Abstract
The ICE framework is a machine-learning-based technique originally introduced for inductive invariant inference over transition systems, and building on the supervised learning paradigm. Recently, we adapted the approach to non-linear Horn clause solving in the context of higher-order program verification. We showed that we could solve more of our benchmarks (extracted from higher-order program verification problems) than other state-of-the-art Horn clause solvers. This paper discusses some of the many improvements we recently implemented in HoIce, our implementation of this generalized ICE framework.
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Notes
- 1.
Available at https://github.com/hopv/hoice.
- 2.
- 3.
HoIce uses the Z3 [12] SMT solver.
- 4.
Available at https://github.com/hopv/benchmarks/tree/master/clauses.
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Acknowledgments
We thank the anonymous referees for useful comments. This work was supported by JSPS KAKENHI Grant Number JP15H05706.
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Appendices
A Input/Output Format Example
This section illustrates HoIce ’s input/output format. For a complete discussion on the format, please refer to the HoIce wiki https://github.com/hopv/hoice/wiki. HoIce takes special SMT-LIB [2] scripts as inputs such as the one on Fig. 3. A script starts with an optional set-logic HORN command, followed by some predicate declarations using the declare-fun command. Only predicate declaration are allowed: all declarations must have codomain Bool.
A legal input script corresponding to Example 1.
The actual clauses are given as assertions which generally start with some universally quantified variables, wrapping the implication between the body and the head of the clause. Negated existential quantification is also supported, for instance the third assertion on Fig. 3 can be written as
The check-sat command asks whether the Horn clauses are satisfiable, which they are, and HoIce answers sat. Otherwise, it would have answered unsat. Since the clauses are satisifiable, it is legal to ask for a model using the get-model command. HoIce provides one in the standard SMT-LIB fashion:
Note that hoice can read scripts from files, but also on its standard input in an interactive manner.
B Arguments
HoIce has no mandatory arguments. Besides options and flags, users can provide a file path argument in which case HoIce reads the file as an SMT-LIB script encoding a Horn clause problem (see Appendix A). When called with no file path argument, HoIce reads the script from its standard input. In both cases, HoIce outputs the result on its standard output.
Running HoIce with -h or --help will display the (visible) options. We do not discuss them here. Instead, let us clarify which options we used for the results presented in Sect. 5. The relevant option for partial samples from Sect. 4.1 is --partial, while --bias_cexs and --assistant activate constraint breaking as discussed in Sect. 4.2. More precisely, --bias_cexs activates constraint breaking during counterexample extraction, while --assistant triggers post-extraction simplification. The commands ran for the variants of Fig. 1 are thus
As far as the experiments are concerned, we ran Z3 4.7.1 with only one option, the one activating Spacer: fixedpoint.engine=spacer.
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Champion, A., Kobayashi, N., Sato, R. (2018). HoIce: An ICE-Based Non-linear Horn Clause Solver. In: Ryu, S. (eds) Programming Languages and Systems. APLAS 2018. Lecture Notes in Computer Science(), vol 11275. Springer, Cham. https://doi.org/10.1007/978-3-030-02768-1_8
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