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 \ednote{zack: buhm! :-P} 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
123 brokers. Other differences will be detailed in Sect. \ref{conclusions}
124 \ednote{CSC: che si fa di sta frase?}.
126 In Sect. \ref{architecture} we present the architecture of \hbugs{}.
127 Further implementation details are given in Sect. \ref{implementation}.
128 Sect. \ref{tutors} is an overview of the tutors that have been implemented.
129 As usual, the paper ends with the conclusions and future works.
131 \section{An \hbugs{} Bird's Eye View}
133 \myincludegraphics{arch}{t}{8cm}{\hbugs{} architecture}{\hbugs{} architecture}
135 The \hbugs{} architecture (depicted in Fig. \ref{arch}) is based on three
136 different kinds of actors: \emph{clients}, \emph{brokers}, and \emph{tutors}.
137 Each actor present one or more \ws{} interfaces to its neighbors \hbugs{}
140 In this section we will detail the role and requirements of each kind of
141 actors and discuss about the correspondences between them and the MONET
142 entities described in \cite{MONET-Overview}.
145 An \hbugs{} client is a software component able to produce \emph{proof
146 status} and to consume \emph{hints}.
148 A proof status is a representation of an incomplete proof and is supposed to
149 be informative enough to be used by an interactive proof assistant. No
150 additional requirements exist on the proof status, but there should be an
151 agreement on its format between clients and tutors. An hint is a
152 representation\ednote{CSC: non c'\'e un sinonimo pi\'u carino? Zack: l'unico
153 decente sembra essere nuovamente "suggestion". CSC: chiamalo sinonimo!}
154 of a step that can be performed in order to proceed in an
155 incomplete proof. Usually it represents a reference to a tactic available
156 on some proof assistant along with an instantiation for its formal
157 parameters. More structured hints can also be used: an hint can be
158 as complex as a whole proof-plan.
160 \myincludegraphics{interfaces}{t}{10cm}{\hbugs{} \wss{} interfaces}{\hbugs{}
163 Using W3C's terminology \cite{ws-glossary}, clients act both as \ws{}
164 providers and requesters, see Fig. \ref{interfaces}.
165 They act as providers for the broker (to receive hints)
166 and as requesters (to submit new status). Clients
167 additionally use broker service to know which tutors are available and to
168 subscribe to one or more of them.
170 Usually, when the role of client is taken by an interactive proof assistant,
171 new status are sent to the broker as soon as the proof change (e.g. when the
172 user applies a tactic or when a new proof is started) and hints are shown to
173 the user be the means of some effect in the user interface (e.g. popping a
174 dialog box or enlightening a tactic button).\ednote{CSC: questo \'e un
175 possibile posto dove mettere una mini-sessione interattiva. L'appendice
178 \hbugs{} clients act as MONET clients and ask brokers to provide access to a
179 set of services (the tutors). \hbugs{} has no actors corresponding to
180 MONET's Broker Locating Service (since the client is supposed to know the
181 URI of at least one broker). The \hbugs{} client and tutors contact the
182 Getter (a MONET Mathematical Object Manager) to locate and retrieve
183 mathematical items in the \helm{} library.
184 The proof status that are exchanged
185 by the \hbugs{} actors, instead, are built on the fly and are neither
186 stored nor are given an unique identifier (URI) to be managed by the
190 Brokers are the key actors of the \hbugs{} architecture since they
191 act as intermediaries between clients and tutors. They behave as \wss{}
192 providers and requesters for \emph{both} clients and tutors, see Fig.
195 With respect to client, a broker act as \ws{} provider, receiving the
196 proof status and forwarding it to one or more tutors.
197 It also acts as a \ws{} requester sending
198 hints to the client as soon as they are available from the tutors.
