**Time:** Tuesday and Thursday, 9:30-10:45

**Place:** Siebel Center 1131

**Professor:** Jose Meseguer

**Office Hours:** Immediately following class in Dr. Meseguer's office.

- (STACS) Set Theory and Algebra in Computer Science.
- Peter Olveczky's book Formal Modeling and Analysis of Distributed Systems.
- Algebraic Semantics of Imperative Programs, by Goguen, Malcolm.
- Maude 2.7 Manual.
- Constructor-Based Reachability Logic for Rewrite Theories, by Skeirik, Stefanescu, Meseguer.
- All About Maude
- Programming and Symbolic Computation in Maude.
- Maude Termination Assistant. The MTA tool and a collection of examples can be found on the MTA website.
- Continuation Semantics and Program Verification for the IMPL Language, by Abir, Meseguer.

The course will consist of weekly homework assignments and one long homework assignment towards the end covering a wide range of the course material. Each problem on each homework will be worth 10 points, and the homeworks all together will be worth 70% of the grade. The final comprehensive homework assignment will be worth 30% of the grade.

If you are taking the class for 4 credits, then there will also be a final project for the course. The final project must be turned in by the assigned final date before a grade for the class will be issued.

Maude will be the primary tool used in this course, but several tools/libraries written in Maude will also be used. Some of these libraries require extra functionality to be added directly to Maude (at the C++ level). You can choose to build from the Maude sources or directly from binary releases of Maude 2.7.1.

```
# Download and unzip Maude 2.7.1 binary release for Linux
$ wget 'https://github.com/maude-team/maude/releases/download/v2.7.1-ext-hooks/maude-2.7.1-linux.tar.gz'
$ tar -xvf maude-2.7.1-linux.tar.gz
# Download Maude 2.7.1 binary release for MacOS
$ wget 'http://maude.cs.illinois.edu/w/images/2/25/Maude-2.7.1-osx.zip'
$ unzip Maude-2.7.1-osx.zip
# Run Maude
$ ./maude
\||||||||||||||||||/
--- Welcome to Maude ---
/||||||||||||||||||\
Maude 2.7.1 built: Aug 31 2016 18:22:38
Copyright 1997-2016 SRI International
Sun Oct 1 11:59:31 2017
Maude>
```

The MFE packages together several tools which will be used in the course. It's recommended that you use the binary releases for anything needing the MFE. To get setup using MFE, you might do:

```
# Clone MFE
$ git clone 'https://github.com/maude-team/mfe'
# Launch Maude with MFE
$ ./maude mfe/src/mfe.maude
\||||||||||||||||||/
--- Welcome to Maude ---
/||||||||||||||||||\
Maude 2.7.1 built: Aug 31 2016 18:22:38
Copyright 1997-2016 SRI International
Sun Oct 1 11:51:55 2017
Full Maude 2.7e September 22th 2016
The Maude Formal Environment 1.0b
Inductive Theorem Prover - July 20th 2010
Sufficient Completeness Checker 2a - August 2010
Church-Rosser Checker 3p - August 30th 2016
Coherence Checker 3p - August 30th 2016
Maude Termination Tool 1.5j - August 11th 2014
Maude>
```

Note that many of the tools in the MFE *cannot* reason with built-in modules (many of the modules which come in `prelude.maude`

). By default though, Maude includes the built-in module `BOOL`

with any user-defined module. If you get the following error:

`Error The use of built-ins is not supported by the checker.`

be sure to disable including `BOOL`

into your module by default:

```
set include BOOL off .
fmod MYMODULE is
...
endfm
```

If you are in loop mode (eg. if you have already loaded the MFE), then you must do this in parentheses:

```
(set include BOOL off .)
(fmod MYMODULE is
...
endfm)
```

