# Inference

In brief, inference in Markov Logic Networks (MLNs), is the process of estimating the marginal probability or the most probable truth state of the groundings of some query atoms, given a MLN theory (i.e., a collection of weighted formulas) and a collection of imput ground evidence atoms (i.e., input ground predicates with known truth state).

## Types of inference in LoMRF

In order to perform inference, we have to define the following: * The type of the inference. In LoMRF two types of inference can be performed: 1. Marginal inference: computes the probabilities for all possible instantiations of query predicates of being true, given the evidence. 2. Maximum a-posteriori inference (MAP): computes the truth values (0 = False and 1 = True) of all possible instantiations (groundings) that together represent the most probable state. * Input theory file, e.g., theory.mln. * Input evidence, e.g., evidence.db. * Output result file, e.g., output.result. * The atomic signatures (identities) that define the query predicates ('-q' option). Please note that for all query predicates LoMRF takes Open-world assumption. * (Optionally) The atomic signatures (identities) that define predicates with Closed-world assumption ('-cwa' option). By default, all evidence predicates have Closed-world assumption. * (Optionally) The atomic signatures (identities) that define predicates with Open-world assumption ('-owa' option). By default, all non-evidence predicates have Open-world assumption.

### Inference using the lomrf command-line tool

To demonstrate the usage of LoMRF from command-line interface, assume that we have one knowledge base file, named as theoy.mln, and one evidence file, named as evidence.db.

In our example knowledge-base we have the following predicates:

Predicate Name Number of arguments Predicate identity Description
Query_A 2 Query_A/2 first query predicate (Open-world assumption)
Query_B 2 Query_B/2 second query predicate (Open-world assumption)
Ev_A 1 EV_A/1 first evidence predicate (Closed-world assumption)
Ev_B 1 EV_B/1 second evidence predicate (Closed-world assumption)
Hidden 1 Hidden/1 non-evidence predicate (Open-world assumption)

As it is presented in the above table, there are two query predicates, Query_A and Query_B, where each one takes two terms as arguments. Therefore their atomic signatures are Query_A/2 and Query_B/2. Similarly, there are two evidence predicates Ev_A and Ev_B that they take one term as argument. Therefore, the atomic signatures of Ev_A and Ev_B are Ev_A/1 and Ev_B/1, respectively. Finally, there is one non-evidence predicate, named as Hidden, that takes one term as argument, thus it's signatures is Hidden/1.

#### Marginal inference

Marginal inference computes the conditional probability of query predicates, given some evidence.

lomrf infer -inferType marginal -i theory.mln -e evidence.db -r marginal-out.result -q Query_A/2,Query_B/2 -cwa Ev_A/1,Ev_B/1 -owa Hidden/1


Since Hidden/1 is a non-evidence atom and thus does not contain any facts in the evidence.db, we can omit the -owa parameter. Furthermore, when the evidence file (evidence.db) contains at least one fact for Ev_A/1 and Ev_B/1 predicates, then we can also omit the parameter -cwa. In that case, LoMRF can automatically infer which predicates have Open-world or Closed-world assumption. As a result, we can simply give the following parameters and take the same results:

lomrf infer -inferType marginal -i theory.mln -e evidence.db -r marginal-out.result -q Query_A/2,Query_B/2


Also the default inference type for LoMRF is marginal, we can further simplify the example given the following parameters:

lomrf infer -i theory.mln -e evidence.db -r marginal-out.result -q Query_A/2,Query_B/2


The results from marginal inference are stored in the marginal-out.result (see parameter -r)

#### MAP inference

MAP inference, on the other hand, identifies the most probable assignment among all query predicate instantiations that are consistent with the given evidence. This task reduces to finding the truth assignment of all query predicate instantiations that maximises the sum of weights of satisfied ground clauses. This is equivalent to the weighted maximum satisfiability problem.

lomrf infer -inferType map -i theory.mln -e evidence.db -r map-out.result -q Query_A/2,Query_B/2 -cwa Ev_A/1,Ev_B/1 -owa Hidden/1


or simply:

lomrf infer -inferType map -i theory.mln -e evidence.db -r map-out.result -q Query_A/2,Query_B/2


The results from MAP inference are stored in the map-out.result (see paramter -r)

## Probabilistic Inference Examples

See Sections Probabilistic Inference Examples and Temporal Probabilistic Inference Examples. Sources from the examples are located in the LoMRF-data project (follow the instructions in Download Example Data).

## Command-line Interface Options

By executing the lomrf infer -h (or lomrf infer --help) command from the command-line interface, we take a print of multiple parameters. Below we explain all LoMRF inference command-line interface parameters:

### Basic inference options

• -i, --input <kb file> [Required] Specify the input knowledge base file, that is the file that contains the theory (see Syntax and Quick Start for further information). You can specify either full or relative path or only the filename (when the file is in the current working path). For example, (1) full path -i /full/path/to/theory.mln in a Unix-based OS or -i c:\full\path\to\theory.mln in Windows, (2) relative path -i path/to/theory.mln in a Unix-based OS or -i path\to\theory.mln in Windows and (3) current working path -i theory.mln.

