Combining decision procedures for the reals
Jeremy Avigad and Harvey Friedman
*
September 15, 2006
Abstract
We address the general problem of determining the validity of boolean
combinations of equalities and inequalities between realvalued expres
sions. In particular, we consider methods of establishing such assertions
using only restricted forms of distributivity. At the same time, we explore
ways in which “local” decision or heuristic procedures for fragments of the
theory of the reals can be amalgamated into global ones.
Let
T
add
[
Q
] be the firstorder theory of the real numbers in the lan
guage with symbols 0
,
1
,
+
,

, <, . . . , f
a
, . . .
where for each
a
∈
Q
,
f
a
de
notes the function
f
a
(
x
) =
ax
.
Let
T
mult
[
Q
] be the analogous theory
for the language with symbols 0
,
1
,
×
,
÷
, <, . . . , f
a
, . . .
. We show that al
though
T
[
Q
] =
T
add
[
Q
]
∪
T
mult
[
Q
] is undecidable, the universal fragment of
T
[
Q
] is decidable. We also show that terms of
T
[
Q
] can fruitfully be put
in a normal form. We prove analogous results for theories in which
Q
is
replaced, more generally, by suitable subfields
F
of the reals. Finally, we
consider practical methods of establishing quantifierfree validities that
approximate our (impractical) decidability results.
1
Introduction
This paper is generally concerned with the problem of determining the valid
ity of boolean combinations of equalities and inequalities between realvalued
expressions. Such computational support is important not only for the formal
verification of mathematical proofs, but, more generally, for any application
which depends on such reasoning about the real numbers.
Alfred Tarski’s proof [23] that the theory of the real numbers as an or
dered field admits quantifierelimination is a striking and powerful response to
the problem. The result implies decidability of the full firstorder theory, not
just the quantifierfree fragment.
George Collins’s [10] method of cylindrical
algebraic decomposition made this procedure feasible in practice, and ongoing
research in computational real geometry has resulted in various optimizations
and alternatives (see e.g. [14, 6, 5]). Recently, a proofproducing version of an
*
Work by both authors has been supported by NSF grant DMS0401042. We are grateful
to three anonymous referees for numerous comments, suggestions, and corrections.
1
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elimination procedure due to Paul Cohen has even been implemented in the
framework of a theorem prover for higherorder logic [20].
There are two reasons, however, that one might be interested in alterna
tives to q.e. procedures for real closed fields. The first is that their generality
means that they can be inefficient in restricted settings. For example, one might
encounter an inference like
0
< x < y
→
(1 +
x
2
)
/
(2 +
y
)
17
<
(1 +
y
2
)
/
(2 +
x
)
10
,
in an ordinary mathematical proof.
Such an inference is easily verified, by
noticing that all the subterms are positive and then chaining through the obvious
inferences. Computing sequences of partial derivatives, which is necessary for
the full decision procedure, seems misguided in this instance. A second, more
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 Fall '08
 JOSHUA
 Math, Inequalities, Rational number, Firstorder logic, Conjunctive normal form

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