Synchronizing rhythms of logic (2024)

1.1 Background

Pursuing a vision experienced by HM in 1985, we provedin 2005 that evidence, while it constrains explanations, necessarily leaves openan infinite range of conflicting explanations [2]. Thus, choosing anexplanation requires an extra-logical act. I (JMM) call such an act a guess. What do I mean by “guess”? As an individual scientist, when I’mpuzzled, diverse thoughts flit into my mind. Some are fleeting, others I pick upand use as starting points for doing something, and these I call guesses. Forme, guessing contrasts with calculating. The logical gap between evidence andits explanations, calling for guesswork to link them, reveals anunpredictability far more drastic than quantum uncertainty, and beyond anyfuzziness due to limited sample sizes.

Although it has important consequences, the proof has been largely overlooked.For one thing, verifying the proof requires knowledge omitted in introductorycourses on quantum mechanics. In these, one learns how a quantum state and ameasurement operator on a vector space imply certain probabilities of outcomes,but not about what is called an inverse problem, the problem of using quantumtheory to explain measured data. The inverse problem is important whensomething surprising is found, as in the Stern–Gerlach experiment that led tohalf-integral spin [3].

In a limited way, the inverse problem is addressed in quantum decision theory[4, 5] and in quantum information theory[6]. However, one usually assumes a finite dimensionalvector space, and that the measurement operators are known (this leads towhat is called quantum tomography), but both the operators and thedimension are invisible to experiment, so the assumption is logicallyindefensible. Thus, solving an inverse problem requires determining aquantum state and the measurement operators from the probabilities ofoutcomes abstracted from experiments, without assuming a dimensionof the vector space. We proved that this problem cannot have a uniquesolution, and indeed that infinitely many conflicting explanations fit whateverevidence is at hand, so that choosing among them is impossible without a guess. Inan application, we produced a second explanation of probabilities involved in aquantum-cryptography protocol, revealing an unsuspected security vulnerability[7, 8].

1.2 Obstacle of global time

Appreciating logical synchronization is impeded by the widespread tacit assumption of “universal time,” as if the production oftime and space coordinates for scientific and other purposes were irrelevant totheoretical physics.²²2“An important task of the theoretical physicistlies in distinguishing between trivial and nontrivial discrepancies betweentheory and experiment” [1, p.3]. Setting aside this assumption allows attention to synchronization in digitalcommunications, which is essential to time distribution, to computation, and indeed toall logical operations.

1.3 Inseparability of computation and communication

The digital communication and computation that pervade today’s science depend on:(1) physically distinct conditions to distinguish between 0 and 1, (2)transitions between these conditions, (3) local timing ordered by local clocks thatare synchronized differently from the usual way in physics, and (4) guesswork.How these work together in computer networks is one of the main topics of this report.

Computations involve the communication of inputs and outputs over networks,including the Internet. Going the other way, digital communication requiresmessage processing, e.g. copying, addressing, searching, coding, and decoding, allof which are instances of computation. We elevate this back-and-forthdependence to the following principle of wide application.

We will speak of neither separately, but instead bundle them together, speakingof computational networks. These are found not only in electroniccomputers but across all life. In section2 we define computationalnetworks embedded in cyclic environments that produce inputs and consumeoutputs, as in process control. Computational networks express two levels ofunpredictability: (1) the inputs supplied by an environment are viewed asunpredictable, and (2) unforeseen happenings change one computational networkinto another.

Computational networks perform logical operations that work in coordinatedrhythms. Without communication-guided adjustments, the rhythms of any twological operations drift, as do any two clocks, so that phase deviationsbetween them increase indefinitely. To limit phase deviations, computationalnetworks depend on a type of synchronization that differs from that whichEinstein made famous in special relativity. This logical synchronizationmaintains the necessary phase relations in the rhythms of logical operations. Topicture logical synchronization, think of playing a game of catch, tossing aball back and forth with another person: you cycle through phases of tossingand catching, and if you are looking the wrong way you don’t catch the ball.Coordinating the phase “ready to catch” with the arrival of the tossedball exemplifies logical synchronization. As tangible examples ofcomputational networks, we introduce “token games” in which you participateby moving tokens over a game board.

