Synthesis for Unbounded Bit-vector Arithmetic

Transcription

1 Synthesis fo Unbounded Bit-vecto Aithmetic Andej Spielmann and Vikto Kuncak School of Compute and Communication Sciences (I&C) École Polytechnique Fédéale de Lausanne (EPFL), Switzeland Abstact. We popose to descibe computations using QFPAbit, a language of quantifie-fee linea aithmetic on unbounded integes with bitvecto opeations. We descibe an algoithm that, given a QFPAbit fomula with input and output vaiables denoting integes, geneates an efficient function fom a sequence of inputs to a sequence of outputs, wheneve such function on integes exists. The stating point fo ou method is a polynomial-time tanslation mapping a QF- PAbit fomula into the sequential cicuit that checks the coectness of the input/output elation. Fom such a cicuit, ou synthesis algoithm poduces solved cicuits fom inputs to outputs that ae no moe than singly exponential in size of the oiginal fomula. In addition to the geneal synthesis algoithm, we pesent techniques that ensue that, fo example, multiplication and division with lage constants do not lead to an exponential blowup, addessing a pactical poblem with a pevious appoach that used the MONA tool to geneate the specification automata. 1 Intoduction Ove the past decades, a numbe of decision pocedues has been developed and integated into satisfiability modulo theoy (SMT) solves. Among the pimay uses of this technology so fa has been veification and eo finding. Recently, eseaches stated using this technology fo softwae synthesis [11]. In the line of wok on complete functional synthesis, eseaches poposed to genealize decision pocedues fo infinite domains to synthesis pocedues [7]. The basic idea is to descibe fagments of code using fomulas in a decidable logic. Such a fomula specifies a elation between inputs and outputs. A synthesis pocedue then compiles this fomula into a pogam that maps inputs into outputs, and whose behavio coesponds to invoking a decision pocedue on that paticula constaint. The esulting pogam is guaanteed to satisfy the specification. Synthesis pocedues have been descibed fo, e.g., paameteized Pesbuge aithmetic [7], using a constuctive vesion of quantifie elimination. Fo domains such as intege aithmetic, automata-based methods can have a numbe of advantages compaed to quantifie elimination, including the ability to suppot opeations on unbounded bitvectos. Motivated by these obsevations, in elated pevious wok [4] eseaches consideed synthesis of specifications expessed in weak monadic second-ode logic of one successo (WS1S), which is equivalent to Pesbuge aithmetic with bitwise logical opeatos. In contast to automata-based appoaches to eactive synthesis [1, 2, 5, 8], this appoach uses automata to encode elations on integes, 1

2 which means that the causality estiction of Chuch s synthesis poblem does not apply. The synthesized function fo this poblem cannot always be given as a one-pass finite-state tansduce. The appoach [4] synthesizes a two pass tansduce, whee the fist pass geneates a sequence that abstacts the tee of possible executions, wheeas the second pass pocesses this sequence backwads to choose an acceptable sequence of outputs. The pevious implementation of this appoach used the MONA tool [6] to tansfom the given specification fomula into an automaton accepting a sequence of bits of combined input/output vectos. This implementation theefoe suffeed fom the explicit-state epesentation used by MONA. The most stiking poblem is multiplication by constants, whee a subfomula x = c y leads to cicuits of size popotional to c and thus exponential in the binay epesentation of c. To ovecome the difficulties with explicit-state epesentation used in the implementation of [4], in this pape we investigate an appoach that diectly uses cicuit epesentations fo both specifications and implementations. To avoid the non-elementay wost-case complexity [12] of tansfoming WS1S fomulas to automata, we use as ou specification language quantifie-fee Pesbuge aithmetic with bitvecto opeations [9], denoted QFPAbit. We descibe a polynomial-time tansfomation between sequential cicuits and QFPAbit. We then pesent an algoithm fo tansfoming sequential cicuit epesentations of input/output elations into systems of sequential cicuits that map inputs into outputs. The wost-case complexity of ou tanslation is bounded by a singly exponential function of the specification cicuit size. Building on this geneal esult, we identify optimizations that exploit the stuctue of specifications to educe the potential fo exponential explosion. Ou pototype implementation confims the impoved asymptotic behavio of this synthesis appoach, and is available fo download fom Additional details of ou constuctions ae available in the technical epot [10]. 2 Peliminaies 2.1 Quantifie-Fee Pesbuge Aithmetic with Bit-vecto Logical Opeatos Pesbuge Aithmetic with Bit-vecto Logical Opeatos is the stuctue of integes with addition and bit-vecto logical opeations acting on the binay two s complement epesentation of the integes. Let V be a finite set of vaiables. Let c Z, x V, and % anges ove =,, <,, >,. The following is the gamma of QFPAbittems and fomulas. T := c x T + T ct T T T T T F := T % T F F F F F F F F F Vaiables ange ove the set of integes Z. The bitvecto logical opeatos act on the two s complement encoding of numbes [9]: x k,...x 0 Z = 2 k x k + Σ k 1 i=0 2i x i. A popety of this encoding is that eplicating the most significant bit does not change the value. This justifies ou definition of the bit-vecto opeatos because fo any two 2

