Event calculus

Domain-independent axiom

The event calculus was developed in part as an alternative to the situation calculus, as a solution to the frame problem, of representing and reasoning about the way in which actions and other events change the state of some world.

There are many variants of the event calculus. But the core axiom of one of the simplest and most useful variants can be expressed as a single, domain-independent axiom:

${\displaystyle \mathit{h}\mathit{o}\mathit{l}\mathit{d}\mathit{s}\mathit{A}\mathit{t}(F,T2)\leftarrow }$

${\displaystyle [\mathit{h}\mathit{a}\mathit{p}\mathit{p}\mathit{e}\mathit{n}\mathit{s}\mathit{A}\mathit{t}(E1,T1)\wedge \mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{i}\mathit{a}\mathit{t}\mathit{e}\mathit{s}(E1,F,T1)\wedge (T1<T2)\wedge }$

${\displaystyle \mathrm{\neg }\mathrm{\exists }E2,T[\mathit{h}\mathit{a}\mathit{p}\mathit{p}\mathit{e}\mathit{n}\mathit{s}\mathit{A}\mathit{t}(E2,T)\wedge \mathit{t}\mathit{e}\mathit{r}\mathit{m}\mathit{i}\mathit{n}\mathit{a}\mathit{t}\mathit{e}\mathit{s}(E2,F,T)\wedge (T1\le T<T2)]}$

The axiom states that

a fluent ${\displaystyle F}$ holds at a time ${\displaystyle T2}$ if

an event ${\displaystyle E1}$ happens at a time ${\displaystyle T1}$ and

${\displaystyle E1}$ initiates ${\displaystyle F}$ at ${\displaystyle T1}$ and

${\displaystyle T1}$ is before ${\displaystyle T2}$ and

it is not the case that there exists an event ${\displaystyle E2}$ and a time ${\displaystyle T}$ such that

${\displaystyle E2}$ happens at ${\displaystyle T}$ and

${\displaystyle E2}$ terminates ${\displaystyle F}$ at ${\displaystyle T}$ and

${\displaystyle T1}$ is before or at the same time as ${\displaystyle T}$ and

${\displaystyle T}$ is before ${\displaystyle T2}$.

The event calculus solves the frame problem by interpreting this axiom in a non-monotonic logic, such as first-order logic with circumscription or, as a logic program, in Horn clause logic with negation as failure. In fact, circumscription is one of the several semantics that can be given to negation as failure, and it is closely related to the completion semantics for logic programs (which interprets if as if and only if).

The core event calculus axiom defines the ${\displaystyle holdsAt}$ predicate in terms of the ${\displaystyle happensAt}$, ${\displaystyle initiates}$, ${\displaystyle terminates}$, ${\displaystyle <}$ and ${\displaystyle \le }$ predicates. To apply the event calculus to a particular problem, these other predicates also need to be defined.

The event calculus is compatible with different definitions of the temporal predicates ${\displaystyle <}$ and ${\displaystyle \le }$. In most applications, times are represented discretely, by the natural numbers, or continuously, by non-negative real numbers. However, times can also be partially ordered.

Domain-dependent axioms

To apply the event calculus in a particular problem domain, it is necessary to define the ${\displaystyle initiates}$ and ${\displaystyle terminates}$ predicates for that domain. For example, in the blocks world domain, an event ${\displaystyle move(Object,Place)}$ of moving an object onto a place initiates the fluent ${\displaystyle on(Object,Place)}$, which expresses that the object is on the place and terminates the fluent ${\displaystyle on(Object,Place1)}$, which expresses that the object is on a different place:

${\displaystyle \mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{i}\mathit{a}\mathit{t}\mathit{e}\mathit{s}(move(Object,Place),on(Object,Place),Time).}$

${\displaystyle \mathit{t}\mathit{e}\mathit{r}\mathit{m}\mathit{i}\mathit{n}\mathit{a}\mathit{t}\mathit{e}\mathit{s}(move(Object,Place),on(Object,Place1),Time)\leftarrow different(Place1,Place).}$

If we want to represent the fact that a ${\displaystyle Fluent}$ holds in an initial state, say at time 1, then with the simple core axiom above we need an event, say ${\displaystyle initialise(Fluent)}$, which initiates the ${\displaystyle Fluent}$ at any time:

${\displaystyle \mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{i}\mathit{a}\mathit{t}\mathit{e}\mathit{s}(initialise(Fluent),Fluent,Time).}$

Problem-dependent axioms

To apply the event calculus, given the definitions of the ${\displaystyle holdsAt}$, ${\displaystyle initiates}$, ${\displaystyle terminates}$, ${\displaystyle <}$ and ${\displaystyle \le }$ predicates, it is necessary to define the ${\displaystyle happensAt}$ predicates that describe the specific context of the problem.