Although some contributions to phase deviations can be predicted, forothers the statistical properties are non-stationary andunpredictable. To maintain logical synchronization, rate adjustments guided byguesswork are necessary.³³3For a pendulum clock, one adjusts the tick rateby changing the length of the pendulum by a small amount. Typically, a screw at thebottom of the pendulum is turned to raise or lower its center ofgravity slightly. Phase deviations in logical synchronization limitthe rate of information processing. The tighter the logically synchronization,the smaller the phase deviations, and so the more information can flow.Controlling drift in clocks is analogous to steering a car in a gustycrosswind: better control depends on better guesses.

We discuss only the logical aspect of computational networks; i.e., we saynothing about energy, weight, shape, visual or auditory output, etc. We discussthe physical mechanisms of electronic computation only to show their “lumpy”character, leaving open whether the lumps are small or large and whether they arelumps in frequencies of oscillations or in spatial extent or some othermodality. Because it is focused exclusively on the logical aspect, our formulationof computational networks and their logical synchronization has applications to notonly man-made systems but also biology, including human thought.Rejecting the assumption that “physical” means “entirely explicable,” werepresent human thought as computing interspersed with unforeseeable reprogrammings.

In section3, we represent the logic of computational networks by means of markedgraphs. The concept of logical distance is introduced and shown to dependon not only the graph but also its family of markings, of which there maybe more than one. Unpredictable changes in computation are expressed by unpredictable changes from one marked graph to another.

As discussed in section4, logical synchronization and its dependence onextra-logical adjustment point to an alternative concept of time and space. Insection4.2, we show how measurements of computational networks that attemptto determine a temporal order of events sometimes lead to nonsense. Analternative “time” based on logical synchronization, rather than that definedin special relativity, carries forward the identification of “time” withcommunication that was begun by Einstein in 1905.

Section5 pulls together the main results relevant to biology, andsection6 reflects on the main points. For example, we see biological evolution as mirroring the human scientific approach: processes involving computation, testing, and punctuated by analogies of guesses. That discussion also draws together some points that appeared as fragments in previous sections.

2.1 Computing by logical operations in contact with unpredictable environments

The gates of a computational network physically perform logicaloperations associated with the mathematical logical connectivessuch as and, not, nor, and nand, where nand is thenegative of the and connective. Nand is defined by the table ofarguments and values in fig.1. We emphasize the distinctionbetween the physical performance of logical operations, which involvesmotion, and mathematical logical connectives which can be written intables that sit still on a page. To distinguish the operations from theconnectives, we capitalize the operations as AND, NOT, NOR, NAND, etc.

Because all the logical operations can be defined as compounds of NAND,that operation plays a major role in what follows.⁵⁵5How the AND, NOT, and NOR gates of computational networks are made from NANDgates (together with FORKs) is shown in fig.21 in AppendixA.3.Each input of any logicaloperation can be either of two distinct physical conditions, and so can theoutput. These conditions are represented by 0 and 1, as indicated infig.1.

2.2 Gates and networks

While the gates mentioned so far deal only with 0’s and 1’s, we also use theterm gate to indicate a mechanism that performs complex logical functions,as occurs in coarsened descriptions of logical operations (see AppendixA.2). We deliberately leave unspecified the various physicalmechanisms that can serve as gates, and we name gates by the operations thatthey perform.

Whatever a computer chip can do can be done by a network of NAND gates wiredtogether. We express the logic of wiring by another connective that we call fork, which as far as we can tell has been overlooked in formallogic. Figure2 illustrates this connective along with the associated gateoperation FORK.

2.3 Graphs and token games

Graphs comprised of nodes connected by arrows are used widely in science. For example,fig.3 illustrates the citric acidcycle that powers most life [13].