The SCC tool works on an older version of Maude, and I only know of Linux binaries for it. Login to EWS or use a virtual machine if you are not on a Linux machine.

```
$ wget 'http://maude.cs.uiuc.edu/tools/scc/releases/Maude-ceta-2.3-linux.tgz'
$ tar -xvf Maude-ceta-2.3-linux.tgz
$ cd Maude-ceta-2.3-linux
$ ./maude-ceta
\||||||||||||||||||/
--- Welcome to Maude ---
/||||||||||||||||||\
Maude-ceta 2.3 built: Feb 18 2007 09:49:47
Copyright 1997-2007 SRI International
With CETA extensions Copyright 2006-2007
University of Illinois at Urbana-Champaign
Mon Oct 2 19:17:22 2017
# Load SCC
Maude> load scc.maude
# Start the SCC
Maude> loop init-cc .
Starting the Maude Sufficient and Canonical Completeness Checker.
# Load the file with module of interest (note that it should NOT be in parens)
Maude> load ../../../PATH/TO/FILE/WITH/MODULE/NOT/IN/PARENTHESES
# Select the CC-LOOP module.
Maude> select CC-LOOP .
# Check Sufficient Completeness (modulo other proof goals)
Maude> (scc MODULE-OF-INTEREST .)
Warning: no loop state.
Advisory: attempting to reinitialize loop.
Starting the Maude Sufficient and Canonical Completeness Checker.
Checking sufficient completeness of MODULE-OF-INTEREST ...
Success: MODULE-OF-INTEREST is sufficiently complete under the assumption that it is ground weakly-normalizing, confluent, and ground sort-decreasing.
```

Go to the Maude ITP Webpage to download the sources for the Maude ITP. The Maude ITP runs on Maude 2.7.

Commands that you may find useful (see the tutorial on the Maude ITP website):

`ind`

for doing structural induction on a variable.`auto`

for discharging goals with rewriting.`cns`

for using the lemma of constants.`imp`

for handling implications.`enable`

and`disable`

for enabling/disabling rewriting with given rules. Note that some of the other commands may add equations to the module in question. For example, when using the`(imp .)`

command on goal`u = v ==> phi`

, the equation`u = v`

will be added to the module (assumed as true) and`phi`

will be the new proof goal. No check is made that these added equations don't loop forever, so it may be necessary to`disable`

those equations from consideration.

Using the example module from lecture 13 (say file `nat-ord.maude`

):

```
fmod NATURAL-ORD is
sort Natural .
op 0 : -> Natural [ctor] .
op s : Natural -> Natural [ctor] .
op _<_ : Natural Natural -> Bool .
op _=<_ : Natural Natural -> Bool .
vars N M : Natural .
eq N < 0 = false .
eq 0 < s(N) = true .
eq s(N) < s(M) = N < M .
ceq N =< M = true if N < M .
ceq N =< M = true if not(M < N) .
ceq N =< M = false if M < N .
endfm
```

And an example interaction:

```
# Download and unpack the Maude itp
$ wget 'http://maude.cs.uiuc.edu/tools/itp/maude_itp.tar.gz'
$ tar -xvf maude_itp.tar.gz
\||||||||||||||||||/
--- Welcome to Maude ---
/||||||||||||||||||\
Maude 2.7 built: Oct 17 2017 22:51:54
Copyright 1997-2014 SRI International
Tue Oct 17 23:40:31 2017
Maude> load maude_itp/itp-tool.maude
Maude> load nat-ord.maude
Maude> select ITP-TOOL .
Maude> loop init-itp .
ITP tool (May 25th, 2006)
--- Introduce linearity goal
--- ------------------------
Maude> (goal linear : NATURAL-ORD |- A{N:Natural ; M:Natural} (((N =< M) or (M =< N)) = (true)) .)
rewrites: 1200 in 18ms cpu (18ms real) (66666 rewrites/second)
=================================
label-sel: linear@0
=================================
A{M:Natural ; N:Natural}
N:Natural =< M:Natural or M:Natural =< N:Natural = true
+++++++++++++++++++++++++++++++++
--- Use lemma of constants
--- ----------------------
Maude> (cns .)
rewrites: 223 in 3ms cpu (2ms real) (74333 rewrites/second)
New goals:
=================================
label-sel: linear@0
=================================
N*Natural =< M*Natural or M*Natural =< N*Natural = true
+++++++++++++++++++++++++++++++++
--- Split on N < M
--- --------------
Maude> (split on (N*Natural < M*Natural) .)
rewrites: 222 in 5ms cpu (5ms real) (44400 rewrites/second)
New goals:
=================================
label-sel: linear@1.0
=================================
N*Natural =< M*Natural or M*Natural =< N*Natural = true
=================================
label: linear@2.0
=================================
N*Natural =< M*Natural or M*Natural =< N*Natural = true
+++++++++++++++++++++++++++++++++
--- Discharge goal 1 with assumptions
--- ---------------------------------
Maude> (auto .)
rewrites: 112 in 0ms cpu (0ms real) (~ rewrites/second)
Eliminated current goal.
New current goal:
=================================
label-sel: linear@2.0
=================================
N*Natural =< M*Natural or M*Natural =< N*Natural = true
+++++++++++++++++++++++++++++++++
--- Discharge goal 2 with assumptions
--- ---------------------------------
Maude> (auto .)
rewrites: 54 in 0ms cpu (0ms real) (~ rewrites/second)
Eliminated current goal.
q.e.d
+++++++++++++++++++++++++++++++++
```