3 numbes we can always find encodings that have the same length. By the two s complement encoding of a numbe, we mean its shotest possible encoding. Given a QF- PAbit fomula F ove the set of vaiables V = {x 1,...x n }, we say that a valuation val : V Z satisfies F if F is tue when each occuence of a vaiable x i evaluates to val(x i ). We say that F is satisfiable if thee exists a valuation that satisfies F. Note that the identity x = x + 1 holds fo all x. We can use QFPAbit fomulae to define languages ove Σ = {0, 1} n. Let F be a QFPAbit fomula ove the vaiables V = {x 1,...x n }. Let w Σ + be a wod of length m. By w(j) denote the j-th lette of w, indexing fom 0, so that the initial lette is denoted w(0). Each w(j) is a vecto of dimension n, let w i (j) denote the the i- th coodinate of w(j). Define a valuation val w : V Z by val w (x i ) = w i (m 1),...w i (0) Z. Thus, in the matix whose columns ae the lettes of w, the i-th ow epesents the encoding of val w (x i ) with the most significant bit coming fist. The language defined by the fomula is L(F ) = {w Σ + val w satisfies F }. 2.2 Sequential Cicuits A combinational boolean cicuit K is a pai (G, σ) whee G is a finite diected acyclic gaph and σ : U {AND, OR, NOT } is a labeling function such that U is the set of vetices of G whose in-degee is geate than zeo. We equie that wheneve σ(x) = NOT then x has in-degee of one. We call the vetices in U the gates. We denote the vetices of in-degee zeo I and call them inputs; we denote the vetices of out-degee zeo O, and call them outputs. Given a boolean valuation i : I {tue, false}, we define a valuation v on all vetices of G as follows: i(x), if x I v(γ(x)), v(x) = if x U σ(x) = NOT y Γ (x) v(y), if x U σ(x) = AND v(y), if x U σ(x) = OR y Γ (x) whee γ(x) denotes the single neighbo of x connected to it by an edge diected towads x and Γ (x) denotes the set of all neighbos of x connected to it by edges diected towads x. We call the values of v on O the output values of K fo input i. The values of the outputs of a combinational boolean cicuit, defined above, depend only on a single set of inputs and can be epesented in a tuth table. We next eview (clocked) sequential cicuits, which ae equivalent to deteministic finite automata but compactly epesent the set of states and the tansition function. A clocked sequential cicuit (o SC, fo shot) is a tuple (K, M, stoe, load, init) whee K is a combinational boolean cicuit with inputs and outputs I and O; M is a set of D-type flip-flops; stoe : M O; load : M I; init : M {tue, false}. 3

4 The load and stoe functions descibe how the data input of each flip-flop is connected to a unique output of K and how the Q-output of each flip-flop is connected to a unique input of K. Such a backwad-connected output-input pai will be denoted as a state vaiable. We call the inputs of K that ae not in the image of load the input vaiables and call the outputs of K that ae not in the image of stoe the output vaiables. The SC woks in clock pulses. It takes as input a steam that fo evey clock pulse contains values fo all input vaiables, and poduces as output a steam that fo evey clock pulse contains values of all the output vaiables, computed by K. In evey clock pulse, K is povided with input values and it computes output values. The values fo K s inputs coesponding to state vaiables ae loaded fom the flip-flops and the values fo its inputs coesponding to input vaiables ae povided by the input steam. Some of the values of K s outputs ae stoed in the flip-flops fo the use in the next clock cycle, as detemined by stoe. The values stoed in the flip-flops at the beginning of the fist clock cycle ae called initial values of the state vaiables and they ae given by init. Notice that a cicuit with n input vaiables and m output vaiables can be viewed as a machine that, given a wod fom ({0, 1} n ) +, poduces a wod of the same length in ({0, 1} m ) +. We can also use a SC to ecognize a language. Definition 1. Let C be a SC with one output vaiable o and n input vaiables. We say that C accepts the wod w {0, 1} n if the value of o in the last cycle is 1 when the cicuit is given w as input, one lette at each clock cycle. The language of C is L(C) = {w {0, 1} n C accepts w}. Some of the standad finite state machine opeations can be efficiently pefomed on the sequential cicuit epesentations. Given a SC C with input vaiables v 1,...v n, state vaiables q 1,...q n and output o, and a SC C that uses the same input vaiables v 1,...v n and has state vaiables q 1,...q n and output o, we can constuct a cicuit C by simply appending a NOT gate at o and making the output of the NOT gate the output of C. Similaly, we can constuct cicuits C C and C C by connecting the outputs of C and C to an appopiate logical gate, whose output will become the output of the composite cicuit. It can easily be seen that 1) L( C) = ({0, 1} n ) + \L(C); 2) L(C C ) = L(C) L(C ); 3) L(C C ) = L(C) L(C ). 3 Tanslations Between QFPAbit and Sequential Cicuits This section establishes coespondence between QFPAbit and sequential cicuits by poviding tanslations in both diections that maintain a close coespondence between the accepted languages. 3.1 Reduction fom QFPAbit to Sequential Cicuits Since we have aleady shown how to constuct boolean combinations of sequential accepto cicuits, it is enough to find a set of basic QFPAbit fomulae out of which all QFPAbit fomulae can be built using logical connectives, and then show how these basic fomulae can be tanslated to SCs. 4

