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You are here: Home Staff M.Eng. Rihan Hai Correctness check with existing literatures Apply the nested tgd generator

Apply the nested tgd generator

To examine the correctness of our approach with complex mapping scenarios, we have designed a second experiment using the nested tgd generator and populated 1,088,230 nested tgds, which we transfered to plain SO tgds by Skolemization and normalization.

Here we provide two results from the set of 1,088,230 nested tgds to showcase the format and complexity of our results.

 

We first show one of the simplest nested tgd generated, to help understand the format. The parameters are incrementally generated, for instance, consider below parameters:

 patternTree=true


 heightPatternTree=2


 numLeavesPatternTree=1


 fullLeavesPatternTree=false


 fixedMode=true


 numSrcRels=1


 aritySrcRels=1


 numTgtRels=1


 arityTgtRels=2

 

The generated plain SO tgd (first generate nested tgd, then Skolemize and normalize it and obtain below):

?f1(x1),?f2(x1,x2) 

*x1,*x2  [S1(x1) --> T1(f1(x1),x1)] ^

[S1(x1)^S2(x2) --> T2(f2(x1,x2),x2)]  

 

 Show the rewrite result:

*x1 S1(x1) --> ?y1 [T1(y1,x1) ^

*x2 S2(x2) --> ?y2 [T2(y2,x2)]  ]

 

Now we another example which is more complex.

 

 

 Show the original nested tgd:

*x1,*x2,*x3,*x4,*x5,*x6,*x7,*x8,*x9,*x10,*x11 S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11) --> ?y1,?y2,?y3,?y4,?y5,?y6,?y7 [T1(y1,y2)^T2(y3,y4,y5,y6,y7)^

*x12,*x13,*x14,*x15,*x16,*x17,*x18,*x19,*x20,*x21,*x22,*x23,*x24,*x25,*x26 S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26) --> ?y8,?y9 [T3(y8,y9,x13,y3)^T4(x21)^

*x27,*x28,*x29,*x30 S9(x27,x28,x29)^S10(x30) --> ?y10,?y11 [T5(y10,y11,x28)^

*x31,*x32,*x33,*x34,*x35,*x36,*x37,*x38,*x39,*x40,*x41,*x42,*x43,*x44,*x45,*x46,*x47,*x48,*x49,*x50 S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50) --> ?y12,?y13

*x51,*x52,*x53,*x54,*x55,*x56,*x57,*x58,*x59,*x60,*x61,*x62,*x63,*x64,*x65,*x66,*x67,*x68,*x69 S18(x51,x52,x53)^S19(x54,x55,x56,x57,x58)^S20(x59,x60,x61,x62)^S21(x63,x64,x65)^S22(x66,x67)^S23(x68)^S24(x69) --> [T6(x67,y6)^T7(x56,x21,x53,x1,x7)^

*x70,*x71,*x72,*x73,*x74,*x75,*x76,*x77,*x78 S25(x70,x71,x72,x73,x74)^S26(x75,x76,x77,x78) --> ?y14,?y15,?y16 [T8(x73)^T9(y14,y15,x74)^T10(y16,x74)^

*x79,*x80,*x81,*x82,*x83,*x84,*x85,*x86 S27(x79)^S28(x80,x81)^S29(x82,x83)^S30(x84,x85,x86) --> ?y17,?y18 [T11(y17,y18,x80,x61)^

*x87,*x88,*x89,*x90,*x91,*x92,*x93,*x94,*x95 S31(x87,x88,x89,x90)^S32(x91,x92,x93,x94,x95) --> ?y19,?y20 [T12(y19,y20,x95,x2)^

*x96,*x97,*x98,*x99,*x100,*x101,*x102,*x103,*x104,*x105,*x106,*x107,*x108,*x109,*x110,*x111,*x112,*x113,*x114,*x115,*x116,*x117,*x118,*x119,*x120,*x121,*x122 S33(x96,x97,x98,x99,x100)^S34(x101,x102)^S35(x103,x104,x105,x106,x107)^S36(x108)^S37(x109,x110,x111,x112,x113)^S38(x114,x115,x116,x117,x118)^S39(x119,x120,x121,x122) --> ?y21,?y22 [T13(y21,y22,x119,x84)^T14(x97)^

