Implicit artificial grammar learning: effects of complexity and usefulness of the structure

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Implicit artificial grammar learning: effects of complexity and usefulness

of the structure

Bos, E.J. van den

Citation

Bos, E. J. van den. (2007, June 6). Implicit artificial grammar learning: effects of complexity

and usefulness of the structure. Department of Cognitive Psychology, Leiden University

Institute for Psychological Research, Faculty of Social Sciences, Leiden University. Retrieved

from https://hdl.handle.net/1887/12037

Version: Corrected Publisher’s Version

License: Licence agreement concerning inclusion of doctoral thesis in the

Institutional Repository of the University of Leiden

Downloaded from: https://hdl.handle.net/1887/12037

Note: To cite this publication please use the final published version (if applicable).

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113 Appendix A

The computation of topological entropy

The topological entropy measure of complexity of finite state grammars

developed by Bollt and Jones (2000) is based on two assumptions. Firstly, the

grammar is assumed to be unambiguous: each arrow leaving a certain state has to

generate a different symbol. Secondly, the measure is developed for infinite strings,

but for practical purposes concatenations of finite strings can be used. The

computation of topological entropy is an extension of a procedure previously

proposed by Chomsky and Miller (1958) to compute the informational capacity of

each symbol generated by an artificial grammar.

According to this basic procedure, the grammar, usually depicted in a graph,

has to be translated into a transition matrix. In Chomsky’s and Miller’s (1958)

approach, the elements of the matrix stand for the number of paths of length 1 that

provide a transition from the state assigned to a column of the matrix to the state

assigned to a row. Subsequently, the largest non-negative eigenvalue of the matrix is

computed and a logarithm is taken. Topological entropy is defined as the natural

logarithm of the largest non-negative eigenvalue of a grammar’s transition matrix

(Bollt & Jones, 2000).

However, in contrast to Chomsky and Miller (1958), who assumed that the

current state of the grammar is always known, Bollt and Jones (2000) assumed that

the current state of the grammar has to be inferred from the string. In their approach,

the elements of the matrix stand for transitions between the grammar’s symbols

instead of transitions between states. Both approaches lead to the same matrix if all

arrows in the graph are associated with a unique symbol. In this case, the current state

of the grammar is directly given by the symbol and therefore it can be unambiguously

established which symbols are legal successors.

However, if a symbol is associated with more than one link in the grammar, it

is ambiguous by which symbols it may be followed. In such cases, according to Bollt

and Jones (2000), the current state of the grammar has to be inferred from the current

symbol and one or more symbols preceding it. Therefore, they proposed to lift the

transition matrix. Instead of transitions between symbols, the lifted matrix consists of

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transitions between groups of symbols, which size equals the minimum number of

symbols needed to unambiguously predict the legal successors of each element.

Because of this lifting technique and the assumptions underlying it, topological

entropy is sensitive to the presence of long-distance dependencies and correlated with

the number of elements required to predict the next.

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115 Appendix B

Stimuli

Chapters 3 and 4: Experiment 1

Grammar A1

Induction phase

MXSJSQJ MXTNXSJ MXTNXTN MWSJWSJ MWSJSQJ MWTNXSJ MSQJWTN MSQJSQJ MSRZPRZ MPQJWSJ MPQJSRZ MPRZPQJ MXSJWSJWTN MXSJWTNXSJ MXTNXSJWTN MXTNXSJSRZ MXTNXTNXSJ MWSJWTNXSJ MWSJSRZPQJ MWTNXSJWTN MWTNXTNXSJ MWTNXTNXTN MSQJWTNXSJ MSQJSRZPQJ MSQJSRZPRZ MSRZPQJWTN MSRZPRZPRZ MPQJWTNXSJ MPQJWTNXTN MPQJSQJWSJ MPRZPQJSRZ MPRZPRZPQJ

Test phase

MXSJWSJ MXSJSRZ MWSJSRZ MSQJWSJ MSRZPQJ MPQJSQJ MXSJWTNXTN MXTNXSJWSJ

Grammar B1

Induction phase

QPMZSMZ QPMZSTJ QPMZWRN RSMZSTJ RSMZWRN RWQZWRN TSTJPTJ TWQZSMZ TWRNXQZ SXQZSTJ SXQZWQZ SXRNXRN QPMZSTJPTJ QPMZWRNXQZ QPTJPMZSMZ QPTJPMZSTJ QPTJPTJPMZ RSTJPMZSMZ RSTJPMZSTJ RWQZSTJPTJ RWQZWRNXQZ RWRNXRNXQZ TSMZSTJPTJ TSMZWRNXRN TSTJPTJPMZ TWQZWRNXRN TWRNXRNXRN SXQZSTJPMZ SXQZWQZWRN SXQZWRNXQZ SXRNXQZSMZ SXRNXQZSTJ