5 Definition 2. Let w Σ + with Σ = {0, 1} n as usually. Suppose w 1 (0) w 1 (1) w 1 (m) w =... w n (0) w n (1) w n (m) Let S {1,...n} be non-empty. We define the pojection of w onto the coodinates S to be the sting w S = w S (0)...w S (m), whee w S (i) is the column vecto (w j (i)) j S {0, 1} S. Fo a language L Σ +, we define the pojection of L onto the coodinates S to be the language L S = {w S w L}. Note that L S is a language ove the alphabet {0, 1} S. Evey QFPAbit fomula is a boolean combination of atomic fomulae of the fom T 1 %T 2 whee T 1 and T 2 ae tems and % {=,, <,, >, }. We will now show how to tansfom any fomula F into a new one whee the atoms will be of a moe esticted fom. The new fomula will have moe vaiables than F, but when pojected onto the vaiables occuing in F thei languages will be the same. We apply the following sequence of tansfomations: 1. Replace all atomic elations by equalities and stict less-than inequalities using the fact that T 1 < T 2 if and only if T 1 + ( 1)T 2 < Remove all instances of multiplication by constants othe than 1 and powes of two by exploiting the fact that any tem of the fom ct is equal to a sum of tems of the fom 2 k T coesponding to c s two s complement encoding. 3. Remove all instances of multiplication by 1 by eplacing evey sub-tem of the fom ( 1)T by T + 1. This equivalence follows easily fom the definition of the two s complement encoding. 4. Move all additions to sepaate conjuncts on the highest level of the fomula by eplacing evey occuence of T 1 + T 2 by a fesh vaiable s and adding conjuncts s = x + y, x = T 1 and y = T 2 to the fomula, whee x and y ae also fesh vaiables. 5. Move all multiplications by a constant 2 k, which ae the only multiplications now left in the fomula, to conjuncts on the highest level of the fomula by eplacing evey occuence of 2 k T by a fesh vaiable x and adding x = 2 k y and y = T as conjuncts to the fomula, whee y is anothe fesh vaiable. 6. Replace evey additive occuence of an intege constant c inside a lage tem by a fesh vaiable y c and add a conjunct y c = c to the fomula. Let us call the fomula that we obtain G. It has size that is polynomial in the size of F and and it consists only of atoms of the following five foms: (i) T < 0; (ii) T 1 = T 2 ; (iii) y = c; (iv) x = 2 k t; (v) s = x + y, whee x, y, s and t ae vaiables, c is an intege constant and T, T 1, T 2 ae tems that contain exclusively vaiables and bit-vecto logical opeatos. It is easy to constuct SCs fo atoms of each of these fou foms. Fo details of these constuctions along with cicuit diagams, see ou technical epot [10]. The geneal flavo of these cicuits is that they compae steams of binay digits. The most complicated case is (iv), whee the cicuit compaes a binay steam to a vesion of itself 5

6 shifted by a constant numbe of bits. Each of the sub-cicuits fo cases (i),(ii) and (v) has only a constant numbe of state vaiables. In case (iii), the numbe of state vaiables is popotional to the logaithm of the constant c and in case (iv) it is popotional to k. Finally, we compose the patial specification cicuits by boolean opeations to find a SC fo G. The coectness of this synthesis pocedue is expessed in the following theoem. Theoem 1. Let C F be the cicuit obtained fom a QFPAbit fomula F using the above synthesis pocedue. Let V be the set of vaiables occuing in F. Then L(C F ) V = L(F ). Moeove, both the the numbe of gates of C F and the unning time of the synthesis pocedue ae polynomial in the numbe of symbols of F. The numbe of input vaiables of C is the same as that of F and the numbe of C s state vaiables is popotional to the numbe of symbols of F. 3.2 Reduction fom Sequential Cicuits to QFPAbit Let C be a sequential cicuit with an undelying combinational cicuit K = (G, σ), n input vaiables {v 1,...v n }, m state vaiables {q 1,...q m } and output vaiables {o 1,...o l }. Let I : {q 1,...q m } {0, 1} be the initial assignment of values to the state vaiables. Let U be the set of all gates of K othe than those coesponding to the output vaiables and state vaiables of C. We will petend that the elements of U can be used as identifies fo QFPAbit vaiables and constuct a QFPAbit fomula with vaiables {v 1,...v n, q 1,...q m, o 1,...o l } U, such that fo evey satisfying assignment, the two s complement encodings of the values of the vaiables descibes the evolution of the values of the coesponding vaiables and gates in a un of C. Although the QFPAbit vaiables have the same names as the vaiables and gates of the cicuit, it should be clea fom the context which ones do we mean. We will efe to the values of the gates and inputs of the automaton in the k-th clock cycle by q 1 (k),...q m (k),v 1 (k),...v n (k), o 1 (k)...o l (k) and x(k) fo all x U. In the cycle when the inputs ae q 1 (i),...q m (i), v 1 (i),...v n (i), the values of all the gates in U will be x(k), the output vaiables will be o 1 (i),...o l (i) and the outputs coesponding to state vaiables at that cycle will be denoted q 1 (i+1),...q m (i+1), because they seve as inputs fo the next cycle. We stat the numbeing of clock cycles fom 0. We will be abusing notation slightly by witing σ(v)(x 1,..., x k ) fo some gate v and boolean values x 1,..., x k to mean the application of the boolean function epesented by σ(v) to x 1,..., x k. Then fo all j {1,..., m}, k {1,..., l}, x U and all i {0,..., N 1} whee N is the length of the input wod, the un of C on that input wod is chaacteized by the following fou equations: q j (0) = I(q j ) (1) x(i) = σ(x)(γ (x)(i)) (2) o k (i) = σ(o k )(Γ (o k )(i)) (3) q j (i + 1) = σ(q j )(Γ (q j )(i)) (4) 6