Here is the command listing from `rltool-lib.maude`

in the tool archive.

```
sort @QFForm@ @QFCTerm@ @NeQFCTermSet@ @Term@ @TermSet@ @ReachFormEx@ @LabelReachFormEx@ .
subsort @QFCTerm@ < @NeQFCTermSet@ .
subsort @Term@ < @TermSet@ .
op ((_)) : @Bubble@ -> @Term@ [ctor] .
op _U_ : @TermSet@ @TermSet@ -> @TermSet@ [assoc comm ctor] .
op true : -> @QFForm@ [ctor] .
op (_)=(_) : @Bubble@ @Bubble@ -> @QFForm@ [ctor] .
op (_)=/=(_) : @Bubble@ @Bubble@ -> @QFForm@ [ctor] .
op _/\_ : @QFForm@ @QFForm@ -> @QFForm@ [prec 58 assoc comm ctor] .
op _\/_ : @QFForm@ @QFForm@ -> @QFForm@ [prec 59 assoc comm ctor] .
op (_)|_ : @Bubble@ @QFForm@ -> @QFCTerm@ [prec 60 ctor] .
op _\/_ : @NeQFCTermSet@ @NeQFCTermSet@ -> @NeQFCTermSet@ [prec 61 assoc comm ctor] .
op _=>_ : @NeQFCTermSet@ @NeQFCTermSet@ -> @ReachFormEx@ [prec 62 ctor] .
op _:_ : @Token@ @ReachFormEx@ -> @LabelReachFormEx@ [prec 63 ctor] .
sort MetaRLTLCommand .
--- module/var/backend setup
op select_. : @Token@ -> MetaRLTLCommand [ctor] .
op select-rls_. : @NeTokenList@ -> MetaRLTLCommand [ctor] .
op declare-vars_. : @TermSet@ -> MetaRLTLCommand [ctor] .
op use`tool_for_on_. : @Token@ @Token@ @Token@ -> MetaRLTLCommand [ctor] .
op use`tool_for_on_with_. : @Token@ @Token@ @Token@ @NeTokenList@ -> MetaRLTLCommand [ctor] .
--- goal/term state setup
op def-term-set_. : @NeQFCTermSet@ -> MetaRLTLCommand [ctor] .
op add-goal_. : @LabelReachFormEx@ -> MetaRLTLCommand [ctor] . --- add as goal and axiom
op add-goal!_. : @LabelReachFormEx@ -> MetaRLTLCommand [ctor] . --- add as goal
op add-axiom_. : @LabelReachFormEx@ -> MetaRLTLCommand [ctor] . --- add as axiom TODO: revisit this command
op inv_to_with_on_. : @Token@ Qid @TermSet@ @NeQFCTermSet@ -> MetaRLTLCommand [ctor] .
op inv_to_on_. : @Token@ Qid @NeQFCTermSet@ -> MetaRLTLCommand [ctor] .
--- finish setup
op start-proof`. : -> MetaRLTLCommand [ctor] .
--- proof manipulation
--- apply basic proof strategy
op auto_. : Nat -> MetaRLTLCommand [ctor] .
op auto*`. : -> MetaRLTLCommand [ctor] .
op auto`. : -> MetaRLTLCommand [ctor] .
--- delete unwanted goals
op focus_. : NeSet{Nat} -> MetaRLTLCommand [ctor] .
op focus_. : @Term@ -> MetaRLTLCommand [ctor] .
op focus_. : @QFCTerm@ -> MetaRLTLCommand [ctor] .
