/export/starexec/sandbox/solver/bin/starexec_run_HigherOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We consider the system theBenchmark. Alphabet: cons : [c * d] --> d f : [a] --> a false : [] --> b filter : [c -> b * d] --> d filter2 : [b * c -> b * c * d] --> d g : [a] --> a h : [a] --> a map : [c -> c * d] --> d nil : [] --> d true : [] --> b Rules: f(g(x)) => g(f(f(x))) f(h(x)) => h(g(x)) map(i, nil) => nil map(i, cons(x, y)) => cons(i x, map(i, y)) filter(i, nil) => nil filter(i, cons(x, y)) => filter2(i x, i, x, y) filter2(true, i, x, y) => cons(x, filter(i, y)) filter2(false, i, x, y) => filter(i, y) This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0). We use rule removal, following [Kop12, Theorem 2.23]. This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]): f(g(X)) >? g(f(f(X))) f(h(X)) >? h(g(X)) map(F, nil) >? nil map(F, cons(X, Y)) >? cons(F X, map(F, Y)) filter(F, nil) >? nil filter(F, cons(X, Y)) >? filter2(F X, F, X, Y) filter2(true, F, X, Y) >? cons(X, filter(F, Y)) filter2(false, F, X, Y) >? filter(F, Y) We orient these requirements with a polynomial interpretation in the natural numbers. The following interpretation satisfies the requirements: cons = \y0y1.1 + y0 + y1 f = \y0.y0 false = 3 filter = \G0y1.2 + 2y1 + G0(0) + 2y1G0(y1) filter2 = \y0G1y2y3.1 + y0 + y2 + 2y3 + G1(0) + 2y3G1(y3) g = \y0.y0 h = \y0.y0 map = \G0y1.2 + 3y1 + 2y1G0(y1) + 2G0(y1) nil = 0 true = 3 Using this interpretation, the requirements translate to: [[f(g(_x0))]] = x0 >= x0 = [[g(f(f(_x0)))]] [[f(h(_x0))]] = x0 >= x0 = [[h(g(_x0))]] [[map(_F0, nil)]] = 2 + 2F0(0) > 0 = [[nil]] [[map(_F0, cons(_x1, _x2))]] = 5 + 3x1 + 3x2 + 2x1F0(1 + x1 + x2) + 2x2F0(1 + x1 + x2) + 4F0(1 + x1 + x2) > 3 + x1 + 3x2 + F0(x1) + 2x2F0(x2) + 2F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] [[filter(_F0, nil)]] = 2 + F0(0) > 0 = [[nil]] [[filter(_F0, cons(_x1, _x2))]] = 4 + 2x1 + 2x2 + F0(0) + 2x1F0(1 + x1 + x2) + 2x2F0(1 + x1 + x2) + 2F0(1 + x1 + x2) > 1 + 2x1 + 2x2 + F0(0) + F0(x1) + 2x2F0(x2) = [[filter2(_F0 _x1, _F0, _x1, _x2)]] [[filter2(true, _F0, _x1, _x2)]] = 4 + x1 + 2x2 + F0(0) + 2x2F0(x2) > 3 + x1 + 2x2 + F0(0) + 2x2F0(x2) = [[cons(_x1, filter(_F0, _x2))]] [[filter2(false, _F0, _x1, _x2)]] = 4 + x1 + 2x2 + F0(0) + 2x2F0(x2) > 2 + 2x2 + F0(0) + 2x2F0(x2) = [[filter(_F0, _x2)]] We can thus remove the following rules: map(F, nil) => nil map(F, cons(X, Y)) => cons(F X, map(F, Y)) filter(F, nil) => nil filter(F, cons(X, Y)) => filter2(F X, F, X, Y) filter2(true, F, X, Y) => cons(X, filter(F, Y)) filter2(false, F, X, Y) => filter(F, Y) We observe that the rules contain a first-order subset: f(g(X)) => g(f(f(X))) f(h(X)) => h(g(X)) Moreover, the system is orthogonal. Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is terminating when seen as a many-sorted first-order TRS. According to the external first-order termination prover, this system is indeed terminating: || proof of resources/system.trs || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616 || || || Termination w.r.t. Q of the given QTRS could be proven: || || (0) QTRS || (1) QTRS Reverse [EQUIVALENT] || (2) QTRS || (3) RFCMatchBoundsTRSProof [EQUIVALENT] || (4) YES || || || ---------------------------------------- || || (0) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || f(g(%X)) -> g(f(f(%X))) || f(h(%X)) -> h(g(%X)) || || Q is empty. || || ---------------------------------------- || || (1) QTRS Reverse (EQUIVALENT) || We applied the QTRS Reverse Processor [REVERSE]. || ---------------------------------------- || || (2) || Obligation: || Q restricted rewrite system: || The TRS R consists of the following rules: || || g(f(x)) -> f(f(g(x))) || h(f(x)) -> g(h(x)) || || Q is empty. || || ---------------------------------------- || || (3) RFCMatchBoundsTRSProof (EQUIVALENT) || Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R. || The following rules were used to construct the certificate: || || g(f(x)) -> f(f(g(x))) || h(f(x)) -> g(h(x)) || || The certificate found is represented by the following graph. || The certificate consists of the following enumerated nodes: || 1, 3, 8, 9, 10, 17, 18, 19 || || Node 1 is start node and node 3 is final node. || || Those nodes are connected through the following edges: || || * 1 to 8 labelled f_1(0)* 1 to 10 labelled g_1(0)* 3 to 3 labelled #_1(0)* 8 to 9 labelled f_1(0)* 9 to 3 labelled g_1(0)* 9 to 17 labelled f_1(1)* 10 to 3 labelled h_1(0)* 10 to 19 labelled g_1(1)* 17 to 18 labelled f_1(1)* 18 to 3 labelled g_1(1)* 18 to 17 labelled f_1(1)* 19 to 3 labelled h_1(1)* 19 to 19 labelled g_1(1) || || || ---------------------------------------- || || (4) || YES || We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]). We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative): Dependency Pairs P_0: Rules R_0: f(g(X)) => g(f(f(X))) f(h(X)) => h(g(X)) Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite. We consider the dependency pair problem (P_0, R_0, minimal, formative). We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows: This graph has no strongly connected components. By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem. As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination. +++ Citations +++ [Kop12] C. Kop. Higher Order Termination. PhD Thesis, 2012. [KusIsoSakBla09] K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui. Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems. In volume 92(10) of IEICE Transactions on Information and Systems. 2007--2015, 2009.