Test phase

QPTJPMZ RSMZSMZ TSTJPMZ TWQZWQZ SXQZSMZ SXRNXQZ QPMZSTJPMZ TSMZSTJPMZ

Grammar A2

Induction phase

RXMJSQJ RXTNXMJ RXTNXTN QWMJWMJ MWMJSQJ MWTNXMJ QSQJWTN MSQJSQJ QSRZPRZ TPQJWMJ TPQJSRZ TPRZPQJ RXMJWMJWTN RXMJWTNXMJ RXTNXMJWTN RXTNXMJSRZ RXTNXTNXMJ MWMJWTNXMJ QWMJSRZPQJ QWTNXMJWTN MWTNXTNXMJ QWTNXTNXTN QSQJWTNXMJ QSQJSRZPQJ MSQJSRZPRZ MSRZPQJWTN MSRZPRZPRZ TPQJWTNXMJ TPQJWTNXTN TPQJSQJWMJ TPRZPQJSRZ TPRZPRZPQJ

Test phase

RXMJWMJ RXMJSRZ QWMJSRZ MSQJWMJ MSRZPQJ MPQJSQJ MXSJWTNXTN MXTNXSJWSJ

Grammar B2

Induction phase

QPMZSMZ QPMZSTJ QPMZWRN RSMZSTJ RSMZWRN RWQZWRN TSTJPTJ TWQZSMZ TWRNXQZ MXQZSTJ MXQZWQZ MXRNXRN QPMZSTJPTJ QPMZWRNXQZ QPTJPMZSMZ QPTJPMZSTJ QPTJPTJPMZ RSTJPMZSMZ RSTJPMZSTJ RWQZSTJPTJ RWQZWRNXQZ RWRNXRNXQZ TSMZSTJPTJ TSMZWRNXRN TSTJPTJPMZ TWQZWRNXRN TWRNXRNXRN MXQZSTJPMZ MXQZWQZWRN MXQZWRNXQZ MXRNXQZSMZ MXRNXQZSTJ

Test phase

QPTJPMZ RSMZSMZ TSTJPMZ TWQZWQZ MXQZSMZ MXRNXQZ QPMZSTJPMZ TSMZSTJPMZ

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Grammar A1

Test phase

(continued)

MWSJSRZPRZ MSQJWTNXTN MPQJSQJSRZ MPRZPQJWSJ _XSJWTN _WSJWTN _WTNXTN _SQJSRZ _PQJWTN _PRZPRZ _XSJSRZPQJ _WSJWTNXTN _WTNXSJWSJ _SRZPRZPQJ _PQJSRZPQJ _PRZPQJWTN

Grammar B1

Test phase

(continued)

RSTJPTJPTJ TWRNXQZSTJ SXQZWQZSMZ SXRNXRNXQZ _PMZWQZ _PTJPTJ _SMZWQZ _WQZSTJ _WRNXRN _XQZWRN _PMZSMZSTJ _PTJPMZWRN _SMZWRNXQZ _WQZSTJPMZ _XQZSTJPTJ _XRNXQZWRN

Grammar A2

Test phase

(continued)

MWSJSRZPRZ MSQJWTNXTN MPQJSQJSRZ MPRZPQJWSJ _XSJWTN _WSJWTN _WTNXTN _SQJSRZ _PQJWTN _PRZPRZ _XSJSRZPQJ _WSJWTNXTN _WTNXSJWSJ _SRZPRZPQJ _PQJSRZPQJ _PRZPQJWTN

Grammar B2

Test phase

(continued)

RSTJPTJPTJ TWRNXQZSTJ MXQZWQZSMZ MXRNXRNXQZ _PMZWQZ _PTJPTJ _SMZWQZ _WQZSTJ _WRNXRN _XQZWRN _PMZSMZSTJ _PTJPMZWRN _SMZWRNXQZ _WQZSTJPMZ _XQZSTJPTJ _XRNXQZWRN