The concept of a computational network recognizes an unpredictable environmentthat generates inputs and receives outputs (for process-control computations,this back-and-forth communication with an environment is evident). Werepresent computational networks by graphs, in which each nodeeither represents a gate or is labeled ENV to represent an unpredictable environment thatissues inputs and consumes outputs. Each arrow points from one node to either thesame or another node, representing a connection in which the output of onenode serves as an input to that or another node. This use of graphs leads totwo developments, one physical and the other mathematical. On the physicalside, one can make a small computational network by using a drawing of thegraph as a “game board” on which to either perform NAND and FORK operations byhand or make free choices to simulate the unpredictable ENV;tokens labeled by 0 or 1 are moved in what is called a token game [15].On the mathematical side, representing computational networks bygraphs leads to an elaboration of graph theory, i.e. marked graphs, which are discussed in section3.

To show the need for the after in the firing rule, fig.4shows snapshots of a token game, before, after, and during a firing. Figure4(a) shows a fireable node $v$, whilefig.4(c) shows the graph after a firing of $v$.Figure4(d) shows what the snapshot in the midst of the move could show, without the “after” in the firing rule: $v$ is no longer fireable but hasnot been completely fired, and $y$ has been made fireable. Node $y$ can fire before completion of the firing of $v$,resulting in a return to the situation of fig.4 (a), without node $w$ever firing. Preventing such behavior necessitates the“after” condition in the firing rule.

Figure5 shows a simple example of a token game:⁶⁶6The drawing infig.5 resembles a digital circuit diagram, from which we borrow thesymbol for NAND.

In a second example, a (simplified) thermostat performs a repetitive computationto control the temperature of a room. As illustrated in fig.6, the“Compute” node compares a desired setting written on a token supplied by theenvironment (ENV) with the temperature on a token from a digital thermometer inthe environment; from the result of this comparison, the Compute node sends acommand on a token to ENV to turn the furnace on or off. The comparison makes use of anarithmetic adder, illustrated in fig.18 in SectionA.2.

2.4 Communication of tokens depends on logical synchronization

Clocks step the gates of electronic computers through phases, and a gate gatecan receive a token in one phase but not in others [16, 17].Maintaining adequate phasing of token arrivals at logic gates constitutes theirlogical synchronization, illustrated in fig.7. Analogous phasemanagement is to be looked for in biochemical cycles and the spiking of neurons.

The difference in hand positions of the two clocks indicates a propagationdelay; e.g. the clock at the arrow head shows a later time than thesending clock at the arrow tail. As discussed in [16, 17],logical synchronization can be maintained between computers without them beingset to standard time broadcasts, so long as their phases are aligned withpropagation delays.

2.5 Unpredictable steering to maintain logical synchronization

Maintaining logical synchronization requires (1) detecting unpredictable deviations in the phasing of token arrivals, and (2) responding to those deviations by adjusting the rhythms of the computational networks to maintain the deviations within allowed tolerances.

Before it can be maintained, logical synchronization must be acquired so that communication can begin. It is often acquired quickly, but there is no deterministic limit on the time required to do so [18, Chaps.4 and 5].

2.6 Disorders of computing from failures of logical synchronization

If logical synchronization fails in the transmission of a token from one part ofa network to another, the incoming token label can be ambiguous, leading to acomputer crash. To reduce this hazard, designers interpose a “decisionelement,” based on the flip-flop shown in fig.10. However, thisdecision element suffers from a half-life of indecision as discussed insection4.2. Thus, even when filtered by the decision element, the ambiguitycan propagate through a FORK to be seen as 0 by one branch 0 and as 1 by theother branch, thereby producing a logical conflict known as glitch, whicheven filtered leads to a crash [19].

2.7 Preview of cognition in living organisms

In living organisms DNA codes are reproduced with astoundingly low error ratesavailable only in digital computing.⁷⁷7For example, “the human mutationrate …is approximately 1 mutation/10¹⁰ nucleotides/cell division”[13]. While living organisms also make graded movements that correspondto analog computation, without digital computation their DNA replication woulddrift enough to make life impossible[20]. Digital computation in livingorganisms confronts a physical problem—the need for logicalsynchronization—that human hardware engineers have solved (well enough) sothat software engineers do not need to know about it [21]. Many models ofliving organisms employ logic gates, but to the best of our knowledge, we arethe first to recognize the need for logical synchronization in biologicalcomputing.

