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 An usage session is shown in Sect. \ref{usage}.
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 final section of this paper is devoted to 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 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 an
152 encoding of a step that can be performed in order to proceed in an
153 incomplete proof. Usually it represents a reference to a tactic available
154 on some proof assistant along with an instantiation for its formal
155 parameters. More structured hints can also be used: an hint can be
156 as complex as a whole proof-plan.
158 Using W3C's terminology \cite{ws-glossary}, clients act both as \ws{}
159 providers and requesters, see Fig. \ref{interfaces}.
160 They act as providers for the broker (to receive hints)
161 and as requesters (to submit new status). Clients
162 additionally use broker service to know which tutors are available and to
163 subscribe to one or more of them.
165 Usually, when the client role is taken by an interactive proof assistant,
166 new status are sent to the broker as soon as the proof change (e.g. when the
167 user applies a tactic or when a new proof is started) and hints are shown to
168 the user be the means of some effect in the user interface (e.g. popping a
169 dialog box or enlightening a tactic button).
171 \hbugs{} clients act as MONET clients and ask brokers to provide access to a
172 set of services (the tutors). \hbugs{} has no actors corresponding to
173 MONET's Broker Locating Service (since the client is supposed to know the
174 URI of at least one broker). The \hbugs{} client and tutors contact the
175 Getter (a MONET Mathematical Object Manager) to locate and retrieve
176 mathematical items in the \helm{} library.
177 The proof status that are exchanged
178 by the \hbugs{} actors, instead, are built on the fly and are neither
179 stored nor given an unique identifier (URI) to be managed by the
183 \myincludegraphics{interfaces}{t!}{10cm}{\hbugs{} \wss{} interfaces}
184 {\hbugs{} \wss{} interfaces}
186 Brokers are the key actors of the \hbugs{} architecture since they
187 act as intermediaries between clients and tutors. They behave as \wss{}
188 providers and requesters for \emph{both} clients and tutors, see Fig.
191 With respect to the client, a broker acts as a \ws{} provider, receiving the
192 proof status and forwarding it to one or more tutors.
193 It also acts as a \ws{} requester sending
194 hints to the client as soon as they are available from the tutors.
196 With respect to the tutors, the \ws{} provider role is accomplished by
197 receiving hints as soon as they are produced; as a requester, it is
198 accomplished by asking for computations (\emph{musings} in \hbugs{}
199 terminology) on status received by clients and by stopping already late but
200 still ongoing \musings{}.
202 Additionally brokers keep track of available tutors and clients
205 \hbugs{} brokers act as MONET brokers implementing the following components:
206 Client Manager, Service Registry Manager (keeping track of available
207 tutors), Planning Manager (choosing the available tutors among the ones to
208 which the client is subscribed), Execution Manager. The Service Manager
209 component is not required since the session handler, that identifies
210 a session between a service and a broker, is provided to the service by
211 the broker instead of being received from the service when the session is
212 initialized. In particular, a session is identified by an unique identifier
213 for the client (its URL) and an unique identifier for the broker (its
216 The MONET architecture specification does not state explicitly whether
217 the service and broker answers can be asynchronous. Nevertheless, the
218 described information flow implicitly suggests a synchronous implementation.
219 On the contrary, in \hbugs{} every request is asynchronous: the connection
220 used by an actor to issue a query is immediately closed; when a service
221 produces an answer, it gives it back to the issuer by calling the
222 appropriate actor's method.
225 Tutors are software component able to consume proof status producing hints.
226 \hbugs{} does not specify by which means hints should be produced: tutors
227 can use any means necessary (heuristics, external theorem prover or CAS,
228 etc.). The only requirement is that there exists an agreement on the
229 formats of proof status and hints.
231 Tutors act both as \ws{} providers and requesters for the broker, see Fig.
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 failure notification).
238 \hbugs{} tutors act as MONET services.
240 \section{An \hbugs{} Session Example}
242 In this section we describe a typical \hbugs{} session. The aim of the
243 session is to solve the following easy exercise:
245 Let $x$ be a generic real number. Using the \helm{} proof-engine,
248 x = \frac{(x+1)*(x+1) - 1 - x*x}{2}
252 Let us suppose that the \hbugs{} broker is already running and that the
253 tutors already registered themselves to the broker.
