1 \documentclass[runningheads]{llncs}
7 % \myincludegraphics{filename}{place}{width}{caption}{label}
8 \newcommand{\myincludegraphics}[5]{
11 \includegraphics[width=#3]{eps/#1.eps}
19 \usepackage{draftstamp}
21 \newcommand{\musing}{\texttt{musing}}
22 \newcommand{\musings}{\texttt{musings}}
23 \newcommand{\ws}{Web-Service}
24 \newcommand{\wss}{Web-Services}
25 \newcommand{\hbugs}{H-Bugs}
26 \newcommand{\helm}{HELM}
27 \newcommand{\Omegapp}{$\Omega$mega}
28 \newcommand{\OmegaAnts}{$\Omega$mega-Ants}
30 \title{Brokers and Web-Services for Automatic Deduction: a Case Study}
32 \author{Claudio Sacerdoti Coen \and Stefano Zacchiroli}
35 Department of Computer Science\\
36 University of Bologna\\
37 Mura Anteo Zamboni 7, 40127 Bologna, ITALY\\
38 \email{sacerdot@cs.unibo.it}
40 Department of Computer Science\\
41 \'Ecole Normale Sup\'erieure\\
42 45, Rue d'Ulm, F-75230 Paris Cedex 05, FRANCE\\
43 \email{zack@cs.unibo.it}
53 We present a planning broker and several Web-Services for automatic deduction.
54 Each Web-Service implements one of the tactics usually available in an
55 interactive proof-assistant. When the broker is submitted a "proof status" (an
56 unfinished proof tree and a focus on an open goal) it dispatches the proof to
57 the Web-Services, collects the successful results, and send them back to the
58 client as "hints" as soon as they are available.
60 In our experience this architecture turns out to be helpful both for
61 experienced users (who can take benefit of distributing heavy computations)
62 and beginners (who can learn from it).
65 \section{Introduction}
66 The \ws{} approach at software development seems to be a working solution for
67 getting rid of a wide range of incompatibilities between communicating
68 software applications. W3C's efforts in standardizing related technologies
69 grant longevity and implementations availability for frameworks based on
70 \wss{} for information exchange. As a direct consequence, the number of such
71 frameworks is increasing and the World Wide Web is moving from a disorganized
72 repository of human-understandable HTML documents to a disorganized repository
73 of applications working on machine-understandable XML documents both for input
76 The big challenge for the next future is to provide stable and reliable
77 services over this disorganized, unreliable and ever-evolving architecture.
78 The standard solution is providing a further level of
79 stable services (called \emph{brokers}) that behave as common gateway/address
80 for client applications to access a wide variety of services and abstract over
83 Since the \emph{Declaration of Linz}, the MONET
84 Consortium\footnote{\url{http://monet.nag.co.uk/cocoon/monet/index.html}}
85 is working on the development of a framework, based on the
86 \wss{}/brokers approach, aimed at providing a set of software tools for the
87 advertisement and the discovery of mathematical \wss{}.
88 %CSC This framework turns out to be strongly based on both \wss{} and brokers.
90 Several groups have already developed software bus and
91 services\footnote{The most part of these systems predate the development of
92 \wss. Those systems whose development is still active are slowly being
93 reimplemented as \wss.} providing both computational and reasoning
94 capabilities \cite{ws1,ws2,ws3,ws4}: the first ones are implemented on top of
95 Computer Algebra Systems; the second ones provide interfaces to well-known
97 Proof-planners, proof-assistants, CAS and
98 domain-specific problem solvers are natural candidates to be client of these
99 services. Nevertheless, so far the number of examples in the literature has
100 been extremely low and the concrete benefits are still to be assessed.
