this paper is about an application of logic to the methodology of science in the spirit of van benthem s .there are several axiomatizations of special relativity theory available in the literature , all looking different but claiming to axiomatize the same physical theory. such are , among many others , the ones in andrka et al . , ax  , goldblatt  , schelb  , schutz  , suppes  .these papers talk about very different kinds of objects : talks about reference frames , talks about particles and signals , looks like a purely geometrical theory about orthogonality , the central notion of is the so - called minkowski - metric , etc . while , as usual , one gets a better picture of this area via a variety of different eyeglasses " , the following questions arise .what are the connections between these theories ?do they all talk about the same thing ?if they do , do they capture it to the same extent , or is one axiomatization more detailed or accurate than some of the others ?in this paper we want to show how , in the framework of mathematical logic , a concrete , tangible comparison / connection can be elaborated between these theories for special relativity and what we can gain from such an investigation .  for this comparison, we have to use a form of logical definability theory in which totally new kinds of entities can be defined as opposed to traditional definability theory where only new relations can be defined on already available entities .the existing methods of definability theory had to be modified and refined for the purposes of the present situation .thus , definability theory , too , profits from such an application . in the present paper ,we elaborate in detail one piece of such a comparison : we construct an interpretation of the relativity theory in talking about reference frames into the theory in which talks about particles emitting and absorbing signals .then we construct an inverse interpretation for showing which versions of the two theories are definitionally equivalent .since this is a case - study showing applicability of the proposed method for comparing / connecting theories , we tried to give all the detail needed .this is why some sections of the paper may look somewhat technical .an insight of last century mathematical logic is that it is important to fix the vocabulary of a first - order logic theory and stay inside the so obtained language while in a specific theory ( see , e.g. , ) .the symbols in the vocabulary are the concepts that are not analyzed further in the given theory , they are called thus basic ( or primitive ) concepts .but this is not a forever frozen state : we may decide to analyze further the basic concepts of this vocabulary and we can do this in the form of building an interpretation ( in the sense of mathematical logic ) into another language the vocabulary of which consists of new basic concepts , and the interpretation gives us the information of how the old basic concepts " are built up from the new basic concepts " as refined ones .the interpretation we construct in this paper thus refines the basic concept of a reference frame in terms of just sending and receiving signals . to refine the basic concepts of this signalling theory, we can interpret it to , say , in a theory of electromagnetism , or in a quantum - mechanical theory .such an interpretation may also be regarded as defining a so - called operational semantics for the basic concepts of the first theory . starting with the vienna circle ,several authors suggest that a physical theory is a more complex object than just a set of first - order logic ( fol for short ) formulas .a physical theory , they propose , is a fol theory together with instructions for how to interpret the basic symbols ( or vocabulary ) of this theory in the real physical world " .( following carnap  , this is often called a ( partially ) interpreted theory " . )we want to show in section [ sec : reduc ] that such an operational semantics " can be taken to be an interpretation in the sense of mathematical logic .  returning to our concrete example, an operational semantics should say something about how we obtain or set up ( in the real world ) the reference frames for special relativity theory .usually , rigid meter - rods and standard clocks are used for this purpose ( e.g. , ) . however , as points out , we can not use these rigid meter - rods in astronomy or cosmology .the interpretation we give in detail in this paper results also in an operational / experimental / observational definition for setting up a reference frame by just relying on sending and receiving light - signals .this method can be used , in principle , in the above mentioned astronomical scale .summing up , the first language in an interpretation has the theoretical concepts while the basic concepts of the second language are the observational ones ( for the observational - theoretical duality see , e.g. , ) .we can look at the same interpretation from the other direction " : in our example , we may imagine someone living in a space - time , exploring his surroundings by sending and receiving signals , and during this process , he devises so - called theoretical concepts which make thinking more efficient .in particular , he may devise the concept of a reference frame , and even the concept of quantities forming a field , as mental constructs having concrete definitions in terms of observations .the tools of mathematical logic , and more closely those of definability theory ( interpretations are among them ) can be used for modeling this emergence of theoretical concepts .a further aim of the present approach of comparing theories is shifting the emphasis from working inside a single huge theory to working in a modularized hierarchy of smaller theories connected in many ways .usually this approach is called theory - hierarchy .we note that this is not so much a hierarchy as rather a category of theories , technically the category of all fol - theories as objects with interpretations as morphisms of the category .this direction of replacing a huge theory with a category of small theories is present in many parts of science . in foundational thinking, emphasizes this . in computer science , it is present in the form of structured programming . putting theories together " of burstall and goguen refers to the act of computing / generating colimits of certain diagrams in this category . even in such practical areas as using a huge medical data - basethe need of modularizing arises : it is necessary to break up " the given data - base and generate many smaller ones according to the query at hand .the interpretation going from special relativity as formalized in into the more observational signalling theory of we build in the present paper is but one morphism of this huge dynamic category of fol theories .the content of this paper can also be viewed as preparing the ground for an application of algebraic logic to relativity theory , as follows .the cylindric algebra of a theory is an abstract representation of the structure of concepts expressible / definable in that theory and a homomorphism between two cylindric algebras corresponds to an interpretation between the corresponding theories .hence the category of all fol theories is basically the same as the category of cylindric algebras as objects and homomorphisms as morphisms .there are , for example , well known and understood methods for how to compute colimits in this category of algebras .there are still many questions and phenomena to be understood in this area of application of logic .for example , what are the desirable or good properties of an interpretation for being informative about the theories in one or other respect ? consider for a second the definability / interpretability picture between scientific theories ( in fol ) in two versions : ( 1 ) in the framework of traditional definability theory , and ( 2 ) in the new , extended theory of definability used in the present paper .what are the characteristic differences ?we think it is useful to keep this picture / issue in mind .in section [ sec : sr ] we briefly recall the relativity theory from , in section [ sec : sigth ] we recall signalling theory from and we try to give a basic feeling for it by sketching the proof of completeness theorem in .section [ sec : algo ] is an important part of the present paper , it contains an algorithm for how to set up a reference frame in signalling theory , this is an operational semantics " for setting up reference frames of . at the end of the sectionwe outline how the same method could be used for space - times other than special relativistic , e.g. , for the schwarzshild space - time of a black hole .this algorithm is at the heart of the interpretation elaborated in section [ sec : reduc ] .section [ sec : defth ] recalls the features of the more refined definability theory that are needed for defining the interpretation of into .section [ sec : defeq ] rounds up the picture between and by interpreting in a slightly reinforced version of and then giving more information about connections between various concrete theories of special relativity .we end the paper with a conclusion .in this section we give a list of basic concepts and axioms of the fol theory in .the basic notions not analyzed further in are observers " having reference frames in which they represent the world - lines of bodies ( or test particles ) , of which signals ( light - particles , or photons ) are special ones .the world - line of a body represents its motion , it is a function that describes the location of the body at each instant . for representing time " and location " , observers use quantities , quantities are endowed with addition and multiplication in order to be able to express whether a motion is uniform " or not . to make life simpler ,we treat also observers as special bodies .