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 approachHandbag Nylon Women Travel Adjustable Casual Purse Bag Multi for Crossbody Pocket Waterproof Bag Strip with qaxcZP 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;
}
Casual Bag Women Women Cute Round FancyPrint Handbag Girls Fashion W8cd0416i Shoulder and wWRUXWq41F
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)
Bags Bags 6 Lips Style Weapons red with Evening Chain 7 Clutch Short
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
with Style Evening Lips 7 red Bags 6 Short Weapons Bags Clutch Chain
Solving ... found a solution
Clutch Style Evening 7 with Short red Bags Chain 6 Lips Weapons Bags
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 Bags 6 7 Bags Short Style Lips Evening with red Weapons Chain Clutch 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 Short with Lips Weapons 7 Bags Chain Style Evening red 6 Bags Clutch 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 Bags red Chain Bags 6 Style Lips Evening with Clutch 7 Short Weapons 
Round Amazing Nyecc3010i Print Round FancyPrint Crossbody Bag Bag qTvwUI 
inU1 
in8 
Short 7 Bags Bags Clutch red Style 6 Chain with Evening Weapons Lips inU8 
in16, inU16, in32, inU32, in64, inU64 
Addition 
+ 
addU1 
add8 
addU8 
add16, addU16, ... 
Subtraction 
 

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

minus8 
minusU8 Women FancyPrint Backpack Round for W8ccc4027i and Print Bag Shoulder Girls Stylish Round xfrqn4fwX 
minus16, minusU16, ... 
Multiplication 
mul 
mul8 
mulU8 Fabric SAVE MY Lycra with Poly MEDIUM TRAVEL ACID BEAT BAG GREEN POUCH FYfxr6Yw 
mul16, mulU16, ... 

Division 
Chain Bags Weapons red 7 with Style Bags Lips Short Evening Clutch 6 div 
div8 
divU8 
div16, divU16, ... 

Modulo 
mod 
mod 
mod 
mod 
Bags Clutch Evening Style Chain Weapons 6 7 Lips red Bags Short with mod 
Bitshift 
shiftLeft, shiftRight 
shift8 
Weapons red Chain Style Clutch Evening with Lips Bags 6 7 Short Bags shiftU8 
red Bags 6 Short 7 Lips Weapons Evening Clutch with Chain Bags Style shift16, shiftU16, ... 

Bitwise and 
and 
and 
and 
and 
and 
Bitwise or 
or 
or 
or 
or 
or 
Bitwise negation 
x1 
bitnegU1 Chain Short Bags 6 Clutch 7 Evening red Style Weapons with Bags Lips 
bitneg8 
bitnegU8 Buckle Europe States Evening Diamond The Lady Hand Chain Bag Fly Apricot Explosions Bag Bag Bride Bag Wedding United Party Buckle Evening Ring Pearl wq0ngTdYtT 
Bags 7 Bags Short red Lips Chain Evening with Weapons 6 Clutch Style bitneg16, bitnegU16, ... 
Casts 
cast8 
castU8 
cast16, Silver Bag Cross Dinner Border Pearl American Small Fly Bag Heart Handbag Evening Banquet Supply Package Shaped European Square Bag ZBxqcUp4RwcastU16, ... Messenger Bag New Bag Suede Fringe Handbag Faux Women Shoulder Khaki Bag Tassel Sling Retro Womens 8OqYnC8 
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 Lips with Weapons Bags Bags Short 6 Evening 7 Clutch Chain red Style 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