102 In this paper we present an architecture, namely \hbugs{}, implementing a
103 \emph{suggestion engine} for the proof assistant developed on behalf of the
104 \helm{}\footnote{Hypertextual Electronic Library of Mathematics,
105 \url{http://helm.cs.unibo.it}} project
106 \cite{helm}. We provide several \wss{} (called \emph{tutors}) able to
107 suggest possible ways to proceed in a proof. The tutors are orchestrated
108 by a broker (a \ws{} itself) that is able to dispatch a proof
109 status from a client (the proof-assistant) to the tutors;
110 each tutor try to make progress in the proof and, in case
111 of success, notify the client that shows an \emph{hint} to the user.
112 The broker is an instance of the homonymous entity of the MONET framework.
113 The tutors are MONET services. Another \ws (which is not described in this
114 paper and which is called Getter \cite{zack}) is used to locate and download
115 mathematical entities; the Getter plays the role of the Mathematical Object
116 Manager in the MONET framework.
118 A precursor of \hbugs{} is the \OmegaAnts{} project
119 \cite{omegaants1,omegaants2}, which provided similar functionalities to the
120 \Omegapp{} proof-planner \cite{omega}. The main architectural difference
121 between \hbugs{} and \OmegaAnts{} are that the latter is based on a
122 black-board architecture and it is not implemented using \wss{} and
125 In Sect. \ref{architecture} we present the architecture of \hbugs{}.
126 Further implementation details are given in Sect. \ref{implementation}.
127 Sect. \ref{tutors} is an overview of the tutors that have been implemented.
128 As usual, the paper ends with the conclusions and future works.
130 \section{An \hbugs{} Bird's Eye View}
132 \myincludegraphics{arch}{t}{8cm}{\hbugs{} architecture}{\hbugs{} architecture}
134 The \hbugs{} architecture (depicted in Fig. \ref{arch}) is based on three
135 different kinds of actors: \emph{clients}, \emph{brokers}, and \emph{tutors}.
136 Each actor present one or more \ws{} interfaces to its neighbors \hbugs{}
139 In this section we will detail the role and requirements of each kind of
140 actors and discuss about the correspondences between them and the MONET
141 entities described in \cite{MONET-Overview}.
144 An \hbugs{} client is a software component able to produce \emph{proof
145 status} and to consume \emph{hints}.
147 A proof status is a representation of an incomplete proof and is supposed to
148 be informative enough to be used by an interactive proof assistant. No
149 additional requirements exist on the proof status, but there should be an
150 agreement on its format between clients and tutors. An hint is an
151 encoding of a step that can be performed in order to proceed in an
152 incomplete proof. Usually it represents a reference to a tactic available
153 on some proof assistant along with an instantiation for its formal
154 parameters. More structured hints can also be used: an hint can be
155 as complex as a whole proof-plan.
157 Using W3C's terminology \cite{ws-glossary}, clients act both as \ws{}
158 providers and requesters, see Fig. \ref{interfaces}.
159 They act as providers for the broker (to receive hints)
160 and as requesters (to submit new status). Clients
161 additionally use broker service to know which tutors are available and to
162 subscribe to one or more of them.
164 Usually, when the client role is taken by an interactive proof assistant,
165 new status are sent to the broker as soon as the proof change (e.g. when the
166 user applies a tactic or when a new proof is started) and hints are shown to
167 the user be the means of some effect in the user interface (e.g. popping a
168 dialog box or enlightening a tactic button).\ednote{CSC: questo \'e un
169 possibile posto dove mettere una reference a una mini-sessione interattiva.
170 \'E l'appendice un posto dove metterla?}
172 \myincludegraphics{interfaces}{t!}{10cm}{\hbugs{} \wss{} interfaces}
173 {\hbugs{} \wss{} interfaces}
175 \hbugs{} clients act as MONET clients and ask brokers to provide access to a
176 set of services (the tutors). \hbugs{} has no actors corresponding to
177 MONET's Broker Locating Service (since the client is supposed to know the
178 URI of at least one broker). The \hbugs{} client and tutors contact the
179 Getter (a MONET Mathematical Object Manager) to locate and retrieve
180 mathematical items in the \helm{} library.