( another , equivalent , option would be to treat them as entities of different kind " , or of different sort " , than bodies and quantities . ) the reference frame or world - view of an observer gives the information which bodies are present at time at location ; thus expresses that body is present at in , according to observer .we treat quantities as entities of different nature , of different kind , than bodies .  according to the above, the vocabulary of the language of is the following : we have two sorts , bodies and quantities , we have two unary relations of sort , we have two binary functions of sort , and we have a six - place relation the first 2 places of which are of sort and the rest of sort .next , we list the five axioms of . concrete formulas and more intuition can be found in , e.g. , .the world - lines of photons are exactly the straight lines of slope 1 , in each reference frame .all observers coordinatize the same physical reality ( i.e. , the same set of events ) .the owner " of a reference frame sits tight ( stays put ) at the origin .the quantities form a euclidean field w.r.t . the operations , this means that form an ordered field in which each positive quantity has a square root .all observers use the same units of measurement : if two events are simultaneous for observers , then the spatial distance between them is the same according to . + . may seem to be a rather weak axiom system . however , this is not so .all the well - known theorems / predictions of the ( kinematics of ) special relativity can be proved even from .below is a sample of theorems that can be proved from ( for proofs , further theorems provable from , and for extensions see the references given earlier as well as ) :   : :  each observer moves uniformly and slower than light in any other  observer s world - view ( i.e. , the world - line of an observer is a  straight line with slope less than 1 ) .: :  assume that are moving relative to each other . : :  events that are separated in s world - view in a  direction of s motion and simultaneous according to   , are not simultaneous according to . : :  events that are simultaneous according to both and   are exactly the ones that are separated orthogonally  to the direction of motion of . : :  assume that and use the same units of  measurement , i.e. , the spatial distance between events that are  simultaneous to both of them is the same according to them .then  a - synchronicity , time - dilation and length - contraction between   and are exactly according to the  known formula of special relativity , see e.g. , . : :  the world - view transformations between observers in   are exactly the bijections  that preserve minkowski - equidistance ; these bijections are the  so - called poincar - transformations composed with dilations and  field - automorphisms . : :  the world - view transformations between observers in   are exactly the bijections  that preserve minkowski - distance ; these bijections are the so - called  poincar - transformations .  in ,a reference frame is a basic ( or primitive ) notion , just an out - of - the - blue " assigning space - and - time coordinates to events , which all together have to satisfy some regularities ( our axioms ) .the theory does not address the question of how an observer sets up his reference frame .as already outlined in the introduction , according to some authors , a physical theory ( a theory about our physical reality ) , should say something about the meaning ( in the real " physical world ) of the basic concepts , if not otherwise , then in natural language one could amend the theory with a set of so - called _ operational rules _ about how the basic concepts ( the reference frames in our case ) are set up ( experimentally ) . hereusually meter - rods and wrist - watches , or standard clocks , are used , see e.g. , taylor and wheeler ( * ? ? ?* fig.s 9,135 ) , l. e. szabo . in section [ sec : algo ] we give a more ambitious algorithm for setting up coordinate systems .the intention of ax s theory is to give an axiom system for special relativity so that its basic symbols and axioms are designed to be observational .the players of this theory are experimenters that can communicate with each other " by sending signals to each other .together , as a team the experimenters can map " ( or explore ) space - time , without having rigid meter rods or clocks . a definition of an introduced ( or defined ) term in this first - order logic theory can be viewed as an experiment designed to establish whether the defined term holds or not .the basic terms of space and time are defined this way .( indeed , in this theory one can define rigid meter rods " and clocks " from signalling experiments . )the results of the experiments we make can be built into axioms then ( which are designed to be observational - oriented ) , and they can tell us in what kind of space - time we live in .special relativistic ?hyperbolic space with relativistic time ?general relativistic ?all this amounts to an implementation of leibnitzian relational notion of space and time .we return to this subject in more detail in the next section .we begin to describe ax s theory which we call signalling theory . in the vocabulary of we have two sorts , for particles " ( or experimenters , or agents ) and for signals " ( or light - signals ) ; and we have two binary relations between particles and signals .the intended meanings of and are  transmits ( or emits , or sends out ) " , and receives ( or absorbs ) " , respectively .ax uses an impersonal terminology of particle physics , particles emit and absorb signals .we are more attracted to a terminology of communication between active experimenters .these experimenters ( players ) of are somewhat analogous to the observers of . in this paperwhen talking about ax s signalling theory , we will use the terms experimenter and particle interchangeably .  the standard " ( or intended )model we have in mind is the following : let us fix a euclidean field .then is the set of all straight lines in with slope less than 1 , and is the set of all directed finite segments ( including the segments of length zero ) of straight lines with slope 1 . a particle transmits a signal iff the beginning point of lies on , and receiving the signal means that the endpoint of lies on . let us denote this structure by .the main result of is a finite set of axioms , our , which characterizes the class of standard models , i.e. , the models of are exactly the standard models over some euclidean field ( thm.1 in ) . of three groups of axioms , altogether it has 23 elements . instead of listing these 23 axioms , in this paper we will use ax s completeness theorem , since that implies that a formula is provable from iff is true in all the standard models .the question immediately arises : what does this theory have to do with special relativity ? do not we lose much expressive power by using such meager resources ? where do a - synchronicity , time - dilation ,length - contraction come into the picture in ?some answers are in the proof of thm.1 which we briefly outline below. we will give more explicit answers to these questions in the coming sections [ sec : algo ] , [ sec : reduc ] .in particular , we will show that everything we can say in the language of can be said in the spartan language of , too .one of the ideas for proving this can be traced back to hilbert , as will be noted below figure [ times - fig ] .  to give a feeling for and the expressive power of its language, we briefly outline the proof of ax s completeness theorem .let s begin by making a little elbow - room for working .we will need to express things such as two signals are received by an experimenter at the same time " , and signal was received by an experimenter just when he transmitted signal " .since we have no notion of time in our language , we have to express these notions just by using the basic concepts of transmitting and receiving signals . herecomes how we can do this .the ( open ) formula  " is true in a standard model just when the beginning points of the segments coincide , we say that expresses this fact . and , respectively . in place of conjunction will simply write a comma . ] similarly ,  " expresses that the endpoint of coincides with the beginning point of , etc .now can we express that two particles / experimenters meet ?well , they meet if there is a signal that both of them transmit . from now on we will use similar statements without translating them to the language of .  to begin outlining the idea of ax s completeness proof , let be any model of , and let be any experimenter in this model .we will construct an isomorphism between and a standard model which takes to the time - axis in . from now on , in this section denotes this fixed experimenter .we define the set of places " or locations " for experimenter to consist of those particles which are motionless w.r.t .  . for expressing that two particles are motionless w.r.t . each other ,any formula expressing this in the standard models will do .ax uses the following formula : is motionless w.r.t . exactly when and do not meet ( if they are not equal ) and there are two other particles which meet them and each other in 5 distinct events .ax then expresses the betweenness relation for such places as well as the equidistance relation with suitable formulas .having all this , the first group of axioms in states tarskis axioms for axiomatizing euclidean geometry over the euclidean fields ( see ) . from tarski s theoremthen ax gets a euclidean field and an isomorphism between and .  having for our experimenter , what is time " for it ?what are the things that we mark with time ? the events . and what are the events ? in the present vocabulary we take them to be particle emits / receives a signal " , more precisely we take the equivalence classes of them described when we made the elbow - room for this proof ( e.g. , particle may send out signal in the same event when it sends out another signal or when it receives ) .then our experimenter s time will be the events that happened to . for simplicity, we will represent events with special signals , as explained below .  