1 \documentclass[runningheads]{llncs}
7 % \myincludegraphics{filename}{place}{width}{caption}{label}
8 \newcommand{\myincludegraphics}[5]{
11 \includegraphics[width=#3]{eps/#1.eps}
18 %\usepackage[show]{ed}
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
55 interactive proof-assistants. When the broker is submitted a "proof status" (an
56 incomplete 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 to provide a further level of
79 stable services (called \emph{brokers}) that behave as common gateways/addresses
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 clients 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 final section of this paper is devoted to 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 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).
170 \hbugs{} clients act as MONET clients and ask brokers to provide access to a
171 set of services (the tutors). \hbugs{} has no actors corresponding to
172 MONET's Broker Locating Service (since the client is supposed to know the
173 URI of at least one broker). The \hbugs{} client and tutors contact the
174 Getter (a MONET Mathematical Object Manager) to locate and retrieve
175 mathematical items in the \helm{} library.
176 The proof status that are exchanged
177 by the \hbugs{} actors, instead, are built on the fly and are neither
178 stored nor given an unique identifier (URI) to be managed by the
182 \myincludegraphics{interfaces}{t!}{10cm}{\hbugs{} \wss{} interfaces}
183 {\hbugs{} \wss{} interfaces}
185 Brokers are the key actors of the \hbugs{} architecture since they
186 act as intermediaries between clients and tutors. They behave as \wss{}
187 providers and requesters for \emph{both} clients and tutors, see Fig.
190 With respect to the client, a broker acts as a \ws{} provider, receiving the
191 proof status and forwarding it to one or more tutors.
192 It also acts as a \ws{} requester sending
193 hints to the client as soon as they are available from the tutors.
195 With respect to the tutors, the \ws{} provider role is accomplished by
196 receiving hints as soon as they are produced; as a requester, it is
197 accomplished by asking for computations (\emph{musings} in \hbugs{}
198 terminology) on status received by clients and by stopping already late but
199 still ongoing \musings{}.
201 Additionally brokers keep track of available tutors and clients
204 \hbugs{} brokers act as MONET brokers implementing the following components:
205 Client Manager, Service Registry Manager (keeping track of available
206 tutors), Planning Manager (choosing the available tutors among the ones to
207 which the client is subscribed), Execution Manager. The Service Manager
208 component is not required since the session handler, that identifies
209 a session between a service and a broker, is provided to the service by
210 the broker instead of being received from the service when the session is
211 initialized. In particular, a session is identified by an unique identifier
212 for the client (its URL) and an unique identifier for the broker (its
215 The MONET architecture specification does not state explicitly whether
216 the service and broker answers can be asynchronous. Nevertheless, the
217 described information flow implicitly suggests a synchronous implementation.
218 On the contrary, in \hbugs{} every request is asynchronous: the connection
219 used by an actor to issue a query is immediately closed; when a service
220 produces an answer, it gives it back to the issuer by calling the
221 appropriate actor's method.
224 Tutors are software component able to consume proof status producing hints.
225 \hbugs{} does not specify by which means hints should be produced: tutors
226 can use any means necessary (heuristics, external theorem prover or CAS,
227 etc.). The only requirement is that there exists an agreement on the
228 formats of proof status and hints.
230 Tutors act both as \ws{} providers and requesters for the broker, see Fig.
232 providers, they wait for commands requesting to start a new \musing{} on
233 a given proof status or to stop an old, out of date, \musing{}. As
234 requesters, they signal to the broker the end of a \musing{} along with its
235 outcome (an hint in case of success or a failure notification).
237 \hbugs{} tutors act as MONET services.
239 \section{An \hbugs{} Session Example}
240 In this section we describe a typical \hbugs{} session. The aim of the
241 session is to solve the following easy exercise:
243 Let $x$ be a generic real numbers. Using the \helm{} proof-engine,
246 x = \frac{(x+1)*(x+1) - 1 - x*x}{2}
250 \myincludegraphics{step1}{t}{12cm}{Example session.}
252 %\myincludegraphics{step2}{t}{4cm}{Example session, snapshot 2.}
253 % {Example session, snapshot 2.}
255 Let us suppose that the \hbugs{} broker is already running and that several
256 tutors already registered themselves to the broker.