• -e, --evidence <db file(s)> [Optional] Specify the input evidence file(s) (see Syntax and Quick Start for further information). Multiple evidence files are allowed by separating them with commas without spaces. For example, -e file1.db,file2.db,file3.db. Similarly with the -i option, you can specify either full or relative path or only the filename (when the file is in the current working path). In case that there is no evidence, this option can be omitted.

• -r, --result <result file> [Required] Specify the output file name to write the results of the inference. For example, -r output.result. Similarly with the -i option, you can specify either full or relative path or only the filename (when the file is in the current working path).

• -q, --query <string> [Required] Specify the atomic signatures or identities (i.e., predicate_name/arity) of predicates that we would like to query. For example, the query predicate Male(person) has name Male and arity 1 (single argument), therefore we should give the argument -q Male/1. Multiple query atoms are allowed in LoMRF and they are defined as comma-separated identities without white-spaces. For example, -q Male/1,Female/1. Please note that LoMRF takes Open-world assumption for all query atoms and this cannot be overridden.

• -cwa, --closed-world-assumption <string> [Optional] Specify the atomic signatures or identities (i.e., predicate_name/arity) of predicates that we would like to have Closed-world assumption. By default, LoMRF takes Closed-world assumption for all evidence atoms, that is atoms that have at least one fact in the specified evidence files (see -e option). All non-evidence atoms are open-world in LoMRF, except when are included in the -cwa option. For example, assume that in the evidence file(s) we have facts about the predicate Foo/1 but not for the predicate Bar/1 and we would like to have Closed-world for both Foo/1 and Bar/1 predicates. In that case, we have to explicitly define that Bar/1 is also closed-world, therefore we should give -cwa Bar/1. Please note that in this example, it is not required to specify that the predicate Foo/1 is closed-world, because it has at least one fact in the evidence file(s). If it is desirable, we can also define explicitly closed-world for Foo/1, i.e., -cwa Foo/1,Bar/1 (multiple identities are defined using comma-separated values, without spaces).

• -owa, --open-world-assumption <string> [Optional] Specify the atomic signatures or identities (i.e., predicate_name/arity) of predicates that we would like to have Open-world assumption. By default, all non-evidence atoms are open-world in LoMRF, except when are included in the -cwa option.

• -inferType, --inference-type <map | marginal> [Optional] Specify the inference type, either MAP or Marginal. By default LoMRF uses marginal inference, in order to estimate the marginal probabilities of all possible query predicate instantiations. MAP inference can be performed using either local-search algorithm (for details see MaxWalkSAT and the works of Selman et. al., 1993 and Kautz et. al., 1997) or using an Integer Linear Programming (ILP) solver (Huynh and Mooney, 2011). Marginal inference is estimated using the MC-SAT (Hoifung and Domingos, 2006) algorithm, that is a Metropolis–Hastings algorithm (a Markov chain Monte Carlo method) with Slice-sampling technique that combines Simulated-Annealing with a local-search SAT solver (MaxWalkSAT).

### Advanced inference options

• -mapType, --map-type <mws | ilp> [Optional] When inference is set to MAP (i.e., -infer map) we can define either MaxWalkSAT (Selman et. al., 1993; Kautz et. al., 1996) or relaxed Integer Linear Programming (Huynh and Mooney, 2011) as MAP inference algorithm to use. By default LoMRF uses a variant of the MaxWalkSAT algorithm (with TABU search).

• -mapOutput, --map-output-type <all | positive> [Optional] When MAP inference is selected (i.e., -infer map) we can specify the format in the result file (i.e., -r option). We can choose either one of the following:

• all, in order to contain all possible groundings of the resulting query predicates associated with their truth value (i.e., 0 for false and 1 for true) or

• positive, in order to contain only the ground atoms that have positive state (true).

By default, LoMRF outputs all groundings with their truth values.

• -satHardUnit, --sat-hard-unit [Optional] Try to trivially satisfy all hard-constrained unit clauses in MaxWalkSAT. After the grounding process, hard-constrained ground unit-clauses may produced. Such clauses contain only one ground literal and therefore can be handled like fact predicates. If the literal is positive, then the state that satisfies this clause is true. Similarly, if the literal is negative then the state that satisfies this clause is false. Consider, for example the ground hard-constrained unit clause Male(Odysseus). The literal is positive (i.e., is not negated), thus the clause is satisfied when Male(Odysseus) = True.