op focus_/_. : @QFCTerm@ @QFCTerm@ -> MetaRLTLCommand [ctor] .
--- change axioms used
op use-axioms_on_. : @NeTokenList@ Nat -> MetaRLTLCommand [ctor] . --- modify goal to use circularities
op use-axioms_._on_. : @NeTokenList@ @NeTokenList@ Nat -> MetaRLTLCommand [ctor] . --- modify goal to use axioms/circularities
--- do case analysis a variable
op case_on_by_. : Nat @Token@ @TermSet@ -> MetaRLTLCommand [ctor] .
op case_on_by_. : @Term@ @Token@ @TermSet@ -> MetaRLTLCommand [ctor] .
op case_on_by_. : @QFCTerm@ @Token@ @TermSet@ -> MetaRLTLCommand [ctor] .
--- split on a formula
op split_by_. : Nat @QFForm@ -> MetaRLTLCommand [ctor] .
op split_by_. : @Term@ @QFForm@ -> MetaRLTLCommand [ctor] .
op split_by_. : @QFCTerm@ @QFForm@ -> MetaRLTLCommand [ctor] .
op replace_by_. : Nat @QFForm@ -> MetaRLTLCommand [ctor] .
op replace_by_. : @Term@ @QFForm@ -> MetaRLTLCommand [ctor] .
op replace_by_. : @QFCTerm@ @QFForm@ -> MetaRLTLCommand [ctor] .
op split_by_and_. : Nat @QFForm@ @QFForm@ -> MetaRLTLCommand [ctor] .
op split_by_and_. : @Term@ @QFForm@ @QFForm@ -> MetaRLTLCommand [ctor] .
op split_by_and_. : @QFCTerm@ @QFForm@ @QFForm@ -> MetaRLTLCommand [ctor] .
--- try to close a set of goals with default strategy---but do nothing if this process fails
op try-finish_. : NeSet{Nat} -> MetaRLTLCommand [ctor] .
op try-finish_. : @QFCTerm@ -> MetaRLTLCommand [ctor] .
op try-finish_/_. : @QFCTerm@ @QFCTerm@ -> MetaRLTLCommand [ctor] .
--- try to subsume a goal by another
op subsume_by_. : Nat Nat -> MetaRLTLCommand [ctor] .
--- resume from a debugging output pause
op continue`. : -> MetaRLTLCommand [ctor] .
--- delete inactive goals
op delete-inactive`. : -> MetaRLTLCommand [ctor] .
--- output commands
op show-goal-table`. : -> MetaRLTLCommand [ctor] .
op show-goal-ids`. : -> MetaRLTLCommand [ctor] .
op show-all-goal-ids`. : -> MetaRLTLCommand [ctor] .
op show-goals_. : NeSet{Nat} -> MetaRLTLCommand [ctor] .
op show-goals`. : -> MetaRLTLCommand [ctor] .
op show-level_. : Nat -> MetaRLTLCommand [ctor] .
op show-levels`. : -> MetaRLTLCommand [ctor] .
op show-ancestors_generation_. : NeSet{Nat} Bound -> MetaRLTLCommand [ctor] .
op show-narrowings_. : Nat -> MetaRLTLCommand [ctor] .
op show-maude-cmd_on_. : @Token@ Nat -> MetaRLTLCommand [ctor] .
```