7 whee, just like in ou definition of a combinational cicuit, Γ (v) denotes the pat of the neighbohood of a gate v connected to it with incoming edges, and Γ (v)(i) denotes a vecto of values of these nodes in clock cycle i. We next build a QFPAbit fomula fo which evey satisfying evaluation is such that the evese of the bit-sequences of the values it assigns to the vaiables confom to the above conditions. Since C teats all numbes as stating with the most significant bit, in ou QFPAbit epesentation this will be evesed and hence x(0), q j (0) and o k (0) will efe to the least significant bits of the encoding of the values of the vaiables. Fo any gate v of K, let σ v be the fomula obtained by applying the bit-vecto logical opeato coesponding to σ(v) to the vaiables in Γ (v). Then the following fomula can be used to descibe the evolution of the digits of q j : q j = 2σ qj I(q j ) The justification is as follows. Taking the bitwise disjunction of a numbe with 1 o 0 peseves all the digits except the least significant one, which is set to 1 o 0 espectively. Multiplication by 2 induces a shift to the left of the two s complement encoding of a numbe. Hence the above fomula establishes that evey bit of q j is equal to the next bit of σ qj except fo the fist (least significant) one, which is equal to I(q j ). This ensues that equations (1) and (4) ae satisfied. Similaly, the fomulas o j = σ oj and x = σ x asset that the evese binay encodings of o j and x, fo some x U, coespond to thei values in the un of C on the given input as descibed by equations (3) and (2). Since the most significant digit in a two s complement encoding can be eplicated without changing the value of the epesented numbe, QFPAbit fomulas have the popety that the last lette of a wod in a fomula s language can be epeated abitaily many times to obtain anothe wod inside the language. In the undelying cicuit, this would tanslate to a blindness towads the epetition of the initial input lette, which is a popety that not all cicuits have. In geneal we cannot find a fomula whose language contains exactly those wods whose evese encodes a un of the cicuit. The way to teat this poblem is to constuct a fomula that contains a clause saying the vaiables ae only simulating the cicuit fo a finite numbe of steps and then ae allowed to deviate. That way we obtain a fomula fo which to evey possible satisfying evaluation coesponds a wod descibing the un of the cicuit. Howeve, each such valuation will also epesent an infinite numbe of longe incoect desciptions of a un of the cicuit. Fo succintness, let qj 2 σ qj I(q j ). Let y be a fesh vaiable and conside the fomula [ m F C 1 + ((y 1) y) = 2y y > 1 j=1 (q j (y ] 1)) = ( qj (y 1)) [ l j=1 (o j (y ] 1)) = ( σ oj (y 1)) [ x U (x (y 1)) = ( σ x (y 1)) ]. The subfomula 1 + ((y 1) y) = 2y y > 1 assets that y is a powe of two, say y = 2 k, and that k is at least 1. Theefoe the two s complement encoding of (y 1) is 0,..., 0, 1,..., 1 Z with an abitay numbe of zeos and exactly k ones. So the clauses 7