254 When the user starts \texttt{gTopLevel}, the system registers itself to
255 the broker, that sends back the list of available tutors. By default,
256 \texttt{gTopLevel} notifies to the broker its intention of subscribing to every
257 tutor available. The user can always open a configuration window where she
258 is presented the list of available tutors and she can independently subscribe
259 and unsubscribe the system to each tutor.
261 \myincludegraphics{step1}{t}{12cm}{Example session.}
263 %\myincludegraphics{step2}{t}{4cm}{Example session, snapshot 2.}
264 % {Example session, snapshot 2.}
266 The user can now insert into the system the statement of the theorem and start
267 proving it. Let us suppose that the first step of the user is proving
268 that the denominator 2 is different from 0. Once that this technical result
269 is proven, the user must prove the goal shown in the upper right corner
270 of the window in background in Fig. \ref{step1}.
272 While the user is wondering how to proceed in the proof, the tutors are
273 trying to progress in the proof. After a while, the tutors' suggestions
274 start to appear in the lower part of the \hbugs{} interface window
275 (the topmost window in Fig. \ref{step1}). In this case, the tutors are able
276 to produce 23 hints. The first and not very useful hint suggests to proceed in
277 the proof by exchanging the two sides of the equality.
278 The second hint suggests to reduce both sides of the equality to their normal
279 form by using only reductions which are justified by the ring structure of the
280 real numbers; the two normal forms, though, are so different that the proof is
281 not really simplified.
282 All the residual 21 hints suggest to apply one lemma from the distributed
283 library of \helm{}. The user can look at the statement of the lemma by clicking
286 The user can now look at the list of suggestions and realize that a good one is
287 applying the lemma \texttt{r\_Rmult\_mult} that allows to multiply both equality
288 members by the same scalar\footnote{Even if she does not receive the hint, the
289 user probably already knows that this is the right way to proceed. The
290 difficult part where the hint helps is guessing what is the name of the lemma
292 Double-clicking on the hint automatically applies
293 the lemma, reducing the proof to closing three new goals. The first one asks
294 the user the scalar to use as an argument of the previous lemma; the second
295 one states that the scalar is different from 0; the third lemma (the main
296 one) asks to prove the equality between the two new members.
297 % is shown in Fig. \ref{step2} where $?_3[H;x]$ stands for
298 % the still unknown scalar argument, which can have only $H$ and $x$ as
301 The user proceeds by instantiating the scalar with the number 2. The
302 \texttt{Assumption} tutor now suggests to close the second goal (that
303 states that $2 \neq 0$) by applying the hypothesis $H$.
304 No useful suggestions, instead, are generated for the main goal
305 $2*x = 2*((x+1)*(x+1)-1-x*x)*2^{-1}$.
306 To proceed in the proof the user needs to simplify the
307 expression using the lemma $Rinv\_r\_simpl\_m$ that states that
308 $\forall x,y.\;y = x * y * x^{-1}$. Since we do not provide yet any tutor
309 suggesting simplifications, the user must find out this simplification by
310 himself. Once she founds it, the goal is reduced to proving that
311 $2*x = (x+1)*(x+1) - 1 - x*x$. This equality is easily solved by the
312 \texttt{Ring} tutor, that suggests\footnote{The \texttt{Ring} suggestion is
313 just one of the 22 hints that the user receives. It is the only hint that
314 does not open new goals, but the user right now does not have any way to know
315 that.} to the user how to directly finish the proof.
317 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
318 % Comandi da dare a gTopLevel %
319 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
321 % !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)))
325 % Simpl (per fare unfold di Rdiv)
327 % (Rmult_assoc (Rplus R1 R1) (Rplus (Rplus (Rmult (Rplus x R1) (Rplus x R1)) (Ropp R1)) (Ropp (Rmult x x))) (Rinv (Rplus R1 R1)))
329 % (Rinv_r_simpl_m (Rplus R1 R1) (Rplus (Rplus (Rmult (Rplus x R1) (Rplus x R1)) (Ropp R1)) (Ropp (Rmult x x))) H)
331 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
333 \section{Implementation's Highlights}
334 \label{implementation}
335 In this section we present some of the most relevant implementation details of
336 the \hbugs{} architecture.
