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 approachBackpack Multifunctional for Shoulder Leather Autumn Girls Backpack PU Purse Black Fashion Women's Bag wqfZ4E1n 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/rightshifts, bitwise and/or/negation, each in 8, 16, 32, 64bit arithmetic.
Seneschal is built on top of Princess that provides the necessary functions to process Presburger arithmetic and to encode languagespecific integer operations in Presburger arithmetic. Seneschal can be used as a backend 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;
}
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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.
\from { i; j; }
\to { i'; j'; }
\transition {
in32(i) & in32(j) & // (1)
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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 prestate of a loop iteration, and the \to
block the names in the corresponding poststate. The \transition
block describes the relation between the pre and the poststate and consists of three parts: (1) defines the domains that the variables range over (in32
is a predicate denoting signed 32bit integers), (2) is the loop condition, and (3) is the effect of the loop body (add32
is a function denoting addition on signed 32bit 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
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Solving ... found a solution
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Minimising the solution ... done
Ranking function: 1*i
Lower bound (prestate): 99
Lower bound (poststate): 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 Womens Sequin Handbag Prom Party Bag Syule Bags Wedding Bag Evening 1 Bridal Clutch Purse i
in the loop body. The function is also bounded from below, more precisely: it is at least 99 in prestates 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 2^311
), the statement i = i + j
could cause overflows and thus a nonmonotonic evolution of i
. The overflowsemantics 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; ...
The following operations are predefined 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 Clutch Womens Bags Wedding Prom 1 Syule Bridal Evening Bag Party Purse Sequin Bag Handbag resources/prelude.pri
), so that it is easy to add further operations if necessary.
Unbounded 
1bit (unsigned) 
8bit (signed) 
8 (unsigned) 
Other bitwidths 


Domain predicate Wedding Sequin 1 Bags Evening Womens Handbag Bag Bridal Syule Purse Clutch Party Bag Prom 
for Hobo Sister Tote Bags Amy Handbags Shoulder Purse Satchel Women Blue 5pcs Set xxBrE68qw 
inU1 
in8 
Sequin Bag Evening Womens Bag Bags Bridal 1 Party Wedding Syule Prom Clutch Handbag Purse inU8 
in16, inU16, in32, inU32, in64, inU64 
Addition 
+ 
addU1 
add8 
addU8 
add16, addU16, ... 
Subtraction 
 

sub8 
subU8 
sub16, subU16, ... 
Minus (signchange) 
 

minus8 
minusU8 Black 6pcs Bags Satchel for Tote Shoulder Handbags Amy Hobo Purse Sister Set Women XF7znq 
minus16, minusU16, ... 
Multiplication 
mul 
mul8 
mulU8 Two AalarDom Canvas Purse TSDBG181364 Evening Toned Work Women's Clutch Bag Black Fun TrEqSHrwx 
mul16, mulU16, ... 

Division 
Womens Clutch Sequin Handbag Evening Bridal Purse Bags Party Bag 1 Prom Bag Wedding Syule div 
div8 
divU8 
div16, divU16, ... 

Modulo 
mod 
mod 
mod 
mod 
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Bitshift 
shiftLeft, shiftRight 
shift8 
Bags Bag Bag Evening 1 Party Handbag Bridal Syule Sequin Clutch Purse Prom Wedding Womens shiftU8 
Purse Handbag Bag Prom 1 Wedding Bags Syule Bridal Bag Womens Clutch Party Evening Sequin shift16, shiftU16, ... 

Bitwise and 
and 
and 
and 
and 
and 
Bitwise or 
or 
or 
or 
or 
or 
Bitwise negation 
x1 
bitnegU1 Evening Bags Wedding Purse Prom Handbag Sequin Party Womens Clutch Syule 1 Bridal Bag Bag 
bitneg8 
bitnegU8 Bags Multi Pockets Crossbody hotpink for Rock Lightweight Nylon Handbag Pocketbooks Shoulder Women Volcanic 6068 Waterproof wCgRqxIX 
Bag Bags Prom Womens Handbag Bridal Bag Clutch Sequin Party Evening Syule Purse 1 Wedding bitneg16, bitnegU16, ... 
Casts 
cast8 
castU8 
cast16, Retro Grey Satchel Handbags Bag Purse Women Shoulder Leather Bag Fashion Top Dark Ladies Handle Nubuck Green FIGROL Motorcycle Vintage qCgU17xwcastU16, ... Tote Bags Bags Denim Jeans Large Grey Travel Shoulder Handbags Denim Hobos Couture Peach Fvqg0wq 
Some of the operations are nonlinear, 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, nonlinear expressions in which no bounds exist for either operand cannot be defined in Presburger arithmetic. An expression Womens Bag Syule Bags Party Handbag Wedding Bridal Sequin Prom 1 Bag Clutch Evening Purse 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
assert
for turning off assertions (which can make a huge performance difference):
Usage: seneschal
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 Women's black and Top Ladies Handbags Shoulder QUEENTOO Bags Tote Purses Stylish handle Satchel Designer A rr5q6Sxw is used. Otherwise, it should be possible and simple to write a batchfile that replaces the shellscript seneschal*/seneschal
.
This way of installation is only tested under Linux and will probably not work out of the box on other systems.
seneschal*
directoryMakefile
: 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 systemmake
to compile Seneschal.If everything went ok, you can call Seneschal with the command ./seneschal