*x123,*x124,*x125,*x126,*x127,*x128,*x129,*x130,*x131,*x132,*x133,*x134,*x135,*x136,*x137,*x138,*x139,*x140 S40(x123,x124)^S41(x125,x126,x127,x128)^S42(x129,x130,x131,x132)^S43(x133,x134)^S44(x135,x136,x137,x138)^S45(x139,x140) --> ?y23  ] ] ] ] ] ] ] ]

 Show the plain SO tgd:

?f1(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),?f2(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),?f3(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),?f4(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),?f5(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),?f6(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),?f7(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),?f8(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26),?f9(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26),?f10(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30),?f11(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30),?f12(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50),?f13(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50),?f14(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78),?f15(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78),?f16(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78),?f17(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86),?f18(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86),?f19(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95),?f20(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95),?f21(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95,x96,x97,x98,x99,x100,x101,x102,x103,x104,x105,x106,x107,x108,x109,x110,x111,x112,x113,x114,x115,x116,x117,x118,x119,x120,x121,x122),?f22(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95,x96,x97,x98,x99,x100,x101,x102,x103,x104,x105,x106,x107,x108,x109,x110,x111,x112,x113,x114,x115,x116,x117,x118,x119,x120,x121,x122),?f23(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95,x96,x97,x98,x99,x100,x101,x102,x103,x104,x105,x106,x107,x108,x109,x110,x111,x112,x113,x114,x115,x116,x117,x118,x119,x120,x121,x122,x123,x124,x125,x126,x127,x128,x129,x130,x131,x132,x133,x134,x135,x136,x137,x138,x139,x140) 

*x1,*x2,*x3,*x4,*x5,*x6,*x7,*x8,*x9,*x10,*x11,*x12,*x13,*x14,*x15,*x16,*x17,*x18,*x19,*x20,*x21,*x22,*x23,*x24,*x25,*x26,*x27,*x28,*x29,*x30,*x31,*x32,*x33,*x34,*x35,*x36,*x37,*x38,*x39,*x40,*x41,*x42,*x43,*x44,*x45,*x46,*x47,*x48,*x49,*x50,*x51,*x52,*x53,*x54,*x55,*x56,*x57,*x58,*x59,*x60,*x61,*x62,*x63,*x64,*x65,*x66,*x67,*x68,*x69,*x70,*x71,*x72,*x73,*x74,*x75,*x76,*x77,*x78,*x79,*x80,*x81,*x82,*x83,*x84,*x85,*x86,*x87,*x88,*x89,*x90,*x91,*x92,*x93,*x94,*x95,*x96,*x97,*x98,*x99,*x100,*x101,*x102,*x103,*x104,*x105,*x106,*x107,*x108,*x109,*x110,*x111,*x112,*x113,*x114,*x115,*x116,*x117,*x118,*x119,*x120,*x121,*x122,*x123,*x124,*x125,*x126,*x127,*x128,*x129,*x130,*x131,*x132,*x133,*x134,*x135,*x136,*x137,*x138,*x139,*x140  [S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11) --> T1(f1(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),f2(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11))^T2(f3(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),f4(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),f5(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),f6(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11),f7(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11))] ^

[S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26) --> T3(f8(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26),f9(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26),x13,f3(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11))^T4(x21)] ^

[S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26)^S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S9(x27,x28,x29)^S10(x30) --> T5(f10(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30),f11(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30),x28)] ^

[S9(x27,x28,x29)^S10(x30)^S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26)^S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50) --> ] ^

[S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50)^S9(x27,x28,x29)^S10(x30)^S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26)^S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S18(x51,x52,x53)^S19(x54,x55,x56,x57,x58)^S20(x59,x60,x61,x62)^S21(x63,x64,x65)^S22(x66,x67)^S23(x68)^S24(x69) --> T6(x67,f6(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11))^T7(x56,x21,x53,x1,x7)] ^