Chapter 3: Experiment 2

Grammar A3

Induction phase

RMXMJSQN RMXTNXMN RMXTNXTZ QMWMJWMN QMWMJSQN QMWTNXMN QMSQJWTZ QMSQJSQN QMSRZPRJ TMPQJWMN TMPQJSRJ TMPRZPQN RMXMJWMJWTZ RMXMJWTNXMN RMXTNXMJWTZ RMXTNXMJSRJ RMXTNXTNXMN QMWMJWTNXMN QMWMJSRZPQN QMWTNXMJWTZ QMWTNXTNXMN QMWTNXTNXTZ QMSQJWTNXMN QMSQJSRZPQN QMSQJSRZPRJ QMSRZPQJWTZ QMSRZPRZPRJ TMPQJWTNXMN TMPQJWTNXTZ TMPQJSQJWMN TMPRZPQJSRJ TMPRZPRZPQN

Test phase

RMXMJWMN RMXMJSRJ QMWMJSRJ QMSQJWMN QMSRZPQN TMPQJSQN RMXMJWTNXTN RMXTNXMJWMN QMWMJSRZPRJ QMSQJWTNXTZ TMPQJSQJSRJ TMPRZPQJWMN R_XMJWTZ Q_WMJWTZ Q_WTNXTZ Q_SQJSRJ T_PQJWTZ T_PRZPRJ R_XMJSRZPQN Q_WMJWTNXTZ Q_WTNXMJWMN Q_SRZPRZPQN T_PQJSRZPQN T_PRZPQJWTZ

Grammar B3

Induction phase

QTPMZSMJ QTPMZSTN QTPMZWRZ RTSMZSTN RTSMZWRZ RTWQZWRZ RTSTJPTN RTWQZSMJ RTWRNXQJ MTXQZSTN MTXQZWQJ MTXRNXRZ QTPMZSTJPTN QTPMZWRNXQJ QTPTJPMZSMJ QTPTJPMZSTN QTPTJPTJPMJ RTSTJPMZSMJ RTSTJPMZSTN RTWQZSTJPTN RTWQZWRNXQJ RTWRNXRNXQJ RTSMZSTJPTN RTSMZWRNXRZ RTSTJPTJPMJ RTWQZWRNXRZ RTWRNXRNXRZ MTXQZSTJPMJ MTXQZWQZWRZ MTXQZWRNXQJ MTXRNXQZSMJ MTXRNXQZSTN

Test phase

QTPTJPMJ RTSMZSMJ RTSTJPMJ RTWQZWQJ MTXQZSMJ MTXRNXQJ QTPMZSTJPMJ RTSMZSTJPMJ RTSTJPTJPTN RTWRNXQZSTN MTXQZWQZSMJ MTXRNXRNXQJ Q_PMZWQJ Q_PTJPTN R_SMZWQJ R_WQZSTN R_WRNXRZ M_XQZWRZ Q_PMZSMZSTN Q_PTJPMZWRZ R_SMZWRNXQJ R_WQZSTJPMJ M_XQZSTJPTN M_XRNXQZWRZ

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117 Chapters 5 and 6

Induction phase

BOG NUP DIR DIR TAF NUP DIR BOG NUP KES TAF BOG DIR DIR DIR KES BOG DIR KES TAF TAF TAF BOG DIR DIR KES TAF TAF BOG DIR TAF BOG NUP KES BOG DIR DIR TAF BOG NUP DIR BOG BOG BOG BOG NUP KES BOG DIR TAF NUP KES TAF TAF BOG DIR DIR KES TAF TAF TAF BOG DIR TAF BOG BOG BOG NUP DIR BOG NUP KES TAF TAF TAF TAF BOG BOG BOG BOG NUP KES TAF TAF BOG DIR TAF BOG NUP KES TAF TAF BOG

Test phase

Grammatical strings

DIR KES BOG

BOG BOG NUP DIR DIR KES TAF BOG BOG BOG BOG NUP DIR DIR DIR KES TAF BOG DIR DIR TAF NUP DIR DIR KES TAF TAF BOG BOG BOG NUP KES TAF BOG DIR DIR DIR KES TAF BOG DIR DIR DIR TAF NUP DIR DIR TAF NUP KES TAF BOG BOG BOG BOG NUP KES TAF BOG DIR DIR DIR TAF BOG NUP DIR DIR DIR TAF BOG NUP KES BOG DIR KES TAF TAF TAF TAF BOG BOG BOG BOG BOG BOG BOG NUP DIR BOG BOG BOG BOG BOG NUP KES BOG BOG BOG NUP KES TAF TAF TAF BOG DIR TAF BOG BOG BOG BOG NUP DIR DIR TAF BOG BOG NUP KES TAF BOG

