A Connectionist Cognitive Model for Temporal Synchronisation and Learning∗ Lu´ıs C. Lamb and Rafael V. Borges Artur S. d’Avila Garcez Institute of Informatics Department of Computing Federal University of Rio Grande do Sul City University London Porto Alegre, RS, 91501-970, Brazil London EC1V 0HB, UK [email protected]; [email protected] [email protected] Abstract lead to more effective and richer cognitive computational models, and to a better understanding of the processes of ff The importance of the e orts towards integrating the sym- artificial intelligence across the field. bolic and connectionist paradigms of artificial intelligence ff has been widely recognised. Integration may lead to more Several e orts have been made in this direction. However, effective and richer cognitive computational models, and to a most of them deal with knowledge expressed as production better understanding of the processes of artificial intelligence rules or logic programming (d’Avila Garcez, Broda, & Gab- across the field. This paper presents a new model for the bay 2002; Shastri 1999; Towell & Shavlik 1994). This work representation, computation, and learning of temporal logic deals with dynamic knowledge, which evolves in time. We in connectionist systems. The model allows for the encod- present a model for representing, computing, and learning ing of past and future temporal logic operators in neural net- temporal logic in connectionist systems. The model allows works, through a neural-symbolic translation algorithms in- for the encoding of past and future temporal logic opera- troduced in the paper. The networks are relatively simple tors in neural networks, through a translation algorithm in- and can be used for reasoning about time and for learning by examples with the use of standard neural learning algo- troduced in the paper. The networks are relatively simple rithms. We validate the model in a well-known application and can be used for reasoning about time and learning by ex- dealing with temporal synchronisation in distributed knowl- amples with the use of standard neural learning algorithms. edge systems. This opens several interesting research paths We apply the model in a number of experiments, dealing in cognitive modelling, with potential applications in agent with learning, reasoning and synchronisation in a distributed technology, learning and reasoning. knowledge environment. Temporal logic has been amply successful in computer Introduction science. It has been used in the formalisation of several com- putational properties and concepts including verification, The construction of rich computational cognitive models has specification and derivation of computing systems. More recently been pointed out as a key research question for com- recently, such techniques have been successfully used in ar- puter science and cognitive computation (Valiant 2003). To tificial intelligence, in particular, for modelling several di- cope with the requirements of constructing a rich intelli- mensions of multi-agent systems, including model checking, gent behaviour model one should integrate expressive rea- coordination, evolution and cooperation (Fisher, Gabbay, & soning and robust learning in a sound way. However, learn- Vila 2005). This work contributes towards the representa- ing, which has been studied typically under experimental, tion of such expressive, highly successful logical languages statistical approaches would then have to be integrated with in a connectionist system. the reasoning component of intelligent systems, which has We will show that, as pointed out in (Smolensky & mostly been studied using logic-based formalisms. In or- Legendre 2006), cognitive models based on neural-symbolic der to respond to this, we seek to incorporate in a single integration can benefit from their complementary nature. model the two fundamental aspects of intelligent behaviour, Human-inspired inference models may lead to more effec- namely reasoning and learning. Although challenging, the tive reasoning systems, as it is known that neural networks construction of such computational cognitive models would are fault-tolerant and generalize robustly (Browne & Sun meet the requirements for a long standing problem in arti- 2001). We will take advantage of a connectionist architec- ficial intelligence: the integration of the connectionist and ture to learn symbolic temporal knowledge based on infer- the symbolic paradigms of artificial intelligence, which has ence mechanisms from logic, which is also used as back- long been recognised as a standing research issue in the ground knowledge in the learning process. Our experi- field (Page 2000; Smolensky & Legendre 2006; Sun 1995; ments suggest that the proposed model is rich enough to deal Touretzky & Hinton 1985; Valiant 2000). Integration may with temporal reasoning and learning in distributed environ- ∗Research supported by the Brazilian Research Council CNPq. ments, meeting two requirements put forward in (Valiant Copyright c 2007, Association for the Advancement of Artificial 2003): learning and reasoning are integrated in the same Intelligence (www.aaai.org). All rights reserved. model and are tractable. 