339 \paragraph{Proof status}
340 In our implementation of the \hbugs{} architecture we used the proof
341 assistant of the \helm{} project (codename ``gTopLevel'') as an \hbugs{}
342 client. Thus we have implemented serialization/deserialization capabilities
343 for its internal status. In order to be able to describe \wss{} that
344 exchange status in WSDL using the XML Schema type system, we have chosen an
345 XML format as the target format for the serialization.
347 % A schematic representation of the gTopLevel internal status is depicted in
349 Each proof is represented by a tuple of four elements:
350 \emph{uri}, \emph{metasenv}, \emph{proof}, \emph{thesis}.
352 % \myincludegraphics{status}{t}{8cm}{gTopLevel proof status}{gTopLevel proof
356 \item[uri]: an URI chosen by the user at the beginning of the proof
357 process. Once (and if) proved, that URI will globally identify the term
358 inside the \helm{} library (given that the user decides to save it).
359 \item[thesis]: the thesis of the ongoing proof
360 \item[proof]: the current incomplete proof tree. It can contain
361 \emph{metavariables} (holes) that stands for the parts of the proof
362 that are still to be completed. Each metavariable appearing in the
363 tree references one element of the metavariables environment
365 \item[metasenv]: the metavariables environment is a list of
366 \emph{goals} (unproved conjectures).
367 In order to complete the proof, the user has to instantiate every
368 metavariable in the proof with a proof of the corresponding goal.
369 Each goal is identified by an unique identifier and has a context
370 and a type (the goal thesis). The context is a list of named
371 hypotheses (declarations and definitions). Thus the context and the goal
372 thesis form a sequent, which is the statement of the proof that will
373 be used to instantiate the metavariable occurrences.
376 Each of these information is represented in XML as described in
377 \cite{mowglicic}. Additionally, an \hbugs{} status carry the unique
378 identifier of the current goal, which is the goal the user is currently
379 focused on. Using this value it is possible to implement different client
380 side strategies: the user could ask the tutors to work on the goal
381 she is considering or to work on the other ``background'' goals.
384 An hint in the \hbugs{} architecture should carry enough information to
385 permit the client to progress in the current proof. In our
386 implementation each hint corresponds to either one of the tactics available
387 to the user in gTopLevel (together with its actual arguments) or a set
388 of alternative suggestions (a list of hints).
390 For tactics that don't require any particular argument (like tactics that
391 apply type constructors or decision procedures)
392 only the tactic name is represented in the hint. For tactics that need
393 terms as arguments (for example the \texttt{Apply} tactic that apply a
394 given lemma) the hint includes a textual representation of them, using the
395 same representation used by the interactive proof assistant when querying
396 user for terms. In order to be transmitted between \wss{}, hints are
399 It is also possible for a tutor to return more hints at once,
400 grouping them in a particular XML element.
401 This feature turns out to be particularly useful for the
402 \emph{searchPatternApply} tutor (see Sect. \ref{tutors}) that
403 query a lemma database and return to the client a list of all lemmas that
404 could be used to complete the proof. This particular hint is encoded as a
405 list of \texttt{Apply} hints, each of them having one of the results as term
408 We would like to stress that the \hbugs{} architecture has no dependency
409 on either the hint or the status representation: the only message parts
410 that are fixed are those representing the administrative messages
411 (the envelops in the \wss{} terminology). In particular, the broker can
412 manage at the same time several sessions working on different status/hints
413 formats. Of course, there must be an agreement between the clients
414 and the tutors on the format of the data exchanged.
416 In our implementation the client does not trust the tutors hints:
417 being encoded as references to available tactics imply
418 that an \hbugs{} client, on receipt of an hint, simply try to reply the work
419 done by a tutor on the local copy of the proof. The application of the hint
420 can even fail to type check and the client copy of the proof can be left
421 undamaged after spotting the error. Note, however, that it is still
422 possible to implement a complex tutor that looks for a proof doing
424 send back to the client an hint whose argument is a witness (a trace) of
425 the proof found: the client applies the hint reconstructing (and checking
426 the correctness of) the proof from the witness, without having to
427 re-discover the proof itself.
