Micromega: tactics for solving arithmetic goals over ordered rings¶
| Authors: | Frédéric Besson and Evgeny Makarov |
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Short description of the tactics¶
The Psatz module (Require Import Psatz.) gives access to several
tactics for solving arithmetic goals over \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{R}\) [1].
It also possible to get the tactics for integers by a Require Import Lia,
rationals Require Import Lqa and reals Require Import Lra.
liais a decision procedure for linear integer arithmetic;niais an incomplete proof procedure for integer non-linear arithmetic;lrais a decision procedure for linear (real or rational) arithmetic;nrais an incomplete proof procedure for non-linear (real or rational) arithmetic;psatzD nwhereDis \(\mathbb{Z}\) or \(\mathbb{Q}\) or \(\mathbb{R}\), andnis an optional integer limiting the proof search depth, is an incomplete proof procedure for non-linear arithmetic. It is based on John Harrison’s HOL Light driver to the external provercsdp[2]. Note that thecsdpdriver is generating a proof cache which makes it possible to rerun scripts even withoutcsdp.
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Flag
Simplex¶ This option (set by default) instructs the decision procedures to use the Simplex method for solving linear goals. If it is not set, the decision procedures are using Fourier elimination.
The tactics solve propositional formulas parameterized by atomic arithmetic expressions interpreted over a domain \(D \in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}\). The syntax of the formulas is the following:
F ::= A ∣ P ∣ True ∣ False ∣ F ∧ F ∣ F ∨ F ∣ F ↔ F ∣ F → F ∣ ¬ F A ::= p = p ∣ p > p ∣ p < p ∣ p ≥ p ∣ p ≤ p p ::= c ∣ x ∣ −p ∣ p − p ∣ p + p ∣ p × p ∣ p ^ n
where \(c\) is a numeric constant, \(x \in D\) is a numeric variable, the
operators \(−, +, ×\) are respectively subtraction, addition, and product;
\(p ^ n\) is exponentiation by a constant \(n\), \(P\) is an arbitrary proposition.
For \(\mathbb{Q}\), equality is not Leibniz equality = but the equality of
rationals ==.
For \(\mathbb{Z}\) (resp. \(\mathbb{Q}\)), \(c\) ranges over integer constants (resp. rational constants). For \(\mathbb{R}\), the tactic recognizes as real constants the following expressions:
c ::= R0 | R1 | Rmul(c,c) | Rplus(c,c) | Rminus(c,c) | IZR z | IQR q | Rdiv(c,c) | Rinv c
where \(z\) is a constant in \(\mathbb{Z}\) and \(q\) is a constant in \(\mathbb{Q}\).
This includes integer constants written using the decimal notation, i.e., c%R.
Positivstellensatz refutations¶
The name psatz is an abbreviation for positivstellensatz – literally
"positivity theorem" – which generalizes Hilbert’s nullstellensatz. It
relies on the notion of Cone. Given a (finite) set of polynomials \(S\),
\(\mathit{Cone}(S)\) is inductively defined as the smallest set of polynomials
closed under the following rules:
\(\begin{array}{l} \dfrac{p \in S}{p \in \mathit{Cone}(S)} \quad \dfrac{}{p^2 \in \mathit{Cone}(S)} \quad \dfrac{p_1 \in \mathit{Cone}(S) \quad p_2 \in \mathit{Cone}(S) \quad \Join \in \{+,*\}} {p_1 \Join p_2 \in \mathit{Cone}(S)}\\ \end{array}\)
The following theorem provides a proof principle for checking that a set of polynomial inequalities does not have solutions [3].
Theorem (Psatz). Let \(S\) be a set of polynomials. If \(-1\) belongs to \(\mathit{Cone}(S)\), then the conjunction \(\bigwedge_{p \in S} p\ge 0\) is unsatisfiable. A proof based on this theorem is called a positivstellensatz refutation. The tactics work as follows. Formulas are normalized into conjunctive normal form \(\bigwedge_i C_i\) where \(C_i\) has the general form \((\bigwedge_{j\in S_i} p_j \Join 0) \to \mathit{False}\) and \(\Join \in \{>,\ge,=\}\) for \(D\in \{\mathbb{Q},\mathbb{R}\}\) and \(\Join \in \{\ge, =\}\) for \(\mathbb{Z}\).
For each conjunct \(C_i\), the tactic calls an oracle which searches for
\(-1\) within the cone. Upon success, the oracle returns a cone
expression that is normalized by the ring tactic (see The ring and field tactic families)
and checked to be \(-1\).
lra: a decision procedure for linear real and rational arithmetic¶
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lra¶ This tactic is searching for linear refutations. As a result, this tactic explores a subset of the Cone defined as
\(\mathit{LinCone}(S) =\left\{ \left. \sum_{p \in S} \alpha_p \times p~\right|~\alpha_p \mbox{ are positive constants} \right\}\)
The deductive power of
lraoverlaps with the one offieldtactic e.g., \(x = 10 * x / 10\) is solved bylra.
lia: a tactic for linear integer arithmetic¶
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lia¶ This tactic offers an alternative to the
omegatactic. Roughly speaking, the deductive power of lia is the combined deductive power ofring_simplifyandomega. However, it solves linear goals thatomegadoes not solve, such as the following so-called omega nightmare [Pug92].