[S18(x51,x52,x53)^S19(x54,x55,x56,x57,x58)^S20(x59,x60,x61,x62)^S21(x63,x64,x65)^S22(x66,x67)^S23(x68)^S24(x69)^S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50)^S9(x27,x28,x29)^S10(x30)^S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26)^S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S25(x70,x71,x72,x73,x74)^S26(x75,x76,x77,x78) --> T8(x73)^T9(f14(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78),f15(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78),x74)^T10(f16(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78),x74)] ^

[S25(x70,x71,x72,x73,x74)^S26(x75,x76,x77,x78)^S18(x51,x52,x53)^S19(x54,x55,x56,x57,x58)^S20(x59,x60,x61,x62)^S21(x63,x64,x65)^S22(x66,x67)^S23(x68)^S24(x69)^S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50)^S9(x27,x28,x29)^S10(x30)^S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26)^S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S27(x79)^S28(x80,x81)^S29(x82,x83)^S30(x84,x85,x86) --> T11(f17(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86),f18(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86),x80,x61)] ^

[S27(x79)^S28(x80,x81)^S29(x82,x83)^S30(x84,x85,x86)^S25(x70,x71,x72,x73,x74)^S26(x75,x76,x77,x78)^S18(x51,x52,x53)^S19(x54,x55,x56,x57,x58)^S20(x59,x60,x61,x62)^S21(x63,x64,x65)^S22(x66,x67)^S23(x68)^S24(x69)^S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50)^S9(x27,x28,x29)^S10(x30)^S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26)^S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S31(x87,x88,x89,x90)^S32(x91,x92,x93,x94,x95) --> T12(f19(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95),f20(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95),x95,x2)] ^

[S31(x87,x88,x89,x90)^S32(x91,x92,x93,x94,x95)^S27(x79)^S28(x80,x81)^S29(x82,x83)^S30(x84,x85,x86)^S25(x70,x71,x72,x73,x74)^S26(x75,x76,x77,x78)^S18(x51,x52,x53)^S19(x54,x55,x56,x57,x58)^S20(x59,x60,x61,x62)^S21(x63,x64,x65)^S22(x66,x67)^S23(x68)^S24(x69)^S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50)^S9(x27,x28,x29)^S10(x30)^S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26)^S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S33(x96,x97,x98,x99,x100)^S34(x101,x102)^S35(x103,x104,x105,x106,x107)^S36(x108)^S37(x109,x110,x111,x112,x113)^S38(x114,x115,x116,x117,x118)^S39(x119,x120,x121,x122) --> T13(f21(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95,x96,x97,x98,x99,x100,x101,x102,x103,x104,x105,x106,x107,x108,x109,x110,x111,x112,x113,x114,x115,x116,x117,x118,x119,x120,x121,x122),f22(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53,x54,x55,x56,x57,x58,x59,x60,x61,x62,x63,x64,x65,x66,x67,x68,x69,x70,x71,x72,x73,x74,x75,x76,x77,x78,x79,x80,x81,x82,x83,x84,x85,x86,x87,x88,x89,x90,x91,x92,x93,x94,x95,x96,x97,x98,x99,x100,x101,x102,x103,x104,x105,x106,x107,x108,x109,x110,x111,x112,x113,x114,x115,x116,x117,x118,x119,x120,x121,x122),x119,x84)^T14(x97)] ^

[S33(x96,x97,x98,x99,x100)^S34(x101,x102)^S35(x103,x104,x105,x106,x107)^S36(x108)^S37(x109,x110,x111,x112,x113)^S38(x114,x115,x116,x117,x118)^S39(x119,x120,x121,x122)^S31(x87,x88,x89,x90)^S32(x91,x92,x93,x94,x95)^S27(x79)^S28(x80,x81)^S29(x82,x83)^S30(x84,x85,x86)^S25(x70,x71,x72,x73,x74)^S26(x75,x76,x77,x78)^S18(x51,x52,x53)^S19(x54,x55,x56,x57,x58)^S20(x59,x60,x61,x62)^S21(x63,x64,x65)^S22(x66,x67)^S23(x68)^S24(x69)^S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50)^S9(x27,x28,x29)^S10(x30)^S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26)^S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11)^S40(x123,x124)^S41(x125,x126,x127,x128)^S42(x129,x130,x131,x132)^S43(x133,x134)^S44(x135,x136,x137,x138)^S45(x139,x140) --> ]  