429 An alternative implementation where the tutors are trusted would simply
430 send back to the client a new proof-status. Upon receiving the
431 proof-status, the client would just override its current proof status with
432 the suggested one. In the case of those clients which are implemented
433 using proof-objects (as the Coq proof-assistant, for instance), it is
434 still possible for the client to type-check the proof-object and reject
435 wrong hints. The systems that are not based on proof-objects
436 (as PVS, NuPRL, etc.), instead, have to trust the new proof-status. In this
437 case the \hbugs{} architecture needs at least to be extended with
438 clients-tutors authentication.
440 \paragraph{Registries}
441 Being central in the \hbugs{} architecture, the broker is also responsible
442 of housekeeping operations both for clients and tutors. These operations are
443 implemented using three different data structures called \emph{registries}:
444 clients registry, tutors registry and \musings{} registry.
446 In order to use the suggestion engine a client should register itself to the
447 broker and subscribe to one or more tutors. The registration phase is
448 triggered by the client using the \texttt{Register\_client} method of the
449 broker to send him an unique identifier and its base URI as a
450 \ws{}. After the registration, the client can use broker's
451 \texttt{List\_tutors} method to get a list of available tutors.
452 Eventually the client can subscribe to one or more of these using broker's
453 \texttt{Subscribe} method. Clients can also unregister from brokers using
454 \texttt{Unregister\_client} method.
456 The broker keeps track of both registered clients and clients' subscriptions
457 in the clients registry.
459 In order to be advertised to clients during the subscription phase, tutors
460 should register to the broker using the broker's \texttt{Register\_tutor}
461 method. This method is really similar to \texttt{Register\_client}:
462 tutors are required to send an unique identify and a base URI for their
464 Additionally tutors are required to send an human readable description of
465 their capabilities; this information could be used by client's user to
466 decide which tutors he needs to subscribe to. As the clients, tutors can
467 unregister from brokers using \texttt{Unregister\_broker} method.
469 Each time the client status change, it get sent sent to the
470 broker using its \texttt{Status} method. Using both clients registry (to
471 lookup client's subscription) and tutors registry (to check if some tutors
472 has unsubscribed), the broker is able to decide to which tutors the
473 new status have to be forwarded.
474 % \ednote{CSC: qui o nei lavori futuri parlare
475 % della possibilit\'a di avere un vero brocker che multiplexi le richieste
476 % dei client localizzando i servizi, etc.}
478 The forwarding operation is performed using the \texttt{Start\_musing}
479 method of the tutors, that is a request to start a new computation
480 (\emph{\musing{}}) on a given status. The return value of
481 \texttt{Start\_musing} is a
482 \musing{} identifier that is saved in the \musings{} registry along with
483 the identifier of the client that triggered the \musing{}.
485 As soon as a tutor completes an \musing{}, it informs the broker
486 using its \texttt{Musing\_completed} method; the broker can now remove the
487 \musing{} entry from the \musings{} registry and, depending on its outcome,
488 inform the client. In case of success one of the \texttt{Musing\_completed}
489 arguments is an hint to be sent to the client, otherwise there's no need to
490 inform him and the \texttt{Musing\_completed} method is called
491 just to update the \musings{} registry.
493 Consulting the \musings{} registry, the broker is able to know, at each
494 time, which \musings{} are in execution on which tutor. This peculiarity is
495 exploited by the broker on invocation of the \texttt{Status} method.
496 Receiving a new status from the client implies indeed that the previous
497 status no longer exists and all \musings{} working on it should be stopped:
498 additionally to the already described behavior (i.e. starting new
499 \musings{} on the received status), the broker takes also care of stopping
500 ongoing computation invoking the \texttt{Stop\_musing} method of the tutors.
503 As already discussed, all \hbugs{} actors act as \wss{} offering one or more
504 services to neighbor actors. To grant as most accessibility as possible to
505 our \wss{} we have chosen to bind them using the HTTP/POST\footnote{Given
506 that our proof assistant was entirely developed in the Objective Caml
507 language, we have chosen to develop also \hbugs{} in that language in order
508 to maximize code reuse. To develop \wss{} in Objective Caml we have
509 developed an auxiliary generic library (\emph{O'HTTP}) that can be used to
510 write HTTP 1.1 Web servers and abstract over GET/POST parsing. This library
511 supports different kinds of Web servers architecture, including
512 multi-process and multi-threaded ones.} bindings described in
516 Each tutor expose a \ws{} interface and should be able to work, not only for
517 many different clients referring to a common broker, but also for many
518 different brokers. The potential high number of concurrent clients imposes
519 a multi-threaded or multi-process architecture.