Without discussing consciousness, we show that computation has a place in humancontrol of physiological processes and in human thinking. In section5 wemodel biological computation by computational networks linked to unpredictableenvironments, understanding that the networks themselves undergo unpredictablechanges, and that maintaining logical synchronization requires somethingbeyond digital computing. As represented by the marked graphs ofsection3.5, changes in computational networks are pictured incorresponding unpredictable changes in the marked graphs.We suggest that logical distance as defined in section3.3 serves as aphysiological underpinning of navigation.

3.1 Multiple families of markings

As illustrated in fig.8, some graphs admit more than onefamily of markings. The three markings belong to three distinct families ofmarkings, as follows from 4, because the token count of the outer(also the inner) circuit differs in the three cases. The simplest such markedgraph is shown in fig.9.)

Remark: Essentially the same topology appears in the flip-flopshown in fig.10, with its two live and safe families of markings andits two conflicting simple circuits, i.e., (A, ENV, B) and (B, ENV, A). Aninteresting question is whether digital designs can avoid multiple markingfamilies, and if so, at what cost in the number of logical operations required.

In many cases, different families of markings on the same graph representcomputations of different functions. The existence of graphs that support morethan one marking family shows that a marking family conveys information notconveyed by the underlying graph. One can ask the following question: whatconditions on a strong graph are necessary and sufficient for the existence ofmore than one family of markings on that graph? See the next subsection forpart of the answer, which may be original, given that we have not seen it elsewhere.

3.2 Conditions for existence of multiple families of markings

The issue arises of characterizing graphs that have a unique marking family as opposed to those that have multiple marking families.

Question:Does the converse of 5hold; i.e., does every strong graph with conflicting simple directed circuits admit more than one family of markings? It does in a dozen or so cases that we have explored, but we do not know whether this is so in all cases.

3.3 Logical distance

In special relativity, distance is defined as a count of clock ticksbetween sending a signal and receiving an echo from a distantpoint[23]. An analogous and purely mathematical, logical distancebetween nodes of a marked graph with a family of markings is the following (which we have not seen elsewhere).

In fig.13, the token count of the path from $v_{0}$ to $v_{5}$ is 1, while the token count of the path from $v_{5}$ to $v_{0}$ is 2, so the logical distance is 3; the token on arrow $a_{2}$ is on both paths and therefore is counted twice.Figure14 shows an added path that eliminates the need for the overlapping paths of fig.13 and reduces the logical distance to 1.

3.4 Coarse representations with generalized nodes and labels

At the finest level of representation, the only nodes are NAND, FORK, and ENV, with token labels 0 or 1, and such graphs can be huge. To modularize them,fragments of a graph can be contracted to make a graph that expresses a coarser level of detail.¹⁰¹⁰10Contractions of graphs are a special case of graph morphisms.In contrast to an unmarked graph, contracting a marked graph requiresattention to the effect of the contraction on markings and their labels.After contracting a fragment to a node of a coarser graph, the labeling rule for the new node can define a complex logical operation on strings of bits rather than just single bits.

Sometimes we omit token labels, and fig.15 shows contractionswith token labels omitted.

Labeling rules for contractions can involve memory in addition toinputs and outputs. The allowance of multigraphs with loops, rather than thesimple graphs usual for Petri nets, simplifies contractions.SectionA.3 uses contractions in showing howmarked graphs can express the “if-then-else” construct used in computer programs.

3.5 Representing the unpredictable evolution of computational networks

Computational networks can encounter unpredictable needs for revisions. Forexample, two computational networks can come into communication, turning into asingle network; going the other way, a computational network can fission intotwo networks. Some revisions involve sequences of fusion and fission, both ofwhich can be represented by before-and-after snapshots expressed by pairs ofmarked graphs. The simplest cases of (1) fusing nodes of differentcomputational networks into a single node and (2) fission of a single node areshown in fig.16.

Other changes are the injection of a node that splits an arrow, or the deletion of a note that separates two arrows along a path.Left for future exploration are morphisms of labeled marked graphs, which are complicated by multiple marking families.

