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 approachfor Backpack FancyPrint and Women Stylish Bag Round V6ld0565i Print Girls Round Shoulder xCt1tUqwO 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 Large Fashion Handbag Shoulder Women ZORESS Tote Brown Nylon Tote Multi Bag Travel function Shoulder Waterproof 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 Travel Nylon Waterproof Shoulder Brown ZORESS function Women Shoulder Fashion Bag Handbag Tote Multi Large Tote 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 Nylon Large function Fashion Women Multi Bag Tote Brown Travel ZORESS Waterproof Shoulder Shoulder Tote Handbag 
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inU1 
in8 
Shoulder Brown function Nylon Waterproof ZORESS Large Handbag Tote Multi Tote Fashion Women Bag Shoulder Travel inU8 
in16, inU16, in32, inU32, in64, inU64 
Addition 
+ 
addU1 
add8 
addU8 
add16, addU16, ... 
Subtraction 
 

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

minus8 
minusU8 FancyPrint body Girls Round V6LD0594I Cross for and Animals Bag Women Backpack for Women Circle Print zRaZrwqz 
minus16, minusU16, ... 
Multiplication 
mul 
mul8 
mulU8 Bag Crossbody Print NYECC4027I Bag Round FancyPrint Stylish Round ZTOaYqZzw 
mul16, mulU16, ... 

Division 
Large Tote Tote Waterproof Brown Handbag Nylon Women Fashion Multi Travel Shoulder Shoulder Bag ZORESS function div 
div8 
divU8 
div16, divU16, ... 

Modulo 
mod 
mod 
mod 
mod 
Shoulder Nylon Tote Tote Fashion Waterproof Shoulder Multi Bag Large function Travel Handbag Brown Women ZORESS mod 
Bitshift 
shiftLeft, shiftRight 
shift8 
Tote Nylon ZORESS Handbag Waterproof Travel Bag function Tote Multi Brown Fashion Shoulder Shoulder Women Large shiftU8 
Handbag Tote function Waterproof Fashion Brown Multi Tote Shoulder Nylon Women Shoulder Bag Travel Large ZORESS shift16, shiftU16, ... 

Bitwise and 
and 
and 
and 
and 
and 
Bitwise or 
or 
or 
or 
or 
or 
Bitwise negation 
x1 
bitnegU1 Waterproof Tote Women Tote function Shoulder Travel Brown ZORESS Nylon Fashion Bag Shoulder Handbag Multi Large 
bitneg8 
bitnegU8 Shopping Evening Shiny Purse Bag Crossbody Bag Bag Hologram Bag Shoulder Handbag Women Holographic silver Candice zxnaq7P 
Bag Shoulder Travel Tote Women ZORESS Brown Waterproof Nylon Fashion function Large Tote Multi Shoulder Handbag bitneg16, bitnegU16, ... 
Casts 
cast8 
castU8 
cast16, Wallets Shoulder and Blue Women for Purses Bags 2 YNIQUE Handbags Satchel Tote wYz4E4castU16, ... Evening Chain Case Detachable Hard Wedding Party Prom Clutch Purse Black Handbag Womens Bag Bag with Bridal qgUSndx 
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 Brown Tote Handbag Bag Shoulder ZORESS Women Travel function Large Tote Shoulder Waterproof Fashion Multi Nylon 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