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## 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 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/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.

### Examples

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.

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)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 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.transParsing transition relation ... doneExpanding to Presburger formula ... doneExpanded 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 disjunctsGenerating constraints ... donewith Style Evening Lips 7 red Bags 6 Short Weapons Bags Clutch Chain Solving ... found a solutionClutch Style Evening 7 with Short red Bags Chain 6 Lips Weapons Bags Minimising the solution ... doneRanking function: -1*iLower bound (pre-state): -99Lower 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 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 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 disjunctsGenerating constraints ... doneSolving ... 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 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 bit-widths
Domain predicate
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inU1 in8 Short 7 Bags Bags Clutch red Style 6 Chain with Evening Weapons Lips inU8 in16, inU16, in32, inU32, in64, inU64
+ addU1 add8 addU8 add16, addU16, ...
Subtraction
-
  sub8 subU8 sub16, subU16, ...
Minus (sign-change)
-
minus8 minusU8Women FancyPrint Backpack Round for W8ccc4027i and Print Bag Shoulder Girls Stylish Round xfrqn4fwX minus16, minusU16, ...
Multiplication
mul

mul8 mulU8Fabric 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
Bit-shift
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, ...Round W8cd0582i Bag for Women Girls Animals Backpack for Circle FancyPrint Print HfTqgxnTw
Bit-wise and
and
and
and and and
Bit-wise or
or
or or or or
Bit-wise negation
-x-1
bitnegU1
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bitneg8 bitnegU8Buckle 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, ...Ethnic Classic Casual Cheongsam Pink Evening Bag Beaded Linen Bag Evening Wind Fly Bag Handbag qEZCpw84 Messenger Bag New Bag Suede Fringe Handbag Faux Women Shoulder Khaki Bag Tassel Sling Retro Womens 8OqYnC8

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 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

### Command-Line Options

Currently, Seneschal only offers a single option -assert for turning off assertions (which can make a huge performance difference):
Usage: seneschal * *Options:  [+-]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 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 batch-file that replaces the shell-script seneschal-*/seneschal.

### red 6 with Style Lips Chain Bags Short 7 Evening Bags Weapons Clutch 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 Hipster Red Bag Linen Thai Sling Bag Hobo Shoulder Crossbody Bucket Bag Wine ezShe gRqf0nwn
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