200 With respect to tutors, the \ws{} provider role is accomplished by receiving
201 hints as soon as they are produced; as a requester, it is accomplished
202 by asking for computations (\emph{musings} in \hbugs{} terminology) on
203 status received by clients and by stopping already late but still
205 \ednote{Zack: io intanto ho aggiunto una figura, vorrei pero' un tuo
206 commento sulla utilita'/quantita' delle figure ... CSC: vanno benissimo}
208 Additionally brokers keep track of available tutors and clients
211 \hbugs{} brokers act as MONET brokers implementing the following components:
212 Client Manager, Service Registry Manager (keeping track of available
213 tutors), Planning Manager (choosing the available tutors among the ones to
214 which the client is subscribed), Execution Manager. The Service Manager
215 component is not required since the session handler, that identifies
216 a session between a service and a broker, is provided to the service by
217 the broker instead of being received from the service when it is
218 initialized. In particular, a session is identified by an unique identifier
219 for the client (its URL) and an unique identifier for the broker (its
220 URL).\ednote{CSC: OK, sto barando: \hbugs{} non \'e ancora cos\'i
221 multi-sessione. Ma mi sembra la strada che prenderemmo, no?}
223 The MONET architecture specification does not state explicitly whether
224 the service and broker answers can be asynchronous. Nevertheless, the
225 described information flow implicitly suggests a synchronous implementation.
226 On the contrary, in \hbugs{} every request is asynchronous: the connection
227 used by an actor to issue a query is immediately closed; when a service
228 produces an answer, it gives it back to the issuer by calling the
229 appropriate actor's method.
232 Tutors are software component able to consume proof status producing hints.
233 \hbugs{} doesn't specify by which means hints should be produced: tutors can
234 use any means necessary (heuristics, external theorem prover or CAS, ...).
235 The only requirement is that there exists an agreement on the formats of
236 proof status and hints.
238 Tutors act both as \ws{} providers and requesters for the broker. As
239 providers, they wait for commands requesting to start a new \musing{} on
240 a given proof status or to stop an old, out of date, \musing{}. As
241 requesters, they signal to the broker the end of a \musing{} along with its
242 outcome (an hint in case of success or a notification of failure).
244 \hbugs{} tutors act as MONET services.
246 \section{Implementation's Highlights}
247 \label{implementation}
248 In this section we present some of the most relevant implementation details of
249 the \hbugs{} architecture.
252 \paragraph{Proof status}
253 In our implementation of the \hbugs{} architecture we used the proof
254 assistant of the \helm{} project (codename ``gTopLevel'') as an \hbugs{}
255 client. Thus we have implemented serialization/deserialization capabilities
256 for its internal status. In order to be able to describe \wss{} that
257 exchange status in WSDL using the XML Schema type system, we have chosen an
258 XML format as the target format for the serialization.
260 A schematic representation of the gTopLevel internal status is depicted in
261 Fig. \ref{status}. Each proof is represented by a tuple of four elements:
262 \emph{uri}, \emph{metasenv}, \emph{proof}, \emph{thesis}.
264 \myincludegraphics{status}{t}{8cm}{gTopLevel proof status}{gTopLevel proof
268 \item[uri]: an URI chosen by the user at the beginning of the proof
269 process. Once (and if) proved, that URI will globally identify the term
270 inside the \helm{} library (given that the user decides to save it).
271 \item[thesis]: the thesis of the ongoing proof
272 \item[proof]: the current incomplete proof tree. It can contain
273 \emph{metavariables} (holes) that stands for the parts of the proof
274 that are still to be completed. Each metavariable appearing in the
275 tree references one element of the metavariables environment
277 \item[metasenv]: the metavariables environment is a list of
278 \emph{goals} (unproved conjectures).
279 In order to complete the proof, the user has to instantiate every
280 metavariable in the proof with a proof of the corresponding goal.
281 Each goal is identified by an unique identifier and has a context
282 and a type ( the goal thesis). The context is a list of named
283 hypotheses (declarations and definitions). Thus the context and the goal
284 thesis form a sequent, which is the statement of the proof that will
285 be used to instantiate the metavariable occurrences.