4.1 When global snapshots fail, local records work

When global snapshots fail, local records can be organized to determine thegraph and the marking family for a computational network. For example, say eachnode makes a record each time if fires. The record includes a count of thenode’s own firings and its own name. It writes these on the labels of thetokens that it produces, so that its own record of a firing can include thenames and counts that come in the labels on its input tokens. Without thecounts, local records (once collected) determine the graph but not the familyof markings. With the counts, the marking class can be determined. For theexample in fig.17 the following display of records suffices todistinguish among the three families of markings shownin fig.8. ¹¹¹¹11When more than one arrow connectstwo nodes, the arrow names must also be included in the local records.

Marking family 2-4 $\Big{|}$ Marking family 3-3
$V_{1}$#1; source $V_{2}\#0,V_{0}\#0$; target $V_{2},V_{0}$ $\Big{|}$$V_{1}$#1; source $V_{2}\#0,V_{0}\#0$; target $V_{2},V_{0}$
$V_{3}$#1; source $V_{2}\#0,V_{4}\#0$; target $V_{2},V_{4}$ $\Big{|}$$V_{3}$#1; source $V_{2}\#0,V_{4}\#0$; target $V_{2},V_{4}$
$V_{2}$#1; source $V_{1}\#1,V_{3}\#1$; target $V_{1},V_{3}$ $\Big{|}$$V_{2}$#1; source $V_{1}\#1,V_{3}\#1$; target $V_{1},V_{3}$
$V_{4}$#1; source $V_{3}\#1,V_{5}\#0$; target $V_{3},V_{5}$ $\Big{|}$$V_{4}$#1; source $V_{3}\#1,V_{5}\#1$; target $V_{3},V_{5}$
$V_{5}\#1$; source $V_{4}\#1,V_{0}\#0$; target $V_{2},V_{0}$ $\Big{|}$$V_{5}\#1$; source $V_{4}\#0,V_{0}\#0$; target $V_{2},V_{0}$
$V_{0}$#1; source $V_{5}\#1,V_{1}\#1$; target $V_{5},V_{1}$ $\Big{|}$$V_{0}$#1; source $V_{5}\#1,V_{1}\#1$;target $V_{5},V_{1}$

The local records link markings only logically and not temporally. The logical links determine the families of markings, shown in the state-transition graphs of figs. 11 and 12.

4.2 Physical undecidability of temporal order of concurrent events

5.1 Modes of operation in computational networks

A description of logic gates and their interconnections covers the staticstructure of a computational network, but more is needed to describe itsfunction. Most computational networks can behave in any of various modes whenthey are first established, even when only a single mode of behavior is desired.An analogy is waking up and being momentarily disoriented: seeing things“wrongly.” While graphs without markings are used widely in biology, it isonly marked graphs that can distinguish among multiple modes of operation.

For example, multiple modes of operation are possible for a computationalnetwork corresponding to a figure 11.3 shown in Rhythms ofthe Brain, [p.286][28] and captioned “Multiple excitatory glutamatergic loops inthe hippocampal formation and associated structures.” That figure represents a partof a computdational network that contains conflicting simple directed circuits(defined in SectionA.1). Inspection shows that that computationalnetwork admits more than one mode of operation, which implies multiple possiblerhythms of node firings.

5.2 Logical distance in biological computing

The marked graphs suggest a biological evolutionthat allows people to develop a concept of distance. In physics, distance isdefined by the time required for to-and-from communication. For example, radarmeasures distance by the time from sending a signal to receiving an echo[29], and this is the electric version of what bats and porpoises do withsound. As described in section3.3, an analogous logical distance is thenumber of firings of a node that must occur for a label produced by the node ofa computational network to circulate to a distant node and come back to thesender. We conjecture that in animal physiology, logical distance is essentialto the capacity to navigate and to the evolution of the concepts of time anddistance in people.

5.3 Logical synchronization in human conversation

I (JMM) like logical synchronization as a metaphor for “connecting” in humanconversation. Sometimes a friend and I are “synched”: we hear each other.At other times, in some conversations on Zoom, no one hears what anyone else is saying.In both digital technology and in human conversation, the“clicking into synchrony” so that communication is possible comes when itcomes; it is unpredictable and can’t be hurried (although the digital time scale isnanoseconds and the human time scale is something else).