For example, in the blocks world domain, we might want to describe an initial state in which there are two blocks, a red block on a green block on a table, like a toy traffic light, followed by moving the red block to the table at time 1 and moving the green block onto the red block at time 3, turning the traffic light upside down:

${\displaystyle \mathit{h}\mathit{a}\mathit{p}\mathit{p}\mathit{e}\mathit{n}\mathit{s}\mathit{A}\mathit{t}(initialise(on(red\mathrm{\_}block,green\mathrm{\_}block),0)}$

${\displaystyle \mathit{h}\mathit{a}\mathit{p}\mathit{p}\mathit{e}\mathit{n}\mathit{s}\mathit{A}\mathit{t}(initialise(on(green\mathrm{\_}block,table),0)}$

${\displaystyle \mathit{h}\mathit{a}\mathit{p}\mathit{p}\mathit{e}\mathit{n}\mathit{s}\mathit{A}\mathit{t}(move(red\mathrm{\_}block,table),1)}$

${\displaystyle \mathit{h}\mathit{a}\mathit{p}\mathit{p}\mathit{e}\mathit{n}\mathit{s}\mathit{A}\mathit{t}(move(green\mathrm{\_}block,red\mathrm{\_}block),3)}$

A Prolog implementation

The event calculus has a natural implementation in pure Prolog (without any features that do not have a logical interpretation). For example, the blocks world scenario above can be implemented (with minor modifications) by the program:

<span id="mw7w"></span><span class="nf" id="mw8A">holdsAt</span><span class="p" id="mw8Q">(</span><span class="nv" id="mw8g">Fluent</span><span class="p" id="mw8w">,</span> <span class="nv" id="mw9A">Time2</span><span class="p" id="mw9Q">)</span> <span class="p" id="mw9g">:-</span> <span class="nf" id="mw9w">before</span><span class="p" id="mw-A">(</span><span class="nv" id="mw-Q">Time1</span><span class="p" id="mw-g">,</span> <span class="nv" id="mw-w">Time2</span><span class="p" id="mw_A">),</span> <span class="nf" id="mw_Q">happensAt</span><span class="p" id="mw_g">(</span><span class="nv" id="mw_w">Event</span><span class="p" id="mwAQA">,</span> <span class="nv" id="mwAQE">Time1</span><span class="p" id="mwAQI">),</span> <span class="nf" id="mwAQM">initiates</span><span class="p" id="mwAQQ">(</span><span class="nv" id="mwAQU">Event</span><span class="p" id="mwAQY">,</span> <span class="nv" id="mwAQc">Fluent</span><span class="p" id="mwAQg">,</span> <span class="nv" id="mwAQk">Time1</span><span class="p" id="mwAQo">),</span> <span class="o" id="mwAQs">not</span><span class="p" id="mwAQw">(</span><span class="nf" id="mwAQ0">clipped</span><span class="p" id="mwAQ4">(</span><span class="nv" id="mwAQ8">Fluent</span><span class="p" id="mwARA">,</span> <span class="nv" id="mwARE">Time1</span><span class="p" id="mwARI">,</span> <span class="nv" id="mwARM">Time2</span><span class="p" id="mwARQ">)).</span> <span class="nf" id="mwARU">clipped</span><span class="p" id="mwARY">(</span><span class="nv" id="mwARc">Fluent</span><span class="p" id="mwARg">,</span> <span class="nv" id="mwARk">Time1</span><span class="p" id="mwARo">,</span> <span class="nv" id="mwARs">Time2</span><span class="p" id="mwARw">)</span> <span class="p" id="mwAR0">:-</span> <span class="nf" id="mwAR4">terminates</span><span class="p" id="mwAR8">(</span><span class="nv" id="mwASA">Event</span><span class="p" id="mwASE">,</span> <span class="nv" id="mwASI">Fluent</span><span class="p" id="mwASM">,</span> <span class="nv" id="mwASQ">Time</span><span class="p" id="mwASU">),</span> <span class="nf" id="mwASY">happensAt</span><span class="p" id="mwASc">(</span><span class="nv" id="mwASg">Event</span><span class="p" id="mwASk">,</span> <span class="nv" id="mwASo">Time</span><span class="p" id="mwASs">),</span> <span class="nf" id="mwASw">before</span><span class="p" id="mwAS0">(</span><span class="nv" id="mwAS4">Time1</span><span class="p" id="mwAS8">,</span> <span class="nv" id="mwATA">Time</span><span class="p" id="mwATE">),</span> <span class="nf" id="mwATI">before</span><span class="p" id="mwATM">(</span><span class="nv" id="mwATQ">Time</span><span class="p" id="mwATU">,</span> <span class="nv" id="mwATY">Time2</span><span class="p" id="mwATc">).</span> <span class="nf" id="mwATg">initiates</span><span class="p" id="mwATk">(</span><span class="nf" id="mwATo">initialise</span><span class="p" id="mwATs">(</span><span class="nv" id="mwATw">Fluent</span><span class="p" id="mwAT0">),</span> <span class="nv" id="mwAT4">Fluent</span><span class="p" id="mwAT8">,</span> <span class="nv" id="mwAUA">Time</span><span class="p" id="mwAUE">).