8 of the fom (T 1 (y 1)) = (T 2 (y 1)) asset that the k least significant digits of T 1 and T 2 ae the same. The est of the digits can be abitay. Theoem 2. Fo any given satisfying valuation of F C, y = 2 k fo some k. The fist k bits, pesented in the evese ode, of the bit-sequences coesponding to two s complement encodings of q 1,...q m and o 1...o l descibe the evolution of the values of those vaiables thoughout the fist k clock cycles of the un of C on the input wod given by the evese two s complement encodings of v 1,...v n, as specified by the equations (1)-(4). Now we show how this tanslates to accepto cicuits defining a language: Theoem 3. Suppose C is a SC with one output o and let F C F C o < 0. Then L(F C ) if and only if L(C) Poof. The clause o < 0 is tue if and only if the fist digit of o is one. It follows fom the above discussion of F C that fo evey wod w of length k poviding encoding of values fo v 1,..., v n, thee exist infinitely many satisfying evaluations fo F C unde which y = 2 k and the evese encoding of the values of the vaiables descibes the un of C on the evese of w. Now suppose that C accepts w. This happens if and only if in the last, k-th, clock cycle the value of the output bit is one. But this is if and only if the fist digit of the value of o in F C is one. Theefoe the descibed evaluations satisfy F C if and only if C accepts w. This means that the language of C is non-empty if and only if F C is satisfiable. To summaize, we have descibed polynomial-size tanslations between QFPAbit and sequential cicuits going both ways. Fo evey QFPAbit fomula we can constuct a sequential cicuit ecognizing the same language. Fo evey sequential cicuit we can constuct a QFPAbit fomula that contains vaiables epesenting inputs, outputs and state vaiables of the cicuit, and it is satisfied only by valuations that assign these vaiables values whose binay encoding in evese descibes an initial potion of the evolution of the cicuit s vaiables duing a un. If the cicuit has only one output then it is an accepto cicuit and in this case we can constuct a QFPAbit fomula which is satisfiable if and only if the language of the SC is non-empty. Moeove, the fomula will accept a language such that fo evey wod w in this language, an initial pat of w pojected onto the input vaiables and evesed is a wod in the language of the SC. 4 Fom Specification Cicuits to Tansduce Cicuits Given a specification witten as a QFPAbit fomula, we have shown how to build a specification cicuit of a size linea in the size of the fomula. Povided that the vaiables of the fomula, and thus the inputs of the automaton ae patitioned into two goups, ī and ō, intepeted as the inputs and the outputs of the synthesized function, we will now show how to constuct a set of cicuits that will wok as a tansduce, i.e. given a wod fom the ī-pojection of the language, poduce an output wod fom the ōpojection of the language such that togethe they satisfy the specification, if such an output wod exists. The stuctue of ou algoithm is simila to the one pesented in [4]. Ou use of the wod tansduce does not efe to the taditional notion of Finite State 8

9 Tansduces, but to a moe complicated machine with the following main featues. Ou tansduce eads the whole input twice. The fist time fom the beginning to the end to geneate the exhaustive un of the pojection of the specification cicuit onto the input vaiables, and the second time backwads, detemining concete states and output lettes within the exhaustive un. In the meantime it uses an amount of memoy popotional to the length of the input. This allows us to expess functions fo which it is not possible to detemine the output befoe eading the entie input, which is needed to obtain complete synthesis fo QFPAbit. In contast to [4], we will be using sequential cicuits instead of automata. This moe concete implementation allows us to pefom an optimization that will ensue that the pesence of lage intege constants in the fomula does not necessaily cause a blow-up in the size of the tansduce popotional to the value of that constant, as was the case with the pevious appoach. Moeove, even if a state-space expansion does occu, the size of ou cicuits is guaanteed to be singly-exponential in the size of the specification fomula. No such bound on the size of the automata was povided in [4]. In Section 4.2 we study two moe optimization techniques - how to exploit the cicumstance when the specification fomula is eithe a conjunction o a disjunction of sub-fomulas to build the tansduce as a composition of smalle tansduces. Definition 3. Given a (non-)deteministic automaton A = (Σ V, Q, init, F, T ) ove vaiables V and a set I V, the pojection of A to I, denoted by A I, is the nondeteministic automaton (Σ I, Q, init, F, T I ) with T I = {(q, σ I, q ) Q Σ I Q σ Σ V.(q, σ, q ) T σ I = σ I }. Since it is natual to view a sequential cicuit as a DFA, we also allow ouselves to talk about pojections of sequential cicuits. Definition 4. The exhaustive un ρ of an automaton A = (Σ, Q, init, F, T ) on a wod w Σ is a sequence of sets of states S 1,...S w +1 such that (i) S 1 = init and (ii) fo all 1 w, S i+1 = {q Q q S i.(q, w i, q ) T }. Suppose the specification cicuit is a sequential cicuit C with input vaiables ī ō, state vaiables q and one output vaiable detemining the acceptance. Hee by each of ī, ō and q we actually mean vectos of vaiables wide n, l and m bits espectively. We will also be using ī, ō and q to denote the sets of individual vaiables compising each of the vectos. We now patition the state vaiables as follows. We let s be the lagest set of state vaiables such that the value of each of them in the (N + 1)-st clock cycle depends only on the values of ī and the state vaiables inside s in the N-th clock cycle. In paticula, they do not depend on the values of ō. We denote all the othe state vaiables as and we will assume that is a vecto of width m 1 and s is of width m 2. The set s can be detemined by exploing the gaph of dependencies amongst the vaiables of q and ō. We can detemine whethe a fomula ϕ(x), fo example one defining the value of a q j in the next clock cycle, depends on a vaiable x, which it contains, by using a SAT-solve to check whethe the fomula ϕ(tue) ϕ(false) is valid. We will now descibe the opeation of ou tansduce, which consists of thee cicuits that we call C, φ and τ. Cicuit φ is a combinational cicuit and the othe two 9