288 Each of these information is represented in XML as described in
289 \cite{mowglicic}. Additionally, an \hbugs{} status carry the unique
290 identifier of the current goal, which is the goal the user is currently
291 focused on. Using this value it is possible to implement different client
292 side strategies: the user could ask the tutors to work on the goal
293 she is considering or to work on the other ``background'' goals.
296 An hint in the \hbugs{} architecture should carry enough information to
297 permit the client to progress in the current proof. In our
298 implementation each hint corresponds to either one of the tactics available
299 to the user in gTopLevel (together with its actual arguments) or a set
300 of alternative suggestions (a list of hints).
302 For tactics that don't require any particular argument (like tactics that
303 apply type constructors or decision procedures)
304 only the tactic name is represented in the hint. For tactics that need
305 terms as arguments (for example the \texttt{Apply} tactic that apply a
306 given lemma) the hint includes a textual representation of them, using the
307 same representation used by the interactive proof assistant when querying
308 user for terms. In order to be transmitted between \wss{}, hints are
311 It is also possible for a tutor to return more hints at once,
312 grouping them in a particular XML element.
313 This feature turns out to be particularly useful for the
314 \emph{searchPatternApply} tutor (see Sect. \ref{tutors}) that
315 query a lemma database and return to the client a list of all lemmas that
316 could be used to complete the proof. This particular hint is encoded as a
317 list of \texttt{Apply} hints, each of them having one of the results as term
320 We would like to stress that the \hbugs{} architecture has no dependency
321 on either the hint or the status representation: the only message parts
322 that are fixed are those representing the administrative messages
323 (the envelops in the \wss{} terminology). In particular, the broker can
324 manage at the same time several sessions working on different status/hints
325 formats. Of course, there must be an agreement between the clients
326 and the tutors on the format of the data exchanged.
328 In our implementation the client does not trust the tutors hints:
329 being encoded as references to available tactics imply
330 that an \hbugs{} client, on receipt of an hint, simply try to reply the work
331 done by a tutor on the local copy of the proof. The application of the hint
332 can even fail to type check and the client copy of the proof can be left
333 undamaged after spotting the error. Note, however, that it is still
334 possible to implement a complex tutor that looks for a proof doing
336 send back to the client an hint whose argument is a witness (a trace) of
337 the proof found: the client applies the hint reconstructing (and checking
338 the correctness of) the proof from the witness, without having to
339 re-discover the proof itself.
341 An alternative implementation where the tutors are trusted would simply
342 send back to the client a new proof-status. Upon receiving the
343 proof-status, the client would just override its current proof status with
344 the suggested one. In the case of those clients which are implemented
345 using proof-objects (as the Coq proof-assistant, for instance), it is
346 still possible for the client to type-check the proof-object and reject
347 wrong hints. The systems that are not based on proof-objects
348 (as PVS, NuPRL, etc.), instead, have to trust the new proof-status. In this
349 case the \hbugs{} architecture needs at least to be extended with
350 clients-tutors authentication.
352 \paragraph{Registries}
353 Being central in the \hbugs{} architecture, the broker is also responsible
354 of housekeeping operations both for clients and tutors. These operations are
355 implemented using three different data structures called \emph{registries}:
356 clients registry, tutors registry and \musings{} registry.
358 In order to use the suggestion engine a client should register itself to the
359 broker and subscribe to one or more tutors. The registration phase is
360 triggered by the client using the \texttt{Register\_client} method of the
361 broker to send him an unique identifier and its base URI as a
362 \ws{}. After the registration, the client can use broker's
363 \texttt{List\_tutors} method to get a list of available tutors.
364 Eventually the client can subscribe to one or more of these using broker's
365 \texttt{Subscribe} method. Clients can also unregister from brokers using
366 \texttt{Unregister\_client} method.
368 The broker keeps track of both registered clients and clients' subscriptions
369 in the clients registry.
