Ranking Functions for Machine Arithmetic

Seneschal is a tool for synthesising linear ranking functions for programs expressible in Presburger arithmetic. The underlying method is an extension of Podelski's and Rybalchenko's approach for programs encoded as systems of linear rational inequalities. Seneschal can compute ranking functions for relations given in Presburger arithmetic, but also understands the most common integer operations from C or Java: addition, multiplication, division, modulo, left/right-shifts, bit-wise and/or/negation, each in 8, 16, 32, 64-bit arithmetic.

Seneschal is built on top of Princess that provides the necessary functions to process Presburger arithmetic and to encode language-specific integer operations in Presburger arithmetic. Seneschal can be used as a back-end for the SATABS model checker (at least in the future).

Seneschal is described in a paper published at TACAS 2010. Some benchmarks are presented here.

Seneschal is free software and distributed under GPL v3.


Suppose we want to prove termination of the following program:

int i = 0;
int j = [...];
while (i < 100 && j > 0 && j < 1000) {
 i = i + j;

We will do this by generating a ranking function, which is a function of the program variables that is bounded from below, and that monotonically decreases in each loop iteration. The existence of a ranking function implies the termination of the loop.

The transition relation of the program (capturing a single iteration of the loop) can be written in the Seneschal format as:
\from { i; j; }
\to { i'; j'; }
\transition {
in32(i) & in32(j) & // (1)
i < 100 & j > 0 & j < 1000 & // (2)
i' = add32(i, j) & j' = j // (3)

The first two lines declare the variables that the program operates on, which are i and j. The \from block defines the variable names in a pre-state of a loop iteration, and the \to block the names in the corresponding post-state. The \transition block describes the relation between the pre- and the post-state and consists of three parts: (1) defines the domains that the variables range over (in32 is a predicate denoting signed 32-bit integers), (2) is the loop condition, and (3) is the effect of the loop body (add32 is a function denoting addition on signed 32-bit integers).

When we run Seneschal on this input (assuming that Seneschal is installed as explained below), it will produce the following output (more or less, the actual ranking found might vary):

Loading file /tmp/test.trans
Parsing transition relation ... done
Expanding to Presburger formula ... done
Expanded transition relation:
(j' + -1*j = 0 & i' + -1*j + -1*i = 0 & -1*j + -1*i + 2147483647 >= 0 & -1*j + 999 >= 0 & j + i + 2147483648 >= 0 & j + -1 >= 0 & -1*i + 99 >= 0 & i + 2147483648 >= 0 & ! ALL (4294967296*_0 + -1*i' + j + i != 0))
Flattening ... 1 disjuncts
Generating constraints ... done
Solving ... found a solution
Minimising the solution ... done
Ranking function: -1*i
Lower bound (pre-state): -99
Lower bound (post-state): -1098

The most interesting part are the last three lines, which give the computed ranking function. This function is simply -i, which decreases in each loop iteration because some positive value is added to i in the loop body. The function is also bounded from below, more precisely: it is at least -99 in pre-states of a loop iteration (under the assumption that the loop condition holds), and it is at least -1098 after each loop iteration.

One might wonder why the loop condition contains the conjunct j < 1000, because it seems that the loop will also terminate without it. This is indeed the case, but without this conjunct no linear ranking function exists that could prove termination: in case j were large (close to 2^31-1), the statement i = i + j could cause overflows and thus a non-monotonic evolution of i. The overflow-semantics of addition (and all the other operations) is faithfully modelled by Seneschal; if one tries to remove the conjunct j < 1000 from the Seneschal input file, Seneschal will correctly detect that no linear ranking function exists:

Flattening ... 2 disjuncts
Generating constraints ... done
Solving ... no solution

Apart from the connectives shown in the example and the operations given in the next section, Seneschal supports all connectives present in Princess, e.g.: and &, or |, negation !, implication ->, equivalence <->, quantifiers \exists int x; ..., \forall int x; ...

Pre-Defined Operations

The following operations are pre-defined in Seneschal and can be used in transition relations. All of them are simply predicates or functions defined by axioms in Princess (in the file resources/prelude.pri), so that it is easy to add further operations if necessary.

1bit (unsigned)
8bit (signed)
8 (unsigned)
Other bit-widths
Domain predicate

inU1 in8 inU8 in16, inU16, in32, inU32, in64, inU64
+ addU1 add8 addU8 add16, addU16, ...

sub8 subU8 sub16, subU16, ...
Minus (sign-change)
minus8 minusU8 minus16, minusU16, ...

mul8 mulU8 mul16, mulU16, ...

divU8 div16, divU16, ...
mod mod mod
shiftLeft, shiftRight

shiftU8 shift16, shiftU16, ...
Bit-wise and
and and and
Bit-wise or
or or or or
Bit-wise negation
bitneg8 bitnegU8 bitneg16, bitnegU16, ...

cast8 castU8 cast16, castU16, ...

Some of the operations are non-linear, e.g., mul. Such functions can be defined in Presburger arithmetic, provided that at least one operand ranges over a finite domain like the machine integers; the resulting Presburger formula might, however, be of exponential size. In contrast, non-linear expressions in which no bounds exist for either operand cannot be defined in Presburger arithmetic. An expression mul(x, y) will in general cause Seneschal to run forever, but will work just fine if assumptions are given that restrict the value of y to some finite domain (the smaller the domain is, the more efficient will the expression be handled).

Division and modulo are defined such that the following formulae hold (unless y = 0):

0 <= mod(x, y) < |y|
mul(div(x, y), y) + mod(x, y) = x

Command-Line Options

Currently, Seneschal only offers a single option -assert for turning off assertions (which can make a huge performance difference):
Usage: seneschal <option>* <inputfile>*

[+-]assert Enable runtime assertions (default: +)

Installation from the binary distribution

Just download one of the binaries from the list of snapshots below and unpack it in your favourite location on the harddisk. Seneschal is invoked by calling the script seneschal-*/seneschal.

This is only tested under Linux, but should work also under Windows if Cygwin is used. Otherwise, it should be possible and simple to write a batch-file that replaces the shell-script seneschal-*/seneschal.

Installation from the source distribution

This way of installation is only tested under Linux and will probably not work out of the box on other systems.

  1. Download a source distribution of Princess and follow the installation instructions. You will have to use a fairly new Princess version, the last version that was verified to work together with Seneschal is 13/01/2010
  2. Download a source distribution of Seneschal from below, unpack it, and change into the seneschal-* directory
  3. Edit the Makefile: the first two lines in the file specify the location of the Princess and Scala installations. You need to change these lines to the correct paths on your system
  4. Run make to compile Seneschal.

If everything went ok, you can call Seneschal with the command ./seneschal <inputfile>