521 Our current implementation is based on a multi threaded architecture
522 exploiting the capabilities of the O'HTTP library. Each tutor is composed
523 by one thread always running plus
524 an additional thread for each running \musing{}. One thread is devoted to
525 listening for incoming \ws{} request; upon correct receiving requests it
526 pass the control to a second thread, created on the fly, to handle the
527 incoming request following the classical one-thread-per-request approach in
529 If the received request is \texttt{Start\_musing}, a new thread is
530 spawned to handle it; the thread in duty to handle the HTTP request
531 returns an HTTP response containing the identifier of the just started
532 \texttt{musing}, and then dyes. If the received request is
533 \texttt{Stop\_musing}, instead, the spawned thread kills the thread
534 responsible for the \texttt{musing} whose identifier is the argument
535 of the \texttt{Stop\_musing} method.
537 This architecture turns out to be scalable and allows the running threads
538 to share the cache of loaded (and type-checked) theorems.
539 As we will explain in Sect. \ref{tutors}, this feature turns out to be
540 really useful for tactics that rely on a huge but fixed set of lemmas,
541 as every reflexive tactic.
543 The implementation of a tutor with the described architecture is not that
544 difficult having a language with good threading capabilities (as OCaml has)
545 and a pool of already implemented tactics (as gTopLevel has).
546 Still working with threads is known to be really error prone due to
547 concurrent programming intrinsic complexity. Moreover, there is a
548 non-neglectable part of code that needs to be duplicated in every tutor:
549 the code to register the tutor to the broker and to handle HTTP requests;
550 the code to manage the creation and termination of threads; and the code for
551 parsing the requests and serializing the answers. As a consequence we
552 have written a generic implementation of a tutor which is parameterized
553 over the code that actually propose the hint and some administrative
554 data (as the port the tutor will be listening to).
556 The generic tutor skeleton is really helpful in writing new tutors.
557 Nevertheless, the code obtained by converting existing tactics into tutors
558 is still quite repetitive: every tutor that wraps a tactic has to
559 instantiate its own copy of the proof-engine kernel and, for each request,
560 it has to override its status, guess the tactic arguments, apply the tactic
561 and, in case of success, send back an hint with the tactic name and the
562 chosen arguments. Of course, the complex part of the work is guessing the
563 right arguments. For the simple case of tactics that do not require
564 any argument, though, we are able to automatically generate the whole
565 tutor code given the tactic name. Concretely, we have written a
566 tactic-based tutor template and a script that parses an XML file with
567 the specification of the tutor and generates the tutor's code.
568 The XML file describes the tutor's port, the code to invoke the tactic,
569 the hint that is sent back upon successful application and a
570 human readable explanation of the tactic implemented by the tutor.
572 \section{The Implemented \hbugs Tutors}
574 To test the \hbugs{} architecture and to assess the utility of a suggestion
575 engine for the end user, we have implemented several tutors. In particular,
576 we have investigated three classes of tutors:
578 \item \emph{Tutors for beginners}. These are tutors that implement tactics
579 which are neither computationally expensive nor difficult to understand:
580 an expert user can always understand if the tactic can be applied or not
581 without having to try it. For example, the following implemented tutors
582 belong to this class:
584 \item \emph{Assumption Tutor}: it ends the proof if the thesis is
585 equivalent\footnote{In our implementation, the equivalence relation
586 imposed by the logical framework is \emph{convertibility}. Two
587 expressions are convertible when they reduce to the same normal form.
588 Two ``equal'' terms depending on free variables can be non-convertible
589 since free variables stop the reduction. For example, $2x$ is convertible
590 with $(3-1)x$ because they both reduce to the same normal form
591 $x + x + 0$; but $2x$ is not convertible to $x2$ since the latter is
592 already in normal form.}
593 to one of the hypotheses\footnote{
594 In some cases, especially when non-trivial computations are involved,
595 the user is totally unable to figure out the convertibility of two terms.
596 In these cases the tutor becomes handy also for expert users.}.
597 \item \emph{Contradiction Tutor}: it ends the proof by \emph{reductio ad
598 adsurdum} if one hypothesis is equivalent to $False$.