181 The proof status that are exchanged
182 by the \hbugs{} actors, instead, are built on the fly and are neither
183 stored nor are given an unique identifier (URI) to be managed by the
187 Brokers are the key actors of the \hbugs{} architecture since they
188 act as intermediaries between clients and tutors. They behave as \wss{}
189 providers and requesters for \emph{both} clients and tutors, see Fig.
192 With respect to the client, a broker acts as a \ws{} provider, receiving the
193 proof status and forwarding it to one or more tutors.
194 It also acts as a \ws{} requester sending
195 hints to the client as soon as they are available from the tutors.
197 With respect to the tutors, the \ws{} provider role is accomplished by
198 receiving hints as soon as they are produced; as a requester, it is
199 accomplished by asking for computations (\emph{musings} in \hbugs{}
200 terminology) on status received by clients and by stopping already late but
201 still ongoing \musings{}.
203 Additionally brokers keep track of available tutors and clients
206 \hbugs{} brokers act as MONET brokers implementing the following components:
207 Client Manager, Service Registry Manager (keeping track of available
208 tutors), Planning Manager (choosing the available tutors among the ones to
209 which the client is subscribed), Execution Manager. The Service Manager
210 component is not required since the session handler, that identifies
211 a session between a service and a broker, is provided to the service by
212 the broker instead of being received from the service when the session is
213 initialized. In particular, a session is identified by an unique identifier
214 for the client (its URL) and an unique identifier for the broker (its
217 The MONET architecture specification does not state explicitly whether
218 the service and broker answers can be asynchronous. Nevertheless, the
219 described information flow implicitly suggests a synchronous implementation.
220 On the contrary, in \hbugs{} every request is asynchronous: the connection
221 used by an actor to issue a query is immediately closed; when a service
222 produces an answer, it gives it back to the issuer by calling the
223 appropriate actor's method.
226 Tutors are software component able to consume proof status producing hints.
227 \hbugs{} does not specify by which means hints should be produced: tutors
228 can use any means necessary (heuristics, external theorem prover or CAS,
229 etc.). The only requirement is that there exists an agreement on the
230 formats of proof status and hints.
232 Tutors act both as \ws{} providers and requesters for the broker. As
233 providers, they wait for commands requesting to start a new \musing{} on
234 a given proof status or to stop an old, out of date, \musing{}. As
235 requesters, they signal to the broker the end of a \musing{} along with its
236 outcome (an hint in case of success or a notification of failure).
238 \hbugs{} tutors act as MONET services.
240 \section{An \hbugs{} Session Example}
242 \section{Implementation's Highlights}
243 \label{implementation}
244 In this section we present some of the most relevant implementation details of
245 the \hbugs{} architecture.
248 \paragraph{Proof status}
249 In our implementation of the \hbugs{} architecture we used the proof
250 assistant of the \helm{} project (codename ``gTopLevel'') as an \hbugs{}
251 client. Thus we have implemented serialization/deserialization capabilities
252 for its internal status. In order to be able to describe \wss{} that
253 exchange status in WSDL using the XML Schema type system, we have chosen an
254 XML format as the target format for the serialization.
256 A schematic representation of the gTopLevel internal status is depicted in
257 Fig. \ref{status}. Each proof is represented by a tuple of four elements:
258 \emph{uri}, \emph{metasenv}, \emph{proof}, \emph{thesis}.
260 \myincludegraphics{status}{t}{8cm}{gTopLevel proof status}{gTopLevel proof
264 \item[uri]: an URI chosen by the user at the beginning of the proof
265 process. Once (and if) proved, that URI will globally identify the term
266 inside the \helm{} library (given that the user decides to save it).
267 \item[thesis]: the thesis of the ongoing proof
268 \item[proof]: the current incomplete proof tree. It can contain
269 \emph{metavariables} (holes) that stands for the parts of the proof
270 that are still to be completed. Each metavariable appearing in the
271 tree references one element of the metavariables environment
273 \item[metasenv]: the metavariables environment is a list of
274 \emph{goals} (unproved conjectures).
275 In order to complete the proof, the user has to instantiate every
276 metavariable in the proof with a proof of the corresponding goal.