</span> <span class="nf" id="mwAUI">initiates</span><span class="p" id="mwAUM">(</span><span class="nf" id="mwAUQ">move</span><span class="p" id="mwAUU">(</span><span class="nv" id="mwAUY">Object</span><span class="p" id="mwAUc">,</span> <span class="nv" id="mwAUg">Place</span><span class="p" id="mwAUk">),</span> <span class="nf" id="mwAUo">on</span><span class="p" id="mwAUs">(</span><span class="nv" id="mwAUw">Object</span><span class="p" id="mwAU0">,</span> <span class="nv" id="mwAU4">Place</span><span class="p" id="mwAU8">),</span> <span class="nv" id="mwAVA">Time</span><span class="p" id="mwAVE">).</span> <span class="nf" id="mwAVI">terminates</span><span class="p" id="mwAVM">(</span><span class="nf" id="mwAVQ">move</span><span class="p" id="mwAVU">(</span><span class="nv" id="mwAVY">Object</span><span class="p" id="mwAVc">,</span> <span class="nv" id="mwAVg">Place</span><span class="p" id="mwAVk">),</span> <span class="nf" id="mwAVo">on</span><span class="p" id="mwAVs">(</span><span class="nv" id="mwAVw">Object</span><span class="p" id="mwAV0">,</span> <span class="nv" id="mwAV4">Place1</span><span class="p" id="mwAV8">),</span> <span class="nv" id="mwAWA">Time</span><span class="p" id="mwAWE">).</span> <span class="nf" id="mwAWI">happensAt</span><span class="p" id="mwAWM">(</span><span class="nf" id="mwAWQ">initialise</span><span class="p" id="mwAWU">(</span><span class="nf" id="mwAWY">on</span><span class="p" id="mwAWc">(</span><span class="s s-Atom" id="mwAWg">green_block</span><span class="p" id="mwAWk">,</span> <span class="s s-Atom" id="mwAWo">table</span><span class="p" id="mwAWs">)),</span> <span class="mi" id="mwAWw">0</span><span class="p" id="mwAW0">).</span> <span class="nf" id="mwAW4">happensAt</span><span class="p" id="mwAW8">(</span><span class="nf" id="mwAXA">initialise</span><span class="p" id="mwAXE">(</span><span class="nf" id="mwAXI">on</span><span class="p" id="mwAXM">(</span><span class="s s-Atom" id="mwAXQ">red_block</span><span class="p" id="mwAXU">,</span> <span class="s s-Atom" id="mwAXY">green_block</span><span class="p" id="mwAXc">)),</span> <span class="mi" id="mwAXg">0</span><span class="p" id="mwAXk">).</span> <span class="nf" id="mwAXo">happensAt</span><span class="p" id="mwAXs">(</span><span class="nf" id="mwAXw">move</span><span class="p" id="mwAX0">(</span><span class="s s-Atom" id="mwAX4">red_block</span><span class="p" id="mwAX8">,</span> <span class="s s-Atom" id="mwAYA">table</span><span class="p" id="mwAYE">),</span> <span class="mi" id="mwAYI">1</span><span class="p" id="mwAYM">).</span> <span class="nf" id="mwAYQ">happensAt</span><span class="p" id="mwAYU">(</span><span class="nf" id="mwAYY">move</span><span class="p" id="mwAYc">(</span><span class="s s-Atom" id="mwAYg">green_block</span><span class="p" id="mwAYk">,</span> <span class="s s-Atom" id="mwAYo">red_block</span><span class="p" id="mwAYs">),</span> <span class="mi" id="mwAYw">3</span><span class="p" id="mwAY0">).</span>

