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Interactive Proof Systems and Alternating Time-Space

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INTERACTIVE PROOF SYSTEMS AND ALTERNATING TIMESPACE COMPLEXITY Lance Fortnow Carsten Lund University of Chicago Dept of Computer Science E th Street Chicago IL Abstract We show a rough equivalence b etween alternating timespace complexity and a publiccoin interactive pro of system with the verier having a p olynomial related timespace complexity Sp ecial cases include All of NC has interactive pro ofs with a logspace p olynomialtime publiccoin verier vastly improving the b est previous lower b ound of LOGCFL for this mo del All languages in P have interactive pro ofs with a p olynomialtime publiccoin verier using olog n space All exp onentialtime languages have interactive pro of systems with publiccoin p olynomialspace exp onentialtime veriers To achieve b etter b ounds we show how to reduce a k tap e alternat ing Turing machine to a tap e alternating Turing machine with only a constant factor increase in time and space Intro duction In Chandra Kozen and Sto ckmeyer intro duced alternating Tur ing machines an extension of nondeterministic computation where the Turing 1 Supp orted by NSF Grant CCR 2 Supp orted by a fellowship from the University of Arhus Fortnow Lund machine can make b oth existential and universal moves In Goldwasser Micali and Racko and Babai intro duced interactive pro of systems an extension of nondeterministic computation consisting of two players an in nitely p owerful prover and a probabilistic p olynomialtime verier The prover will try to convince the verier of the validity of some statement However the verier do es not trust the prover and will only accept if the prover manages to convince the verier of the validity of the statement There are some obvious similarities b etween alternating Turing machines and interactive pro of systems In fact Goldwasser and Sipser show the equivalence of interactive pro of systems to a Turing machine alternating b e tween nondeterministic and probabilistic moves However until recently com puter scientists generally b elieved that alternating Turing machines had far more p ower than the interactive pro of systems A series of results by Lund Fortnow Karlo and Nisan and Shamir show that the set of languages accepted by an interactive pro of system equals the class of languages accepted in deterministic p olynomial space Since Chandra Kozen and Sto ckmeyer have shown PSPACE equivalent to the languages accepted by a p olynomialtime alternating Turing machine in this case alternating Turing machine and interactive pro of systems have identical p ower We generalize the work of to show a broader equivalence b etween alternating Turing machines and interactive pro of systems We lo ok at time space complexity rst studied for alternating Turing machines by Ruzzo and for interactive pro of systems by Condon We show a general relationship b etween time and space b ounded alternating Turing machines and time and space b ounded veriers We show that all languages accepted by an interactive pro of system with a tntime snspace b ounded verier can also b e accepted by an alternating Turing machine using tn log tntime and snspace Conversely we show that an interactive pro of system can simulate any tntime snspace alternating Turing machine using a p olynp olytntime and p olysnspace b ounded verier We use this close relationship b etween alternating Turing machine and in teractive pro of systems to show a publiccoin interactive pro of system for all languages in NC with a p olynomialtime logspace verier and a pro of sys tem for all languages in P with a p olynomialtime verier using less then log squared space The previous b est known result by Fortnow and Sipser shows LOGCFL has a publiccoin interactive pro of system with a p olynomial time logspace verier LOGCFL consists of all languages logspace reducible to contextfree languages and is known to lie in NC Interactive Pro ofs We also use these theorems to get strong relationships b etween interac tive pro of systems and deterministic computation similar to the relationships b etween alternating Turing machine and deterministic computation found in With the notable exception of Theorem most of the results in this pap er do not relativize Fortnow and Sipser have shown an oracle A such A that some language in coNP do es not have interactive pro ofs relative to A However our result implies every language in PSPACE has interactive pro ofs Background and Denitions An interactive pro of system consists of a proververier pair P V The prover and verier share a reliable communication tap e and access to an input tap e The verier also has access to his own work tap es and a random bit generator such as a fair coin The prover