 

 Show the new nested tgd generated by our approach:

*x1,*x2,*x3,*x4,*x5,*x6,*x7,*x8,*x9,*x10,*x11 S1(x1,x2,x3,x4,x5)^S2(x6,x7,x8,x9)^S3(x10)^S4(x11) --> ?y1,?y2,?y3,?y4,?y5,?y6,?y7 [T1(y1,y2)^T2(y3,y4,y5,y6,y7) ^

*x12,*x13,*x14,*x15,*x16,*x17,*x18,*x19,*x20,*x21,*x22,*x23,*x24,*x25,*x26 S5(x12)^S6(x13,x14,x15,x16,x17)^S7(x18,x19,x20,x21,x22)^S8(x23,x24,x25,x26) --> ?y8,?y9 [T3(y8,y9,x13,y3)^T4(x21) ^

*x27,*x28,*x29,*x30 S9(x27,x28,x29)^S10(x30) --> ?y10,?y11 [T5(y10,y11,x28) ^

*x31,*x32,*x33,*x34,*x35,*x36,*x37,*x38,*x39,*x40,*x41,*x42,*x43,*x44,*x45,*x46,*x47,*x48,*x49,*x50 S11(x31,x32,x33)^S12(x34,x35)^S13(x36,x37,x38)^S14(x39,x40,x41,x42)^S15(x43)^S16(x44,x45,x46)^S17(x47,x48,x49,x50) --> ?y12,?y13 ^

*x51,*x52,*x53,*x54,*x55,*x56,*x57,*x58,*x59,*x60,*x61,*x62,*x63,*x64,*x65,*x66,*x67,*x68,*x69 S18(x51,x52,x53)^S19(x54,x55,x56,x57,x58)^S20(x59,x60,x61,x62)^S21(x63,x64,x65)^S22(x66,x67)^S23(x68)^S24(x69) --> [T6(x67,y6)^T7(x56,x21,x53,x1,x7) ^

*x70,*x71,*x72,*x73,*x74,*x75,*x76,*x77,*x78 S25(x70,x71,x72,x73,x74)^S26(x75,x76,x77,x78) --> ?y14,?y15,?y16 [T8(x73)^T9(y14,y15,x74)^T10(y16,x74) ^

*x79,*x80,*x81,*x82,*x83,*x84,*x85,*x86 S27(x79)^S28(x80,x81)^S29(x82,x83)^S30(x84,x85,x86) --> ?y17,?y18 [T11(y17,y18,x80,x61) ^

*x87,*x88,*x89,*x90,*x91,*x92,*x93,*x94,*x95 S31(x87,x88,x89,x90)^S32(x91,x92,x93,x94,x95) --> ?y19,?y20 [T12(y19,y20,x95,x2) ^

*x96,*x97,*x98,*x99,*x100,*x101,*x102,*x103,*x104,*x105,*x106,*x107,*x108,*x109,*x110,*x111,*x112,*x113,*x114,*x115,*x116,*x117,*x118,*x119,*x120,*x121,*x122 S33(x96,x97,x98,x99,x100)^S34(x101,x102)^S35(x103,x104,x105,x106,x107)^S36(x108)^S37(x109,x110,x111,x112,x113)^S38(x114,x115,x116,x117,x118)^S39(x119,x120,x121,x122) --> ?y21,?y22 [T13(y21,y22,x119,x84)^T14(x97) ^

*x123,*x124,*x125,*x126,*x127,*x128,*x129,*x130,*x131,*x132,*x133,*x134,*x135,*x136,*x137,*x138,*x139,*x140 S40(x123,x124)^S41(x125,x126,x127,x128)^S42(x129,x130,x131,x132)^S43(x133,x134)^S44(x135,x136,x137,x138)^S45(x139,x140) --> ?y23]]]]]]]  ]

 

 

 
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