257 When the user starts \texttt{gTopLevel}, the system registers itself to
258 the broker, that sends back the list of available tutors. By default,
259 \texttt{gTopLevel} notifies the broker its intention of subscribing to every
260 tutor available. The user can always open a configuration window where she
261 is presented the list of available tutors and she can independently subscribe
262 and unsubscribe each tutor.
264 The user can now insert into the system the statement of the theorem and start
265 proving it. Let us suppose that the first step of the user is proving
266 that the denominator 2 is different from 0. Once that this technical result
267 is proven, the user must prove the goal shown in the upper left corner
268 of the window in background in Fig. \ref{step1}.
270 While the user is wondering how to proceed in the proof, the tutors are
271 trying to progress in the proof. After a while, the tutors' suggestions
272 start to appear in the lower part of the \hbugs{} interface window
273 (the topmost window in Fig. \ref{step1}). In this case, the tutors are able
274 to produce 23 hints. The first and not very useful hint suggests to proceed in
275 the proof by exchanging the two sides of the equality.
276 The second hint suggests to reduce both sides of the equality to their normal
277 form by using only reductions which are justified by the ring structure of the
278 real numbers. The two normal forms, though, are so different that the proof is
279 not really simplified.
280 All the residual 21 hints suggest to apply one lemma from the distributed
283 The user can look at the list of suggestions and realize that a good one is
284 that of applying the lemma \texttt{r\_Rmult\_mult} which
285 allow\footnote{The user can always look at
286 the statement of a theorem by clicking on its URI.} to multiply both equality
287 members by the same scalar\footnote{Even if she does not receive the hint, the
288 user probably already knows that this is the right way to proceed. The
289 difficult part where the hint helps is guessing what is the name of the lemma
291 Double-clicking on the hint automatically applies
292 the lemma, reducing the proof to closing three new goals. The first one asks
293 the user the scalar to use as an argument of the previous lemma; the second
294 one states that the scalar is different from 0; the third lemma (the main
295 one) asks to prove the equality between the products of the two old members
297 % is shown in Fig. \ref{step2} where $?_3[H;x]$ stands for
298 % the still unkown scalar argument, which can have only $H$ and $x$ as
301 The user proceeds by istantiating the scalar with the number 2. The
302 \texttt{Assumption} tutor now suggests to close the second goal by applying
303 the hypothesis $H$. No useful suggestions, instead, are generated for the
304 main goal $2x = 2*\frac{((x+1)*(x+1)-1-x*x)}{2}$.
305 To proceed in the proof, indeed, the user needs to symplify the
306 expression using the lemma $Rinv\_r\_simpl\_m$ that states that
307 $\forall x,y.\;y = x * y * x^{-1}$. Since we do not provide yet any tutor
308 suggesting symplifications, the user must find out this symplication by
309 himself. Once she founds it, the goal is reduced to proving that
310 $2x = (x+1)*(x+1) - 1 - x*x$. This equality is easily solved by the
311 \texttt{Ring} tutor, that suggests\footnote{The \texttt{Ring} suggestion is
312 just one of the 22 hints that the user receives. It is the only hint that
313 does not open new goals.} to the user how to directly finish the proof.
315 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
316 % Comandi da dare a gTopLevel %
317 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
319 % !x.(not (eqT ? (Rplus R1 R1) R0)) -> (eqT ? x (Rdiv (Rminus (Rminus (Rmult (Rplus x R1) (Rplus x R1)) R1) (Rmult x x)) (Rplus R1 R1)))
323 % Simpl (per fare unfold di Rdiv)
325 % (Rmult_assoc (Rplus R1 R1) (Rplus (Rplus (Rmult (Rplus x R1) (Rplus x R1)) (Ropp R1)) (Ropp (Rmult x x))) (Rinv (Rplus R1 R1)))
327 % (Rinv_r_simpl_m (Rplus R1 R1) (Rplus (Rplus (Rmult (Rplus x R1) (Rplus x R1)) (Ropp R1)) (Ropp (Rmult x x))) H)
329 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
331 \section{Implementation's Highlights}
332 \label{implementation}
333 In this section we present some of the most relevant implementation details of
334 the \hbugs{} architecture.