The estimation of the relative efficiency of lia vs omega is under evaluation.
High level view of lia¶
Over \(\mathbb{R}\), positivstellensatz refutations are a complete proof
principle [4]. However, this is not the case over \(\mathbb{Z}\). Actually,
positivstellensatz refutations are not even sufficient to decide
linear integer arithmetic. The canonical example is \(2 * x = 1 -> \mathtt{False}\)
which is a theorem of \(\mathbb{Z}\) but not a theorem of \({\mathbb{R}}\). To remedy this
weakness, the lia tactic is using recursively a combination of:
- linear positivstellensatz refutations;
- cutting plane proofs;
- case split.
Cutting plane proofs¶
are a way to take into account the discreteness of \(\mathbb{Z}\) by rounding up (rational) constants up-to the closest integer.
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Theorem
Bound on the ceiling function Let \(p\) be an integer and \(c\) a rational constant. Then \(p \ge c \rightarrow p \ge \lceil{c}\rceil\).
For instance, from 2 x = 1 we can deduce
- \(x \ge 1/2\) whose cut plane is \(x \ge \lceil{1/2}\rceil = 1\);
- \(x \le 1/2\) whose cut plane is \(x \le \lfloor{1/2}\rfloor = 0\).
By combining these two facts (in normal form) \(x − 1 \ge 0\) and \(-x \ge 0\), we conclude by exhibiting a positivstellensatz refutation: \(−1 \equiv x−1 + −x \in \mathit{Cone}({x−1,x})\).
Cutting plane proofs and linear positivstellensatz refutations are a complete proof principle for integer linear arithmetic.
Case split¶
enumerates over the possible values of an expression.
Theorem. Let \(p\) be an integer and \(c_1\) and \(c_2\) integer constants. Then:
\(c_1 \le p \le c_2 \Rightarrow \bigvee_{x \in [c_1,c_2]} p = x\)
Our current oracle tries to find an expression \(e\) with a small range \([c_1,c_2]\). We generate \(c_2 − c_1\) subgoals which contexts are enriched with an equation \(e = i\) for \(i \in [c_1,c_2]\) and recursively search for a proof.
nra: a proof procedure for non-linear arithmetic¶
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nra¶ This tactic is an experimental proof procedure for non-linear arithmetic. The tactic performs a limited amount of non-linear reasoning before running the linear prover of
lra. This pre-processing does the following:
- If the context contains an arithmetic expression of the form \(e[x^2]\) where \(x\) is a monomial, the context is enriched with \(x^2 \ge 0\);
- For all pairs of hypotheses \(e_1 \ge 0\), \(e_2 \ge 0\), the context is enriched with \(e_1 \times e_2 \ge 0\).
After this pre-processing, the linear prover of lra searches for a
proof by abstracting monomials by variables.
nia: a proof procedure for non-linear integer arithmetic¶
psatz: a proof procedure for non-linear arithmetic¶
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psatz¶ This tactic explores the Cone by increasing degrees – hence the depth parameter n. In theory, such a proof search is complete – if the goal is provable the search eventually stops. Unfortunately, the external oracle is using numeric (approximate) optimization techniques that might miss a refutation.
To illustrate the working of the tactic, consider we wish to prove the following Coq goal:
As shown, such a goal is solved by intro x. psatz Z 2.. The oracle returns the
cone expression \(2 \times (x-1) + (\mathbf{x-1}) \times (\mathbf{x−1}) + -x^2\)
(polynomial hypotheses are printed in bold). By construction, this expression
belongs to \(\mathit{Cone}({−x^2,x -1})\). Moreover, by running ring we
obtain \(-1\). By Theorem Psatz, the goal is valid.
| [1] | Support for nat and N is obtained by pre-processing the goal with
the zify tactic. |
| [2] | Support for Z.div and Z.modulo may be obtained by
pre-processing the goal with the Z.div_mod_to_equations tactic (you may
need to manually run zify first). |
| [3] | Support for Z.quot and Z.rem may be obtained by pre-processing
the goal with the Z.quot_rem_to_equations tactic (you may need to manually
run zify first). |
| [4] | Note that support for Z.div, Z.modulo, Z.quot, and
Z.rem may be simultaneously obtained by pre-processing the goal with the
Z.to_euclidean_division_equations tactic (you may need to manually run
zify first). |
| [5] | Sources and binaries can be found at https://projects.coin-or.org/Csdp |
| [6] | Variants deal with equalities and strict inequalities. |
| [7] | In practice, the oracle might fail to produce such a refutation. |