599 \item \emph{Symmetry Tutor}: if the goal thesis is an equality, it
600 suggests to apply the commutative property.
601 \item \emph{Left/Right/Exists/Split/Reflexivity/Constructor Tutors}:
602 the Constructor Tutor suggests to proceed in the proof by applying one
603 or more constructors when the goal thesis is an inductive type or a
604 proposition inductively defined according to the declarative
605 style\footnote{An example of a proposition that can be given in
606 declarative style is the $\le$ relation over natural numbers:
607 $\le$ is the smallest relation
608 such that $n \le n$ for every $n$ and $n \le m$ for every $n,m$ such
609 that $n \le p$ where $p$ is the predecessor of $m$. Thus, a proof
610 of $n \le n$ is simply the application of the first constructor to
611 $n$ and a proof of $n \le m$ is the application of the second
612 constructor to $n,m$ and a proof of $n \le m$.}.
613 Since disjunction, conjunction, existential quantification and
614 Leibniz equality are particular cases of inductive propositions,
615 all the other tutors of this class are instantiations of the
616 the Constructor tactic. Left and Right suggest to prove a disjunction
617 by proving its left/right member; Split reduces the proof of a
618 conjunction to the two proof of its members; Exists suggests to
619 prove an existential quantification by providing a
620 witness\footnote{This task is left to the user.}; Reflexivity proves
621 an equality whenever the two sides are convertible.
623 Beginners, when first faced with a tactic-based proof-assistant, get
624 lost quite soon since the set of tactics is large and their names and
625 semantics must be remembered by heart. Tutorials are provided to guide
626 the user step-by-step in a few proofs, suggesting the tactics that must
627 be used. We believe that our beginners tutors can provide an auxiliary
628 learning tool: after the tutorial, the user is not suddenly left alone
629 with the system, but she can experiment with variations of the proof given
630 in the tutorial as much as she like, still getting useful suggestions.
631 Thus the user is allowed to focus on learning how to do a formal proof
632 instead of wasting efforts trying to remember the interface to the system.
633 \item{Tutors for Computationally Expensive Tactics}. Several tactics have
634 an unpredictable behavior, in the sense that it is unfeasible to understand
635 whether they will succeed or they will fail when applied and what will be
636 their result. Among them, there are several tactics either computationally
637 expensive or resources consuming. In the first case, the user is not
638 willing to try a tactic and wait for a long time just to understand its
639 outcome: she would prefer to keep on concentrating on the proof and
640 have the tactic applied in background and receive out-of-band notification
641 of its success. The second case is similar, but the tactic application must
642 be performed on a remote machine to avoid overloading the user host
643 with several concurrent resource consuming applications.
645 Finally, several complex tactics and in particular all the tactics based
646 on reflexive techniques depend on a pretty large set of definitions, lemmas
647 and theorems. When these tactics are applied, the system needs to retrieve
648 and load all the lemmas. Pre-loading all the material needed by every
649 tactic can quickly lead to long initialization times and to large memory
650 footstamps. A specialized tutor running on a remote machine, instead,
651 can easily pre-load the required theorems.
653 As an example of computationally expensive task, we have implemented
654 a tutor for the \emph{Ring} tactic \cite{ringboutin}.
655 The tutor is able to prove an equality over a ring by reducing both members
656 to a common normal form. The reduction, which may require some time in
658 is based on the usual commutative, associative and neutral element properties
659 of a ring. The tactic is implemented using a reflexive technique, which
660 means that the reduction trace is not stored in the proof-object itself:
661 the type-checker is able to perform the reduction on-the-fly thanks to
662 the conversion rules of the system. As a consequence, in the library there
663 must be stored both the algorithm used for the reduction and the proof of
664 correctness of the algorithm, based on the ring axioms. This big proof
665 and all of its lemmas must be retrieved and loaded in order to apply the
666 tactic. The Ring tutor loads and cache all the required theorems the
667 first time it is contacted.
668 \item{Intelligent Tutors}. Expert users can already benefit from the previous
669 class of tutors. Nevertheless, to achieve a significative production gain,
670 they need more intelligent tutors implementing domain-specific theorem
671 provers or able to perform complex computations. These tutors are not just
672 plain implementations of tactics or decision procedures, but can be
673 more complex software agents interacting with third-parties software,
674 such as proof-planners, CAS or theorem-provers.