337 \paragraph{Proof status}
338 In our implementation of the \hbugs{} architecture we used the proof
339 assistant of the \helm{} project (codename ``gTopLevel'') as an \hbugs{}
340 client. Thus we have implemented serialization/deserialization capabilities
341 for its internal status. In order to be able to describe \wss{} that
342 exchange status in WSDL using the XML Schema type system, we have chosen an
343 XML format as the target format for the serialization.
345 % A schematic representation of the gTopLevel internal status is depicted in
347 Each proof is represented by a tuple of four elements:
348 \emph{uri}, \emph{metasenv}, \emph{proof}, \emph{thesis}.
350 % \myincludegraphics{status}{t}{8cm}{gTopLevel proof status}{gTopLevel proof
354 \item[uri]: an URI chosen by the user at the beginning of the proof
355 process. Once (and if) proved, that URI will globally identify the term
356 inside the \helm{} library (given that the user decides to save it).
357 \item[thesis]: the thesis of the ongoing proof
358 \item[proof]: the current incomplete proof tree. It can contain
359 \emph{metavariables} (holes) that stands for the parts of the proof
360 that are still to be completed. Each metavariable appearing in the
361 tree references one element of the metavariables environment
363 \item[metasenv]: the metavariables environment is a list of
364 \emph{goals} (unproved conjectures).
365 In order to complete the proof, the user has to instantiate every
366 metavariable in the proof with a proof of the corresponding goal.
367 Each goal is identified by an unique identifier and has a context
368 and a type (the goal thesis). The context is a list of named
369 hypotheses (declarations and definitions). Thus the context and the goal
370 thesis form a sequent, which is the statement of the proof that will
371 be used to instantiate the metavariable occurrences.
374 Each of these information is represented in XML as described in
375 \cite{mowglicic}. Additionally, an \hbugs{} status carry the unique
376 identifier of the current goal, which is the goal the user is currently
377 focused on. Using this value it is possible to implement different client
378 side strategies: the user could ask the tutors to work on the goal
379 she is considering or to work on the other ``background'' goals.
382 An hint in the \hbugs{} architecture should carry enough information to
383 permit the client to progress in the current proof. In our
384 implementation each hint corresponds to either one of the tactics available
385 to the user in gTopLevel (together with its actual arguments) or a set
386 of alternative suggestions (a list of hints).
388 For tactics that don't require any particular argument (like tactics that
389 apply type constructors or decision procedures)
390 only the tactic name is represented in the hint. For tactics that need
391 terms as arguments (for example the \texttt{Apply} tactic that apply a
392 given lemma) the hint includes a textual representation of them, using the
393 same representation used by the interactive proof assistant when querying
394 user for terms. In order to be transmitted between \wss{}, hints are
397 It is also possible for a tutor to return more hints at once,
398 grouping them in a particular XML element.
399 This feature turns out to be particularly useful for the
400 \emph{searchPatternApply} tutor (see Sect. \ref{tutors}) that
401 query a lemma database and return to the client a list of all lemmas that
402 could be used to complete the proof. This particular hint is encoded as a
403 list of \texttt{Apply} hints, each of them having one of the results as term
406 We would like to stress that the \hbugs{} architecture has no dependency
407 on either the hint or the status representation: the only message parts
408 that are fixed are those representing the administrative messages
409 (the envelops in the \wss{} terminology). In particular, the broker can
410 manage at the same time several sessions working on different status/hints
411 formats. Of course, there must be an agreement between the clients
412 and the tutors on the format of the data exchanged.
414 In our implementation the client does not trust the tutors hints:
415 being encoded as references to available tactics imply
416 that an \hbugs{} client, on receipt of an hint, simply try to reply the work
417 done by a tutor on the local copy of the proof. The application of the hint
418 can even fail to type check and the client copy of the proof can be left
419 undamaged after spotting the error. Note, however, that it is still
420 possible to implement a complex tutor that looks for a proof doing
422 send back to the client an hint whose argument is a witness (a trace) of
423 the proof found: the client applies the hint reconstructing (and checking
424 the correctness of) the proof from the witness, without having to
425 re-discover the proof itself.