in the standard models, there are special signals that are received by everyone who transmitted them , we call these signals _ events _ , we will denote them by variants of :   .  in the standard models ,events are the light - like segments of zero length , so they correspond to elements of .( these zero - length signals may look counter - intuitive to some readers .it is just handy and not important that we use or have these at hand , everything works with a slight modification if we omit these short signals from the standard models . ) we say that event happened to experimenter , or in other words , experimenter _ participated in event _ , if transmitted ( and then also received ) .the events that happened to will constitute s _ world - line_.  we can then express simultaneity of events by using that the speeds of light - signals are the same ( see figure [ sim - fig ] ) .ax then states an axiom to the effect that signals make a one - to - one correspondence between s world - line and the simultaneous events on any given line in .this makes s world - line isomorphic to , we take this to be the time - axis . from now on it is more or less straightforward what we have to include to in order to make isomorphic to .e.g. , we can state that for any event there is a simultaneous event on the world - line of , and there is a particle that participates in and is motionless w.r.t .this concludes the proof - idea .the purpose of this section is twofold .firstly , in section [ sec : reduc ] we want to give an interpretation of into , and for this we need concrete formulas representing the proof - idea given in the previous section .for example , ax used tarski s theorem for getting the euclidean field , but we will need to exhibit concrete formulas defining this . secondly , we want to make the previous proof - idea into an algorithm for setting up coordinate systems ( i.e. , reference frames ) with the use of just light - signals and freely moving particles .this could also be viewed as providing _operational semantics _ to the basic notion of a coordinate system of .what we give in this section will not be an algorithm in the strict sense , it will be more like a recipe for how to design experiments / measurements for assigning coordinates to events .these experiments will also be suitable for finding out / confirming that we live in a special relativistic space - time ( if we do ) .for this reason , we will try to make the formulas executable " when possible .there will be plenty of room for improving on this aspect , the reader is invited to design more practical experiments .assume that we are given a model of , and is an experimenter in this model .just as in the previous section , this experimenter is fixed throughout this section .we are going to give a recipe for defining a field of quantities and for assigning four quantities to each event .such an assignment is called a _ coordinate system _ ( or reference frame ) .these coordinate systems will satisfy the axioms of .a _ location _ for was defined as a particle that is motionless w.r.t . . in the previous section we recalled a formula , from , expressing whether is motionless w.r.t . ( in symbols , ). however , the algorithm suggested by that formula is not very convenient since it involves deciding whether meets or not , and for this has to know all the events that happened and will happen to him .this is not very practical as an experiment , since may need to wait " for an infinity of time before he could know the result . using theaffine desargues property ( adp for short , see , e.g. , ) one can design a more realistic experiment which decides  in a finite time " , we are going to describe it now .we note that in the standard models the adp is true , because it is true in the affine space , for any field .  for a while, it will be easier to think in 4-dimensional space - time than tracing motion in 3-dimensional space .geometrically , is motionless w.r.t . iff the world - line of is parallel to that of .the conclusion of the adp is that two lines are parallel , but in the hypothesis part parallelism of two other sets of lines are used .we are lucky : we have light - signals and their speeds are the same in both directions , thus we can use parallelism of world - lines of two sets of light - signals in the hypothesis part of the adp . the experiment is depicted in geometrical form in the left - hand part of figure [ des - fig ] . here is the non - geometrical " description of the experiment : assume wants to decide whether is motionless w.r.t . him or not .he asks a brother ( another experimenter ) to throw towards him three test " particles ( balls " ) at once ( in one event ) , faster than and faster than in such a way that when meets , the latter sends out a signal towards that reflects back and the reflected signal reaches just when reaches .( the brother and have to experiment a little while till finding the right velocities for such three particles . ) after checking that have the desired property , asks to do the same : when reaches , he should send a signal towards that reflects this signal back towards . if the reflected signal reaches just when reaches , then is motionless w.r.t . ; otherwise is not motionless w.r.t .it is best to imagine this experiment to take place in outer space , far from heavy heavenly objects so that gravity and friction do not bend the world - lines of the balls " . from now on ,we will use locations " and places " as being particles / experimenters motionless w.r.t .our fixed experimenter .[ b][b] [ b][b] [ b][b][b][b][b][b ] [ b][b] [ m][b ] ? [ b][b][b][b][b][b] [ sim - fig ] on the left : experiment for checking whether is motionless w.r.t . . in the middle : experiment to make sure that are simultaneous w.r.t. . on the right : time - equidistance of events .,title="fig:"]1.5 cm [ b][b] [ b][b][b][b] [ sim - fig ] on the left : experiment for checking whether is motionless w.r.t .  . in the middle : experiment to make sure that are simultaneous w.r.t. . on the right : time - equidistance of events .,title="fig:"]2.1 cm [ b][b] [ b][b][b][b] [ b][b][b][b] [ sim - fig ] on the left : experiment for checking whether is motionless w.r.t .  . in the middle : experiment to make sure that are simultaneous w.r.t. . on the right : time - equidistance of events .,title="fig : " ]  two events are defined to be _ simultaneous _w.r.t. iff there is a place such that from two signals can be sent at the same event towards the locations of and respectively such that if these signals are sent back from and right away , they will arrive back to at the same event , see middle of figure [ sim - fig ] .formally : , where means that are the events of sending and receiving , respectively , formally : where expresses that are sent out at the same event , formally : and a similar definition for .( note that if we want a more experiment - friendly formula for , then we can use the following : . )we even can provide instructions for where to look for such a place : it can be chosen to be the midpoint of the line - segment connecting the locations of and .( we can use this experiment for setting two clocks at the places of which tick simultaneously " . )we get an ordering on all the events from the fact that we send a signal earlier than receiving it , namely is _ earlier _ than iff we can send a signal at to an event from where it bounces back to (  ] ,  and  . )note that implies that happens earlier than iff happens earlier than . by using time - equidistance, we can define _ addition _ by selecting a _ zero _ " time as parameter , namely . now that we have addition, we do not stop before having _multiplication_. for this we have to choose a _ unit _ time , distinct from and happening later than , as another parameter .for defining multiplication , we will need the _ collinearity _ relation on locations , we will get this by noticing that the space - trajectories of signals are ( 3-dimensional ) straight lines in the standard models : iff exist signals and events such that and , , for some permutation of .we define for the case when happened later than , and happened later than .see the left - hand part of figure [ times - fig ] .( the other cases are similar , we leave them out . ) here is how we find out whether is : we find two places and collinear with and we find a particle such that if and send towards , simultaneously , at time zero , a light - signal and , and another light - signal and another particle  with the same speed " as , then these four arrive ( to ) at times , respectively . formally :  ] , and  ] ) .by now we defined a structure and we defined distance .setting up a coordinate system needs three more parameters , the three space - axes .let be such that , and are pairwise orthogonal .we have everything for defining the usual spatial coordinates of the place .see the right - hand part of figure [ times - fig ] .the spatial coordinates of a location are defined the usual way by projecting " to the three coordinate axes , along lines parallel with some of the axes , and measuring the distance of the projected points from the origin ( our experimenter in our case ) .see the formula below .we can now round up the definition of the coordinate system our experimenter is setting up .we already defined the time - coordinate of an event , and we define the space - coordinates of to be the spatial coordinates just defined for the location of " , the latter being the unique particle participating in and motionless w.r.t .our experimenter .the formula defined below expresses that the coordinates of the event are in the coordinate system specified by .: :   ],for each  formula in .  