The Prolog program differs from the earlier formalisation in the following ways:

• The core axiom has been rewritten, using an auxiliary predicate clipped(Fact, Time1, Time2). This rewriting enables the elimination of existential quantifiers, conforming to the Prolog convention that all variables are universally quantified.
• The order of the conditions in the body of the core axiom(s) has been changed, to generate answers to queries in temporal order.
• The equality in the condition ${\displaystyle T1\le T}$ has been removed from the corresponding condition before(Time1, Time). This builds in a simplifying assumption that events do not simultaneously initiate and terminate the same fluent. As a consequence, the definition of the ${\displaystyle terminates}$ predicate has been simplified by eliminating the condition that ${\displaystyle different(Place1,Place)}$.

Given an appropriate definition of the predicate before(Time1, Time2), the Prolog program generates all answers to the query what holds when? in temporal order:

<span id="mwAbU"></span><span class="s s-Atom" id="mwAbY">?-</span> <span class="nf" id="mwAbc">holdsAt</span><span class="p" id="mwAbg">(</span><span class="nv" id="mwAbk">Fluent</span><span class="p" id="mwAbo">,</span> <span class="nv" id="mwAbs">Time</span><span class="p" id="mwAbw">).</span> <span class="nv" id="mwAb0">Fluent</span> <span class="o" id="mwAb4">=</span> <span class="nf" id="mwAb8">on</span><span class="p" id="mwAcA">(</span><span class="s s-Atom" id="mwAcE">green_block</span><span class="p" id="mwAcI">,</span><span class="s s-Atom" id="mwAcM">table</span><span class="p" id="mwAcQ">),</span> <span class="nv" id="mwAcU">Time</span> <span class="o" id="mwAcY">=</span> <span class="mf" id="mwAcc">1.</span> <span class="nv" id="mwAcg">Fluent</span> <span class="o" id="mwAck">=</span> <span class="nf" id="mwAco">on</span><span class="p" id="mwAcs">(</span><span class="s s-Atom" id="mwAcw">red_block</span><span class="p" id="mwAc0">,</span><span class="s s-Atom" id="mwAc4">green_block</span><span class="p" id="mwAc8">),</span> <span class="nv" id="mwAdA">Time</span> <span class="o" id="mwAdE">=</span> <span class="mf" id="mwAdI">1.</span> <span class="nv" id="mwAdM">Fluent</span> <span class="o" id="mwAdQ">=</span> <span class="nf" id="mwAdU">on</span><span class="p" id="mwAdY">(</span><span class="s s-Atom" id="mwAdc">green_block</span><span class="p" id="mwAdg">,</span><span class="s s-Atom" id="mwAdk">table</span><span class="p" id="mwAdo">),</span> <span class="nv" id="mwAds">Time</span> <span class="o" id="mwAdw">=</span> <span class="mf" id="mwAd0">2.</span> <span class="nv" id="mwAd4">Fluent</span> <span class="o" id="mwAd8">=</span> <span class="nf" id="mwAeA">on</span><span class="p" id="mwAeE">(</span><span class="s s-Atom" id="mwAeI">red_block</span><span class="p" id="mwAeM">,</span><span class="s s-Atom" id="mwAeQ">table</span><span class="p" id="mwAeU">),</span> <span class="nv" id="mwAeY">Time</span> <span class="o" id="mwAec">=</span> <span class="mf" id="mwAeg">2.</span> <span class="nv" id="mwAek">Fluent</span> <span class="o" id="mwAeo">=</span> <span class="nf" id="mwAes">on</span><span class="p" id="mwAew">(</span><span class="s s-Atom" id="mwAe0">green_block</span><span class="p" id="mwAe4">,</span><span class="s s-Atom" id="mwAe8">table</span><span class="p" id="mwAfA">),</span> <span class="nv" id="mwAfE">Time</span> <span class="o" id="mwAfI">=</span> <span class="mf" id="mwAfM">3.</span> <span class="nv" id="mwAfQ">Fluent</span> <span class="o" id="mwAfU">=</span> <span class="nf" id="mwAfY">on</span><span class="p" id="mwAfc">(</span><span class="s s-Atom" id="mwAfg">red_block</span><span class="p" id="mwAfk">,</span><span class="s s-Atom" id="mwAfo">table</span><span class="p" id="mwAfs">),</span> <span class="nv" id="mwAfw">Time</span> <span class="o" id="mwAf0">=</span> <span class="mf" id="mwAf4">3.</span> <span class="nv" id="mwAf8">Fluent</span> <span class="o" id="mwAgA">=</span> <span class="nf" id="mwAgE">on</span><span class="p" id="mwAgI">(</span><span class="s s-Atom" id="mwAgM">red_block</span><span class="p" id="mwAgQ">,</span><span class="s s-Atom" id="mwAgU">table</span><span class="p" id="mwAgY">),</span> <span class="nv" id="mwAgc">Time</span> <span class="o" id="mwAgg">=</span> <span class="mf" id="mwAgk">4.</span> <span class="nv" id="mwAgo">Fluent</span> <span class="o" id="mwAgs">=</span> <span class="nf" id="mwAgw">on</span><span class="p" id="mwAg0">(</span><span class="s s-Atom" id="mwAg4">green_block</span><span class="p" id="mwAg8">,</span><span class="s s-Atom" id="mwAhA">red_block</span><span class="p" id="mwAhE">),</span> <span class="nv" id="mwAhI">Time</span> <span class="o" id="mwAhM">=</span> <span class="mf" id="mwAhQ">4.</span> <span class="nv" id="mwAhU">Fluent</span> <span class="o" id="mwAhY">=</span> <span class="nf" id="mwAhc">on</span><span class="p" id="mwAhg">(</span><span class="s s-Atom" id="mwAhk">red_block</span><span class="p" id="mwAho">,</span><span class="s s-Atom" id="mwAhs">table</span><span class="p" id="mwAhw">),</span> <span class="nv" id="mwAh0">Time</span> <span class="o" id="mwAh4">=</span> <span class="mf" id="mwAh8">5.</span> <span class="nv" id="mwAiA">Fluent</span> <span class="o" id="mwAiE">=</span> <span class="nf" id="mwAiI">on</span><span class="p" id="mwAiM">(</span><span class="s s-Atom" id="mwAiQ">green_block</span><span class="p" id="mwAiU">,</span><span class="s s-Atom" id="mwAiY">red_block</span><span class="p" id="mwAic">),</span> <span class="nv" id="mwAig">Time</span> <span class="o" id="mwAik">=</span> <span class="mf" id="mwAio">5.</span>

The program can also answer negative queries, such as which fluents do not hold at which times? However, to work correctly, all variables in negative conditions must first be instantiated to terms containing no variables. For example:

<span id="mwAi8"></span><span class="nf" id="mwAjA">timePoint</span><span class="p" id="mwAjE">(</span><span class="mi" id="mwAjI">1</span><span class="p" id="mwAjM">).</span> <span class="nf" id="mwAjQ">timePoint</span><span class="p" id="mwAjU">(</span><span class="mi" id="mwAjY">2</span><span class="p" id="mwAjc">).</span> <span class="nf" id="mwAjg">timePoint</span><span class="p" id="mwAjk">(</span><span class="mi" id="mwAjo">3</span><span class="p" id="mwAjs">).</span> <span class="nf" id="mwAjw">timePoint</span><span class="p" id="mwAj0">(</span><span class="mi" id="mwAj4">4</span><span class="p" id="mwAj8">).</span> <span class="nf" id="mwAkA">timePoint</span><span class="p" id="mwAkE">(</span><span class="mi" id="mwAkI">5</span><span class="p" id="mwAkM">).</span> <span class="nf" id="mwAkQ">fluent</span><span class="p" id="mwAkU">(</span><span class="nf" id="mwAkY">on</span><span class="p" id="mwAkc">(</span><span class="s s-Atom" id="mwAkg">red_block</span><span class="p" id="mwAkk">,</span> <span class="s s-Atom" id="mwAko">green_block</span><span class="p" id="mwAks">)).</span> <span class="nf" id="mwAkw">fluent</span><span class="p" id="mwAk0">(</span><span class="nf" id="mwAk4">on</span><span class="p" id="mwAk8">(</span><span class="s s-Atom" id="mwAlA">green_block</span><span class="p" id="mwAlE">,</span> <span class="s s-Atom" id="mwAlI">red_block</span><span class="p" id="mwAlM">)).</span> <span class="nf" id="mwAlQ">fluent</span><span class="p" id="mwAlU">(</span><span class="nf" id="mwAlY">on</span><span class="p" id="mwAlc">(</span><span class="s s-Atom" id="mwAlg">red_block</span><span class="p" id="mwAlk">,</span> <span class="s s-Atom" id="mwAlo">table</span><span class="p" id="mwAls">)).</span> <span class="nf" id="mwAlw">fluent</span><span class="p" id="mwAl0">(</span><span class="nf" id="mwAl4">on</span><span class="p" id="mwAl8">(</span><span class="s s-Atom" id="mwAmA">green_block</span><span class="p" id="mwAmE">,</span> <span class="s s-Atom" id="mwAmI">table</span><span class="p" id="mwAmM">)).</span> <span class="s s-Atom" id="mwAmQ">?-</span> <span class="nf" id="mwAmU">timePoint</span><span class="p" id="mwAmY">(</span><span class="nv" id="mwAmc">T</span><span class="p" id="mwAmg">),</span> <span class="nf" id="mwAmk">fluent</span><span class="p" id="mwAmo">(</span><span class="nv" id="mwAms">F</span><span class="p" id="mwAmw">),</span> <span class="o" id="mwAm0">not</span><span class="p" id="mwAm4">(</span><span class="nf" id="mwAm8">holdsAt</span><span class="p" id="mwAnA">(</span><span class="nv" id="mwAnE">F</span><span class="p" id="mwAnI">,</span> <span class="nv" id="mwAnM">T</span><span class="p" id="mwAnQ">)).</span> <span class="nv" id="mwAnU">F</span> <span class="o" id="mwAnY">=</span> <span class="nf" id="mwAnc">on</span><span class="p" id="mwAng">(</span><span class="s s-Atom" id="mwAnk">green_block</span><span class="p" id="mwAno">,</span><span class="s s-Atom" id="mwAns">red_block</span><span class="p" id="mwAnw">),</span> <span class="nv" id="mwAn0">T</span> <span class="o" id="mwAn4">=</span> <span class="mf" id="mwAn8">1.</span> <span class="nv" id="mwAoA">F</span> <span class="o" id="mwAoE">=</span> <span class="nf" id="mwAoI">on</span><span class="p" id="mwAoM">(</span><span class="s s-Atom" id="mwAoQ">red_block</span><span class="p" id="mwAoU">,</span><span class="s s-Atom" id="mwAoY">table</span><span class="p" id="mwAoc">),</span> <span class="nv" id="mwAog">T</span> <span class="o" id="mwAok">=</span> <span class="mf" id="mwAoo">1.