427 An alternative implementation where the tutors are trusted would simply
428 send back to the client a new proof-status. Upon receiving the
429 proof-status, the client would just override its current proof status with
430 the suggested one. In the case of those clients which are implemented
431 using proof-objects (as the Coq proof-assistant, for instance), it is
432 still possible for the client to type-check the proof-object and reject
433 wrong hints. The systems that are not based on proof-objects
434 (as PVS, NuPRL, etc.), instead, have to trust the new proof-status. In this
435 case the \hbugs{} architecture needs at least to be extended with
436 clients-tutors authentication.
438 \paragraph{Registries}
439 Being central in the \hbugs{} architecture, the broker is also responsible
440 of housekeeping operations both for clients and tutors. These operations are
441 implemented using three different data structures called \emph{registries}:
442 clients registry, tutors registry and \musings{} registry.
444 In order to use the suggestion engine a client should register itself to the
445 broker and subscribe to one or more tutors. The registration phase is
446 triggered by the client using the \texttt{Register\_client} method of the
447 broker to send him an unique identifier and its base URI as a
448 \ws{}. After the registration, the client can use broker's
449 \texttt{List\_tutors} method to get a list of available tutors.
450 Eventually the client can subscribe to one or more of these using broker's
451 \texttt{Subscribe} method. Clients can also unregister from brokers using
452 \texttt{Unregister\_client} method.
454 The broker keeps track of both registered clients and clients' subscriptions
455 in the clients registry.
457 In order to be advertised to clients during the subscription phase, tutors
458 should register to the broker using the broker's \texttt{Register\_tutor}
459 method. This method is really similar to \texttt{Register\_client}:
460 tutors are required to send an unique identify and a base URI for their
462 Additionally tutors are required to send an human readable description of
463 their capabilities; this information could be used by client's user to
464 decide which tutors he needs to subscribe to. As the clients, tutors can
465 unregister from brokers using \texttt{Unregister\_broker} method.
467 Each time the client status change, it get sent sent to the
468 broker using its \texttt{Status} method. Using both clients registry (to
469 lookup client's subscription) and tutors registry (to check if some tutors
470 has unsubscribed), the broker is able to decide to which tutors the
471 new status have to be forwarded.
472 % \ednote{CSC: qui o nei lavori futuri parlare
473 % della possibilit\'a di avere un vero brocker che multiplexi le richieste
474 % dei client localizzando i servizi, etc.}
476 The forwarding operation is performed using the \texttt{Start\_musing}
477 method of the tutors, that is a request to start a new computation
478 (\emph{\musing{}}) on a given status. The return value of
479 \texttt{Start\_musing} is a
480 \musing{} identifier that is saved in the \musings{} registry along with
481 the identifier of the client that triggered the \musing{}.
483 As soon as a tutor completes an \musing{}, it informs the broker
484 using its \texttt{Musing\_completed} method; the broker can now remove the
485 \musing{} entry from the \musings{} registry and, depending on its outcome,
486 inform the client. In case of success one of the \texttt{Musing\_completed}
487 arguments is an hint to be sent to the client, otherwise there's no need to
488 inform him and the \texttt{Musing\_completed} method is called
489 just to update the \musings{} registry.
491 Consulting the \musings{} registry, the broker is able to know, at each
492 time, which \musings{} are in execution on which tutor. This peculiarity is
493 exploited by the broker on invocation of the \texttt{Status} method.
494 Receiving a new status from the client implies indeed that the previous
495 status no longer exists and all \musings{} working on it should be stopped:
496 additionally to the already described behavior (i.e. starting new
497 \musings{} on the received status), the broker takes also care of stopping
498 ongoing computation invoking the \texttt{Stop\_musing} method of the tutors.
501 As already discussed, all \hbugs{} actors act as \wss{} offering one or more
502 services to neighbor actors. To grant as most accessibility as possible to
503 our \wss{} we have chosen to bind them using the HTTP/POST\footnote{Given
504 that our proof assistant was entirely developed in the Objective Caml
505 language, we have chosen to develop also \hbugs{} in that language in order
506 to maximize code reuse. To develop \wss{} in Objective Caml we have
507 developed an auxiliary generic library (\emph{O'HTTP}) that can be used to
508 write HTTP 1.1 Web servers and abstract over GET/POST parsing. This library
509 supports different kinds of Web servers architecture, including
510 multi-process and multi-threaded ones.} bindings described in
514 Each tutor expose a \ws{} interface and should be able to work, not only for
515 many different clients referring to a common broker, but also for many
516 different brokers. The potential high number of concurrent clients imposes
517 a multi-threaded or multi-process architecture.