in the new , non - traditional " or generalized " definability theory we will use a notion of interpretation that does the same thing , except that the universe of will not necessarily be a subset of the universe of , therefore its definition and the property analogous to ( ) above will be more involved .we will define new entities as elements of new sorts " .using many - sorted fol is not an essential feature of this generalized definability theory , just it is convenient in many cases , as it is in our present task .we illustrate the idea of defining new sorts with a simple example . the language of affine planes in , e.g. , is two - sorted , we have two sorts and we have a binary relation between them , the relation of incidence ( or membership ) between a point and a line . another language in use forthe same is one - sorted , see , e.g. , , we have one sort and we have a three - place relation of collinearity " between three points .everyone can connect the two ways of thinking about affine planes immediately : a line is the set of all points collinear with given two distinct points .thus a line is a subset of the old universe , given two distinct points the line going through them is defined by  : :   .but the new sort stands for the set of all these subsets! we can specify one line with the open formula with one free variable , but how can we define the set of all lines ?well , we will define the set of the parameters specifying the individual lines : we identify the set of all lines with the set of pairs of distinct points .thus the formula defining the new sort will have two free variables and it will state .we are almost there , except that different pairs of distinct points may specify the same line , and we have to take this into account when talking about equality of lines , i.e. , when interpreting the equality symbol on the sort .we can do this again with a formula using 4 free variables stating when the lines specified by and coincide . in our casethis formula can be taken to be .so far we have defined the universe of the new sort and the equality relation of this new sort by two formulas in the old " language , i.e. , in the language talking about and .having defined a universe means that we have variables ranging over this universe ( and we can quantify over them ) . in other words , we have to introduce variables of sort .then , in order to be able to use the definition of the new sort , we need to connect to variables used in the definition for , i.e. , to .we can state this connection by matching a variable of sort with variables denoting its defining parameters " , e.g. , we can state that denote parameters that define . after thiswe can define the incidence relation , too : , where is a variable of sort and is a variable of sort .  summing up : definingthe new sort goes by defining the variables of the new sort and matching them to the variables of the old sort occurring in the defining formula of the sort , defining the equality on the sort , and defining the non - logical symbol of incidence which involves the sort .thus we can interpret the 2-sorted language of affine planes into the one - sorted one by the following data :  : :     , : :   , : :   , : :   .the above data then define a translation function from the 2-sorted language of affine planes to their one - sorted language as follows :  : :   , +   , +   , +  the rest of the definition of is  more or less straightforward .the new feature in this translation function , over the traditional one , is that we translate the quantifiers according to the defining formula and variable - matching of the new sort and we translate equality on the new sort , too . throughout, we will use the above variable matching without recalling it .this translation is not only recursive and structural , it is also meaning preserving in the sense analogous to ( ) . in more detail : let be a model of the one - sorted language .we will construct its translation " , a model of the two - sorted language .let  : :   ,  and let be defined by : :   .assume that is an equivalence relation on , then define  : :   +   +   .let denote the sets of variables in the 2-sorted language of the affine planes and let be the variables of the one - sorted language .now , let be any evaluation of the variables of the 2-sorted language , and let be an evaluation of the variables of the one - sorted language such that if , and if then is an arbitrary element of .then the following is true for each formula of the 2-sorted language :  ( ) : :  \quad\mbox { if and only if } \quad  { \mathfrak{m}}\models{\mbox{\small \sf tr}}(\psi)[{\mbox{\small \sf tr}}(k)] ] , : :   .recall from section [ sec : algo ] that the formula expresses that the coordinates of the event are , in the coordinate system specified by .  : :   .it remains to define the world - view relation .the intuitive meaning of the formula will be that is an observer , and the event at place in s coordinate system is on the world - line of .let be variables of sort and let be variables of sort .assume that is an observer , i.e. , .let us denote the concrete value of an abstract quantity in s coordinate system by  : :   .we can now define as follows :  : :   .  by the above, we gave definitions for all the sort and relation symbols of the language of in the language of .this defines a translation function between the two languages .let and stand for the equality relations between terms of sort and , respectively .in the next theorem we state , without proof , that we indeed obtained an interpretation .[ interpret ] as given in this section is an interpretation of into , that is , the following are true :  : :   , : :   , : :    all .having defined the desired interpretation of into , in the next section we extend this interpretation to a definitional equivalence between a slightly stronger version of and .in this section we investigate interpretability and definitional equivalence between some of the fol theories formalizing special relativity .we show that a slightly reinforced version of is definitionally equivalent to .we mean interpretability and definitional equivalence in the sense of the generalized definability theory of outlined in section [ sec : defth ] .the interpreted theory is stronger than the original one in the sense that there are sentences in the language of such that while .such a sentence is , e.g. , all lines of slope less than 1 are world - lines of observers " .we can express exactly how much more is true in the translated models by amending with some existence , extensionality , and time - orientation axioms ( see below ) and showing that the so obtained theory is definitionally equivalent with .this is what we are going to do now .the formulas describing the difference " between and are as follows .formulas expressing that we have all kinds of possible observers ( from each point , in each direction , for each velocity less than the speed of light there is an observer moving in that direction with that speed , each observer can re - coordinatize its coordinate - system with any space - isometry , each observer can set the unit of its clock arbitrarily ) , and otherwise we are as economic as possible ( at most one photon through any two distinct events , only one observer with the same coordinate - system , only photons and observers as bodies , only one time - orientation for each observer ) .  these additional axioms , except the one about setting the clocks , are denoted as axthex , axcoord , axextob , axextph , axnobody , in .let axclock formulate that each observer can set the unit of its clock arbitrarily ( in the spirit of the above axioms ) .let denote the set of these axioms and let denote the theory amended with these formulas :  : :   , : :   .  to statedefinitional equivalence between and , we now define an interpretation of into .we have to define the universes of particles and signals and the relations of transmitting and receiving , inside .intuitively , particles are defined to be observers , with two particles being equal if their world - lines coincide :  : :     ,  where   . : :   , : :   .signals are defined to be photons with two events on their world - lines representing the beginning and end - points of the signal .we represent the two events with observers meeting the photon .the following formulae express in that in s world - view , meets at time " , and meet in one event " , respectively :  : :   , : :   .now we are ready to interpret signals in :  : :     ,  where   . : :   . : :   .  finally ,  : :   , : :   .the above define a translation function as indicated in section [ sec : defth ] .we state in the next theorem , without proof , that this interprets in , and moreover , together with the interpretation defined in the previous section it forms a definitional equivalence between and .this is the main theorem of this paper .[ mainthm ] is definitionally equivalent to , the pair of interpretations forms a definitional equivalence between them .we can read the above theorem as saying that what the theory tells about special relativity is exactly what the theory says .since no axiom in follows from , we can conclude that tells more than , the amount of more " is exactly .however , we did not include the axioms of into , because we do not need them in proving the main predictions of relativity theory ; we feel that they do not belong to the core of the physical theory .moreover , of the axioms of , we consider only as having a physically ( or even philosophically ) relevant content , namely it says that time is oriented " .on the other hand , we will see that has a content that does not say about special relativity theory .this is the axiom of .so , what is the connection between and ?below we answer this question .the interpretation we defined in the previous section does not interpret in , because does not follow from ( i.e. , it is not true in the standard models of ) .can be interpreted in in the way that we interpret in the field . ]the reason for this is the following . states that any two observers use the same units of measurement .we can express in the language of that two observers use the same units of measurement " , and this defines an equivalence relation on the set of all observers .for to be true , we should select any one of the blocks of this equivalence relation ( since states that any two observers use the same units of measurements ) .but which one should we select ?the question might sound familiar . in the previous analogous case ( that concerned the various incarnations of the field ) we took all the classes up to isomorphism " .however , in the present case there are no definable bijections between the blocks of this equivalence relation .we can get around this problem by adding to the models of a unit of measurement " .we can do this , e.g. , the following way .we add a new basic two - place relation symbol ( short for time unit " ) of sort to the language of signalling theory . in each standard model interpret as the set of pairs of events with minkowski - distance 1 .