371 In order to be advertised to clients during the subscription phase, tutors
372 should register to the broker using the broker's \texttt{Register\_tutor}
373 method. This method is really similar to \texttt{Register\_client}:
374 tutors are required to send an unique identify and a base URI for their
376 Additionally tutors are required to send an human readable description of
377 their capabilities; this information could be used by client's user to
378 decide which tutors he needs to subscribe to. Like clients, tutors can
379 unregister from brokers using \texttt{Unregister\_broker} method.
381 Each time the client status change, the status is sent to the
382 broker using its \emph{Status} method. Using both clients registry (to
383 lookup client's subscription) and tutors registry (to check if some tutors
384 has unsubscribed), the broker is able to decide to which tutors the
385 new status must be forwarded.\ednote{CSC: qui o nei lavori futuri parlare
386 della possibilit\'a di avere un vero brocker che multiplexi le richieste
387 del tutor localizzando i servizi, etc.}
389 The forwarding operation is performed using the \texttt{Start\_musing}
390 method of the tutors, that is a request to start a new computation
391 (\emph{\musing{}}) on a given status. The return value of
392 \texttt{Start\_musing} is a
393 \musing{} identifier that is saved in the \musings{} registry along with
394 the identifier of the client that triggered the \musing{}.
396 As soon as a tutor completes an \musing{}, it informs the broker
397 using its \texttt{Musing\_completed} method; the broker can now remove the
398 \musing{} entry from the \musings{} registry and, depending on its outcome,
399 inform the client. In case of success one of the \texttt{Musing\_completed}
400 arguments is an hint to be sent to the client, otherwise there's no need to
401 inform him and the \texttt{Musing\_completed} method is called
402 just to update the \musings{} registry.
404 Consulting the \musings{} registry, the tutor\ednote{CSC: ma \'e vero o
405 stai delirando? Zack: e' vero, non ti fidi? :-) Up to delay di rete
406 ovviamente ... CSC: ma a che serve???} is able to know, at each time,
407 which \musings{} are in execution on which tutor. This peculiarity is
408 exploited by the broker on invocation of Status method. Receiving a new
409 status from the client implies indeed that the previous status no longer
410 exists and all \musings{} working on it should be stopped: additionally to
411 the already described behavior (i.e. starting new \musings{} on the
412 received status), the tutor\ednote{CSC: Ma sei veramente veramente sicuro?}
413 takes also care of stopping ongoing computation invoking
414 \texttt{Stop\_musing} tutors' method.
417 As already discussed, all \hbugs{} actors act as \wss{} offering one or more
418 services to neighbor actors. To grant as most accessibility as possible to
419 our \wss{} we have chosen to bind them using the HTTP/POST bindings
420 described in \cite{wsdlbindings}\footnote{Given that our proof assistant was
421 entirely developed in the Objective Caml language, we have chosen to
422 develop also \hbugs{} in that language in order to maximize code reuse. To
423 develop \wss{} in Objective Caml we have developed an auxiliary generic
424 library (\emph{O'HTTP}) that can be used to write HTTP 1.1 Web servers and
425 abstract over GET/POST parsing. This library supports different kinds of Web
426 servers architecture, including multi-process and multi-threaded ones.}.
429 Each tutor expose a \ws{} interface and should be able to work, not only for
430 many different clients referring to a common broker, but also for many
431 different brokers. The potential high number of concurrent clients imposes
432 a multi-threaded or multi-process architecture.
434 Our current implementation is based on a multi threaded architecture
435 exploiting the capabilities of the O'HTTP library. Each tutor is composed
436 by two thread always running plus
437 an additional thread for each running \musing{}. One thread is devoted to
438 listening for incoming \ws{} request; upon correct receiving requests it
439 pass the control to the second always-running thread which handle the pool
440 of running \musings{}. When a new \musing{} is requested, a new thread is
441 spawned to work them out; when a request to interrupt an old \musing{} is
442 received, the thread actually running them is killed freeing its
443 resources.\ednote{CSC: A cosa dobbiamo questa architettura delirante? Se non
444 ricordo male al problema dell'uccisione dei thread. Ora o si spiega
445 il motivo di questa architettura o si glissa/bluffa. Zack: cosa ti sembra
446 delirante? che i thread vengono uccisi? ... non mi e' molto chiaro ...