277 Each goal is identified by an unique identifier and has a context
278 and a type ( the goal thesis). The context is a list of named
279 hypotheses (declarations and definitions). Thus the context and the goal
280 thesis form a sequent, which is the statement of the proof that will
281 be used to instantiate the metavariable occurrences.
284 Each of these information is represented in XML as described in
285 \cite{mowglicic}. Additionally, an \hbugs{} status carry the unique
286 identifier of the current goal, which is the goal the user is currently
287 focused on. Using this value it is possible to implement different client
288 side strategies: the user could ask the tutors to work on the goal
289 she is considering or to work on the other ``background'' goals.
292 An hint in the \hbugs{} architecture should carry enough information to
293 permit the client to progress in the current proof. In our
294 implementation each hint corresponds to either one of the tactics available
295 to the user in gTopLevel (together with its actual arguments) or a set
296 of alternative suggestions (a list of hints).
298 For tactics that don't require any particular argument (like tactics that
299 apply type constructors or decision procedures)
300 only the tactic name is represented in the hint. For tactics that need
301 terms as arguments (for example the \texttt{Apply} tactic that apply a
302 given lemma) the hint includes a textual representation of them, using the
303 same representation used by the interactive proof assistant when querying
304 user for terms. In order to be transmitted between \wss{}, hints are
307 It is also possible for a tutor to return more hints at once,
308 grouping them in a particular XML element.
309 This feature turns out to be particularly useful for the
310 \emph{searchPatternApply} tutor (see Sect. \ref{tutors}) that
311 query a lemma database and return to the client a list of all lemmas that
312 could be used to complete the proof. This particular hint is encoded as a
313 list of \texttt{Apply} hints, each of them having one of the results as term
316 We would like to stress that the \hbugs{} architecture has no dependency
317 on either the hint or the status representation: the only message parts
318 that are fixed are those representing the administrative messages
319 (the envelops in the \wss{} terminology). In particular, the broker can
320 manage at the same time several sessions working on different status/hints
321 formats. Of course, there must be an agreement between the clients
322 and the tutors on the format of the data exchanged.
324 In our implementation the client does not trust the tutors hints:
325 being encoded as references to available tactics imply
326 that an \hbugs{} client, on receipt of an hint, simply try to reply the work
327 done by a tutor on the local copy of the proof. The application of the hint
328 can even fail to type check and the client copy of the proof can be left
329 undamaged after spotting the error. Note, however, that it is still
330 possible to implement a complex tutor that looks for a proof doing
332 send back to the client an hint whose argument is a witness (a trace) of
333 the proof found: the client applies the hint reconstructing (and checking
334 the correctness of) the proof from the witness, without having to
335 re-discover the proof itself.
337 An alternative implementation where the tutors are trusted would simply
338 send back to the client a new proof-status. Upon receiving the
339 proof-status, the client would just override its current proof status with
340 the suggested one. In the case of those clients which are implemented
341 using proof-objects (as the Coq proof-assistant, for instance), it is
342 still possible for the client to type-check the proof-object and reject
343 wrong hints. The systems that are not based on proof-objects
344 (as PVS, NuPRL, etc.), instead, have to trust the new proof-status. In this
345 case the \hbugs{} architecture needs at least to be extended with
346 clients-tutors authentication.
348 \paragraph{Registries}
349 Being central in the \hbugs{} architecture, the broker is also responsible
350 of housekeeping operations both for clients and tutors. These operations are
351 implemented using three different data structures called \emph{registries}:
352 clients registry, tutors registry and \musings{} registry.
354 In order to use the suggestion engine a client should register itself to the
355 broker and subscribe to one or more tutors. The registration phase is
356 triggered by the client using the \texttt{Register\_client} method of the
357 broker to send him an unique identifier and its base URI as a
358 \ws{}. After the registration, the client can use broker's
359 \texttt{List\_tutors} method to get a list of available tutors.