( we note that these relations are not definable in in the language of signalling theory . )let us denote the so expanded standard models by , and let denote an axiom system for their theory ( in the extended language ) .now , the interpretation we gave in this section can be extended to interpret in .moreover , it also can be made into a definitional equivalence between a stronger version of and , that we obtain from by exchanging axclock with :  : :   ,  to our minds , the following theorem clarifies the connection between and .it says that the content of is to set the time - unit : the difference between and is that in we can express minkowski - distance , while in we have only minkowski equidistance .[ symthm ] the following ( i),(ii ) hold :  ( i ) : :   is definitionally equivalent  to . (ii ) : :   is not definitionally  equivalent to .  the proof of part ( i ) of the above theorem goes by extending the interpretation to , this amounts to defining the new relation in ; and also making some ( minor ) changes in the definition of .the proof of part ( ii ) of the above theorem goes by showing that the automorphism groups of members of and differ from each other , this technique is elaborated in , e.g. , .we included into as a tool for convenience , it seems to carry no philosophical or physical importance . is only a simplifying assumption .  concerning some of the other theories for special relativity , we mention that is definitionally equivalent to goldblatt s theory for special relativity in ( * ? ?* appendix a ) amended with time - orientation .i.e. , the two theories are almost the same , the only difference is that assumes time - orientation while goldblatt s theory does not . the proof of this last statement can be put together from the definitions and ideas in sections [ sec : algo ] , [ sec : reduc ] . also , ( a slight variant of ) our is definitionally equivalent to ( a slight variant of ) suppes s axiomatization of special relativity in .we intended to show in this paper some results the methods of mathematical logic can provide for other branches of science , in particular , for physics and the methodology of science . using the tools of definability theory of first - order logic , we compared in detail two rather different axiom systems for special relativity theory .one of these , of , is coordinate - system- , or reference frame - oriented , while the other , of , uses meager resources and talks about particles emitting and absorbing signals .the two theories use disjoint languages and talk about different kinds of entities . yet, a precise comparison was made possible by using mathematical logic , and we obtained the following : can express and states everything that does , except for the relativistic ( minkowski ) distance between events ( implied by in ) , while in addition it states time - orientation for space - time together with some auxiliary simplifying axioms ( ) . informally ,     a byproduct of these investigations is a concrete operational semantics for special relativity theory .we believe that interpreting one theory in another is a flexible methodology for connecting physical theories with each other as well as with the physical reality " .andrka , h. , madarsz , j. x. , nmeti , i. : on the logical structure of relativity theories .alfrd rnyi institute of mathematics , hungar .sci . , budapest , research report , july 5 , 2002 , with contributions from a. andai , g. sgi , i. sain and cs .contents.html[http://www.math-inst.hu/pub/algebraiclogic/ contents.html ] .1312 pp .andrka , h. , madarsz , j. x. , nmeti , i. : logic of space - time and relativity theory . in : handbook of spatial logics .eds : aiello , m. pratt - hartmann , i. , and van benthem , j. springer verlag , 2007 .pp.607 - 711 .andrka , h. , madarsz , j. x. , nmeti , i. , nmeti , p. , szkely , g. : vienna circle and logical analysis of relativity theory . in : the vienna circle in hungary ( der wiener kreis in ungarn ) .eds : mte , a. , rdei , m. , stadler , f. veroffentlichungen des instituts wiener kreis , band 16 , springer verlag , 2011 , pp.247 - 268 .burstall , r. , goguen , j. : putting theories together to make specifications . in : proc .ijcai77 ( proceedings of the 5th international joint conference on artificial intelligence , vol 2 , pp.1045 - 1058 .morgan kaufmann puboishers inc .san francisco , ca , usa , 1977 .harnik , v. : model theory vs. categorical logic : two approaches to pretopos completion ( a.k.a . ) . centre de recherches mathmatiques crm proceedings and lecture notes vol 53 .american mathematical society 2011 .pp.79 - 106 .konev , b. , lutz , c. , ponomaryov , d. , wolter , f. : decomposing description logic ontologies . in : proceedings of 12th conf . on the principles of knowledge representation and reasoning , association for the advancement of artificial intelligence , 2010 .pp.236 - 246 .szab , l. e. : empirical foundation of space and time . in : epsa07 : launch of the european philosophy of science association , surez , m. , dorato , mm . ,rdei , m. ( eds . ) , springer , 2009 .http://phil.elte.hu/leszabo/preprints/leszabo-madrid2007-preprint.pdf  szczerba , l. w. : interpretability of elementary theories . in : butts , r. e. , hintikka , j. ( eds )logic , foundations of mathematics and computability theory ( proc .fifth internat.congr . of logic ,methodology and philos . of sci .western ontario , london , ont . , 1975 ) , part i , 129145 , reidel , dordrecht ( 1977 )  szkely , g. : first - order logic investigation of relativity theory with an emphasis on accelerated observers .phd dissertation , etvs lorand university , faculty of sciences , institute of mathematics , budapest 2009 .arxiv:1005.0973[gr - qc ] .suppes , p. : axioms for relativistic kinematics with or without parity . in l. henkin ,a. taarski , and p. suppes , editors , symposium on the axiomatic method with special reference to physics , north holland , 1959 .pp.297 - 307 .

Abstract: there are several first - order logic ( fol ) axiomatizations of special relativity theory in the literature , all looking different but claiming to axiomatize the same physical theory . in this paper , we elaborate a comparison , in the framework of mathematical logic , between these fol theories for special relativity . for this comparison , we use a version of mathematical definability theory in which new entities can also be defined besides new relations over already available entities . in particular , we build an interpretation ( in tarski s sense ) of the reference - frame oriented theory of into the observationally oriented signalling theory of james ax  . this interpretation provides with an operational / experimental semantics . then we make precise , quantitative " comparisons between these two theories via using the notion of definitional equivalence . this is an application of logic to the philosophy of science and physics in the spirit of van benthem s .   [ section ]


n this paper , we consider the problem of detecting , localizing and tracking people not wearing/ carrying an electronic device , thus not actively participating in the localization effort , in large and heavily obstructed outdoor environments . to this purpose , we use an rf sensor network , _i.e. _ , a wireless system composed of low - power , inexpensive commercial of - the - shelf radio transceivers operating in the ghz ism band .these devices form a wireless mesh network by continuously broadcasting packets and measuring the received signal strength ( rss ) of the links of the network .radio tomographic imaging ( rti ) is applied to process the rss measurements collected by the rf sensors on multiple frequency channels and estimate 2d images of the change in the electromagnetic field of the deployment area due to the presence and movements of animates ( _ e.g. _ , people , large animals ) or metallic objects ( _ e.g. _ , cars ) . by further processing these images, the targets can be accurately detected and tracked .our work introduces an accurate and energy - efficient rti method that successfully deals with the nonstationary noise introduced in the rss measurements by environmental factors , such as rainfalls or wind - driven foliage , tipically encountered in real - world outdoor environments .rf sensor networks represent an appealing technology for detecting and tracking people outdoors : due to the small size and low cost of the rf units , they are less invasive and better conceivable than video camera networks , and considerably less expensive than ultra - wideband ( uwb ) transceivers. they also work in the dark and through smoke and non - metallic walls .however , an rf sensor network deployed outdoors faces challenging conditions , different from the more static ones typically found in real - world indoor environments .the main challenges encountered in outdoor environments are :  * the presence of what we refer to as _ environmental noise _ , _i.e. _ , the significant variation in rss observed when no person is located in the monitored area due to the time - varying multipath introduced by _e.g. _ wind - driven foliage , rainfalls or snow . in homes and buildings , on the contrary ,the rss measured in quasi stationary conditions does not vary significantly with time : the spurious variation observed in stationary conditions can be introduced by overlapping wi - fi networks increasing the floor noise level of the radio channel . * the lack of an initial calibration of the system performed in stationary conditions . due to the intrinsic nonstationarity