519 Our current implementation is based on a multi threaded architecture
520 exploiting the capabilities of the O'HTTP library. Each tutor is composed
521 by one thread always running plus
522 an additional thread for each running \musing{}. One thread is devoted to
523 listening for incoming \ws{} request; upon correct receiving requests it
524 pass the control to a second thread, created on the fly, to handle the
525 incoming request following the classical one-thread-per-request approach in
527 If the received request is \texttt{Start\_musing}, a new thread is
528 spawned to handle it; the thread in duty to handle the HTTP request
529 returns an HTTP response containing the identifier of the just started
530 \texttt{musing}, and then dyes. If the received request is
531 \texttt{Stop\_musing}, instead, the spawned thread kills the thread
532 responsible for the \texttt{musing} whose identifier is the argument
533 of the \texttt{Stop\_musing} method.
535 This architecture turns out to be scalable and allows the running threads
536 to share the cache of loaded (and type-checked) theorems.
537 As we will explain in Sect. \ref{tutors}, this feature turns out to be
538 really useful for tactics that rely on a huge but fixed set of lemmas,
539 as every reflexive tactic.
541 The implementation of a tutor with the described architecture is not that
542 difficult having a language with good threading capabilities (as OCaml has)
543 and a pool of already implemented tactics (as gTopLevel has).
544 Still working with threads is known to be really error prone due to
545 concurrent programming intrinsic complexity. Moreover, there is a
546 non-neglectable part of code that needs to be duplicated in every tutor:
547 the code to register the tutor to the broker and to handle HTTP requests;
548 the code to manage the creation and termination of threads; and the code for
549 parsing the requests and serializing the answers. As a consequence we
550 have written a generic implementation of a tutor which is parameterized
551 over the code that actually propose the hint and some administrative
552 data (as the port the tutor will be listening to).
554 The generic tutor skeleton is really helpful in writing new tutors.
555 Nevertheless, the code obtained by converting existing tactics into tutors
556 is still quite repetitive: every tutor that wraps a tactic has to
557 instantiate its own copy of the proof-engine kernel and, for each request,
558 it has to override its status, guess the tactic arguments, apply the tactic
559 and, in case of success, send back an hint with the tactic name and the
560 chosen arguments. Of course, the complex part of the work is guessing the
561 right arguments. For the simple case of tactics that do not require
562 any argument, though, we are able to automatically generate the whole
563 tutor code given the tactic name. Concretely, we have written a
564 tactic-based tutor template and a script that parses an XML file with
565 the specification of the tutor and generates the tutor's code.
566 The XML file describes the tutor's port, the code to invoke the tactic,
567 the hint that is sent back upon successful application and a
568 human readable explanation of the tactic implemented by the tutor.
570 \section{The Implemented \hbugs Tutors}
572 To test the \hbugs{} architecture and to assess the utility of a suggestion
573 engine for the end user, we have implemented several tutors. In particular,
574 we have investigated three classes of tutors:
576 \item \emph{Tutors for beginners}. These are tutors that implement tactics
577 which are neither computationally expensive nor difficult to understand:
578 an expert user can always understand if the tactic can be applied or not
579 without having to try it. For example, the following implemented tutors
580 belong to this class:
582 \item \emph{Assumption Tutor}: it ends the proof if the thesis is
583 equivalent\footnote{In our implementation, the equivalence relation
584 imposed by the logical framework is \emph{convertibility}. Two
585 expressions are convertible when they reduce to the same normal form.
586 Two ``equal'' terms depending on free variables can be non-convertible
587 since free variables stop the reduction. For example, $2x$ is convertible
588 with $(3-1)x$ because they both reduce to the same normal form
589 $x + x + 0$; but $2x$ is not convertible to $x2$ since the latter is
590 already in normal form.}
591 to one of the hypotheses\footnote{
592 In some cases, especially when non-trivial computations are involved,
593 the user is totally unable to figure out the convertibility of two terms.
594 In these cases the tutor becomes handy also for expert users.}.
595 \item \emph{Contradiction Tutor}: it ends the proof by \emph{reductio ad
596 adsurdum} if one hypothesis is equivalent to $False$.