360 Eventually the client can subscribe to one or more of these using broker's
361 \texttt{Subscribe} method. Clients can also unregister from brokers using
362 \texttt{Unregister\_client} method.
364 The broker keeps track of both registered clients and clients' subscriptions
365 in the clients registry.
367 In order to be advertised to clients during the subscription phase, tutors
368 should register to the broker using the broker's \texttt{Register\_tutor}
369 method. This method is really similar to \texttt{Register\_client}:
370 tutors are required to send an unique identify and a base URI for their
372 Additionally tutors are required to send an human readable description of
373 their capabilities; this information could be used by client's user to
374 decide which tutors he needs to subscribe to. Like clients, tutors can
375 unregister from brokers using \texttt{Unregister\_broker} method.
377 Each time the client status change, the status is sent to the
378 broker using its \texttt{Status} method. Using both clients registry (to
379 lookup client's subscription) and tutors registry (to check if some tutors
380 has unsubscribed), the broker is able to decide to which tutors the
381 new status must be forwarded.\ednote{CSC: qui o nei lavori futuri parlare
382 della possibilit\'a di avere un vero brocker che multiplexi le richieste
383 dei client localizzando i servizi, etc.}
385 The forwarding operation is performed using the \texttt{Start\_musing}
386 method of the tutors, that is a request to start a new computation
387 (\emph{\musing{}}) on a given status. The return value of
388 \texttt{Start\_musing} is a
389 \musing{} identifier that is saved in the \musings{} registry along with
390 the identifier of the client that triggered the \musing{}.
392 As soon as a tutor completes an \musing{}, it informs the broker
393 using its \texttt{Musing\_completed} method; the broker can now remove the
394 \musing{} entry from the \musings{} registry and, depending on its outcome,
395 inform the client. In case of success one of the \texttt{Musing\_completed}
396 arguments is an hint to be sent to the client, otherwise there's no need to
397 inform him and the \texttt{Musing\_completed} method is called
398 just to update the \musings{} registry.
400 Consulting the \musings{} registry, the broker is able to know, at each
401 time, which \musings{} are in execution on which tutor. This peculiarity is
402 exploited by the broker on invocation of the \texttt{Status} method.
403 Receiving a new status from the client implies indeed that the previous
404 status no longer exists and all \musings{} working on it should be stopped:
405 additionally to the already described behavior (i.e. starting new
406 \musings{} on the received status), the broker takes also care of stopping
407 ongoing computation invoking the \texttt{Stop\_musing} method of the tutors.
410 As already discussed, all \hbugs{} actors act as \wss{} offering one or more
411 services to neighbor actors. To grant as most accessibility as possible to
412 our \wss{} we have chosen to bind them using the HTTP/POST\footnote{Given
413 that our proof assistant was entirely developed in the Objective Caml
414 language, we have chosen to develop also \hbugs{} in that language in order
415 to maximize code reuse. To develop \wss{} in Objective Caml we have
416 developed an auxiliary generic library (\emph{O'HTTP}) that can be used to
417 write HTTP 1.1 Web servers and abstract over GET/POST parsing. This library
418 supports different kinds of Web servers architecture, including
419 multi-process and multi-threaded ones.} bindings described in
423 Each tutor expose a \ws{} interface and should be able to work, not only for
424 many different clients referring to a common broker, but also for many
425 different brokers. The potential high number of concurrent clients imposes
426 a multi-threaded or multi-process architecture.
428 Our current implementation is based on a multi threaded architecture
429 exploiting the capabilities of the O'HTTP library. Each tutor is composed
430 by one thread always running plus
431 an additional thread for each running \musing{}. One thread is devoted to
432 listening for incoming \ws{} request; upon correct receiving requests it
433 pass the control to a second thread, created on the fly, to handle the
434 incoming request following the classical one-thread-per-request approach in
436 If the received request is \texttt{Start\_musing}, a new thread is
437 spawned to handle it; the thread in duty to handle the HTTP request
438 returns an HTTP response containing the identifier of the just started
439 \texttt{musing}, and then dyes. If the received request is
440 \texttt{Stop\_musing}, instead, the spawned thread kills the thread
441 responsible for the \texttt{musing} whose identifier is the argument
442 of the \texttt{Stop\_musing} method.