827 Next, we introduce the basics of connectionist and tem- true if and only if there is a clause in P of the form A ← , , ..., n poral models used in the paper. We then present a repre- L1 L2 Ln and i=1 IP(Li) is true. sentation of temporal formalisms in connectionist systems. An algorithm that translates temporal theories including past Temporal Reasoning and future operators into neural networks is introduced and We start by defining a language that extends (propositional) we prove that the translation and the computation of tempo- logic programs with a unary temporal operator that repre- ral knowledge in our model is sound. We then validate the sent the immediately previous timepoint. α denotes that α approach with experiments using well-known testbeds for is true at the previous timepoint. The syntax of -based pro- temporal knowledge synchronisation in distributed systems, grams can be defined as a set of clauses α ← λ1,λ2, ..., λn, and show that empirical learning benefits from using tempo- where α is an (temporal) atom and λi, for 1 ≤ i ≤ n and ral background knowledge. Finally, we conclude and point n ≥ 0, are literals. An atom is defined as any expression out directions for future research. mA, where m is a chain of m previous time operators, with m ≥ 0, and A is a propositional variable. A literal is an atom Preliminaries or the negation of an atom. We characterize the semantics of a -based program through the use of a fixed point defini- This section introduces the basics of connectionist models T and symbolic temporal reasoning used in the paper. We tion. We define the immediate consequence operator P of a -based program P as a mapping between interpretations assume familiarity with neural networks models and only t summarise used concepts. A neural network can be seen IP at timepoint t. t as a massively parallel distributed processor that stores ex- Definition 2 TP(IP)(α) is true if and only if one of the periential knowledge (Haykin 1999). A multilayer percep- following holds: (i) there is a clause in P of the form α ← λ ,λ , ..., λ t n λ α tron (MLP) is composed of several layers of simple process- 1 2 n where IP( i=1 i) is true; (ii) is an atom t−1 t ing units, the artificial neurons. There are several methods of the form β, and FP (β) is true, where FP is the fixed for representing time and symbolic knowledge in MLPs. t t point of P at time t, i.e., TP(FP)(α) = FP(α). d’Avila Garcez & Lamb (2006) consider a parallel repre- Following (Gelfond & Lifschitz 1988), we can show that sentation of time, using an ensemble of MLPs, where each the TP operator converges to a unique stable state for a network represents a specific timepoint. Elman (1990) de- large class of propositional logic programs. Such stable state scribes the use of recurrent links and delay units to propagate represents the fixed point semantics of the program. The ap- values through time. Nonlinear Auto Regressive with eX- proach used in (d’Avila Garcez, Broda, & Gabbay 2002) to ogenous inputs (NARX) networks (Siegelmann, Horne, & compute the semantics of a logic program P consists in gen- Giles 1995) are based on a recurrent multi-layer architecture erating an input neuron to represent each atom in P, a hidden where recurrent links are allowed only from output to input neuron for each clause C of P (computing the conjunction neurons. In such models, each timepoint is considered as of the body literals in C), and an output neuron for each the application of an input pattern and the subsequent prop- atom α, computing the disjunction of all the clauses where agation of values through the network. Each recurrent link α is the head. To recursively compute the TP operator, re- implies in a delay on the propagated value i.e., the activation current links are set up from the output to the input neuron value of an output neuron N at time t is applied to an input representing the same atom, in such a way that the resulting neuron N at t + 1. Also, delay units can be inserted before interpretation of one computation of TP is applied as input the input neurons in order to allow a greater delay for both for the next one. input and recurrent values. In order to build a connectionist computational architec- In order to represent rich symbolic knowledge in con- ture for representing -based programs, we will add recur- nectionist models, such as modal and temporal knowledge rent links from output units representing an atom α to the (which have been shown adequate in modelling multi-agent input neuron representing nα, with a chain of n delay units. cognition (Fisher, Gabbay, & Vila 2005)), one typically These units simulate short term memory, holding the activa- makes use of a hybrid approach, translating symbolic knowl- tion value of a neuron, relative to a time t − n, during the edge into a neural network, e.g.