_2(minus^#(x, plus(y, z))) , minus^#(0(), y) -> c_3() , minus^#(s(x), s(y)) -> c_4(minus^#(x, y)) , plus^#(0(), y) -> c_5(y) , plus^#(s(x), y) -> c_6(plus^#(x, s(y))) , plus^#(s(x), y) -> c_7(plus^#(y, x)) , zero^#(0()) -> c_8() , zero^#(s(x)) -> c_9() , p^#(0()) -> c_10() , p^#(s(x)) -> c_11(x) , div^#(x, y) -> c_12(quot^#(x, y, 0())) , quot^#(0(), s(y), z) -> c_13(z) , quot^#(s(x), s(y), z) -> c_14(quot^#(minus(p(ack(0(), x)), y), s(y), s(z))) , ack^#(0(), x) -> c_15(plus^#(x, s(0()))) , ack^#(0(), x) -> c_16(x) , ack^#(s(x), 0()) -> c_17(ack^#(x, s(0()))) , ack^#(s(x), s(y)) -> c_18(ack^#(x, ack(s(x), y))) } Strict Trs: { minus(x, 0()) -> x , minus(minus(x, y), z) -> minus(x, plus(y, z)) , minus(0(), y) -> 0() , minus(s(x), s(y)) -> minus(x, y) , plus(0(), y) -> y , plus(s(x), y) -> plus(x, s(y)) , plus(s(x), y) -> s(plus(y, x)) , zero(0()) -> true() , zero(s(x)) -> false() , p(0()) -> 0() , p(s(x)) -> x , div(x, y) -> quot(x, y, 0()) , quot(0(), s(y), z) -> z , quot(s(x), s(y), z) -> quot(minus(p(ack(0(), x)), y), s(y), s(z)) , ack(0(), x) -> plus(x, s(0())) , ack(0(), x) -> s(x) , ack(s(x), 0()) -> ack(x, s(0())) , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,8,9,10} by applications of Pre({3,8,9,10}) = {1,2,4,5,11,13,16}. Here rules are labeled as follows: DPs: { 1: minus^#(x, 0()) -> c_1(x) , 2: minus^#(minus(x, y), z) -> c_2(minus^#(x, plus(y, z))) , 3: minus^#(0(), y) -> c_3() , 4: minus^#(s(x), s(y)) -> c_4(minus^#(x, y)) , 5: plus^#(0(), y) -> c_5(y) , 6: plus^#(s(x), y) -> c_6(plus^#(x, s(y))) , 7: plus^#(s(x), y) -> c_7(plus^#(y, x)) , 8: zero^#(0()) -> c_8() , 9: zero^#(s(x)) -> c_9() , 10: p^#(0()) -> c_10() , 11: p^#(s(x)) -> c_11(x) , 12: div^#(x, y) -> c_12(quot^#(x, y, 0())) , 13: quot^#(0(), s(y), z) -> c_13(z) , 14: quot^#(s(x), s(y), z) -> c_14(quot^#(minus(p(ack(0(), x)), y), s(y), s(z))) , 15: ack^#(0(), x) -> c_15(plus^#(x, s(0()))) , 16: ack^#(0(), x) -> c_16(x) , 17: ack^#(s(x), 0()) -> c_17(ack^#(x, s(0()))) , 18: ack^#(s(x), s(y)) -> c_18(ack^#(x, ack(s(x), y))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { minus^#(x, 0()) -> c_1(x) , minus^#(minus(x, y), z) -> c_2(minus^#(x, plus(y, z))) , minus^#(s(x), s(y)) -> c_4(minus^#(x, y)) , plus^#(0(), y) -> c_5(y) , plus^#(s(x), y) -> c_6(plus^#(x, s(y))) , plus^#(s(x), y) -> c_7(plus^#(y, x)) , p^#(s(x)) -> c_11(x) , div^#(x, y) -> c_12(quot^#(x, y, 0())) , quot^#(0(), s(y), z) -> c_13(z) , quot^#(s(x), s(y), z) -> c_14(quot^#(minus(p(ack(0(), x)), y), s(y), s(z))) , ack^#(0(), x) -> c_15(plus^#(x, s(0()))) , ack^#(0(), x) -> c_16(x) , ack^#(s(x), 0()) -> c_17(ack^#(x, s(0()))) , ack^#(s(x), s(y)) -> c_18(ack^#(x, ack(s(x), y))) } Strict Trs: { minus(x, 0()) -> x , minus(minus(x, y), z) -> minus(x, plus(y, z)) , minus(0(), y) -> 0() , minus(s(x), s(y)) -> minus(x, y) , plus(0(), y) -> y , plus(s(x), y) -> plus(x, s(y)) , plus(s(x), y) -> s(plus(y, x)) , zero(0()) -> true() , zero(s(x)) -> false() , p(0()) -> 0() , p(s(x)) -> x , div(x, y) -> quot(x, y, 0()) , quot(0(), s(y), z) -> z , quot(s(x), s(y), z) -> quot(minus(p(ack(0(), x)), y), s(y), s(z)) , ack(0(), x) -> plus(x, s(0())) , ack(0(), x) -> s(x) , ack(s(x), 0()) -> ack(x, s(0())) , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) } Weak DPs: { minus^#(0(), y) -> c_3() , zero^#(0()) -> c_8() , zero^#(s(x)) -> c_9() , p^#(0()) -> c_10() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..