444 This architecture turns out to be scalable and allows the running threads
445 to share the cache of loaded (and type-checked) theorems.
446 As we will explain in Sect. \ref{tutors}, this feature turns out to be
447 really useful for tactics that rely on a huge but fixed set of lemmas,
448 as every reflexive tactic.
450 The implementation of a tutor with the described architecture is not that
451 difficult having a language with good threading capabilities (as OCaml has)
452 and a pool of already implemented tactics (as gTopLevel has).
453 Still working with threads is known to be really error prone due to
454 concurrent programming intrinsic complexity. Moreover, there is a
455 non-neglectable part of code that needs to be duplicated in every tutor:
456 the code to register the tutor to the broker and to handle HTTP requests;
457 the code to manage the creation and termination of threads; and the code for
458 parsing the requests and serializing the answers. As a consequence we
459 have written a generic implementation of a tutor which is parameterized
460 over the code that actually propose the hint and some administrative
461 data (as the port the tutor will be listening to).
463 The generic tutor skeleton is really helpful in writing new tutors.
464 Nevertheless, the code obtained by converting existing tactics into tutors
465 is still quite repetitive: every tutor that wraps a tactic has to
466 instantiate its own copy of the proof-engine kernel and, for each request,
467 it has to override its status, guess the tactic arguments, apply the tactic
468 and, in case of success, send back an hint with the tactic name and the
469 chosen arguments. Of course, the complex part of the work is guessing the
470 right arguments. For the simple case of tactics that do not require
471 any argument, though, we are able to automatically generate the whole
472 tutor code given the tactic name. Concretely, we have written a
473 tactic-based tutor template and a script that parses an XML file with
474 the specification of the tutor and generates the tutor's code.
475 The XML file describes the tutor's port, the code to invoke the tactic,
476 the hint that is sent back upon successful application and a
477 human readable explanation of the tactic implemented by the tutor.
479 \section{The Implemented \hbugs Tutors}
481 To test the \hbugs{} architecture and to assess the utility of a suggestion
482 engine for the end user, we have implemented several tutors. In particular,
483 we have investigated three classes of tutors:
485 \item \emph{Tutors for beginners}. These are tutors that implement tactics
486 which are neither computationally expensive nor difficult to understand:
487 an expert user can always understand if the tactic can be applied or not
488 without having to try it. For example, the following implemented tutors
489 belong to this class:
491 \item \emph{Assumption Tutor}: it ends the proof if the thesis is
492 equivalent\footnote{In our implementation, the equivalence relation
493 imposed by the logical framework is \emph{convertibility}. Two
494 expressions are convertible when they reduce to the same normal form.
495 Two ``equal'' terms depending on free variables can be non-convertible
496 since free variables stop the reduction. For example, $2x$ is convertible
497 with $(3-1)x$ because they both reduce to the same normal form
498 $x + x + 0$; but $2x$ is not convertible to $x2$ since the latter is
499 already in normal form.}
500 to one of the hypotheses\footnote{
501 In some cases, especially when non-trivial computations are involved,
502 the user is totally unable to figure out the convertibility of two terms.
503 In these cases the tutor becomes handy also for expert users.}.
504 \item \emph{Contradiction Tutor}: it ends the proof by \emph{reductio ad
505 adsurdum} if one hypothesis is equivalent to $False$.
506 \item \emph{Symmetry Tutor}: if the goal thesis is an equality, it
507 suggests to apply the commutative property.