of the monitored environment , rti methods to be applied foroutdoor localization can not rely on an initial calibration to estimate the average rss of the links of the network , which is then used as a reference to calculate the change in the electromagnetic field introduced by the presence of people in the monitored area . *the need to address the energy efficiency of the system .the remote locations and harsh weather conditions drive the need for a battery powered capability of the rf sensors , which , in turn , requires the adoption of an adaptive radio duty cycling mechanism to extend the lifetime of the system or , when energy harvesting systems are enforced , reduce their cost and size .several application scenarios could take advantage of an rf sensor network for outdoor people detection and localization .for example , in natural parks populated by protected animal species , a system composed of small and covert devices could be used to detect and locate poachers even in areas characterized by a dense canopy , which can not be monitored _ e.g. _ with air surveillance .an outdoor rti system could be used to monitor long international borders running across heavily forested areas from people trying to _ e.g. _ smuggle drugs or weapons , to protect large vineyards from vandalism or theft , or to create a virtual fence at a pasture site that would not require the cattle to wear gps collars .besides security and perimeter surveillance applications , an rti system would greatly extend the capabilities of _ smart spaces _ such as university campuses , malls and hospitals . in this work, we introduce a novel rti method for outdoor environments achieving high detection and localization accuracy and improving the overall system s energy efficiency .the main contributions of our work can be summarized as follows :  * we characterize the relationship existing in nonstationary outdoor environments between the variation in rss due to environmental noise and the link s _ fade level _ , _ i.e. _ , the difference between the measured and theoretical rss of a link .* we propose a method for selecting those link - frequency channel combinations ( which we refer to as _ link - channel pairs _ ) that , even in nonstationary environmental conditions , appear to be the most reliable for detecting the presence of a person on the link line , _ i.e. _ , the imaginary straight line connecting transmitter and receiver .we verify that this selection method allows enhancing both the detection and localization accuracy while simultaneously reducing the system s energy consumption .* we introduce an adaptive method to recalibrate on - line the reference rss of the selected link - channel pairs , even in presence of environmental noise and people in the monitored area , and apply background subtraction on the estimated radio tomographic images to further increase the robustness of the system to the time - varying environmental noise .the performance of the outdoor rti method was evaluated in a set of experiments carried out in a challenging outdoor environment , _i.e _ , a heavily forested area with trees and bushes of various height , shape and size .tests were performed in different environmental conditions ( _ e.g. _ , with no wind , light breeze , or gusts of wind ) .false alarm rate , localization accuracy and energy efficiency associated with the novel method are compared to those of the rti methods originally introduced in , to date the most accurate rti methods used in indoor environments , which we suitably adapt to the considered outdoor scenario in order to make a fair comparison .experimental results demonstrate that the novel outdoor rti method keeps under control the false positive and negative rates ( % and % , respectively ) , and reduces both the localization error ( from % to % ) and the energy consumption of the whole system ( from % to % ) compared to state - of - the - art methods .the paper is organized as follows . in section[ sec : related_literature ] , we survey the related literature on people detection and localization in outdoor environments and on state - of - the - art rti methods . the novel outdoor rti method is presented in section [ sec : multichannel_rti ] , while the experimental setup is described in section [ sec : setup ] .section [ sec : eval_metrics ] lists the metrics used to evaluate the performance of the system .the results of the tests are presented in section [ sec : results ] .conclusions are given in section [ sec : conclusion ] .the problem of localizing and tracking people in large outdoor areas has been addressed by several works .the system in was composed of spatially distributed rfid readers measuring the rss of rfid tags carried by people moving in the monitored area .the position of the targets was estimated by using a rss - distance model whose parameters were calibrated on - line to make the system more robust to environmental changes .the work in presented a heterogeneous system composed of low - quality wireless camera nodes devoted to people localization and rfid readers devoted to identification .both the systems in and assumed that the targets to be located were carrying an rfid tag .device - free localization and tracking of people has often been carried out by using visual sensor networks , _i.e. _ , wireless systems composed of a large number of low - power camera nodes .however , cameras produce a large amount of image data for the limited network s resources . in addition, cameras suffer from poor light conditions and occlusions in cluttered outdoor environments ( such as _ e.g. _ forested areas ) .for these reasons , other works have proposed using radars to detect personnel in heavily wooded areas . in , micro - doppler signals , _i.e. _ , doppler scattering returns not due to gross translation of the target but are instead produced by the periodic movements ( _ e.g. _ , of legs or arms ) of a walking human target , are exploited for detecting humans and distinguishing them from other animals .however , stand - alone low - frequency radars have a poor angular resolution . to overcome this limitation, the work in proposes a multiple - input multiple - output ( mimo ) radar exploiting the angular diversity of spaced antennas to detect the changes in the rf channel due to personnel in the deployment area . in this work, we present an energy - efficient rti method that enables device - free detection , localization and tracking of people in large and cluttered outdoor areas covered by a small number of low - cost and low - power rf sensors .the impact of environmental factors on wireless sensor networks deployed outdoors has been previously investigated .the works in analyze how the density and seasonal variations of vegetation , and daily temperature / humidity fluctuations affect the rss and the links connectivity .the work in studies how wind , _i.e. _ , wind - driven vegetation movement , affects the propagation of radio signals at different frequencies ( , , , and ghz ) .experimental data demonstrate that the variation in rss increases with the wind speed ( more consistently at higher frequencies ) and that the fade distribution goes from being rician to being rayleigh with an increasing wind speed .however , these works did not consider the combined effect of environmental factors and animates on the measured rss .different rti methods were introduced in . in ,frequency diversity was exploited to improve the localization accuracy .the same principle was later applied in . in all these works ,the rti systems were deployed in stationary indoor environments ( _ i.e. _ , no changes in the environment nor time - varying environmental noise ) .the system described in was deployed in a real - world domestic environment over a three months period . to recalibrate in an on - line fashion the reference rss values of the links of the network ,the rss signals were low - pass filtered . by doing this , the system maintained an high localization accuracy ( m approximately ) despite the frequent changes in the environment . in ,the authors carried out tests in a through - wall scenario having some of the rf sensors located in the proximity of trees whose branches were swaying in the wind .they proposed a subspace decomposition method to separate the change in rss due to human motion from that introduced by environmental noise .however , the amount of variation in the rss signals due to the environment was estimated during an initial calibration performed with no person in the deployment area .thus , the method assumed that the spatial and temporal characteristics of the environmental noise were constant .such an assumption can not be made in a forested outdoor environment , where the environmental noise is highly time - varying .the rti methods described above did not consider the issue of energy efficiency : the nodes radio , _i.e. _ , the most energy - hungry component , was always on , being the nodes in receive mode at all times , except for when they were transmitting a packet .energy efficient methods were presented in . in ,the authors apply compressed sensing techniques to rti , reducing the number of links that have to be sampled in order to reconstruct the whole image . in ,an accurate time synchronization protocol is used to enable radio duty cycling . in ,only the links near the current location of the tracked targets are measured . in our work ,we present a method to select those link - channel pairs that are robust to environmental noise ( thus , the most informative and reliable for rti ) .this approach allows increasing the energy efficiency of the system while simultaneously improving its detection and localization performance .this section describes the novel outdoor rti method . we provide the basics of rti , focusing on the solutions developed to address the challenges posed by multipath - rich and time - varying outdoor environments .the reader is invited to refer to for a detailed description of the principles of rti .algorithm [ alg : outdoor_rti_algorithm ] details the novel outdoor rti method .calculate the projection matrix    first , we analyze the characteristics of the links in nonstationary outdoor environments .to do this , we use the concept of fade level introduced in , which defines the relationship between steady - state , narrow - band fading and the changes in rss due to a person crossing the link line . the fade level of link on channel , ,can be estimated as : where : in ( [ eq : fade_level ] ) , is the average rss of link on channel measured with no person in the proximity of the link line , and is the theoretical rss , predicted by using the log - distance path loss model , for two nodes at distance . the equation in ( [ eq : pd ] ) represents a _ node - specific _ log - distance path loss model whose parameters , and ( _ i.e. _ , the path loss exponent , reference path loss , and reference distance , respectively ) , are derived by fitting the average rss of those links having node as transmitter . in other works , _e.g. _ , , a global path loss exponent was estimated by fitting the average rss of all the links of the network . in this work , by generating for each rf sensor an individual distance - rss model , hardware variability factors ( such as _ e.g. _ antenna impedance matching or relative antenna orientation between transmitter and receiver ) and local environmental differences ( such as _ e.g. _ the proximity to the node of dense foliage ) are taken into consideration . in section [ sec : results ] , we show that , by deriving an individual model for each node instead of a global one , the system achieves better detection and localization performance .  the fade level can be interpreted as a measure of whether a link - channel pair is experiencing destructive or constructive multipath interference ( or not ) : in the first case , the fade level is negative and the link - channel pair is said to be in _ deep fade _ ; in the latter case , the fade level is positive and the link - channel pair is said to be in _ anti - fade_. however , even in multipath - rich indoor environments , the measured rss does not vary significantly unless , _e.g. _ , a human body affects the propagation of one ( or more ) of the multipath components . on the contrary , outdoor environmental factors , such as wind , rainfall or snow ,introduce a significant variation in rss even when no person is found in the deployment area .we measure the effect of the environmental noise on a link - channel pair as the variance of the rss measurements collected when no person is in the proximity of the link line .the relationship between fade level and environmental noise is depicted in figure [ fig : fl_var_stationary ] .the points in figure [ fig : fl_var_stationary](a ) are obtained by measuring the rss of all the link - channel pairs of the system described in section [ sec : setup ] during four different intervals of minutes with wind of varying intensity blowing in the deployment area .figure [ fig : fl_var_stationary](b ) shows the boxplot of the rss variance of all the link - channel pairs , divided in six sub - groups ( a - f ) based on their fade level .the data show that link - channel pairs in deep fade , _i.e. _ , those having negative fade level , have a higher rss variance when the wind becomes stronger .thus , those links hardly contribute to the detection and localization problems .data in figure [ fig : fl_var_stationary ] show that link - channel pairs with a positive fade level ( _ i.e. _ , in anti - fade ) are more robust to environmental noise .moreover , link - channel pairs in anti - fade are characterized by a smaller sensitivity area , _i.e. _ , the area where a person affects the rss .this area is modeled as an ellipse having transmitter and receiver at the foci : for link - channel pairs in anti - fade , it can be modeled as a _narrow _ ellipse around the link line .thus , link - channel pairs with a positive fade level are not only more robust to environmental noise but can also be considered more reliable indicators of the presence of a person in the proximity of the link line .we provide an example of this in figure [ fig : link_example ] , which shows the rss , measured on different frequency channels , of a m link cutting through several branches of the trees found in the area . during the test , carried out in a day with wind of moderate intensity, a person crossed the link line at s. the only channel in anti - fade , _i.e. _ , channel having dbm , measured a consistent attenuation ( dbm ) in rss at the crossing event , while exhibiting small variation in rss due to the movements of the wind - driven foliage . on the contrary ,the other two channels in deep fade show a continuous , significant variation in rss due to the action of the wind , and this does not allow to unequivocally detect the crossing event .we now present a method to select a subset of link - channel pairs to be used for rti .the reason for doing this is twofold : on one hand , we want to select link - channel pairs that are more robust to environmental noise and , at the same time , more informative about the position of the targets in order to enhance the detection and localization performance of the system . on the other ,we want to increase the energy efficiency of the system , since the radio of the rf sensors can be turned off during the tdma slots originally allocated to the discarded link - channel pairs .define to be the set of link - channel pairs with positive fade level where is the link and the frequency channel .due to the low nodes density typical of outdoor deployments and the presence of several obstructions , some link - channel pairs may measure a rss close to the sensitivity threshold of the radio modules ( dbm at typical ambient temperatures for the radios described in section [ hardware_protocol ] ) .links with an average rss close to the sensitivity threshold belong to a _ grey region _ in which their connectivity performance becomes highly unpredictable and their rss measurements include noise and interference due to very weak signals .moreover , the connectivity of these links is highly affected by changes in the environment .thus , we discard all link - channel pairs with an average rss lower than a pre - defined threshold ( we set dbm ) .define to be the set of link - channel pairs with average rss higher than  now define to be the set of link - channel pairs to be considered . both the fade level and the average rss of the link - channel pairs are estimated when no person is found in the deployment area .  for each link - channel pair ,we calculate a weight as follows where is the rss variance measured when no person is found in the deployment area .consequently , link - channel pairs having low rss variance are assigned a higher weight , since the variations in rss measured on such links have a higher probability of being human - induced .finally , define the set as _i.e. _ , as the set in which , for each link in , only the frequency channel in ( where represents the set of measured frequency channels ) characterized by the largest weight is included .the change in rss at time for each link - channel pair in is estimated as : where and are the _ measured _ and _ reference _ rss of the link - channel pair , both considered at time , respectively . inmost of the previous works ( _ e.g. _ , ) , the reference rss was calculated as the average rss measured during an initial calibration of the system carried out in absence of people in the monitored area .this approach showed to be effective in time invariant indoor environments . in ,the reference rss was estimated in real - time by low - pass filtering the measured rss .this approach was effective in a dynamic domestic environment characterized by frequent , bursty events modifying the attenuation field of the deployment area .however , this approach was not effective when the person to be localized was not moving for an extended period of time ( _ e.g. _ , sleeping ) : in that case , the reference rss quickly converged to the measured rss , bringing the corresponding change in rss to zero despite the presence of the person in the deployment area . due to the presence of a time - varying environmental noise affecting the rss measurements , an outdoor rti system must continuously update the reference rss of the selected link - channel pairs , even when people are not moving inside the deployment area .  to estimate the reference rss in real - time without losing people who stop moving , we use the following procedure .we first create , for each selected link - channel pair , a fifo buffer of elements . is the length of the considered time window ( _ e.g. _ , s ) and is the sampling interval of the rti system , _i.e. _ , the interval of time required by the system to complete all the tdma rounds of communication on the frequency channels in ( see section [ hardware_protocol ] ) . define to be the set of estimated positions at time of the people located in the deployment area ( if no person is in the deployment area ) .since the change in rss of a link is assumed to be a spatial integral of the attenuation field of the monitored area , only the attenuation in the elliptical sensitivity area of the link will affect its rss .thus , at time , only the fifo buffers of link - channel pairs whose sensitivity areas do not include one of the positions in are updated .for each link - channel pair in , the reference rss is calculated as the average of the rss measurements contained in its fifo buffer . in this way, the system does not require an initial calibration to calculate the reference rss .moreover , the reference rss does not converge to the measured rss even when the person stops moving for an extended period of time .when we consider all the links of the network , the change in the electromagnetic field of the deployment area is estimated as : where and are vector of size of the rss change and noise of the links of the network , respectively .for the links belonging to , the rss change is calculated as in ( [ eq : rss_change ] ) .for all the others , the rss change is zero .vector