</span> <span class="nv" id="mwAos">F</span> <span class="o" id="mwAow">=</span> <span class="nf" id="mwAo0">on</span><span class="p" id="mwAo4">(</span><span class="s s-Atom" id="mwAo8">red_block</span><span class="p" id="mwApA">,</span><span class="s s-Atom" id="mwApE">green_block</span><span class="p" id="mwApI">),</span> <span class="nv" id="mwApM">T</span> <span class="o" id="mwApQ">=</span> <span class="mf" id="mwApU">2.</span> <span class="nv" id="mwApY">F</span> <span class="o" id="mwApc">=</span> <span class="nf" id="mwApg">on</span><span class="p" id="mwApk">(</span><span class="s s-Atom" id="mwApo">green_block</span><span class="p" id="mwAps">,</span><span class="s s-Atom" id="mwApw">red_block</span><span class="p" id="mwAp0">),</span> <span class="nv" id="mwAp4">T</span> <span class="o" id="mwAp8">=</span> <span class="mf" id="mwAqA">2.</span> <span class="nv" id="mwAqE">F</span> <span class="o" id="mwAqI">=</span> <span class="nf" id="mwAqM">on</span><span class="p" id="mwAqQ">(</span><span class="s s-Atom" id="mwAqU">red_block</span><span class="p" id="mwAqY">,</span><span class="s s-Atom" id="mwAqc">green_block</span><span class="p" id="mwAqg">),</span> <span class="nv" id="mwAqk">T</span> <span class="o" id="mwAqo">=</span> <span class="mf" id="mwAqs">3.</span> <span class="nv" id="mwAqw">F</span> <span class="o" id="mwAq0">=</span> <span class="nf" id="mwAq4">on</span><span class="p" id="mwAq8">(</span><span class="s s-Atom" id="mwArA">green_block</span><span class="p" id="mwArE">,</span><span class="s s-Atom" id="mwArI">red_block</span><span class="p" id="mwArM">),</span> <span class="nv" id="mwArQ">T</span> <span class="o" id="mwArU">=</span> <span class="mf" id="mwArY">3.</span> <span class="nv" id="mwArc">F</span> <span class="o" id="mwArg">=</span> <span class="nf" id="mwArk">on</span><span class="p" id="mwAro">(</span><span class="s s-Atom" id="mwArs">red_block</span><span class="p" id="mwArw">,</span><span class="s s-Atom" id="mwAr0">green_block</span><span class="p" id="mwAr4">),</span> <span class="nv" id="mwAr8">T</span> <span class="o" id="mwAsA">=</span> <span class="mf" id="mwAsE">4.</span> <span class="nv" id="mwAsI">F</span> <span class="o" id="mwAsM">=</span> <span class="nf" id="mwAsQ">on</span><span class="p" id="mwAsU">(</span><span class="s s-Atom" id="mwAsY">green_block</span><span class="p" id="mwAsc">,</span><span class="s s-Atom" id="mwAsg">table</span><span class="p" id="mwAsk">),</span> <span class="nv" id="mwAso">T</span> <span class="o" id="mwAss">=</span> <span class="mf" id="mwAsw">4.</span> <span class="nv" id="mwAs0">F</span> <span class="o" id="mwAs4">=</span> <span class="nf" id="mwAs8">on</span><span class="p" id="mwAtA">(</span><span class="s s-Atom" id="mwAtE">red_block</span><span class="p" id="mwAtI">,</span><span class="s s-Atom" id="mwAtM">green_block</span><span class="p" id="mwAtQ">),</span> <span class="nv" id="mwAtU">T</span> <span class="o" id="mwAtY">=</span> <span class="mf" id="mwAtc">5.</span> <span class="nv" id="mwAtg">F</span> <span class="o" id="mwAtk">=</span> <span class="nf" id="mwAto">on</span><span class="p" id="mwAts">(</span><span class="s s-Atom" id="mwAtw">green_block</span><span class="p" id="mwAt0">,</span><span class="s s-Atom" id="mwAt4">table</span><span class="p" id="mwAt8">),</span> <span class="nv" id="mwAuA">T</span> <span class="o" id="mwAuE">=</span> <span class="mf" id="mwAuI">5.</span>

Reasoning tools

In addition to Prolog and its variants, several other tools for reasoning using the event calculus are also available:

• Abductive Event Calculus Planners
• Discrete Event Calculus Reasoner
• Event Calculus Answer Set Programming
• Reactive Event Calculus
• Run-Time Event Calculus (RTEC)
• Epistemic Probabilistic Event Calculus (EPEC)

Extensions

Notable extensions of the event calculus include Markov logic networks–based variants probabilistic, epistemic and their combinations.