447 CSC: la motivazione per avere due thread always running e non due}
449 This architecture turns out to be scalable and allows the running threads
450 to share the cache of loaded (and type-checked) theorems.
451 As we will explain in Sect. \ref{tutors}, this feature turns out to be
452 really useful for tactics that rely on a huge but fixed set of lemmas,
453 as every reflexivite tactic.
455 The implementation of a tutor with the described architecture is not that
456 difficult having a language with good threading capabilities (as OCaml has)
457 and a pool of already implemented tactics (as gTopLevel has).
458 Still working with threads is known to be really error prone due to
459 concurrent programming intrinsic complexity. Moreover, there is a
460 non-neglectable part of code that needs to be duplicated in every tutor:
461 the code to register the tutor to the broker and to handle HTTP requests;
462 the code to manage the creation and termination of threads; and the code for
463 parsing the requests and serializing the answers. As a consequence we
464 have written a generic implementation of a tutor which is parameterized
465 over the code that actually propose the hint and some administrative
466 data (as the port the tutor will be listening to).
468 The generic tutor skeleton is really helpful in writing new tutors.
469 Nevertheless, the code obtained by converting existing tactics into tutors
470 is still quite repetitive: every tutor that wraps a tactic has to
471 instantiate its own copy of the proof-engine kernel and, for each request,
472 it has to override its status, guess the tactic arguments, apply the tactic
473 and, in case of success, send back an hint with the tactic name and the
474 chosen arguments. Of course, the complex part of the work is guessing the
475 right arguments. For the simple case of tactics that do not require
476 any argument, though, we are able to automatically generate the whole
477 tutor code given the tactic name. Concretely, we have written a
478 tactic-based tutor template and a script that parses an XML file with
479 the specification of the tutor and generates the tutor's code.
480 The XML file describes the tutor's port, the code to invoke the tactic,
481 the hint that is sent back upon successful application and a
482 human readable explanation of the tactic implemented by the tutor.
484 \section{The Implemented \hbugs Tutors}
486 To test the \hbugs{} architecture and to assess the utility of a suggestion
487 engine for the end user, we have implemented several tutors. In particular,
488 we have investigated three classes of tutors:
490 \item \emph{Tutors for beginners}. These are tutors that implement tactics
491 which are neither computationally expensive nor difficult to understand:
492 an expert user can always understand if the tactic can be applied or not
493 without having to try it. For example, the following implemented tutors
494 belong to this class:
496 \item \emph{Assumption Tutor}: it ends the proof if the thesis is
497 equivalent\footnote{In our implementation, the equivalence relation
498 imposed by the logical framework is \emph{convertibility}. Two
499 expressions are convertible when they reduce to the same normal form.
500 Two ``equal'' terms depending on free variables can be non-convertible
501 since free variables stop the reduction. For example, $2x$ is convertible
502 with $(3-1)x$ because they both reduce to the same normal form
503 $x + x + 0$; but $2x$ is not convertible to $x2$ since the latter is
504 already in normal form.}
505 to one of the hypotheses\footnote{
506 In some cases, especially when non-trivial computations are involved,
507 the user is totally unable to figure out the convertibility of two terms.
508 In these cases the tutor becomes handy also for expert users.}.
509 \item \emph{Contradiction Tutor}: it ends the proof by \emph{reductio ad
510 adsurdum} if one hypothesis is equivalent to $False$.
511 \item \emph{Symmetry Tutor}: if the goal thesis is an equality, it
512 suggests to apply the commutative property.
