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SMTtoTPTP - A Converter for Theorem Proving Formats Peter Baumgartner Introduction TPTP (Thousands of Problems for Theorem Proving) Languages : clause logic, [typed]FOL[+arithmetics], HOL Problem library : > 20k problems Infrastructure :

  1. SMTtoTPTP - A Converter for Theorem Proving Formats Peter Baumgartner

  2. Introduction TPTP (Thousands of Problems for Theorem Proving) Languages : clause logic, [typed]FOL[+arithmetics], HOL Problem library : > 20k problems Infrastructure : utilities, solutions to problems SMT-LIB Language : sorted FOL + background theories (e.g., arithmetics, arrays) Problem library : > 100k problems Infrastructure : utilities SMTtoTPTP Translation SMT-LIB problems ⇒ TPTP problems Who bene fi ts? (Remark: “sort” = “type” in this talk) 2

  3. Who Bene fi ts? Maintainers of TPTP problem collections SMTtoTPTP makes it easy to add existing SMT-LIB benchmarks to TPTP Developers of TPTP theorem provers SMTtoTPTP provides a front-end for problems written in SMT-LIB Users of SMT solvers SMTtoTPTP provides the link to (also) use TPTP theorem provers Rest of this talk Example SMT-LIB ⇒ TPTP transformation SMTtoTPTP algorithm 3

  4. SMT-LIB Scripts (set-logic UFLIA) (declare-sort Color 0) (declare-fun red () Color) (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  5. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) (declare-sort Color 0) (declare-fun red () Color) (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  6. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) (declare-fun red () Color) (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  7. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  8. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  9. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) Macro Sort ↦ Sort (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) Int) (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  10. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) Macro Sort ↦ Sort (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) get-int : Int) ( Pair Int Color ) ↦ Int (declare-fun int-color-pair (Int Color) (Pair Int Color)) (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  11. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) Macro Sort ↦ Sort (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) get-int : Int) ( Pair Int Color ) ↦ Int (declare-fun int-color-pair (Int Color) (Pair Int Color)) (Well-sorted) input formula (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4

  12. SMT-LIB Scripts Uninterpreted function symbols + LIA (set-logic UFLIA) 0-ary sort Color (declare-sort Color 0) Color -constant red (declare-fun red () Color) 2-ary sort Pair (declare-sort Pair 2) Macro Sort ↦ Sort (define-sort Int-Pair (S) (Pair Int S)) (declare-fun get-int ((Int-Pair Color)) get-int : Int) ( Pair Int Color ) ↦ Int (declare-fun int-color-pair (Int Color) (Pair Int Color)) (Well-sorted) input formula (assert (forall ((i Int) (c Color)) (= (get-int (int-color-pair i c)) i))) (check-sat) 4 Translation into TPTP? Compatibility with TPTP format?

  13. SMT-LIB ⇒ TPTP: (In)Compatibilities ( ✓ = compatible ✗ = incompatible) Sorts ✓ SMT-LIB arithmetic sorts ≈ TPTP arithmetic sorts ✗ SMT-LIB: n-ary user sorts ≠ TPTP: 0-ary user sorts Overloaded operators ✓ SMT-LIB equality = TPTP equality = : S × S ↦ Bool for any sort S ✓ SMT-LIB arithmetic operators ≈ TPTP arithmetic operators ✗ SMT-LIB overloaded array operators (prede fi ned) (declare-sort Array 2) select : ( Array S T) × S ↦ T for any sorts S and T store : ( Array S T) × S × T ↦ ( Array S T) ⇒ It is the types that require the most attention in the transformation 5

  14. Example SMT-LIB ⇒ TPTP (set-logic UFLIA) tff('Color', type, 'Color': $tType). (declare-sort Color 0) (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

  15. Example SMT-LIB ⇒ TPTP Color ↝ ‘Color’ (set-logic UFLIA) tff('Color', type, 'Color': $tType). (declare-sort Color 0) (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

  16. Example SMT-LIB ⇒ TPTP Color ↝ ‘Color’ (set-logic UFLIA) Constant red : unused hence forget tff('Color', type, 'Color': $tType). (declare-sort Color 0) (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

  17. Example SMT-LIB ⇒ TPTP Color ↝ ‘Color’ (set-logic UFLIA) Constant red : unused hence forget tff('Color', type, 'Color': $tType). (declare-sort Color 0) Instance (Pair Int Color) ↝ ‘Pair[Int,Color]’ (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

  18. Example SMT-LIB ⇒ TPTP Color ↝ ‘Color’ (set-logic UFLIA) Constant red : unused hence forget tff('Color', type, 'Color': $tType). (declare-sort Color 0) Instance (Pair Int Color) ↝ ‘Pair[Int,Color]’ (declare-fun red () Color) tff('Pair', type, (declare-sort Pair 2) 'Pair[Int,Color]': $tType). (define-sort Int-Pair (S) (Pair Int S)) tff(get_int, type, get_int: 'Pair[Int,Color]' > $int). (declare-fun get-int ((Int-Pair Color)) Int) tff(int_color_pair, type, int_color_pair: (declare-fun int-color-pair ($int * 'Color') > 'Pair[Int,Color]'). (Int Color) (Pair Int Color)) (assert tff(formula, axiom, (forall ((i Int) (c Color)) ( ! [I:$int, C:'Color'] : (= (get-int (get_int(int_color_pair(I, C)) (int-color-pair i c)) = I))). i))) (check-sat) 6

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