Ungrammatical strings

DIR KES DIR BOG BOG BOG KES NUP BOG DIR BOG TAF NUP DIR BOG BOG NUP BOG KES BOG BOG KES NUP TAF TAF BOG DIR BOG TAF BOG NUP DIR DIR DIR DIR KES DIR BOG DIR DIR NUP TAF KES BOG BOG BOG BOG BOG NUP BOG DIR BOG BOG KES NUP TAF TAF BOG BOG NUP TAF KES TAF TAF BOG DIR BOG TAF NUP KES TAF BOG DIR DIR DIR NUP TAF KES BOG DIR DIR KES DIR TAF TAF BOG DIR TAF DIR BOG BOG NUP DIR DIR TAF DIR NUP KES TAF BOG BOG BOG BOG NUP BOG KES TAF BOG DIR TAF BOG BOG BOG KES NUP BOG DIR TAF KES TAF TAF TAF TAF BOG DIR TAF NUP TAF KES TAF TAF BOG

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Chapter 5: Completion test

Grammatical

DIR ___ BOG DIR ___ TAF BOG BOG BOG ___ NUP DIR DIR DIR KES ___ BOG DIR TAF ___ NUP DIR BOG BOG ___ NUP KES BOG BOG ___ KES TAF TAF BOG DIR DIR DIR ___ KES BOG DIR DIR DIR TAF ___ DIR DIR ___ NUP KES TAF BOG BOG BOG BOG NUP KES ___ BOG BOG BOG ___ KES TAF TAF BOG DIR DIR DIR KES ___ TAF BOG DIR DIR DIR TAF ___ KES BOG DIR DIR ___ BOG BOG NUP DIR DIR DIR TAF ___ NUP KES BOG BOG ___ BOG BOG BOG BOG NUP DIR BOG BOG NUP KES TAF ___ TAF BOG DIR TAF BOG BOG BOG NUP ___ BOG DIR TAF NUP ___ TAF TAF TAF BOG

Ungrammatical

BOG ___ NUP DIR DIR DIR ___ BOG BOG BOG ___ KES BOG DIR DIR TAF ___ DIR DIR KES TAF ___ BOG BOG BOG NUP ___ TAF BOG DIR ___ DIR KES TAF BOG DIR DIR TAF NUP ___ BOG DIR TAF ___ BOG NUP DIR BOG BOG BOG BOG BOG ___ DIR BOG NUP KES ___ TAF TAF BOG DIR DIR DIR TAF ___ NUP DIR DIR DIR ___ NUP KES TAF BOG DIR KES ___ TAF TAF TAF BOG DIR ___ BOG NUP KES TAF BOG BOG BOG BOG BOG BOG ___ KES BOG BOG ___ BOG BOG NUP KES TAF BOG DIR ___ TAF TAF TAF TAF TAF BOG DIR TAF BOG BOG ___ BOG NUP DIR DIR TAF BOG BOG NUP KES ___ BOG

Chapter 7 Simple grammar

Induction phase

NMSQJSQN NMSQJWMN NMSQPMPMN NMSQPTZ NMSRSQN NMSRZPRJ NMWMJSRJ NMWMJWMN NMWMN NMWMPMN NMWMPMPMN NMWMPTJTZ NMWTJMN NMWTNXTZ NRXMJSRJ NRXMJWMN NRXMN NRXMPMPTZ NRXMPTJTZ NRXMPTNXTZ NRXMPTZ

Complex grammar

Induction phase

NMSQJSQN NMSQJWMN NMSRZNQWMN NMSRZPRJ NMSWQSQN NMSWTPRJ NMWMJSRJ NMWMJWMN NMWMJWTNXTZ NMWMN

NMWTNXMJWMN NMWTNXTNXTZ NMWTNXTZ NRXMJSRJ NRXMJWMJWMN NRXMJWMN NRXMJWTNXTZ NRXMN

NRXTNXMJSRJ NRXTNXMJWMN NRXTNXTNXTZ

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119 Induction phase (continued)