676 To test the productivity impact of intelligent tutors, we have implemented
677 a tutor that is interfaced with the \helm{}
678 Search-Engine\footnote{\url{http://helm.cs.unibo.it/library.html}} and that
679 is able to look for every theorem in the distributed library that can
680 be applied to proceed in the proof. Even if the tutor deductive power
681 is extremely limited\footnote{We do not attempt to check if the new goals
682 obtained applying a lemma can be automatically proved or, even better,
683 automatically disproved to reject the lemma.}, it is not unusual for
684 the tutor to come up with precious hints that can save several minutes of
685 work that would be spent in proving again already proven results or
686 figuring out where the lemmas could have been stored in the library.
689 \section{Conclusions and Future Work}
691 In this paper we described a suggestion engine architecture for
692 proof-assistants: the client (a proof-assistant) sends the current proof
693 status to several distributed \wss{} (called tutors) that try to progress
694 in the proof and, in case of success, send back an appropriate hint
695 (a proof-plan) to the user. The user, that in the meantime was able to
696 reason and progress in the proof, is notified with the hints and can decide
697 to apply or ignore them. A broker is provided to decouple the clients and
698 the tutors and to allow the client to locate and invoke the available remote
699 services. The whole architecture is an instance of the MONET architecture
700 for Mathematical \wss{}.
702 A running prototype has been implemented as part of the
703 \helm{} project \cite{helm}
704 and we already provide several tutors. Some of them are simple tutors that
705 try to apply one or more tactics of the \helm{} Proof-Engine, which is also
706 our client. We also have a much more complex tutor that is interfaced
707 with the \helm{} Search-Engine and looks for lemmas that can be directly
710 We have many plans for further developing both the \hbugs{} architecture and
711 our prototype. Interesting results could be obtained
712 augmenting the informative content of each suggestion. We can for example
713 modify the broker so that also negative results are sent back to the client.
714 Those negative suggestions could be reflected in the user interface by
715 deactivating commands to narrow the choice of tactics available to the user.
716 This approach could be interesting especially for novice users, but require
717 the client to trust other actors a bit more than in the current approach.
719 We plan also to add some rating mechanism to the architecture. A first
720 improvement in this direction could be distinguishing between hints that, when
721 applied, are able to completely close one or more goals and
722 tactics that progress in the proof by reducing one or more goals to new goals:
723 the new goals could be false and the proof can be closed only by backtracking.
725 Other heuristics and/or measures could be added to rate
726 hints and show them to the user in a particular order: an interesting one
727 could be a measure that try to minimize the size of the generated proof,
728 privileging therefore non-overkilling solutions \cite{ring}.
730 We are also considering to follow the \OmegaAnts{} path more closely adding
731 ``recursion'' to the system so that the proof status resulting from the
732 application of old hints are cached somewhere and could be used as a starting
733 point for new hint searches. The approach is interesting, but it represents
734 a big shift towards automatic theorem proving: thus we must consider if it is
735 worth the effort given the increasing availability of automation in proof
736 assistants tactics and the ongoing development of \wss{} based on
737 already existent and well developed theorem provers.
739 Even if not strictly part of the \hbugs{} architecture, the graphical user
740 interface (GUI) of our prototype needs a lot of improvement if we would like
741 it to be really usable by novices. In particular, the user is too easily
742 distracted by the tutor's hints that are ``pushed'' to her.
744 Our \wss{} still lack a real integration in the MONET architecture,
745 since we do not provide the different ontologies to describe our problems,
746 solutions, queries and services. In the short term, completing this task
747 could provide a significative feedback to the MONET consortium and would
748 enlarge the current set of available MONET actors on the Web. In the long
749 term, new more intelligent tutors could be developed on top of already
750 existent MONET \wss{}.
752 To conclude, \hbugs{} is a nice experiment meant to understand whether the
753 current \wss{} technology is mature enough to have a concrete and useful
754 impact on the daily work of proof-assistants users. So far, only the tutor
755 that is interfaced with the \helm{} Search-Engine has effectively increased
756 the productivity of experts users. The usefulness of the tutors developed for
757 beginners, instead, need further assessment.
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