represents the change in the _ discretized _ electromagnetic field of the monitored area , _i.e. _ , the intensity of each pixel of the radio tomographic image to be estimated .thus , is a vector of size , where is the total number of pixels of the radio tomographic image .element of the weight matrix indicates how pixel affects the change in rss of link . since the sensitivity area of a linkis modeled as an ellipse , is computed as : where is the area of the ellipse , _i.e. _ , the sensitivity area , of link , and are the distances of pixel from the transmitter and receiver , respectively , and is the parameter defining the width of the sensitivity area . the linear model in ( [ eq : matrix_rti ] ) is based on the correlated shadowing models in .regularization is required to solve the ill - posed problem of estimating the intensity of the _ many _ pixels in from the _ few _ links measurements in .we apply the regularized least square approach used also in : where :  is the regularization parameter ( _ e.g. _ , ) .the a priori covariance matrix is calculated using an exponential spatial decay model : =\sigma_{x}^{2}e^{-d_{i , j } /\delta_{c}},\ ] ] where is the variance at each pixel , is the euclidean distance between pixels and , and is a pixels correlation distance parameter .the inversion matrix , of size , has to be calculated only once after the deployment of the rf sensors , so that in ( [ eq : linear_transformation ] ) can be estimated in real - time .background subtraction is often used in machine vision in order to detect moving objects in videos recorded by static cameras . by iterating the process described in section [ image_estimation ] and exploiting the temporal continuity of the estimated images ,these can be considered as the _ frames _ of a video in which the targets to be detected and tracked are the blobs corresponding to people entering and moving in the monitored area .over the years , several background subtraction methods have been proposed .motion detection is performed by calculating the pixel - wise difference between the current frame and a reference ( or background ) image , which in our case is the estimated electromagnetic field of the monitored area when no person is found in it .only the pixels whose intensity is significantly different from that of the background image are labeled as _ foreground _ pixels , thus considered as potential moving objects , _i.e. _ , people . in a rti system deployed outdoor , the background image can change , gradually or suddenly , due to the environmental noise in the rss signals . thus , it must be continuously updated to increase the robustness of the motion detection process .we model each pixel of the background image as a realization of a gaussian distribution , where and are the expectation and standard deviation of the intensity of pixel at time .these parameters are computed using the corresponding fifo buffers containing elements for each pixel .the set of background pixels is defined as : where is a threshold defining the confidence interval of the intensity of the pixels ( we set ) and and are the mean and standard deviation of the gaussian distribution of pixel estimated at time , respectively . at time ,only the fifo buffers of the pixels in the background are updated with the current intensity value .  at time , the background image is calculated as a vector whose elements are the mean of the pixels fifo buffers : .\ ] ] finally , the background subtracted radio tomographic image , , is calculated as : the background subtraction allows increasing the difference in intensity between the pixels of the blobs corresponding to real people and those of the blobs introduced by the environmental noise , making the blobs tracking more accurate .   to detect and track the blobs corresponding to real people , we apply on the method in , providing high accuracy ( m tracking error approximately ) and real - time performance with multiple people even when these have intersecting trajectories .we refer the reader to for a detailed description of the applied multiple target tracking method . by applying the same detection and tracking methods , we can fairly evaluate the performance of the novel outdoor rti method presented in this work and compare it to previous works .the parameters of the method presented in section [ sec : multichannel_rti ] are summarized in table [ t : table_parameters ] ..rti parameters ( default values ) [ cols="^,^,^",options="header " , ]   [ t : indoor_rti_methods ]  we now consider the rti method in .there , the _ relative _ fade level of a link - channel pair is estimated as the difference between its average rss estimated in stationary conditions , and the lowest average rss measured on the frequency channels in : thus , , and for one channel on each link .the rss variance of the link - channel pairs estimated in stationary conditions is not taken into consideration .we refer to the rti method in as _ relative fade level _ , or _ rfl _ , and consider three different versions :  * the link - channel pairs with are selected .the change in rss of the links is estimated by applying the weighted average approach in ( [ eq : weighted_rsschange ] ) . * the link - channel pairs with an average rss , measured in stationary conditions , lower than discarded . among the remaining ones ,the link - channel pairs with are selected .the weighted average approach in ( [ eq : weighted_rsschange ] ) is applied .* the link - channel pairs with an average rss , measured in stationary conditions , lower than are discarded . for each link , only the frequency channel with the maximum fade level is selected .the results are summarized in table [ t : indoor_rti_methods ] . despite selecting more link - channel pairs ,the three considered versions of the method in have a localization accuracy from % ( with _ _ ) to % ( with _ _ ) worse than that of the most energy efficient outdoor rti method _. in addition , the method in has a significantly higher false alarm rate ( % with _ _ , % with _ _ , % with_ _ ) than the proposed outdoor rti method ( % with _ _ ) .we now analyze the effects of the background subtraction algorithm detailed in section [ sec : background_sub ] on the localization accuracy of the outdoor rti method ( _ _ ) .figure [ fig : back_sub ] shows the cumulative distribution functions of and , both when background subtraction is applied ( solid lines ) or not applied ( dashed lines ) .background subtraction increases the robustness of the motion detection and localization process to the rss variation due to environmental factors , which would introduce spurious blobs in the estimated radio tomographic images .consequently , by applying the background subtraction algorithm , the rti system is overall more accurate , and does not incur in very high ( _ i.e. _ , above m ) localization errors when links located far away from the current position of the target are affected by a consistent environmental noise .we analyze the effect of three parameters in table [ t : table_parameters ] on the localization accuracy of the system .figure [ fig : sens_analysis ] shows how and are affected by , _i.e. _ , the width of the elliptical sensitivity area of the links , , _i.e. _ , the length of the time window used to determine the reference rss of the links , and , _i.e. _ , the length of the background image training period .simulations are performed by post - processing the rss measurements collected during the deployment of the system when the person was located inside the monitored area . in the simulations , we set .the results in figure [ fig : sens_analysis ] indicate that the proposed outdoor rti method is robust to variations of the three considered parameters , as both and have a % maximum variation from the values reported in table [ t : rti_methods_table ] .in this paper , we present and evaluate a novel rti method for accurately detecting and tracking people in nonstationary outdoor environments .an rti system is composed of static rf sensors , communicating on multiple frequency channels in the ghz ism band , which continuously measure the rss of the links of the mesh network to estimate the change in the electromagnetic field of the monitored area introduced by the presence and movements of people .the outdoor rti method described in this work makes the system robust to the time - varying environmental noise typical of harsh outdoor environments , _i.e. _ , the variation in rss introduced by environmental factors such as wind - 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Abstract: rf sensor networks are used to localize people indoor without requiring them to wear invasive electronic devices . these wireless mesh networks , formed by low - power radio transceivers , continuously measure the received signal strength ( rss ) of the links . radio tomographic imaging ( rti ) is a technique that generates 2d images of the change in the electromagnetic field inside the area covered by the radio transceivers to spot the presence and movements of animates ( _ e.g. _ , people , large animals ) or large metallic objects ( _ e.g. _ , cars ) . here , we present a rti system for localizing and tracking people outdoors . differently than in indoor environments where the rss does not change significantly with time unless people are found in the monitored area , the outdoor rss signal is time - variant , _ e.g. _ , due to rainfalls or wind - driven foliage . we present a novel outdoor rti method that , despite the nonstationary noise introduced in the rss data by the environment , achieves high localization accuracy and dramatically reduces the energy consumption of the sensing units . experimental results demonstrate that the system accurately detects and tracks a person in real - time in a large forested area under varying environmental conditions , significantly reducing false positives , localization error and energy consumption compared to state - of - the - art rti methods .   radio tomography , device - free localization , wireless sensor networks , adaptive systems