597 \item \emph{Symmetry Tutor}: if the goal thesis is an equality, it
598 suggests to apply the commutative property.
599 \item \emph{Left/Right/Exists/Split/Reflexivity/Constructor Tutors}:
600 the Constructor Tutor suggests to proceed in the proof by applying one
601 or more constructors when the goal thesis is an inductive type or a
602 proposition inductively defined according to the declarative
603 style\footnote{An example of a proposition that can be given in
604 declarative style is the $\le$ relation over natural numbers:
605 $\le$ is the smallest relation
606 such that $n \le n$ for every $n$ and $n \le m$ for every $n,m$ such
607 that $n \le p$ where $p$ is the predecessor of $m$. Thus, a proof
608 of $n \le n$ is simply the application of the first constructor to
609 $n$ and a proof of $n \le m$ is the application of the second
610 constructor to $n,m$ and a proof of $n \le m$.}.
611 Since disjunction, conjunction, existential quantification and
612 Leibniz equality are particular cases of inductive propositions,
613 all the other tutors of this class are instantiations of the
614 the Constructor tactic. Left and Right suggest to prove a disjunction
615 by proving its left/right member; Split reduces the proof of a
616 conjunction to the two proof of its members; Exists suggests to
617 prove an existential quantification by providing a
618 witness\footnote{This task is left to the user.}; Reflexivity proves
619 an equality whenever the two sides are convertible.
621 Beginners, when first faced with a tactic-based proof-assistant, get
622 lost quite soon since the set of tactics is large and their names and
623 semantics must be remembered by heart. Tutorials are provided to guide
624 the user step-by-step in a few proofs, suggesting the tactics that must
625 be used. We believe that our beginners tutors can provide an auxiliary
626 learning tool: after the tutorial, the user is not suddenly left alone
627 with the system, but she can experiment with variations of the proof given
628 in the tutorial as much as she like, still getting useful suggestions.
629 Thus the user is allowed to focus on learning how to do a formal proof
630 instead of wasting efforts trying to remember the interface to the system.
631 \item{Tutors for Computationally Expensive Tactics}. Several tactics have
632 an unpredictable behavior, in the sense that it is unfeasible to understand
633 whether they will succeed or they will fail when applied and what will be
634 their result. Among them, there are several tactics either computationally
635 expensive or resources consuming. In the first case, the user is not
636 willing to try a tactic and wait for a long time just to understand its
637 outcome: she would prefer to keep on concentrating on the proof and
638 have the tactic applied in background and receive out-of-band notification
639 of its success. The second case is similar, but the tactic application must
640 be performed on a remote machine to avoid overloading the user host
641 with several concurrent resource consuming applications.
643 Finally, several complex tactics and in particular all the tactics based
644 on reflexive techniques depend on a pretty large set of definitions, lemmas
645 and theorems. When these tactics are applied, the system needs to retrieve
646 and load all the lemmas. Pre-loading all the material needed by every
647 tactic can quickly lead to long initialization times and to large memory
648 footstamps. A specialized tutor running on a remote machine, instead,
649 can easily pre-load the required theorems.
651 As an example of computationally expensive task, we have implemented
652 a tutor for the \emph{Ring} tactic \cite{ringboutin}.
653 The tutor is able to prove an equality over a ring by reducing both members
654 to a common normal form. The reduction, which may require some time in
656 is based on the usual commutative, associative and neutral element properties
657 of a ring. The tactic is implemented using a reflexive technique, which
658 means that the reduction trace is not stored in the proof-object itself:
659 the type-checker is able to perform the reduction on-the-fly thanks to
660 the conversion rules of the system. As a consequence, in the library there
661 must be stored both the algorithm used for the reduction and the proof of
662 correctness of the algorithm, based on the ring axioms. This big proof
663 and all of its lemmas must be retrieved and loaded in order to apply the
664 tactic. The Ring tutor loads and cache all the required theorems the
665 first time it is contacted.
666 \item{Intelligent Tutors}. Expert users can already benefit from the previous
667 class of tutors. Nevertheless, to achieve a significative production gain,
668 they need more intelligent tutors implementing domain-specific theorem
669 provers or able to perform complex computations. These tutors are not just
670 plain implementations of tactics or decision procedures, but can be
671 more complex software agents interacting with third-parties software,
672 such as proof-planners, CAS or theorem-provers.