10 ae sequential. Thei oles ae analogous to those of the deteministic automaton A and functions φ and τ in [4]. Ou specification cicuit C fulfills the esponsibility of the specification automaton A used in [4]. C pefoms two tasks. Fist, it uns the pat of C that computes the sequence of values of s as C consumes ī. In paallel with this, C also simulates the exhaustive un of the pojection of C onto the input vaiables ī. So unning C with the sequence of values fo ī as input will geneate a sequence of values fo s togethe with a sequence of sets of possible values fo the est of the state vaiables, which ae. We will stoe this tace in a memoy fom which it can late be ead in the evesed ode. This sepaation of sets s and is one of the main impovements in ou appoach ove pevious wok. It takes advantage of the simple idea that when pojecting a deteministic automaton onto a subset of its input vaiables, it is possible that the tansitions within a subset of the states of the automaton emain deteministic even with the esticted alphabet, and hence that pat of the automaton does not need to go though an exponential expansion due to the pojection. This optimization applies in paticula in the case when the specification fomula contains division of a tem that is completely detemined by ī-vaiables by a powe of 2. An intuitive explanation is the following. The specification cicuit fo the fomula x = 2 k t veifies whethe the encoding of x is a copy of the encoding of t shifted to the left by k bits. Theefoe it needs k state vaiables to emembe the past k bits of x. The values of these k state vaiables ae independent of t and hence if x is an ī-vaiable, which means that we ae pefoming division, then these k state vaiables will belong to s and they will not paticipate in the state-space explosion of C. On the othe hand, this optimization does not apply if x is an ō-vaiable, i.e. when we ae pefoming multiplication. The pupose of φ is to find inside the last stoed set of possible states fo one which is, combined with the last stoed value of s, an accepting state of C. Eventually, we un τ, which econstucts a whole accepting un of C by tacing backwads though the stoed exhaustive un of its pojection onto the input vaiable set ī, using the accepting state detemined by φ as a stating point. Duing this backwad un it constucts a sequence of ō lettes that is the final output of the tansduce. 4.1 Implementation of C, φ and τ as Cicuits Fo C, conside the cicuit in the figue in Appendix A, which has state vaiables R 1,...R 2 m 1 and s, and no outputs. Let C 1 and C 2 denote the sub-cicuits of C fo computing and s espectively. In the figue, we denote the coesponding combinational cicuits behind these SCs by K 1 and K 2. We let C ī be the pojection automaton obtained fom C 1 by pojecting it onto the ī-vaiables. The intended meaning of the state vaiables R 1,...R 2 m 1 of C is that R k is set to tue if and only if at that point the non-deteministic automaton C ī could be in the state numbe k. Since thee ae exactly 2 m1 possible states of C ī, we can make some abitay assignment of the possible states of C ī to the R k s. Initially, A is in a state whee all vaiables R k ae 0 except fo one, coesponding to the initial state of C ī. The initial value of s is also detemined by the given initial state of C. The i and ō j denoted in italics epesent constant bit-vectos given as input to each of the 2 ml copies of C 1. The indexes ae assigned so that j is the assignment of state 10

11 vaiables of C ī coesponding to the state which is epesented by R j. Hence each of the C 2 -subcicuits poduces an outcome -state fo a given combination of a pevious state and values fo the ō-vaiables. Each of the AND-like-gates with an R k insciption is undestood to have negations at an appopiate combination of its inputs, so that it etuns tue if and only if its input epesents the -state coesponding to R k and also the incoming signal fom the state vaiable R j is tue. This last condition has the effect of consideing the output only of those sub-cicuits fo which the input state j is actually one of the possible states in the exhausting un of C ī at the moment. The last laye of odinay OR-gates just has the effect that if any of the possible combinations of an active pevious state and an ō-lette poduces the state coesponding to R k then R k is set to one in the next cycle. The main idea of this cicuit is that fo evey state of C that is possible at the pesent clock cycle, it ties evey possible ō-lette to poduce the set of all possible states in the next clock cycle. Now assume that the sequence of states this cicuit goes though while eading an input wod is saved in a memoy fom whee it can eadily be ead in the evese ode. Recall that φ is supposed to find an accepting state of C amongst the possible states encoded in the last state of C - that is, in the combination of the exhaustive state of C ī encoded by R 1,...R 2 m 1 and the deteministic pat of the state, s. A slight divegence between deteministic automata used in [4] and ou vaiant of sequential cicuits is that whethe the cicuit accepts depends not only on the cuent value of its state vaiables but also on the value of all its inputs - the cicuit accepts simply when it outputs a 1. To account fo this, ou φ cicuit has to choose both a state fom amongst the states possible in the penultimate clock cycle of the un of C, and a suitable ō- lette, such that the esulting state is accepting. If such state and ō-lette do not exist, the use is notified that fo the given sequence of values fo the ī-vaiables thee exists no satisfying sequence of values fo the ō-vaiables. The implementation of φ is a cicuit vey simila to that fo C, also containing 2 m1+l copies of K 1. Howeve, since it only needs to be un fo one clock cycle, it is a combinational cicuit athe than a sequential one. Finally, we use a vey simila cicuit fo the function τ. In each clock cycle, it takes as input a tansition S, ī, S of C and a state q S and geneates a state q S and an output symbol ō such that thee is a valid tansition in C fom q to q while eading the lette obtained by combining ī with ō. This is again implemented by guessing combinations of an appopiate ō-lette and -state, so τ consists of 2 m1+l copies of K 1 and some sevicing cicuity. The output of τ and also the final output of the tansduce is the sequence of ō lettes. Notice that it comes in the evese ode, espective to ī. 4.2 Constucting Tansduce as a Composition of Tansduces fo Sub-fomulas Suppose that the specification fomula F on its highest level is a disjunction of subfomulas ϕ 1,...ϕ k. Then we can build a tansduce fo each of them sepaately and un them in paallel. If fo a given input any of the tansduces finds an output satisfying the sub-fomula coesponding to that tansduce, say ϕ i, then this output can be taken 11