• -satHardPriority, --sat-hard-priority [Optional] This option applies to MaxWalkSAT and MC-SAT algorithms. Enables satisfiability priority to hard-constrained clauses. Specifically, when the local-search algorithm (either MaxWalkSAT or MC-SAT) requires to select an unsatisfied clause and the satHardPriority is enabled, then the algorithm picks a recently unsatisfied hard-constrained clause (if any exists). In any other case the selection is random. Therefore, satHardPriority adds an additional bias to satisfy hard-constrained clauses.

• -ilpRounding, --ilp-rounding <roundup | mws> [Optional] Specify which rounding algorithm to use in ILP. LoMRF uses the algorithm proposed by Huynh and Mooney (2011), in which the ILP problem is relaxed as a standard LP problem. The resulting LP solution, however, may not integral. As a result the solution may contain ground query predicates in which their state may be any number in the interval from 0 to 1, exclusive. Since, in MAP inference the solution should contain only 0/1 states for each resulting ground predicate, LoMRF employs a rounding algorithm to change the non-integral solutions to 0/1 state values. By default LoMRF uses the RoundUp algorithm of Huynh and Mooney (2011) (i.e., -ilpRounding roundup). Alternatively, LoMRF can run the local-search MaxWalkSAT only for the non-integral part (i.e., -ilpRounding mws).

• -ilpSolver, --ilp-solver <lpsolve | ojalgo | gurobi> [Optional] When MAP inference is chosen to be solved using an ILP solver, we can specify which solver to use. We can choose between the open-source solvers LPSolve and ojAlgo, as well as the commercial solver Gurobi. By default LoMRF uses the open-source solver LPSolve.

• -samples, --num-samples <value> [Optional] Specify the number of samples to take in MC-SAT (default is 1000).

• -pSA, --probability-simulated-annealing <value> [Optional] Specify the probability of performing a simulated annealing step (default is 0.5), when using MC-SAT for marginal inference. Every step in MC-SAT may involve multiple iterations, in which the state of ground atoms is flipped. At each iteration MC-SAT performs either simulated annealing or a standard MaxWalkSAT step. This option adjusts the probability to choose a simulated annealing step.

• -pBest, --probability-best-search <value> [Optional] The probability of performing a greedy search (default is 0.5) in MaxWalkSAT or MC-SAT. The algorithm performs either a noisy or a greedy move, in particular:

• In a noisy move it takes a random unsatisfied clause and it chooses a random atom to flip.
• In a greedy move it chooses to flip the atom having the best score, that is, it satisfies more clauses.

• -saTemperature, --simulated-annealing-temperature <value> [Optional] Adjust the temperature of the simulated annealing step in the MC-SAT algorithm. The value should range between 0 and 1. The default value is 0.8.

• -maxFlips, --maximum-flips <value> [Optional] Specify the maximum number of flips taken to reach a solution in MaxWalkSAT and MC-SAT (default is 1000000).

• -targetCost, --target-cost <value> [Optional] In MaxWalkSAT and MC-SAT, any possible world having cost below this threshold will be considered as a solution (default is 0.0001).

• -maxTries, --maximum-tries <value> [Optional] Specify the maximum number of attempts to find a solution in MaxWalkSAT and MC-SAT (default is 1)

• -numSolutions, --number-of-solutions <value> [Optional] Give the n-th solution (i.e., cost < target cost) in MC-SAT (default is 10).

• -tabuLength, --tabu-length <value> [Optional] Specify the minimum number of flips between flipping the same atom in MaxWalkSAT and MC-SAT (default is 10)

• -unitProp, --use-unit-propagation <value> [Optional] Perform unit-propagation in MC-SAT (default is true). Performs unit propagation across the constraints in order to trivially satisfy as many as possible. When enabled, the search space in MC-SAT is minimized, increases sampling performance and accuracy.

• -lateSA, --late-simulated-annealing <value> [Optional] When enabled (i.e., true), simulated annealing is performed only when MC-SAT reaches a plateau (i.e. a world with cost <= 'targetCost'). Disabling lateSA (= false) causes MC-SAT to converge slower, since in every iteration simulated annealing is performed (with probability = 'pSA'). By default lateSA is 'true'.

• -noNegWeights, --eliminate-negative-weights [Optional] When it is enabled, LoMRF eliminates negative weight values from (ground) clauses. Specifically, the sign of negative weights in clauses is inverted, as well as all disjunctions become conjunctions (due to de Morgan's law). For example, using the de Morgan's law, the weighted clause -2 A(x) v B(x) is equivalent to -2 !(!A(x) ^ !B(x)). In MLN this is also equivalent to 2 !A(x) ^ !B(x), which produces the following two unit clauses: 1 A(x) and 1 B(x).

• -noNegatedUnit, --eliminate-negated-unit [Optional] When it is enabled, unit clauses with negative literal become unit clauses with positive literal and inverted sign in their corresponding weight.

• -dynamic, --dynamic-implementations <string> [Optional] Comma separated paths to search recursively for dynamic predicates/functions implementations (.class and .jar files).