508 \item \emph{Left/Right/Exists/Split/Reflexivity/Constructor Tutors}:
509 the Constructor Tutor suggests to proceed in the proof by applying one
510 or more constructors when the goal thesis is an inductive type or a
511 proposition inductively defined according to the declarative
512 style\footnote{An example of a proposition that can be given in
513 declarative style is the $\le$ relation: $\le$ is the smallest relation
514 such that $n \le n$ for every $n$ and $n \le m$ for every $n,m$ such
515 that $n \le p$ where $p$ is the predecessor of $m$. Thus, a proof
516 of $n \le n$ is simply the application of the first constructor to
517 $n$ and a proof of $n \le m$ is the application of the second
518 constructor to $n,m$ and a proof of $n \le m$.}.
519 Since disjunction, conjunction, existential quantification and
520 Leibniz equality are particular cases of inductive propositions,
521 all the other tutors of this class are instantiations of the
522 the Constructor tactic. Left and Right suggest to prove a disjunction
523 by proving its left/right member; Split reduces the proof of a
524 conjunction to the two proof of its members; Exists suggests to
525 prove an existential quantification by providing a
526 witness\footnote{This task is left to the user.}; Reflexivity proves
527 an equality whenever the two sides are convertible.
529 Beginners, when first faced with a tactic-based proof-assistant, get
530 lost quite soon since the set of tactics is large and their names and
531 semantics must be remembered by heart. Tutorials are provided to guide
532 the user step-by-step in a few proofs, suggesting the tactics that must
533 be used. We believe that our beginners tutors can provide an auxiliary
534 learning tool: after the tutorial, the user is not suddenly left alone
535 with the system, but she can experiment with variations of the proof given
536 in the tutorial as much as she like, still getting useful suggestions.
537 Thus the user is allowed to focus on learning how to do a formal proof
538 instead of wasting efforts trying to remember the interface to the system.
539 \item{Tutors for Computationally Expensive Tactics}. Several tactics have
540 an unpredictable behavior, in the sense that it is unfeasible to understand
541 whether they will succeed or they will fail when applied and what will be
542 their result. Among them, there are several tactics either computationally
543 expensive or resources consuming. In the first case, the user is not
544 willing to try a tactic and wait for a long time just to understand its
545 outcome: she would prefer to keep on concentrating on the proof and
546 have the tactic applied in background and receive out-of-band notification
547 of its success. The second case is similar, but the tactic application must
548 be performed on a remote machine to avoid overloading the user host
549 with several concurrent resource consuming applications.
551 Finally, several complex tactics and in particular all the tactics based
552 on reflexive techniques depend on a pretty large set of definitions, lemmas
553 and theorems. When these tactics are applied, the system needs to retrieve
554 and load all the lemmas. Pre-loading all the material needed by every
555 tactic can quickly lead to long initialization times and to large memory
556 footstamps. A specialized tutor running on a remote machine, instead,
557 can easily pre-load the required theorems.
559 As an example of computationally expensive task, we have implemented
560 a tutor for the \emph{Ring} tactic \cite{ringboutin}.
561 The tutor is able to prove an equality over a ring by reducing both members
562 to a common normal form. The reduction, which may require some time in
564 is based on the usual commutative, associative and neutral element properties
565 of a ring. The tactic is implemented using a reflexive technique, which
566 means that the reduction trace is not stored in the proof-object itself:
567 the type-checker is able to perform the reduction on-the-fly thanks to
568 the conversion rules of the system. As a consequence, in the library there
569 must be stored both the algorithm used for the reduction and the proof of
570 correctness of the algorithm, based on the ring axioms. This big proof
571 and all of its lemmas must be retrieved and loaded in order to apply the
572 tactic. The Ring tutor loads and cache all the required theorems the
573 first time it is contacted.
574 \item{Intelligent Tutors}. Expert users can already benefit from the previous
575 class of tutors. Nevertheless, to achieve a significative production gain,
576 they need more intelligent tutors implementing domain-specific theorem
577 provers or able to perform complex computations. These tutors are not just
578 plain implementations of tactics or decision procedures, but can be
579 more complex software agents interacting with third-parties software,
580 such as proof-planners, CAS or theorem-provers.