can b e any function from the messages previously sent to the prover to a p olynomiallength resp onse P and V form an interactive proto col for a language L if If x L then PrP V x accepts If x L then for all P PrP V x accepts Goldwasser Micali and Racko and Babai require the verier com putes in probabilistic p olynomial time and space Lund Fortnow Karlo and Nisan showed an interactive pro of system for every language in the p olynomialtime hierarchy Using their techniques Shamir showed that the set of languages accepted by these interactive pro of systems coincides with the class of languages decidable in deterministic p olynomial space In this pa p er we will examine the complexity of interactive pro of systems with veriers having diering restrictions on time and space In general the verier may use private coins where the prover do es not know what the coin tosses were A publiccoin interactive pro of system allows the prover access to the veriers coin Equivalently we require the veriers messages to consist of exactly the veriers coin tosses since the previous round Goldwasser and Sipser show the class of languages accepted by interactive pro ofs with a p olynomialtime verier do es not dep end on whether the verier uses public or private coins However a dierence b etween private and public coins do es seem to hold for time and space b ounded veriers Condon show that interactive pro of systems with private coins and p olynomialtime logspace veriers can simu late any standard interactive proto col and thus accept any PSPACE language Fortnow Lund However a deterministic p olynomialtime Turing machine can simulate any publiccoin interactive pro of system with a p olynomialtime logspace verier Corollary Thus assuming P PSPACE private coins are strictly more p owerful than public coins in an interactive pro of system with a time and space b ounded verier In this pap er we study the complexity of the publiccoin interactive pro of system mo del In this pap er we contrast the p ower of interactive pro of systems with al ternating Turing machines as develop ed by Chandra Kozen and Sto ckmeyer An alternating Turing machine is a generalization of a nondeterministic machine where the machine may make b oth existential and universal choices A string is accepted by an alternating Turing machine M if there exists a rst existential choice such that for all rst universal choices there exists a second existential choice such that M accepts See for a complete technical denition Let n represent the length of the input string Let ATIMEtn b e the set of languages accepted by an alternating Turing machine running in time O tn Let ASPACEsn b e the analogous class for space Chandra Kozen and Sto ckmeyer show the following relationships S sn For sn log n ASPACEsn DTIMEc c For tn n ATIMEtn DSPACEtn ATIMEtn This implies for example that P ASPACElog n and PSPACE S k ATIMEn k We assume throughout this pap er that tn sn log n nondecreasing and fully time and space constructible in the following strong sense There exists a deterministic Turing machine M such that given m written in binary sm will output the pair tm where tm is written in binary Furthermore M uses O tm time and O sm space Note most natural functions fulll these conditions We also assume all inputs are elements of f g We say that t s is timespace constructible We dene the following time and space classes generalizing the TISP ter minology intro duced by Bruss and Meyer to describ e deterministic compu tation b ounded in b oth time and space A language L is in ATISPtn sn if some alternating Turing machine M accepts L and M runs in time O tn and space O sn on every computation path A language L is in IPTISPtn sn if there exists an publiccoin in teractive pro of for L such that the verier uses at most O tn time and Interactive Pro ofs O sn space on every computation path with every p ossible prover We de ne IPTIMEtn and IPSPACEsn analogously For IPSPACEsn we restrict the interactive pro ofs systems to having nite computation paths If an alternating Turing machine or a verier ever enters the same cong uration twice then it will have an innite computation path Thus we may always assume sn log tn Ruzzo rst studied time and space b ounded alternating Turing machine k k k complexity showing ATISPlog n log n NC for all k where NC is the set of languages accepted by a logspace uniform circuit family of p olynomial k size and log n depth Condon rst studied the complexity class IPTISPt s under the name BCTIMESPACE In she showed that IPTISPpol y tn log tn DTIMEpol y tn S k IPTISPn log n Fortnow and Sipser studied the class BPNL k They show LOGCFL BPNL P where LOGCFL NC is the class of languages logspace reducible to contextfree languages Restricted Alternating Turing Machines We will
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