674 To test the productivity impact of intelligent tutors, we have implemented
675 a tutor that is interfaced with the \helm{}
676 Search-Engine\footnote{\url{http://helm.cs.unibo.it/library.html}} and that
677 is able to look for every theorem in the distributed library that can
678 be applied to proceed in the proof. Even if the tutor deductive power
679 is extremely limited\footnote{We do not attempt to check if the new goals
680 obtained applying a lemma can be automatically proved or, even better,
681 automatically disproved to reject the lemma.}, it is not unusual for
682 the tutor to come up with precious hints that can save several minutes of
683 work that would be spent in proving again already proven results or
684 figuring out where the lemmas could have been stored in the library.
687 \section{Conclusions and Future Work}
689 In this paper we described a suggestion engine architecture for
690 proof-assistants: the client (a proof-assistant) sends the current proof
691 status to several distributed \wss{} (called tutors) that try to progress
692 in the proof and, in case of success, send back an appropriate hint
693 (a proof-plan) to the user. The user, that in the meantime was able to
694 reason and progress in the proof, is notified with the hints and can decide
695 to apply or ignore them. A broker is provided to decouple the clients and
696 the tutors and to allow the client to locate and invoke the available remote
697 services. The whole architecture is an instance of the MONET architecture
698 for Mathematical \wss{}.
700 A running prototype has been implemented as part of the
701 \helm{} project \cite{helm}
702 and we already provide several tutors. Some of them are simple tutors that
703 try to apply one or more tactics of the \helm{} Proof-Engine, which is also
704 our client. We also have a much more complex tutor that is interfaced
705 with the \helm{} Search-Engine and looks for lemmas that can be directly
708 We have many plans for further developing both the \hbugs{} architecture and
709 our prototype. Interesting results could be obtained
710 augmenting the informative content of each suggestion. We can for example
711 modify the broker so that also negative results are sent back to the client.
712 Those negative suggestions could be reflected in the user interface by
713 deactivating commands to narrow the choice of tactics available to the user.
714 This approach could be interesting especially for novice users, but require
715 the client to trust other actors a bit more than in the current approach.
717 We plan also to add some rating mechanism to the architecture. A first
718 improvement in this direction could be distinguishing between hints that, when
719 applied, are able to completely close one or more goals and
720 tactics that progress in the proof by reducing one or more goals to new goals:
721 the new goals could be false and the proof can be closed only by backtracking.
723 Other heuristics and/or measures could be added to rate
724 hints and show them to the user in a particular order: an interesting one
725 could be a measure that try to minimize the size of the generated proof,
726 privileging therefore non-overkilling solutions \cite{ring}.
728 We are also considering to follow the \OmegaAnts{} path more closely adding
729 ``recursion'' to the system so that the proof status resulting from the
730 application of old hints are cached somewhere and could be used as a starting
731 point for new hint searches. The approach is interesting, but it represents
732 a big shift towards automatic theorem proving: thus we must consider if it is
733 worth the effort given the increasing availability of automation in proof
734 assistants tactics and the ongoing development of \wss{} based on
735 already existent and well developed theorem provers.
737 Even if not strictly part of the \hbugs{} architecture, the graphical user
738 interface (GUI) of our prototype needs a lot of improvement if we would like
739 it to be really usable by novices. In particular, the user is too easily
740 distracted by the tutor's hints that are ``pushed'' to her.
742 Our \wss{} still lack a real integration in the MONET architecture,
743 since we do not provide the different ontologies to describe our problems,
744 solutions, queries and services. In the short term, completing this task
745 could provide a significative feedback to the MONET consortium and would
746 enlarge the current set of available MONET actors on the Web. In the long
747 term, new more intelligent tutors could be developed on top of already
748 existent MONET \wss{}.
750 To conclude, \hbugs{} is a nice experiment meant to understand whether the
751 current \wss{} technology is mature enough to have a concrete and useful
752 impact on the daily work of proof-assistants users. So far, only the tutor
753 that is interfaced with the \helm{} Search-Engine has effectively increased
754 the productivity of experts users. The usefulness of the tutors developed for
755 beginners, instead, need further assessment.
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