513 \item \emph{Left/Right/Exists/Split/Reflexivity/Constructor Tutors}:
514 the Constructor Tutor suggests to proceed in the proof by applying one
515 or more constructors when the goal thesis is an inductive type or a
516 proposition inductively defined according to the declarative
517 style\footnote{An example of a proposition that can be given in
518 declarative style is the $\le$ relation: $\le$ is the smallest relation
519 such that $n \le n$ for every $n$ and $n \le m$ for every $n,m$ such
520 that $n \le p$ where $p$ is the predecessor of $m$. Thus, a proof
521 of $n \le n$ is simply the application of the first constructor to
522 $n$ and a proof of $n \le m$ is the application of the second
523 constructor to $n,m$ and a proof of $n \le m$.}.
524 Since disjunction, conjunction, existential quantification and
525 Leibniz equality are particular cases of inductive propositions,
526 all the other tutors of this class are instantiations of the
527 the Constructor tactic. Left and Right suggest to prove a disjunction
528 by proving its left/right member; Split reduces the proof of a
529 conjunction to the two proof of its members; Exists suggests to
530 prove an existential quantification by providing a
531 witness\footnote{This task is left to the user.}; Reflexivity proves
532 an equality whenever the two sides are convertible.
534 Beginners, when first faced with a tactic-based proof-assistant, get
535 lost quite soon since the set of tactics is large and their names and
536 semantics must be remembered by heart. Tutorials are provided to guide
537 the user step-by-step in a few proofs, suggesting the tactics that must
538 be used. We believe that our beginners tutors can provide an auxiliary
539 learning tool: after the tutorial, the user is not suddenly left alone
540 with the system, but she can experiment with variations of the proof given
541 in the tutorial as much as she like, still getting useful suggestions.
542 Thus the user is allowed to focus on learning how to do a formal proof
543 instead of wasting efforts trying to remember the interface to the system.
544 \item{Tutors for Computationally Expensive Tactics}. Several tactics have
545 an unpredictable behavior, in the sense that it is unfeasible to understand
546 whether they will succeed or they will fail when applied and what will be
547 their result. Among them, there are several tactics either computationally
548 expensive or resources consuming. In the first case, the user is not
549 willing to try a tactic and wait for a long time just to understand its
550 outcome: she would prefer to keep on concentrating on the proof and
551 have the tactic applied in background and receive out-of-band notification
552 of its success. The second case is similar, but the tactic application must
553 be performed on a remote machine to avoid overloading the user host
554 with several concurrent resource consuming applications.
556 Finally, several complex tactics and in particular all the tactics based
557 on reflexive techniques depend on a pretty large set of definitions, lemmas
558 and theorems. When these tactics are applied, the system needs to retrieve
559 and load all the lemmas. Pre-loading all the material needed by every
560 tactic can quickly lead to long initialization times and to large memory
561 footstamps. A specialized tutor running on a remote machine, instead,
562 can easily pre-load the required theorems.
564 As an example of computationally expensive task, we have implemented
565 a tutor for the \emph{Ring} tactic \cite{ringboutin}.
566 The tutor is able to prove an equality over a ring by reducing both members
567 to a common normal form. The reduction, which may require some time in
569 is based on the usual commutative, associative and neutral element properties
570 of a ring. The tactic is implemented using a reflexive technique, which
571 means that the reduction trace is not stored in the proof-object itself:
572 the type-checker is able to perform the reduction on-the-fly thanks to
573 the conversion rules of the system. As a consequence, in the library there
574 must be stored both the algorithm used for the reduction and the proof of
575 correctness of the algorithm, based on the ring axioms. This big proof
576 and all of its lemmas must be retrieved and loaded in order to apply the
577 tactic. The Ring tutor loads and cache all the required theorems the
578 first time it is contacted.
579 \item{Intelligent Tutors}. Expert users can already benefit from the previous
580 class of tutors. Nevertheless, to achieve a significative production gain,
581 they need more intelligent tutors implementing domain-specific theorem
582 provers or able to perform complex computations. These tutors are not just
583 plain implementations of tactics or decision procedures, but can be
584 more complex software agents interacting with third-parties software,
585 such as proof-planners, CAS or theorem-provers.
