Modular Primal-Dual Fixpoint Logic Solving for Temporal Verification
We present a novel approach to deciding the validity of formulas in first-order fixpoint logic with background theories and arbitrarily nested inductive and co-inductive predicates defining least and greatest fixpoints. Our approach is constraint-based, and reduces the validity checking problem of t...
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| Vydáno v: | Proceedings of ACM on programming languages Ročník 7; číslo POPL; s. 2111 - 2140 |
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New York, NY, USA
ACM
09.01.2023
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| ISSN: | 2475-1421, 2475-1421 |
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| Abstract | We present a novel approach to deciding the validity of formulas in first-order fixpoint logic with background theories and arbitrarily nested inductive and co-inductive predicates defining least and greatest fixpoints. Our approach is constraint-based, and reduces the validity checking problem of the given first-order-fixpoint logic formula (formally, an instance in a language called µCLP) to a constraint satisfaction problem for a recently introduced predicate constraint language. Coupled with an existing sound-and-relatively-complete solver for the constraint language, this novel reduction alone already gives a sound and relatively complete method for deciding µCLP validity, but we further improve it to a novel modular primal-dual method. The key observations are (1) µCLP is closed under complement such that each (co-)inductive predicate in the original primal instance has a corresponding (co-)inductive predicate representing its complement in the dual instance obtained by taking the standard De Morgan’s dual of the primal instance, and (2) partial solutions for (co-)inductive predicates synthesized during the constraint solving process of the primal side can be used as sound upper-bounds of the corresponding (co-)inductive predicates in the dual side, and vice versa. By solving the primal and dual problems in parallel and exchanging each others’ partial solutions as sound bounds, the two processes mutually reduce each others’ solution spaces, thus enabling rapid convergence. The approach is also modular in that the bounds are synthesized and exchanged at granularity of individual (co-)inductive predicates. We demonstrate the utility of our novel fixpoint logic solving by encoding a wide variety of temporal verification problems in µCLP, including termination/non-termination, LTL, CTL, and even the full modal µ-calculus model checking of infinite state programs. The encodings exploit the modularity in both the program and the property by expressing each loops and (recursive) functions in the program and sub-formulas of the property as individual (possibly nested) (co-)inductive predicates. Together with our novel modular primal-dual µCLP solving, we obtain a novel approach to efficiently solving a wide range of temporal verification problems. |
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| AbstractList | We present a novel approach to deciding the validity of formulas in first-order fixpoint logic with background theories and arbitrarily nested inductive and co-inductive predicates defining least and greatest fixpoints. Our approach is constraint-based, and reduces the validity checking problem of the given first-order-fixpoint logic formula (formally, an instance in a language called µCLP) to a constraint satisfaction problem for a recently introduced predicate constraint language. Coupled with an existing sound-and-relatively-complete solver for the constraint language, this novel reduction alone already gives a sound and relatively complete method for deciding µCLP validity, but we further improve it to a novel modular primal-dual method. The key observations are (1) µCLP is closed under complement such that each (co-)inductive predicate in the original primal instance has a corresponding (co-)inductive predicate representing its complement in the dual instance obtained by taking the standard De Morgan’s dual of the primal instance, and (2) partial solutions for (co-)inductive predicates synthesized during the constraint solving process of the primal side can be used as sound upper-bounds of the corresponding (co-)inductive predicates in the dual side, and vice versa. By solving the primal and dual problems in parallel and exchanging each others’ partial solutions as sound bounds, the two processes mutually reduce each others’ solution spaces, thus enabling rapid convergence. The approach is also modular in that the bounds are synthesized and exchanged at granularity of individual (co-)inductive predicates. We demonstrate the utility of our novel fixpoint logic solving by encoding a wide variety of