I picture my own thoughts as comprising fragments of computational networks in a flux of unpredictable modifications and interactions, coming into and out of logical synchronization. As emphasized in section4.2, there is no universal time step to the rhythms of logical operations and their modifications. At unpredictable moments, fragments of networks of thought can meet, fuse with one another, possibly to fission later, responding sometimes to spontaneous inner life and other times to external stimuli.

A.1 Glossary and properties of marked graphs

The definitions of terms for graphs vary among authors. Herein we use terms adapted mostly from [31].

We say that an arrow connects the first node of its pair to the second node.The first node is the source of the arrow, and the second node is thetarget of the arrow. An arrow is an out-arrow of its sourcenode and an in-arrow of its target node. The source and target nodes ofan arrow need not be distinct, in which case the arrow is a loop.More than one arrow can connect a given pair of nodes. In the jargon, our graphs are “finite, directed multi-graphswith loops.”
A node that is a source or a target (or both) of an arrow is incident with the arrow.
A directed path in a graph is a sequence of arrows joined head to tail at nodes, with no repeated arrows.
A simple directed path in a graph is a directed path in which all nodes are distinct.
A circuit is a closed directed path (which can be a loop)
A simple directed circuit is a closed simple directed path.
A graph is strongif it contains a directed path from every node to every other node.
An induced subgraph of a graph $G$ is any graph consisting of a subset of nodes of $G$ and all the arrows of $G$ that connect pairs of those nodes.
We call a simple path from a node $v_{1}$ to a node $v_{2}$ a 1-1 path if (a) it consists of a single arrow or (b) its nodes other than its end points each have one in-arrow and one out-arrow.
We call a node with one in-arrow and one out-arrow a1-1 node.

A node is fireable in a marking if the token number on each in-arrow of the node is at least 1.
For an unlabeled marked graph, a firing of a node is a relation between two markings: markings $M$ and $M^{\prime}$ are related by the firing of node $v$ if the token number of $M^{\prime}$ for each in-arrow of node $a$ 1 lower than the corresponding token number for $M$, and if the token number $M^{\prime}$ for each out-arrow of node $v$ is 1 higher than the corresponding token number for $M$.
A marking is called live if every node is either fireable, or can be made so through some sequence of firings.

From [22, Theorem 12] we have the following proposition.

Hence, “live markings of a strongly connected graph partition into equivalence classes. …Let us refer to each equivalence class as afamily” [22, Thm. 12]. It follows that any marking of a family can be produced from any other by some sequence of firings.
We call the sum of token numbers over each arrow of a circuit the token count of the circuit. Because the firing of a node belonging to a circuit balances the decrease in the token number on the in-arrow of the node with the increase in the token number on the circuit out-arrow, we have the following proposition.

To use marked-graphs to represent interconnected logical operations one must restrict the graphs and the markings to avoid the piling up more than one token on any arrow; that means requiring safe markings.

To represent computing mathematically, we invoke strong graphs with live and safe families of markings. Only for strong graphs are live and safe markings possible.¹⁴¹⁴14Among graphs that are connected in the weak sense that ignores arrow directions, a graph can have a live and safe marking if and only if it is strongly connected.

A.2 Fragments of graphs for addition, illustrating graph contractions

Numerical comparisons (as in the thermostat of fig.6) involveaddition. Figure18 shows addition performed by gates[34], and itillustrates coarse descriptions using contracted fragments of graphs.

A.3 Program control in marked graphs

While our purpose is not to express general-purpose computers, this can be doneby marked graphs that have paths of token propagation under program control.For instance, fig.19(a) pictures a contracted fragment of a graphrepresenting a switching element controlled by input c. If c is 0,then the token label of $X$ goes to $A$ and the token label of $Y$ goes to $B$, whileif c is 1, then the token label of $X$ goes to $B$ and the token label of $Y$goes to $A$. Figure19(b) shows the switching element as an uncontractedfragment of a graph.

Figure20 shows how the input ${\bf c}$ from the environment controls the routing of other inputs x and y through a computation.

A.4 Counting by variation in token labels

Asymptotically, all nodes fire equally frequently, but token labels allowthe expression of rhythms of bit values. For example, fig.21shows a marked graph in which node $V_{7}$ produces a 1 bit every third cycleand a 0 bit on the two cycles in between.