NRXTJMN NRXTJMPMN NRXTJTJMN NRXTJTJTZ NRXTJTNXMN NRXTNXMPMN NRXTNXTJTZ NRXTNXTNXTZ NRXTNXTZ NRXTZ ZQSQJSQN ZQSQJWTZ ZQSQN ZQSQPMN ZQSQPMPTZ ZQSRSQN ZQSRZPQN ZQWMJSQN ZQWMJSRJ ZQWMJWMN ZQWMPTZ ZQWTJMPMN ZQWTJTZ ZQWTNXMN ZQWTZ ZTPQJSQN ZTPQJWTZ ZTPQPMN ZTPQPMPMN ZTPQPTJTZ ZTPQPTNXTZ ZTPRJ ZTPRSQPTZ ZTPRSRJ ZTPRSRSRJ ZTPRSRZPQN ZTPRZPQPMN ZTPRZPRSRJ ZTPRZPRZPRJ

Induction phase (continued)

NRXTNXTZ

NRXTZ ZQSQJSQN ZQSQJWTZ ZQSQN ZQSRZNQWTZ ZQSRZPQN ZQSWQSQN ZQSWQSRJ ZQSWTPRJ ZQWMJSQN ZQWMJSRJ ZQWMJWMN ZQWMJWTNXTZ ZQWTNXMJWMN ZQWTNXMN ZQWTNXTNXMN ZQWTZ

ZTNQSQJWTZ ZTNQSQN ZTNQSWQWTZ ZTNQWMJWMN ZTNQWTNXTZ ZTNQWTZ ZTNTNQSQN ZTNTNQWTZ ZTNTNTPRJ ZTNTPQJWTZ ZTNTPQN ZTNTPWQWMN ZTPQJSQN ZTPQJSRJ ZTPRJ ZTPRZPQN ZTPWQSQN ZTPWQWTZ ZTPWTNQWMN ZTPWTPQN ZTPWTPRJ

Test phase

Grammatical

NMSQN NMSRJ ZTPQN NMSRSRJ NMWTJTZ NRXTJTZ ZQSRSRJ ZTPQPTZ NMSQJSRJ NMWMJWTZ NRXMJWTZ ZQSQJSRJ ZQWMJWTZ ZQWTNXTZ

Grammatical

NMSQN NMSRJ ZTPQN ZTNTPRJ NMSQJSRJ NMSRZPQN NMWMJSQN NMWMJWTZ NRXMJWTZ ZQSQJSRJ ZQSQJWMN ZQSWQWMN ZQWTNXTZ ZTPQJWMN

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Grammatical (continued)

NMSQPMPTZ

NRXMPMPMN ZQSQPMPMN ZTPQPTJMN ZTPRSRSQN NRXTJTNXTZ NRXTNXTJMN ZTPQPTNXMN ZTPRSRZPRJ NRXTNXTNXMN

Grammatical (continued)

ZTNTNQWMN

ZTNTNTPQN ZTNQWMJWTZ ZTNTPQJWMN ZTPWTNQWTZ NMWTNXMJWTZ NRXTNXMJWTZ NRXTNXTNXMN ZQWMJWTNXMN ZQWTNXTNXTZ

Ungrammatical

NWMTZ

ZQMWN NMQSPMN ZQSPQTZ ZWQMPMN ZQWTMJN NMSQWJTZ NMWJMSQN NMTWNXMN NRXJMSQN ZQSQJMWN ZQSRZRPJ ZPTQJSRJ ZTPQWJMN NMWPMTJMN NRMXPTJMN NRTXJMPTZ ZQSQTPJTZ ZWQTJMPTZ ZPTRSQPMN NXRMPTNXMN NRXNTXMPTZ ZTPRPZRSQN ZTPRZPRZQPN

Ungrammatical

ZTNQRSJ NSMQJWTZ NMSTWPQN NMTWNXMN ZQSRZRPJ ZQWMJTWZ ZTNPTWQWTZ NRMXJWMJWTZ NWMTZ

ZQMWN ZTPJQWTZ ZTNTQNSRJ NMWMJWTNMXN NMWTNXTXNMN NRXTNXJMSQN ZQWTNMXJWTZ ZNTQWMN NMSQWSRJ NMWSQWMN ZQWSQWTZ ZSQWTPQN ZTPQWSRJ ZQSRZQNWMN ZTNSQWQWMN