12 to be the global output. If ϕ i mentions only a subset of the output vaiables then values fo the emaining ones can be picked abitaily. If the specification fomula, on the othe hand, is a conjunction of sub-fomulas, then we also have to mind dependencies between the vaiables. Definition 5. We say that a QFPAbit fomula ψ ove vaiables V uniquely detemines a set of vaiables x as a function of a set of vaiables ȳ, if fo any patial valuation valȳ : ȳ Z that only assigns values to the vaiables of ȳ, the set of satisfying valuations of ψ that extend val is non-empty and all of them give all the vaiables in ō the same values. If the specification fomula F is a conjunction of sub-fomulas ϕ 1,...ϕ k, we can apply the following easoning. Suppose that thee exists ō ō such that some ϕ j uniquely detemines the values of ō as a function of ī. Now suppose that val : ī ō Z is a satisfying valuation fo F. Then, in paticula, it is a satisfiing valuation fo ϕ j and it assigns ō the same values as any satisfying valuation of ϕ j that gives the ī s the same values as val. This means that we can build an independent tansduce fo ϕ j and use its output to fix the values of ō in F, allowing us to build a smalle tansduce fo the est of the vaiables. Notice that the values that the tansduce fo ϕ j computes fo those vaiables that have not been poven to be uniquely detemined by ī must be ignoed, because thei values need not be a pat of a satisfying valuation fo the est of F. In pactice, we can use this fact to constuct a sequence of tansduces with inceasing numbe of ī vaiables and deceasing numbe of ō vaiables. We scan though the list of conjuncts of F and wheneve we find one, say ϕ j, in which some subset of ō vaiables is uniquely detemined by the ī vaiables, we build a tansduce fo it, eclassify the uniquely detemined ō-vaiables to ī-vaiables in F and epeat the pocess, wiing the appopiate outputs of the tansduce fo ϕ j to become the inputs of the next tansduce. If it tuns out that in a paticula conjunct, all the occuing ō-vaiables ae uniquely detemined, this whole conjunct can be emoved fom F. Notice that fo egulaly occuing conjuncts of a standad fom, like fo example equality assetions involving standad aithmetical opeations, we will not have to invoke the geneal tansduce-synthesis method descibed at the beginning of this section. Instead, we can use potentially moe efficient pe-computed cicuits loaded fom a libay. This can, fo example, be applied in the case when the conjunct assets that an ō-vaiable is a constant multiple of a tem that is uniquely detemined by the ī vaiables. The length of the esulting sequence of tansduces is at most quadatic in the numbe of ō vaiables, which can be seen by inspecting the unning time of the tivial algoithm that loops though the cojuncts in an abitay fixed ode and halts when duing an iteation examining all the conjuncts it can not eclassify any new ō-vaiables to ī-vaiables. Obviously, this optimization is useful only if the specification fomula F is in fact a conjunction containing conjuncts that do have the popety of uniquely detemining some of the ō-vaiables as a function of the ī vaiables. As discussed in Section 3.1, befoe building the specification cicuit we fist pe-pocess the input fomula, so that the fomula that is eventually used fo building the cicuit is G F ϕ 1... ϕ n 12

