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K.l. Mcmillan - One of the best experts on this subject based on the ideXlab platform.
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A Structural Induction Theorem for Processes
Information and Computation, 1995Co-Authors: R. P. Kurshan, K.l. McmillanAbstract:This paper deals with the formal verification of finite state systems that hav an arbitrary number of isomorphic components. We present a technique for inductively generalizing tests on a system of fixed size in order to show that a system of arbitrary size satisfies a given specification. This makes it possible to use finite state verification systems, such as COSPAN, to verify parameterized protocols. The method also may be useful for verifying systems of fixed but large size, since it reduces the size of the system that must be checked automatically. The basis of the method is a Structural Induction theorem for processes, which is stated and proved in this paper. The theorem applies to a variety of process formalisms satisfying simple algebraic laws. We give examples of proofs using the calculus of communicating systems (CCS) and the s/r model.
R. P. Kurshan - One of the best experts on this subject based on the ideXlab platform.
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A Structural Induction Theorem for Processes
Information and Computation, 1995Co-Authors: R. P. Kurshan, K.l. McmillanAbstract:This paper deals with the formal verification of finite state systems that hav an arbitrary number of isomorphic components. We present a technique for inductively generalizing tests on a system of fixed size in order to show that a system of arbitrary size satisfies a given specification. This makes it possible to use finite state verification systems, such as COSPAN, to verify parameterized protocols. The method also may be useful for verifying systems of fixed but large size, since it reduces the size of the system that must be checked automatically. The basis of the method is a Structural Induction theorem for processes, which is stated and proved in this paper. The theorem applies to a variety of process formalisms satisfying simple algebraic laws. We give examples of proofs using the calculus of communicating systems (CCS) and the s/r model.
Narayanan, Krishna Shankara - One of the best experts on this subject based on the ideXlab platform.
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SD-Regular Transducer Expressions for Aperiodic Transformations
'Institute of Electrical and Electronics Engineers (IEEE)', 2021Co-Authors: Dartois Luc, Gastin Paul, Narayanan, Krishna ShankaraAbstract:International audienceFO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star-free expressions = aperiodic languages) result of Schützenberger. Our result also generalizes a lesser known characterization by Schützenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite words captured by an aperiodic deterministic two-way transducer can be described with an SD-regular transducer expression (SDRTE). An SDRTE is a regular expression where Kleene stars are used in a restricted way: they can appear only on aperiodic languages which are prefix codes of bounded synchronization delay. SDRTEs are constructed from simple functions using the combinators unambiguous sum (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned. In order to construct an SDRTE associated with an aperiodic deterministic two-way transducer, (i) we concretize Schützenberger's SD=AP result, by proving that aperiodic languages are captured by SD-regular expressions which are unambiguous and stabilising; (ii) by Structural Induction on the unambiguous, stabilising SD-regular expressions describing the domain of the transducer, we construct SDRTEs. Finally, we also look at various formalisms equivalent to SDRTEs which use the function composition, allowing to trade the k-chained star for a 1-star
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Regular Transducer Expressions for Regular Transformations
2018Co-Authors: Dave Vrunda, Gastin Paul, Narayanan, Krishna ShankaraAbstract:Functional MSO transductions, deterministic two-way transducers, as well as streaming string transducers are all equivalent models for regular functions. In this paper, we show that every regular function, either on finite words or on infinite words, captured by a deterministic two-way transducer, can be described with a regular transducer expression (RTE). For infinite words, the transducer uses Muller acceptance and $\omega$-regular look-ahead. \RTEs are constructed from constant functions using the combinators if-then-else (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product, the 2-chained Kleene-iteration and the 2-chained omega-iteration. Our proof works for transformations of both finite and infinite words, extending the result on finite words of Alur et al.\ in LICS'14. In order to construct an RTE associated with a deterministic two-way Muller transducer with look-ahead, we introduce the notion of transition monoid for such two-way transducers where the look-ahead is captured by some backward deterministic B\"uchi automaton. Then, we use an unambiguous version of Imre Simon's famous forest factorization theorem in order to derive a "good" ($\omega$-)regular expression for the domain of the two-way transducer. "Good" expressions are unambiguous and Kleene-plus as well as $\omega$-iterations are only used on subexpressions corresponding to \emph{idempotent} elements of the transition monoid. The combinator expressions are finally constructed by Structural Induction on the "good" ($\omega$-)regular expression describing the domain of the transducer
Gastin Paul - One of the best experts on this subject based on the ideXlab platform.