582 To test the productivity impact of intelligent tutors, we have implemented
583 a tutor that is interfaced with the \helm{}
584 Search-Engine\footnote{\url{http://mowgli.cs.unibo.it/library.html}} and that
585 is able to look for every theorem in the distributed library that can
586 be applied to proceed in the proof. Even if the tutor deductive power
587 is extremely limited\footnote{We do not attempt to check if the new goals
588 obtained applying a lemma can be automatically proved or, even better,
589 automatically disproved to reject the lemma.}, it is not unusual for
590 the tutor to come up with precious hints that can save several minutes of
591 work that would be spent in proving again already proven results or
592 figuring out where the lemmas could have been stored in the library.
595 \section{Conclusions and Future Work}
597 In this paper we described a suggestion engine architecture for
598 proof-assistants: the client (a proof-assistant) sends the current proof
599 status to several distributed \wss{} (called tutors) that try to progress
600 in the proof and, in case of success, send back an appropriate hint
601 (a proof-plan) to the user. The user, that in the meantime was able to
602 reason and progress in the proof, is notified with the hints and can decide
603 to apply or ignore them. A broker is provided to decouple the clients and
604 the tutors and to allow the client to locate and invoke the available remote
605 services. The whole architecture is an instance of the MONET architecture
606 for Mathematical \wss{}.
608 A running prototype has been implemented as part of the
609 \helm{} project \cite{helm}
610 and we already provide several tutors. Some of them are simple tutors that
611 try to apply one or more tactics of the \helm{} Proof-Engine, which is also
612 our client. We also have a much more complex tutor that is interfaced
613 with the \helm{} Search-Engine and looks for lemmas that can be directly
616 We have many plans for further developing both the \hbugs{} architecture and
617 our prototype. Interesting results could be obtained
618 augmenting the informative content of each suggestion. We can for example
619 modify the broker so that also negative results are sent back to the client.
620 Those negative suggestions could be reflected in the user interface by
621 deactivating commands to narrow the choice of tactics available to the user.
622 This approach could be interesting especially for novice users, but require
623 the client to trust other actors a bit more than in the current approach.
625 We plan also to add some rating mechanism to the architecture. A first
626 improvement in this direction could be distinguishing between hints that, when
627 applied, are able to completely close one or more goals and
628 tactics that progress in the proof by reducing one or more goals to new goals:
629 the new goals could be false and the proof can be closed only by backtracking.
631 Other heuristics and/or measures could be added to rate
632 hints and show them to the user in a particular order: an interesting one
633 could be a measure that try to minimize the size of the generated proof,
634 privileging therefore non-overkilling solutions \cite{ring}.
636 We are also considering to follow the \OmegaAnts{} path more closely adding
637 ``recursion'' to the system so that the proof status resulting from the
638 application of old hints are cached somewhere and could be used as a starting
639 point for new hint searches. The approach is interesting, but it represents
640 a big shift towards automatic theorem proving: thus we must consider if it is
641 worth the effort given the increasing availability of automation in proof
642 assistants tactics and the ongoing development of \wss{} based on
643 already existent and well developed theorem provers.
645 Even if not strictly part of the \hbugs{} architecture, the graphical user
646 interface (GUI) of our prototype needs a lot of improvement if we would like
647 it to be really usable by novices. In particular, the user is too easily
648 distracted by the tutor's hints that are ``pushed'' to her.
650 Our \wss{} still lack a real integration in the MONET architecture,
651 since we do not provide the different ontologies to describe our problems,
652 solutions, queries and services. In the short term, completing this task
653 could provide a significative feedback to the MONET consortium and would
654 enlarge the current set of available MONET actors on the Web. In the long
655 term, new more intelligent tutors could be developed on top of already
656 existent MONET \wss{}.
658 To conclude, \hbugs{} is a nice experiment meant to understand whether the
659 current \wss{} technology is mature enough to have a concrete and useful
660 impact on the daily work of users of proof-assistants. So far, only the tutor
661 that is interfaced with the \helm{} Search-Engine has effectively increased
662 the productivity of experts users. The usefulness of the tutors developed for
663 beginners, instead, need further assessment.
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