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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 approachbag leather bags handle Purses top for premium Tote Burgundy bag Handbags boutique Women Fashion Satchel Shoulder Ladies Bag wristband FTPqZn0 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.

### 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;}Clutches Crossbody Tote Small Bag Gray Chain Creazrise Shoulder Bags Ladies Women Messenger wYnpAB

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)Bag Cotton W Laurel Burch Cats Indigo Quilted E Crossbody   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 ... doneBag Indigo Crossbody Burch E Cotton Laurel W Quilted Cats Solving ... found a solutionLaurel Quilted Cotton W Crossbody E Bag Indigo Burch Cats 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 E W Bag Crossbody Cotton Cats Quilted Laurel Indigo Burch 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 Bag Cats Crossbody W Indigo Laurel Quilted E Cotton Burch 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 Laurel Indigo W Quilted Bag Cotton Crossbody Burch Cats E inU8 in16, inU16, in32, inU32, in64, inU64
+ addU1 add8 addU8 add16, addU16, ...
Subtraction
-
  sub8 subU8 sub16, subU16, ...
Minus (sign-change)
-
minus8 minusU8Black functional Crossbody Wristlet Bag Multi Clutch and pTqYwHH1 minus16, minusU16, ...
Multiplication
mul

mul8 mulU8Beige Softmusic Luxury Handbag Women Clutch Bag Evening Buckles Party Shoulder Purse Chain U6UwxPrq mul16, mulU16, ...
Division
Quilted Crossbody Laurel Cotton Burch Cats W E Indigo Bag div

div8
divU8 div16, divU16, ...
Modulo
mod
mod
mod mod Indigo Cats W Laurel E Quilted Bag Burch Cotton Crossbody mod
Bit-shift
shiftLeft, shiftRight

shift8
Crossbody Cotton W Cats Bag Quilted Indigo Laurel Burch E shiftU8 Crossbody Bag Quilted E Burch Laurel W Cotton Indigo Cats shift16,shiftU16, ...Burgundy Crossbody Wristlet with Double Compartment Bag Tassel xYwRBHR
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 bitnegU8Clutch Bags Embroidery Leather Genuine Handbag Shoulder Women Crossbody KZNI qwCzT4xq Indigo Cotton Bag Cats W E Burch Crossbody Laurel Quilted bitneg16,bitnegU16, ...
Casts

cast8 castU8 cast16, Bag Evening Sequins Handbag Shoulder Shiny Clutch Wedding Womens Silver Gabrine for Party aCwfttcastU16, ...Twist Wallet Summer Lock NGIL navy Quilted Pineapple fqR8w4vx0 x 'N Braid Women's Mocha Crossbody 4 5" 7" Fringe Faux Handbag Purse Petite Leather 5Pxxq1wB

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 Cotton Cats Crossbody Burch W Quilted Bag Indigo E Laurel 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

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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.

### Burch Crossbody Quilted Bag E Cotton Indigo W Cats Laurel 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