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SD-Regular Transducer Expressions for Aperiodic Transformations
'Institute of Electrical and Electronics Engineers (IEEE)', 2021Co-Authors: Dartois Luc, Gastin Paul, Narayanan, Krishna ShankaraAbstract:International audienceFO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star-free expressions = aperiodic languages) result of Schützenberger. Our result also generalizes a lesser known characterization by Schützenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite words captured by an aperiodic deterministic two-way transducer can be described with an SD-regular transducer expression (SDRTE). An SDRTE is a regular expression where Kleene stars are used in a restricted way: they can appear only on aperiodic languages which are prefix codes of bounded synchronization delay. SDRTEs are constructed from simple functions using the combinators unambiguous sum (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned. In order to construct an SDRTE associated with an aperiodic deterministic two-way transducer, (i) we concretize Schützenberger's SD=AP result, by proving that aperiodic languages are captured by SD-regular expressions which are unambiguous and stabilising; (ii) by Structural Induction on the unambiguous, stabilising SD-regular expressions describing the domain of the transducer, we construct SDRTEs. Finally, we also look at various formalisms equivalent to SDRTEs which use the function composition, allowing to trade the k-chained star for a 1-star
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SD-Regular Transducer Expressions for Aperiodic Transformations
2021Co-Authors: Dartois Luc, Gastin Paul, Krishna, Shankara NarayananAbstract:FO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the long standing open problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star free expressions = aperiodic languages) result of Sch\"utzenberger. Our result also generalizes a lesser known characterization by Sch\"utzenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite words captured by an aperiodic deterministic two-way transducer can be described with an SD-regular transducer expression (SDRTE). An SDRTE is a regular expression where Kleene stars are used in a restricted way: they can appear only on aperiodic languages which are prefix codes of bounded synchronization delay. SDRTEs are constructed from simple functions using the combinators unambiguous sum (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned. In order to construct an SDRTE associated with an aperiodic deterministic two-way transducer, (i) we concretize Sch\"utzenberger's SD=AP result, by proving that aperiodic languages are captured by SD-regular expressions which are unambiguous and stabilising; (ii) by Structural Induction on the unambiguous, stabilising SD-regular expressions describing the domain of the transducer, we construct SDRTEs. Finally, we also look at various formalisms equivalent to SDRTEs which use the function composition, allowing to trade the k-chained star for a 1-star
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Regular Transducer Expressions for Regular Transformations
2018Co-Authors: Dave Vrunda, Gastin Paul, Narayanan, Krishna ShankaraAbstract:Functional MSO transductions, deterministic two-way transducers, as well as streaming string transducers are all equivalent models for regular functions. In this paper, we show that every regular function, either on finite words or on infinite words, captured by a deterministic two-way transducer, can be described with a regular transducer expression (RTE). For infinite words, the transducer uses Muller acceptance and $\omega$-regular look-ahead. \RTEs are constructed from constant functions using the combinators if-then-else (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product, the 2-chained Kleene-iteration and the 2-chained omega-iteration. Our proof works for transformations of both finite and infinite words, extending the result on finite words of Alur et al.\ in LICS'14. In order to construct an RTE associated with a deterministic two-way Muller transducer with look-ahead, we introduce the notion of transition monoid for such two-way transducers where the look-ahead is captured by some backward deterministic B\"uchi automaton. Then, we use an unambiguous version of Imre Simon's famous forest factorization theorem in order to derive a "good" ($\omega$-)regular expression for the domain of the two-way transducer. "Good" expressions are unambiguous and Kleene-plus as well as $\omega$-iterations are only used on subexpressions corresponding to \emph{idempotent} elements of the transition monoid. The combinator expressions are finally constructed by Structural Induction on the "good" ($\omega$-)regular expression describing the domain of the transducer
Dartois Luc - One of the best experts on this subject based on the ideXlab platform.
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SD-Regular Transducer Expressions for Aperiodic Transformations
2021Co-Authors: Dartois Luc, Gastin Paul, Krishna, Shankara NarayananAbstract:FO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the long standing open problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star free expressions = aperiodic languages) result of Sch\"utzenberger. Our result also generalizes a lesser known characterization by Sch\"utzenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite words captured by an aperiodic deterministic two-way transducer can be described with an SD-regular transducer expression (SDRTE). An SDRTE is a regular expression where Kleene stars are used in a restricted way: they can appear only on aperiodic languages which are prefix codes of bounded synchronization delay. SDRTEs are constructed from simple functions using the combinators unambiguous sum (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned. In order to construct an SDRTE associated with an aperiodic deterministic two-way transducer, (i) we concretize Sch\"utzenberger's SD=AP result, by proving that aperiodic languages are captured by SD-regular expressions which are unambiguous and stabilising; (ii) by Structural Induction on the unambiguous, stabilising SD-regular expressions describing the domain of the transducer, we construct SDRTEs. Finally, we also look at various formalisms equivalent to SDRTEs which use the function composition, allowing to trade the k-chained star for a 1-star
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SD-Regular Transducer Expressions for Aperiodic Transformations
'Institute of Electrical and Electronics Engineers (IEEE)', 2021Co-Authors: Dartois Luc, Gastin Paul, Narayanan, Krishna ShankaraAbstract:International audienceFO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star-free expressions = aperiodic languages) result of Schützenberger. Our result also generalizes a lesser known characterization by Schützenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite words captured by an aperiodic deterministic two-way transducer can be described with an SD-regular transducer expression (SDRTE). An SDRTE is a regular expression where Kleene stars are used in a restricted way: they can appear only on aperiodic languages which are prefix codes of bounded synchronization delay. SDRTEs are constructed from simple functions using the combinators unambiguous sum (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned. In order to construct an SDRTE associated with an aperiodic deterministic two-way transducer, (i) we concretize Schützenberger's SD=AP result, by proving that aperiodic languages are captured by SD-regular expressions which are unambiguous and stabilising; (ii) by Structural Induction on the unambiguous, stabilising SD-regular expressions describing the domain of the transducer, we construct SDRTEs. Finally, we also look at various formalisms equivalent to SDRTEs which use the function composition, allowing to trade the k-chained star for a 1-star