587 To test the productivity impact of intelligent tutors, we have implemented
588 a tutor that is interfaced with the \helm{}
589 Search-Engine\footnote{\url{http://mowgli.cs.unibo.it/library.html}} and that
590 is able to look for every theorem in the distributed library that can
591 be applied to proceed in the proof. Even if the tutor deductive power
592 is extremely limited\footnote{We do not attempt to check if the new goals
593 obtained applying a lemma can be automatically proved or, even better,
594 automatically disproved to reject the lemma.}, it is not unusual for
595 the tutor to come up with precious hints that can save several minutes of
596 work that would be spent in proving again already proven results or
597 figuring out where the lemmas could have been stored in the library.
600 \section{Conclusions and Future Work}
602 In this paper we described a suggestion engine architecture for
603 proof-assistants: the client (a proof-assistant) sends the current proof
604 status to several distributed \wss{} (called tutors) that try to progress
605 in the proof and, in case of success, send back an appropriate hint
606 (a proof-plan) to the user. The user, that in the meantime was able to
607 reason and progress in the proof, is notified with the hints and can decide
608 to apply or ignore them. A broker is provided to decouple the clients and
609 the tutors and to allow the client to locate and invoke the available remote
610 services. The whole architecture is an instance of the MONET architecture
611 for Mathematical \wss{}.
613 A running prototype has been implemented as part of the
614 \helm{} project \cite{helm}
615 and we already provide several tutors. Some of them are simple tutors that
616 try to apply one or more tactics of the \helm{} Proof-Engine, which is also
617 our client. We also have a much more complex tutor that is interfaced
618 with the \helm{} Search-Engine and looks for lemmas that can be directly
621 We have many plan for further developing both the \hbugs{} architecture and
622 our prototype. Interesting results could be obtained
623 augmenting the informative content of each suggestion. We can for example
624 modify the broker so that also negative results are sent back to the client.
625 Those negative suggestions could be reflected in the user interface by
626 deactivating commands to narrow the choice of tactics available to the user.
627 This approach could be interesting especially for novice users, but require
628 the client to trust other actors a bit more than in the current approach.
630 We plan also to add some rating mechanism to the architecture. A first
631 improvement in this direction could be to distinguish between hints that, when
632 applied, are able to completely close one or more goals and
633 tactics that progress in the proof by reducing one or more goals to new goals:
634 the new goals could be false and the proof can be closed only by backtracking.
636 Other heuristics and/or measures could be added to rate
637 hints and show them to the user in a particular order: an interesting one
638 could be a measure that try to minimize the size of the generated proof,
639 privileging therefore non-overkilling solutions \cite{ring}.
641 We are also considering to follow the \OmegaAnts{} path more closely adding
642 ``recursion'' to the system so that proof status resulting from the
643 application of old hints are cached somewhere and could be used as a starting
644 point for new hint searches. The approach is interesting, but it represents
645 a big shift towards automatic theorem proving: thus we must consider if it is
646 worth the effort given the increasing availability of automation in proof
647 assistants' tactics and the ongoing development of \wss{} based on
648 already existent and well developed theorem provers.
650 Even if not strictly part of the \hbugs{} architecture, the graphical user
651 interface (GUI) of our prototype needs a lot of improvement if we would like
652 it to be really usable by novices. In particular, the user is too easily
653 distracted by the tutor's hints that are ``pushed'' to her.
655 Our \wss{} still lack a real integration in the MONET architecture,
656 since we do not provide the different ontologies to describe our problems,
657 solutions, queries and services. In the short term, completing this task
658 could provide a significative feedback to the MONET consortium and would
659 enlarge the current set of available MONET actors on the Web. In the long
660 term, new more intelligent tutors could be developed on top of already
661 existent MONET \wss{}.
663 To conclude, \hbugs{} is a nice experiment meant to understand whether the
664 current \wss{} technology is mature enough to have a concrete and useful
665 impact on the daily work of users of proof-assistants. So far, only the tutor
666 that is interfaced with the \helm{} Search-Engine has effectively increased
667 the productivity of experts users. The usefulness of the tutors developed for
668 beginners, instead, need further assessment.
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