temporal verification problems in µCLP, including termination/non-termination, LTL, CTL, and even the full modal µ-calculus model checking of infinite state programs. The encodings exploit the modularity in both the program and the property by expressing each loops and (recursive) functions in the program and sub-formulas of the property as individual (possibly nested) (co-)inductive predicates. Together with our novel modular primal-dual µCLP solving, we obtain a novel approach to efficiently solving a wide range of temporal verification problems. We present a novel approach to deciding the validity of formulas in first-order fixpoint logic with background theories and arbitrarily nested inductive and co-inductive predicates defining least and greatest fixpoints. Our approach is constraint-based, and reduces the validity checking problem of the given first-order-fixpoint logic formula (formally, an instance in a language called µCLP) to a constraint satisfaction problem for a recently introduced predicate constraint language. Coupled with an existing sound-and-relatively-complete solver for the constraint language, this novel reduction alone already gives a sound and relatively complete method for deciding µCLP validity, but we further improve it to a novel modular primal-dual method. The key observations are (1) µCLP is closed under complement such that each (co-)inductive predicate in the original primal instance has a corresponding (co-)inductive predicate representing its complement in the dual instance obtained by taking the standard De Morgan’s dual of the primal instance, and (2) partial solutions for (co-)inductive predicates synthesized during the constraint solving process of the primal side can be used as sound upper-bounds of the corresponding (co-)inductive predicates in the dual side, and vice versa. By solving the primal and dual problems in parallel and exchanging each others’ partial solutions as sound bounds, the two processes mutually reduce each others’ solution spaces, thus enabling rapid convergence. The approach is also modular in that the bounds are synthesized and exchanged at granularity of individual (co-)inductive predicates. We demonstrate the utility of our novel fixpoint logic solving by encoding a wide variety of temporal verification problems in µCLP, including termination/non-termination, LTL, CTL, and even the full modal µ-calculus model checking of infinite state programs. The encodings exploit the modularity in both the program and the property by expressing each loops and (recursive) functions in the program and sub-formulas of the property as individual (possibly nested) (co-)inductive predicates. Together with our novel modular primal-dual µCLP solving, we obtain a novel approach to efficiently solving a wide range of temporal verification problems. |
| ArticleNumber | 72 |
| Author | Gu, Yu Unno, Hiroshi Terauchi, Tachio Koskinen, Eric |
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| Cites_doi | 10.1145/1133255.1134029 10.1017/S1471068411000627 10.2307/2275338 10.1145/1328897.1328459 10.1145/3209108.3209204 10.1145/2499370.2491969 10.1145/3373718.3394766 10.1007/s100090100049 10.1016/0743-1066(94)90033-7 10.1145/1925844.1926431 10.1145/2813885.2737993 10.1145/2629488 10.1007/s10817-019-09532-0 10.1007/s10703-016-0249-4 10.1007/s10817-016-9388-y 10.1145/2345156.2254112 |
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| Keywords | fixpoint logics constraint logic programming temporal verification |
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Maher. 1994. Constraint logic programming: a survey. The Journal of Logic Programming, 19 (1994), 503 – 581. Cook Byron (e_1_2_1_13_1) 2014 Cook Byron (e_1_2_1_15_1) Henzinger Thomas A. (e_1_2_1_33_1) 2004 Cook Byron (e_1_2_1_19_1) Gurfinkel Arie (e_1_2_1_31_1) 2015 Godefroid Patrice (e_1_2_1_28_1) Komuravelli Anvesh (e_1_2_1_40_1) Unno Hiroshi (e_1_2_1_56_1) Urban Caterina (e_1_2_1_58_1) e_1_2_1_20_1 e_1_2_1_41_1 McMillan Kenneth L. (e_1_2_1_45_1) 2014; 259 e_1_2_1_43_1 Urban Caterina (e_1_2_1_60_1) Ball Thomas (e_1_2_1_2_1) 2002 Kahsai Temesghen (e_1_2_1_37_1) Terauchi Tachio (e_1_2_1_51_1) 2015 Kobayashi Naoki (e_1_2_1_39_1) Chen Hong Yi (e_1_2_1_11_1) 2014 Cook Byron (e_1_2_1_14_1) Champion Adrien (e_1_2_1_9_1) Jhala Ranjit (e_1_2_1_36_1) 2006 Padon Oded (e_1_2_1_48_1) 2022 Fedyukovich Grigory (e_1_2_1_24_1) Unno Hiroshi (e_1_2_1_57_1) Cook Byron (e_1_2_1_16_1) 2017; 64 Bradley Aaron R. 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| SubjectTerms | Automated reasoning Constraint and logic programming Logic and verification Program verification Theory of computation |
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| Title | Modular Primal-Dual Fixpoint Logic Solving for Temporal Verification |
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