13 whee each of the ϕ i has one of the following foms: (i) x = 2 k t; (ii) x = c; (iii) s = x + y; (iv) T 1 = T 2, whee x, y, t, s ae vaiables, c is an intege constant and T 1, T 2 ae tems built out of vaiables and bit-vecto logical opeations. F is a boolean combination of atoms of simila foms, but at the pesent time we do not have methods fo investigating vaiable dependencies in non-atomic fomulas. On the othe hand, fo each of the ϕ j s we can exactly detemine which ō-vaiables ae uniquely detemined by the ī-vaiables. In case (i), if at least one of the vaiables pesent is an ī-vaiable then the othe is detemied. In case (i), vaiable x is detemined, and in case (iii), if at least two of the vaiables ae ī-vaiables then the last one is detemined. In case (iv), since T 1 and T 2 contain only vaiables and bit-vecto logical opeations, the equality holds exactly if the popositional fomulas coesponding to T 1 and T 2 evaluate to the same boolean value in evey clock cycle. Theefoe it is enough to investigate which ō vaiables ae uniquely detemined by the ī vaiables in the popositional fomula ˆT 1 ˆT 2, whee ˆT 1 and ˆT 2 ae popositional fomulas obtained fom T 1 and T 2 by eplacing the bit-vecto logical opeatos by standad boolean opeatos and teating the QFPAbit vaiables as popositional vaiables. Example. We demonstate the usefulness of this optimization technique on an example. Let us foget fo a moment that ou language contains an out-of-the-box plus opeato and suppose we would like to synthesize a function fo pefoming addition and outputting the sequence of cay bits at the same time. It can be specified in QFPAbit as follows. (s = x y c) (c = 2((x y) (x c) (y c))) whee x and y ae designated as inputs and s and c ae outputs epesenting the sum and the sequence of cay bits espectively. Clealy, the ight-hand conjunct detemines c uniquely, given values fo x and y. Ou pototype implementation is able to detect this and builds a tansduce which is a composition of two pats - one fo the ighthand conjunct, which computes the value of c given values fo x and y, and one fo the left-hand conjunct that computes the value of s given values fo x, y and c. Due to this factoisation, the total numbe of gates in all the cicuits involved is 7.2 smalle than when we enfoce the building of a single monolithic tansduce fo the whole fomula. To conclude the discussion of this optimization technique, let us look close at how it applies to those ϕ j s that ae of fom x = 2 k t. Because of the way how these conjuncts oiginate duing the pe-pocessing of the specification fomula, often both x and t ae output vaiables. If afte inspecting some othe conjuncts we manage to specify one of them as an input vaiable, the othe is immediately detemined by it and we will be able to emove this conjunct fom the fomula and constuct an efficient tansduce fo it. We can summaize this in the following lemma. Lemma 1. Suppose that the oiginal fomula, befoe pe-pocessing, contains multiplication by a constant c in a context of the fom T 1 [ct ] = T 2 such that eithe all the ō-vaiables occuing in T ae uniquely detemined by the ī-vaiables, o the ō-vaiables of T occu nowhee else in T 1 and T 2 and the value of a fesh vaiable x is uniquely detemined in the fomula T 1 [x] = T 2. Then the total size of all the cicuits of the tansduce obtained by the pocedue descibed in this section will be popotional to the logaithm of c. 13

14 5 Conclusion We have pesented a synthesis pocedue that stats fom QFPAbit desciption of an input/output elation, geneates a sequential cicuit of a polynomial size, and then tansfoms this cicuit into a synthesized system of sequential cicuits that maps a sequence of inputs into a sequence of outputs. The descibed synthesis pocedue impoves the pevious wok by two independent optimizations. We have built a pototype implementation that allowed us to show on examples that these techniques wok and ae impotant. Acknowledgements The idea of eplacing synthesis fom WS1S with synthesis fom QFPAbit as well as a polynomial tanslation fom QFPAbit into cicuits oiginated in a discussion between Babaa Jobstmann and Vikto Kuncak in Octobe Babaa Jobstmann was also suggesting decomposing specifications and pefoming synthesis modulaly. We thank Aati Gupta fo pointing to the elated wok in he PhD thesis [3] as well as Shaad Malik and Paolo Ienne fo useful discussions. Refeences 1. R. Bloem, B. Jobstmann, N. Piteman, A. Pnueli, and Y. Sa a. Synthesis of eactive(1) designs. J. Comput. Syst. Sci., 78(3): , J. Buchi and L. Landwebe. Solving sequential conditions by finite-state stategies. Tansactions of the Ameican Mathematical Society, 138( ):5, A. Gupta. Inductive Boolean Function Manipulation: A Hadwae Veification Methodology fo Automatic Induction. PhD thesis, CMU, J. Hamza, B. Jobstmann, and V. Kuncak. Synthesis fo egula specifications ove unbounded domains. In Fomal Methods in Compute-Aided Design (FMCAD), 2010, pages IEEE, B. Jobstmann and R. Bloem. Optimizations fo LTL synthesis. In FMCAD, N. Klalund, A. Mølle, and M. I. Schwatzbach. MONA implementation secets. In Poc. 5th Int. Conf. Implementation and Application of Automata. LNCS, V. Kuncak, M. Maye, R. Piskac, and P. Sute. Complete functional synthesis. In ACM SIGPLAN Conf. Pogamming Language Design and Implementation (PLDI), M. Rabin. Automata on infinite objects and Chuch s poblem. Numbe 13 in Regional Confeence Seies in Mathematics. Ameican Mathematical Society, T. Schuele and K. Schneide. Veification of data paths using unbounded integes: Automata stike back. Hadwae and Softwae, Veification and Testing, pages 65 80, A. Spielmann and V. Kuncak. On synthesis fo unbounded bit-vecto aithmetic. Technical Repot EPFL-REPORT , EPFL, S. Sivastava, S. Gulwani, and J. S. Foste. Fom pogam veification to pogam synthesis. In POPL, L. Stockmeye and A. R. Meye. Cosmological lowe bound on the cicuit complexity of a small poblem in logic. J. ACM, 49(6): ,

15 A Schema of Cicuit C s K 2 R... 1 R2 R2 m1 i 1 o1 2 o1 2 m1 o 1 1 o2 2 o2 2 m1 o 2 1 o2 l 2 o2 l 2 m1 o 2 l K 1 K 1... K 1 K 1 K 1... K 1 K 1 K 1... K Q D Q D Q D Q D 15